mm-1073 >> For example, for an arbitrary 3+1-dimension Lorentzian >> manifold to be isometrically embedded into an N+M-dimension >> flat manifold, the lowest known dimension is N=88,M=2 (!). > That is terrible. Is this lower bound considered > conservative, or are there particular examples that are known to need > this? I don't know. There are definitely 3+1-d manifolds that cannot be embedded isometrically in a 4+1-d flat manifold. But of course there are also 2-d manifolds that cannot be embedded in a 3-d flat Euclidean space (Klein bottle). As this last example shows, you need to understand issues of orientability.... > Part of the problem is that I am still trying to determine the > more rigerous definition. I'm not knowledgeable enough to help. And not interested enough to pursue it. >Basically I am trying to build an intuitive > view of GR, by picturing metrics as surfaces in higher dimensional > space. I suspect that is either useless or doomed. AFAICT the important thing about GR is that you really want to discuss and understand it intrinsically. After all, we clearly have no way to go outside the entire universe to examine any such embedding. In my experience, modern tensor analysis goes a long way here, and the old component notation is obscure at best, and often misleading. > For instance I know that the robertson walker metric at a > particular time slice has constant curvature everywhere. I can > picture this as a sphere in a n+1 dimensional space. Hmmm. That completely misses the time evolution, which is the interesting part. In fact, for all but trivial manifolds the time dependence is the intriguing part. And to handle that you must inherently use semi-Riemannian techniques. Tom Roberts tjroberts@lucent.com === Subject: Re: Embedding theorem and uniqueness? > Hi- > According to the embedding theorem, any manifold can be > represented as a surface in a higher dimensional space (ie- the > 2-manifold with constant curvature a^2 can be represented as the > sphere of radius a in three dimensional space). My question is- is > the surface unique (up to trivial transformations such as translation, > rotation, and curling up in a yet higher dimensional space such as > when one rolls up a plane to a cylinder). For instance, is there > another surface in three dimensional space that has constant curvature > a^2? Are there other solutions which have constant curvature but > eventually have a singularity at global locations? > -I I'm not sure what you mean by the embedding theorem. There are several embedding theorems for manifolds, from Whitney's theorem embedding n- dimensional manifolds in R^(2n), to Nash's isometric embedding theorem, to a number of refinements. You probably meant the sphere's radius to be 1/a: a larger sphere has smaller curvature. The Gauss-Bonnet Theorem expresses the Euler characteristic of a manifold in terms of the integral of the Gaussian curvature. This is sufficient to guarantee that the surface has Euler characteristic equal to 2, since it must be positive, and all compact connected surfaces have Euler characteristic given by 2 - 2g, where g is the genus (or number of handles, for a sphere with n handles). For higher-dimensional manifolds, a manifold with sectional curvature strictly between 1 and 4 is homeomorphic to a sphere. Dale === Subject: Re: Embedding theorem and uniqueness? Hi- > I'm not sure what you mean by the embedding theorem. There are several > embedding theorems for manifolds, from Whitney's theorem embedding n- > dimensional manifolds in R^(2n), to Nash's isometric embedding theorem, > to a number of refinements. Sorry, I was referring to isometric embedding. > You probably meant the sphere's radius to be 1/a: a larger sphere has > smaller curvature. Sorry again, I was being stupid. I definitely meant 1/a. > The Gauss-Bonnet Theorem expresses the Euler characteristic of a > manifold in terms of the integral of the Gaussian curvature. This > is sufficient to guarantee that the surface has Euler characteristic > equal to 2, since it must be positive, and all compact connected > surfaces have Euler characteristic given by 2 - 2g, where g is the > genus (or number of handles, for a sphere with n handles). > For higher-dimensional manifolds, a manifold with sectional curvature > strictly between 1 and 4 is homeomorphic to a sphere. If I am correct this is only applicable to topological embedding (am I misreading?). What about isometric? Is it clear for example that any 2 manifold with constant curvature a^2 embedded in 3 space is exactly a sphere of radius 1/a? I suspect that there might be other surfaces with constant curvature a^2 at local patches but not necessarily ones that globally so (I suspect so because curvature of a 2 dimensional manifold in 3 space = product of curvature of a line going through the point in two orthogonal directions, so a point can have curvature a^2 not only by having curvature a for two lines going through the point, but also b and c such that b*c=a). I am interested in this because I am trying to build a more intuitive view of GR, and in many cases I can better visualize a manifold as a surface in euclidean space than in the original lower dimensional curved space (my brain evolved to like euclidean space). Of course if the euclidean space is 40 dimensions I might be better off not doing this but there are many cases where the euclidean space is of low dimension (Robertson-Walker, even the Schwarzschild can be embedded in low dimensional space). I am even considering for my own purposes looking at how the Einstein equation looks from this paradigm, but if there is no unique surface in a higher dimensional space corresponding to a metric, the whole process might be doomed. One of the reasons I like this approach is because of the ugly gauge invariance of the manifold metric (ie- the same manifold can be described by many different metrics all differing by simple change of coordinates). If the surface of the embedded manifold is unique I can visualize a manifold in a very specific way. -I === Subject: Re: Embedding theorem and uniqueness? > Hi- > I'm not sure what you mean by the embedding theorem. There are several > embedding theorems for manifolds, from Whitney's theorem embedding n- > dimensional manifolds in R^(2n), to Nash's isometric embedding theorem, > to a number of refinements. > Sorry, I was referring to isometric embedding. > You probably meant the sphere's radius to be 1/a: a larger sphere has > smaller curvature. > Sorry again, I was being stupid. I definitely meant 1/a. > The Gauss-Bonnet Theorem expresses the Euler characteristic of a > manifold in terms of the integral of the Gaussian curvature. This > is sufficient to guarantee that the surface has Euler characteristic > equal to 2, since it must be positive, and all compact connected > surfaces have Euler characteristic given by 2 - 2g, where g is the > genus (or number of handles, for a sphere with n handles). > For higher-dimensional manifolds, a manifold with sectional curvature > strictly between 1 and 4 is homeomorphic to a sphere. > If I am correct this is only applicable to topological > embedding (am I misreading?). What about isometric? Is it clear for > example that any 2 manifold with constant curvature a^2 embedded in 3 > space is exactly a sphere of radius 1/a? I suspect that there might > be other surfaces with constant curvature a^2 at local patches but not > necessarily ones that globally so (I suspect so because curvature of a > 2 dimensional manifold in 3 space = product of curvature of a line > going through the point in two orthogonal directions, so a point can > have curvature a^2 not only by having curvature a for two lines going > through the point, but also b and c such that b*c=a). > I am interested in this because I am trying to build a more > intuitive view of GR, and in many cases I can better visualize a > manifold as a surface in euclidean space than in the original lower > dimensional curved space (my brain evolved to like euclidean space). > Of course if the euclidean space is 40 dimensions I might be better > off not doing this but there are many cases where the euclidean space > is of low dimension (Robertson-Walker, even the Schwarzschild can be > embedded in low dimensional space). I am even considering for my own > purposes looking at how the Einstein equation looks from this > paradigm, but if there is no unique surface in a higher dimensional > space corresponding to a metric, the whole process might be doomed. You may be interested in what Wesson's theory STM has to say on the issue of embeddings. See http://arxiv.org/abs/gr-qc/0302015. It is interesting to note that in such 5D embeddings FRW models are computationally found to be a hypersuface of a flat 5D space - http://arxiv.org/abs/gr-qc/0202010. Bill > One of the reasons I like this approach is because of the ugly gauge > invariance of the manifold metric (ie- the same manifold can be > described by many different metrics all differing by simple change of > coordinates). If the surface of the embedded manifold is unique I can > visualize a manifold in a very specific way. > -I === Subject: Re: Embedding theorem and uniqueness? > Hi- > According to the embedding theorem, any manifold can be > represented as a surface in a higher dimensional space (ie- the > 2-manifold with constant curvature a^2 can be represented as the > sphere of radius a in three dimensional space). My question is- is > the surface unique (up to trivial transformations such as translation, > rotation, and curling up in a yet higher dimensional space such as > when one rolls up a plane to a cylinder). For instance, is there > another surface in three dimensional space that has constant curvature > a^2? Are there other solutions which have constant curvature but > eventually have a singularity at global locations? > -I > I'm not sure what you mean by the embedding theorem. There are several > embedding theorems for manifolds, from Whitney's theorem embedding n- > dimensional manifolds in R^(2n), to Nash's isometric embedding theorem, > to a number of refinements. I am aware of the Campbell-Magaard theorem from an interest I have in Wesson's STM theory. Would you have references for the other theorems? Bill > You probably meant the sphere's radius to be 1/a: a larger sphere has > smaller curvature. > The Gauss-Bonnet Theorem expresses the Euler characteristic of a > manifold in terms of the integral of the Gaussian curvature. This > is sufficient to guarantee that the surface has Euler characteristic > equal to 2, since it must be positive, and all compact connected > surfaces have Euler characteristic given by 2 - 2g, where g is the > genus (or number of handles, for a sphere with n handles). > For higher-dimensional manifolds, a manifold with sectional curvature > strictly between 1 and 4 is homeomorphic to a sphere. > Dale === Subject: Re: Embedding theorem and uniqueness? >Hi- > According to the embedding theorem, any manifold can be >represented as a surface in a higher dimensional space (ie- the >2-manifold with constant curvature a^2 can be represented as the >sphere of radius a in three dimensional space). My question is- is >the surface unique (up to trivial transformations such as translation, >rotation, and curling up in a yet higher dimensional space such as >when one rolls up a plane to a cylinder). For instance, is there >another surface in three dimensional space that has constant curvature >a^2? Are there other solutions which have constant curvature but >eventually have a singularity at global locations? > -I >>I'm not sure what you mean by the embedding theorem. There are several >>embedding theorems for manifolds, from Whitney's theorem embedding n- >>dimensional manifolds in R^(2n), to Nash's isometric embedding theorem, >>to a number of refinements. > I am aware of the Campbell-Magaard theorem from an interest I have in > Wesson's STM theory. Would you have references for the other theorems? > Bill The Whitney and Nash results are global results, that is, the manifold in question is embedded in R^N as a submanifold, without self- intersections. As far as I have been able to glean from a quick search, having been uninformed about Campbell-Magaard, that result is a local result, meaning that a neighborhood of a point in the manifold is embedded. This is a *far* weaker result than Nash's (no offense intended: one is comparing local and global results: local embedding is simpler than global embedding, no issue about it) : Campbell- Magaard produce a local embedding of an n-dimensional manifold in R^(n+1). If one were to ignore the issue of isometry, this would be a simple application of the implicit function theorem. In addition, it is easy to come up with 2-dimensional examples where such a global theorem cannot hold, even without requiring isometry (every closed n-manifold in R^(n+1) must be orientable, for instance). For references, do this: Whitney's embedding theorem is of considerable importance, for two reasons: the general position argument that trivially obtains embeddings in dimensions > 2n, and the Whitney trick, which enables one to eliminate self-intersections in dimension 2n, as long as n>=3. The absence of a Whitney trick in dimension 4 is of monumental importance in low-dimensional topology. For example, it is responsible for the apparent disparity between what techniques can be applied for high-dimensional (>=5) manifolds, and what works in low dimensions. R(only) H(only) Bing had a Texasism regarding this phenomenon (which I'll paraphrase, not having been present at the creation): Dimensions >4 are big enough to reason with; Dimension 2 is small enough to spank; Dimensions 3 and 4 are too big to spank, but to little to reason with John Nash proved that any Riemannian manifold can be embedded in Euclidean space (with its standard metric) by an isometry. The dimension requirements seem extravagant, but then one needs to satisfy a set of PDEs that are anything but well-behaved; for that, one needs slack (to paraphrase J.R. Bob Dobbs). This last one shows a number of hits that reveal an improved proof of the Nash-Moser isometric theorem; Nash's result requires a dimension of n(3n + 11)/2, whereas a 1987 result of Gunther requires only max{ n(n+3)/2 + 5, n(n+5)/2 } while both are quadratic in n (the dimension of the embedded manifold), the former is roughly 3n^2/2 , whereas the latter is roughly n^2/2. With the exception of the Whitney result, that's more than I really know. I'm familiar with Whitney's embedding theorem, since I used to pretend to do differential topology, and have never pretended to be a differential geometer. >>You probably meant the sphere's radius to be 1/a: a larger sphere has >>smaller curvature. >>The Gauss-Bonnet Theorem expresses the Euler characteristic of a >>manifold in terms of the integral of the Gaussian curvature. This >>is sufficient to guarantee that the surface has Euler characteristic >>equal to 2, since it must be positive, and all compact connected >>surfaces have Euler characteristic given by 2 - 2g, where g is the >>genus (or number of handles, for a sphere with n handles). >>For higher-dimensional manifolds, a manifold with sectional curvature >>strictly between 1 and 4 is homeomorphic to a sphere. >>Dale That's all I know for now. Dale. === Subject: Re: Embedding theorem and uniqueness? > [... interesting stuff] Beware: the semi-Riemannian manifolds of GR are not subject to the same theorems as Riemannian geometry. There are additional difficulties.... Tom Roberts tjroberts@lucent.com === Subject: Re: Embedding theorem and uniqueness? > > [... interesting stuff] I just want to confirm it is interesting stuff indeed. I am trying to follow through to the best of my ability. My background in math is not topology - it was functional analysis, distribution theory and non standard analysis although I did take standard courses such as introduction to algebraic topology (largely forgotten). Bill > Beware: the semi-Riemannian manifolds of GR are not subject to the same > theorems as Riemannian geometry. There are additional difficulties.... > Tom Roberts tjroberts@lucent.com === Subject: Re: Embedding theorem and uniqueness? > > [... interesting stuff] > I just want to confirm it is interesting stuff indeed. I am trying to > follow through to the best of my ability. My background in math is not > topology - it was functional analysis, distribution theory and non standard > analysis although I did take standard courses such as introduction to > algebraic topology (largely forgotten). Do you think anyone really gives a , you arrogant ass? You must think this whole NG is about you. === Subject: Re: Embedding theorem and uniqueness? > > [... interesting stuff] > I just want to confirm it is interesting stuff indeed. I am trying to > follow through to the best of my ability. My background in math is not > topology - it was functional analysis, distribution theory and non standard > analysis although I did take standard courses such as introduction to > algebraic topology (largely forgotten). > Do you think anyone really gives a , you arrogant ass? When people go to the trouble of posting detailed references those that have courtesy thank them and give them feedback. Of course such is lost on a troll like you whose only jollies in life is posting drivel on what should be serious newsgroups and watching what ensures. > You must think this whole NG is about you. Of course not - nor is it about your semantically motivated idiotic spew concerning time and motion that is a pretext for your troll antics. Bill === Subject: Re: Embedding theorem and uniqueness? And the obsession continues........ === Subject: Re: Embedding theorem and uniqueness? >> [... interesting stuff] > Beware: the semi-Riemannian manifolds of GR are not subject to the > same theorems as Riemannian geometry. There are additional > difficulties.... > Tom Roberts tjroberts@lucent.com Actually, these manifolds are (also) Riemannian manifolds, only with a different (i.e., Riemannian) metric, and so the underlying topological spaces *are* subject to all the same theorems. That is to say, given a semi-Riemannian manifold M, there is a Riemannian manifold M', with M' homeomorphic to M; any fact regarding the topology of M' will also hold for M. Metric-specific facts about M' are just that: metric-specific, and so may be irrelevant when discussing M. However, your point is well taken, that the semi-Riemannian category is subject to additional constraints due to the presence of the mixed metric. Dale. === Subject: Re: Embedding theorem and uniqueness? A notational correction. Feh. ... a buncha stuff ... > The Gauss-Bonnet Theorem expresses the Euler characteristic of a > manifold in terms of the integral of the Gaussian curvature. This > is sufficient to guarantee that the surface has Euler characteristic > equal to 2, since it must be positive, and all compact connected > surfaces have Euler characteristic given by 2 - 2g, where g is the > genus (or number of handles, for a sphere with n handles). > ^^^ Of course, that would be a sphere with g handles. ... the rest deleted ... Dale. === Subject: Re: Surprising Pattern of Florida's Election Results > I cannot say anything about US politics, but isn't it obvious that the > *only* way to prevent vote fraud is by using machines that produce > cryptographic paper trails, which can be publicly verified afterwards? Try this: You vote, machine prints a ballot that says ATTILLA THE HUN or whatever your choice was, voter puts the ballot in a box. The ballots are the official vote. Hard to cheat without getting caught. As far identifying the voter I think there is always some way to do that with machines. So the simple hand ballot is better. Computers alone is the worst system in every way except getting a quick result. === Subject: Re: Surprising Pattern of Florida's Election Results > I cannot say anything about US politics, but isn't it obvious that the > *only* way to prevent vote fraud is by using machines that produce > cryptographic paper trails, which can be publicly verified afterwards? > Try this: You vote, machine prints a ballot that says ATTILLA THE HUN > or whatever your choice was, voter puts the ballot in a box. The > ballots are the official vote. Hard to cheat without getting caught. > As far identifying the voter I think there is always some way to do > that with machines. So the simple hand ballot is better. > Computers alone is the worst system in every way except getting a > quick result. --------------------------------------------------------- <<<< Newsclip Autopsy >> FOCUS: VOTERGATE MIS-TRUTH: Major Newspapers STILL Not Telling The Whole Story About Ohio Recount (One is even lying about it). http://newsclipautopsy.blogspot.com/ ---------------------------------------------------------- === Subject: Re: Surprising Pattern of Florida's Election Results > I cannot say anything about US politics, but isn't it obvious that the > *only* way to prevent vote fraud is by using machines that produce > cryptographic paper trails, which can be publicly verified afterwards? > It's a no-brainer, but why can't US, arguably the biggest IT industry > in the world, manage it? After all, you've got some of the brightest > crypto researchers in the world. They could solve the theory part in > one day. And IBM could design the machine in two months. These > discussions are quite unfortunate. > Doesn't that open the door to someone (either a hacker or insider) being > able to determine how every person voted? Well gosh, isn't that true with regular paper ballots anyway? It's true - the only way to have fair elections is to have some sort of verifiable paper trail (if not paper, then some other medium, but definitely not bit bucket in computer terminology). === Subject: Re: Surprising Pattern of Florida's Election Results > Doesn't that open the door to someone (either a hacker or insider) being > able to determine how every person voted? > Well gosh, isn't that true with regular paper ballots anyway? It's true - > the > only way to have fair elections is to have some sort of verifiable paper > trail > (if not paper, then some other medium, but definitely not bit bucket in > computer terminology). How are you going to link a paper ballot to an individual voter -- finger prints? I would rather trust my anonymity to a box full of paper ballots than to some encrypted ID. With the encryption system, if the decryption method and key are leaked, stolen or sold, then every vote by every voter becomes immediately accessible. I can see a web page Find out how your neighbors, family and co-workers voted. It is essential that there is no method that can be used to determine how individuals voted. -- Phil Sherrod (phil.sherrod 'at' sandh.com) http://www.dtreg.com (decision tree modeling) http://www.nlreg.com (nonlinear regression) http://www.NewsRover.com (Usenet newsreader) http://www.LogRover.com (Web statistics analysis) === Subject: Re: Surprising Pattern of Florida's Election Results > Doesn't that open the door to someone (either a hacker or insider) being > able to determine how every person voted? > Well gosh, isn't that true with regular paper ballots anyway? It's true - > the > only way to have fair elections is to have some sort of verifiable paper > trail > (if not paper, then some other medium, but definitely not bit bucket in > computer terminology). > How are you going to link a paper ballot to an individual voter -- finger > prints? > I would rather trust my anonymity to a box full of paper ballots than to > some encrypted ID. I agree, sort of. But it would be relatively simple to figure out who voted with the paper ballots. In fact, that's why I'd prefer it. With the encryption, it's more sophisticated, so it's harder to know who's spoofing who. But anyone with access to the ballot box has access to who voted for who. You go there and check in with your vote card or id, so they know who you are. Someone makes a little note about what you look like. Or in sparsely populated or unbusy precincts, they merely pay attention. Then when you put in your ballot, they simply keep a list of the order that people put their ballots in. At the end of the day, simply take out the ballots and voila - a list in order with your list of names. Very very easy. === Subject: Re: Surprising Pattern of Florida's Election Results <90b9p0dtd91jl0t2nht8gt6niom4tehuaj@4ax.com> <41ab9d84_4@news1.prserv.net> <8cCdncH31tZwCTHcRVn-3w@giganews.com> <41ac9fd3_4@news1.prserv.net> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> How are you going to link a paper ballot to an individual voter -- >> finger prints? >> I would rather trust my anonymity to a box full of paper ballots >> than to some encrypted ID. > I agree, sort of. But it would be relatively simple to figure out > who voted with the paper ballots. In fact, that's why I'd prefer > it. With the encryption, it's more sophisticated, so it's harder to > know who's spoofing who. But anyone with access to the ballot box > has access to who voted for who. You go there and check in with > your vote card or id, so they know who you are. Someone makes a > little note about what you look like. Or in sparsely populated or > unbusy precincts, they merely pay attention. Then when you put in > your ballot, they simply keep a list of the order that people put > their ballots in. At the end of the day, simply take out the > ballots and voila - a list in order with your list of names. Very > very easy. You have never seen a ballot box, right? No, the ballots are not neatly stacked. You have also never participated as a voting official, right? The idea that any single person has a chance to get a cleanly sorted stack of ballots out while the other officials are in the same room is a bit ridiculous. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Surprising Pattern of Florida's Election Results > You have never seen a ballot box, right? No, the ballots are not > neatly stacked. You have also never participated as a voting > official, right? The idea that any single person has a chance to get > a cleanly sorted stack of ballots out while the other officials are in > the same room is a bit ridiculous. 1) It's not clear whether or not I've seen a ballot box, but one thing is for sure - you've never seen the ones we use around here 2) I didn't say it would take just one person 3) haven't you ever heard of people sneaking into files on their own with no one looking? === Subject: Re: Surprising Pattern of Florida's Election Results :> You have never seen a ballot box, right? No, the ballots are not :> neatly stacked. You have also never participated as a voting :> official, right? The idea that any single person has a chance to get :> a cleanly sorted stack of ballots out while the other officials are in :> the same room is a bit ridiculous. : 1) It's not clear whether or not I've seen a ballot box, but one thing is for : sure - you've never seen the ones we use around here And it seems on one else here has seen the sort of ballot box you are talking about. : 2) I didn't say it would take just one person You strongly implied it. Your first response in this thread was as follows: >> Doesn't that open the door to someone (either a hacker or insider) being >> able to determine how every person voted? > Well gosh, isn't that true with regular paper ballots anyway? It's true - > the only way to have fair elections is to have some sort of > verifiable paper trail (if not paper, then some other medium, but > definitely not bit bucket in computer terminology). The complaint about electronic voting was that *someone* could determine how every person voted. You responded that this was true of paper ballots. Now you are claiming that you did not mean 'someone' but a group of people with presumably unlimited and unsupervised access to the ballots. : 3) haven't you ever heard of people sneaking into files on their own with no one : looking? What files are you talking about? There are no files that contain the results of your vote when paper ballots are used. Stephen === Subject: Re: Surprising Pattern of Florida's Election Results > The complaint about electronic voting was that *someone* could > determine how every person voted. You responded that this was > true of paper ballots. Now you are claiming that you did not mean > 'someone' but a group of people with presumably unlimited and > unsupervised access to the ballots. What I said was that it didn't rule out getting someone else to turn a blind eye. And saying presumably unlimited is ridiculous. > : 3) haven't you ever heard of people sneaking into files on their own with no one > : looking? > What files are you talking about? There are no files that contain > the results of your vote when paper ballots are used. That's not the point. The point - which is not difficult to follow - is that people have many many many many times throughout history gotten access to things they weren't supposed to. === Subject: Re: Surprising Pattern of Florida's Election Results > Doesn't that open the door to someone (either a hacker or insider) being > able to determine how every person voted? > Well gosh, isn't that true with regular paper ballots anyway? It's true - > the > only way to have fair elections is to have some sort of verifiable paper > trail > (if not paper, then some other medium, but definitely not bit bucket in > computer terminology). > How are you going to link a paper ballot to an individual voter -- finger > prints? > I would rather trust my anonymity to a box full of paper ballots than to > some encrypted ID. > I agree, sort of. But it would be relatively simple to figure out who voted > with the paper ballots. In fact, that's why I'd prefer it. With the > encryption, it's more sophisticated, so it's harder to know who's spoofing who. > But anyone with access to the ballot box has access to who voted for who. You > go there and check in with your vote card or id, so they know who you are. > Someone makes a little note about what you look like. Or in sparsely populated > or unbusy precincts, they merely pay attention. Then when you put in your > ballot, they simply keep a list of the order that people put their ballots in. > At the end of the day, simply take out the ballots and voila - a list in order > with your list of names. Very very easy. Ballot boxes do not hold ballots in the order they came in. There is usually more than one box, too. With a computer, all you have to do is record the time of each vote and security is broken. === Subject: Re: Surprising Pattern of Florida's Election Results > Ballot boxes do not hold ballots in the order they came in. There is > usually more than one box, too. > With a computer, all you have to do is record the time of each vote > and security is broken. I agree on both points. If there were even a couple of out-of-order ballots in the box, then the entire order for all voters would be shifted. Also, with paper ballots, voter identification would be a slow, tedious, error-prone process. However, with a computer and the encryption key, you could access votes for all voters in a fraction of a second. -- Phil Sherrod (phil.sherrod 'at' sandh.com) http://www.dtreg.com (decision tree modeling) http://www.nlreg.com (nonlinear regression) http://www.NewsRover.com (Usenet newsreader) http://www.LogRover.com (Web statistics analysis) === Subject: Re: Surprising Pattern of Florida's Election Results Gentlemen, it is time to define ballot box in this discussion. Let us call low-tech ballot box one that cannot preserve order of ballots, which are counted manually by the local committee. (I have participated twice in such committees. Btw, our ballots were in envelopes, the box was made of transparent plexiglass, and it was shaken well before opening.) A high-tech ballot box involves conveniences. AfaIcs, they might conceivably serve to learn what one has voted (or didn't vote). If we cannot agree on which kind of voting system we propose or criticize, let us have two different threads! ~George Kahrimanis === Subject: Re: Surprising Pattern of Florida's Election Results > If we cannot agree on which kind of voting system we propose or > criticize, let us have two different threads! Gosh, it wouldn't be Usenet if we were all on the same page.... === Subject: Re: Surprising Pattern of Florida's Election Results > Ballot boxes do not hold ballots in the order they came in. There is > usually more than one box, too. Many do, and often there is not more than one. === Subject: Re: Surprising Pattern of Florida's Election Results > I would rather trust my anonymity to a box full of paper ballots than to > some encrypted ID. > I agree, sort of. But it would be relatively simple to figure out who > voted > with the paper ballots. In fact, that's why I'd prefer it. With the > encryption, it's more sophisticated, so it's harder to know who's spoofing > who. > But anyone with access to the ballot box has access to who voted for who. > You > go there and check in with your vote card or id, so they know who you are. > Someone makes a little note about what you look like. Or in sparsely > populated > or unbusy precincts, they merely pay attention. Then when you put in your > ballot, they simply keep a list of the order that people put their ballots > in. > At the end of the day, simply take out the ballots and voila - a list in > order > with your list of names. Very very easy. If they stacked them in a pile you could do that. But ballot boxes are just that -- boxes. They have a slot in the top, and they are large enough to hold all of the ballots that are cast at the polling place. Hence, the order of the ballots in the box is relatively random. A mechanical lever machine is better: There is just a counter that gives the totals for each candidate. The computer system has the most serious flaws. If the vote record for each voter is stored in some sort of encrypted form, then if the decryption system is leaked, stolen or sold, an accurate list of votes cast by ALL voters instantly becomes available. I don't care how good the encryption system is; if a person has the key, then they can be bribed or the key can be stolen. -- Phil Sherrod (phil.sherrod 'at' sandh.com) http://www.dtreg.com (decision tree modeling) http://www.nlreg.com (nonlinear regression) http://www.NewsRover.com (Usenet newsreader) http://www.LogRover.com (Web statistics analysis) === Subject: Re: Surprising Pattern of Florida's Election Results >If they stacked them in a pile you could do that. But ballot boxes are just >that -- boxes. They have a slot in the top, and they are large enough to >hold all of the ballots that are cast at the polling place. Hence, the >order of the ballots in the box is relatively random. It depends on the box and the slot. Elections for the Mayor of London and Greater London Assembly had boxes with slots at the side so you could slide your (not folded) ballot paper in, making it easier for the counting machine to scan later. My guess is that in the majority of cases the votes just piled up neatly in order. === Subject: Re: Surprising Pattern of Florida's Election Results > Someone makes a little note about what you look like. Or in sparsely > populated > or unbusy precincts, they merely pay attention. Then when you put in your > ballot, they simply keep a list of the order that people put their ballots > in. > At the end of the day, simply take out the ballots and voila - a list in > order > with your list of names. Very very easy. > If they stacked them in a pile you could do that. But ballot boxes are just > that -- boxes. They have a slot in the top, and they are large enough to > hold all of the ballots that are cast at the polling place. Hence, the > order of the ballots in the box is relatively random. If you're talking about ballot papers that are small relative to the size of the box, then I agree. Where I vote, the ballots are very large (11x14) and stack right up in the machine. === Subject: Re: Surprising Pattern of Florida's Election Results <90b9p0dtd91jl0t2nht8gt6niom4tehuaj@4ax.com> <41ab9d84_4@news1.prserv.net> <8cCdncH31tZwCTHcRVn-3w@giganews.com> <41ac9fd3_4@news1.prserv.net> <41acee3b_2@news1.prserv.net> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > If you're talking about ballot papers that are small relative to the > size of the box, then I agree. Where I vote, the ballots are very > large (11x14) and stack right up in the machine. That sounds completely lunatic. While there are some ballots with an even quite larger size in some regions here, you have to fold them small enough to make them fit the slot in the ballot box. They don't stack neatly from there. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Surprising Pattern of Florida's Election Results > If you're talking about ballot papers that are small relative to the > size of the box, then I agree. Where I vote, the ballots are very > large (11x14) and stack right up in the machine. > That sounds completely lunatic. While there are some ballots with an > even quite larger size in some regions here, you have to fold them > small enough to make them fit the slot in the ballot box. They don't > stack neatly from there. Ours go into an optical reader - they don't get dropped in a slot like back in the 1800s! === Subject: Re: Surprising Pattern of Florida's Election Results <90b9p0dtd91jl0t2nht8gt6niom4tehuaj@4ax.com> <41ab9d84_4@news1.prserv.net> <8cCdncH31tZwCTHcRVn-3w@giganews.com> <41ac9fd3_4@news1.prserv.net> <41acee3b_2@news1.prserv.net> <41ae0031_1@news1.prserv.net> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> If you're talking about ballot papers that are small relative to >> the size of the box, then I agree. Where I vote, the ballots are >> very large (11x14) and stack right up in the machine. >> That sounds completely lunatic. While there are some ballots with >> an even quite larger size in some regions here, you have to fold >> them small enough to make them fit the slot in the ballot box. >> They don't stack neatly from there. > Ours go into an optical reader - they don't get dropped in a slot > like back in the 1800s! In a lot of ways, the American democracy is not what is was back in the 1800s. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Surprising Pattern of Florida's Election Results > I agree, sort of. But it would be relatively simple to figure out who voted > with the paper ballots. In fact, that's why I'd prefer it. With the > encryption, it's more sophisticated, so it's harder to know who's spoofing who. > But anyone with access to the ballot box has access to who voted for who. You > go there and check in with your vote card or id, so they know who you are. > Someone makes a little note about what you look like. Or in sparsely populated > or unbusy precincts, they merely pay attention. Then when you put in your > ballot, they simply keep a list of the order that people put their ballots in. > At the end of the day, simply take out the ballots and voila - a list in order > with your list of names. Very very easy. Not with Ukrainian disappearing ink! Nobody can tell who you voted for, not even the tellers. NigelH === Subject: Re: Surprising Pattern of Florida's Election Results >> I cannot say anything about US politics, but isn't it obvious that the >> *only* way to prevent vote fraud is by using machines that produce >> cryptographic paper trails, which can be publicly verified afterwards? >> It's a no-brainer, but why can't US, arguably the biggest IT industry >> in the world, manage it? After all, you've got some of the brightest >> crypto researchers in the world. They could solve the theory part in >> one day. And IBM could design the machine in two months. These >> discussions are quite unfortunate. >> Doesn't that open the door to someone (either a hacker or insider) being >> able to determine how every person voted? > Well gosh, isn't that true with regular paper ballots anyway? I don't see how. Perhaps you are thinking of doing fingerprint analysis on each and every ballot? How do you sort out the prints of the poll workers? Besides, I don't think we have yet reached the point where every voter has fingerprints on file with the government. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Surprising Pattern of Florida's Election Results > Doesn't that open the door to someone (either a hacker or insider) being >> able to determine how every person voted? > Well gosh, isn't that true with regular paper ballots anyway? > I don't see how. Perhaps you are thinking of doing fingerprint analysis > on each and every ballot? Fingerprints? What are you talking about? All they need to do is look at the votes on the paper and then look at the face of the person who just handed them the paper. === Subject: Re: Surprising Pattern of Florida's Election Results > Well gosh, isn't that true with regular paper ballots anyway? > I don't see how. Perhaps you are thinking of doing fingerprint analysis > on each and every ballot? > Fingerprints? What are you talking about? All they need to do is look at > the > votes on the paper and then look at the face of the person who just handed > them the paper. Where do you vote? Every place I've ever seen that uses paper ballots (including Russia) has a box with a slot where you insert your ballot. -- Phil Sherrod (phil.sherrod 'at' sandh.com) http://www.dtreg.com (decision tree modeling) http://www.nlreg.com (nonlinear regression) http://www.NewsRover.com (Usenet newsreader) http://www.LogRover.com (Web statistics analysis) === Subject: Re: Surprising Pattern of Florida's Election Results > Well gosh, isn't that true with regular paper ballots anyway? > I don't see how. Perhaps you are thinking of doing fingerprint analysis > on each and every ballot? > Fingerprints? What are you talking about? All they need to do is look at > the > votes on the paper and then look at the face of the person who just handed > them the paper. > Where do you vote? Every place I've ever seen that uses paper ballots > (including Russia) has a box with a slot where you insert your ballot. Sure, but what happens to the box? It could easily be figured out, if someone really wanted to. === Subject: Re: Surprising Pattern of Florida's Election Results <90b9p0dtd91jl0t2nht8gt6niom4tehuaj@4ax.com> <41ab9d84_4@news1.prserv.net> <41ac8f94_1@news1.prserv.net> <41ac9eef_3@news1.prserv.net> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> Fingerprints? What are you talking about? All they need to do >> is look at the votes on the paper and then look at the face of >> the person who just handed them the paper. >> Where do you vote? Every place I've ever seen that uses paper >> ballots (including Russia) has a box with a slot where you insert >> your ballot. > Sure, but what happens to the box? It could easily be figured out, > if someone really wanted to. This application of the word easily in a new one to me. The people supervising the vote are community members from different parties, and the process is so transparent that tampering with it would be obvious, and have consequences just in a single voting district. Do you think you could design a ballot box that would tamper with votes, maybe by swallowing them and replacing them with prebuilt votes, getting the counts exactly right? Those boxes are checked by the voting officials from several parties. If some manufacturer would be caught with such stuff, he'd go to jail. In contrast, the people building vote machines are from a single source with vested interests, and there is no way that officials can check the results. If votes go missing, there is no way to take apart the machine and look for hidden vaults. There are no records, so you can't go to jail. In particular, if people swallow technical error as an explanation for stuff like a voting district having a negative count of Democratic votes. With wood boxes, any effective manipulation would convince any judge of premeditated malice. With computers, everybody accepts complete junk as plausible. Even when registered voting software gets replaced unofficially before an election (if that is not premeditated malice, I don't know what is), the responsible persons for that do not even get a slap on the wrist. It's just unbelievable. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Surprising Pattern of Florida's Election Results > Sure, but what happens to the box? It could easily be figured out, > if someone really wanted to. > This application of the word easily in a new one to me. The people > supervising the vote are community members from different parties, and > the process is so transparent that tampering with it would be obvious, > and have consequences just in a single voting district. > Do you think you could design a ballot box that would tamper with > votes, maybe by swallowing them and replacing them with prebuilt > votes, getting the counts exactly right? Those boxes are checked by > the voting officials from several parties. If some manufacturer would > be caught with such stuff, he'd go to jail. *Anyone* committing voting fraud will go to jail. That's what you don't seem to be acknowledging. Any type of voter fraud is easy, or difficult, depending on your point of view. But certainly just looking at ballots to see who is casting them is not one of the more difficult tasks, relatively. > In contrast, the people building vote machines are from a single > source with vested interests, and there is no way that officials can > check the results. If votes go missing, there is no way to take apart > the machine and look for hidden vaults. There are no records, so you > can't go to jail. Hmmm. I can't agree with that. > With computers, everybody accepts complete junk as plausible. Even > when registered voting software gets replaced unofficially before an > election (if that is not premeditated malice, I don't know what is), > the responsible persons for that do not even get a slap on the wrist. Computer crimes are just as bad. Just recently a guy was arrested (convicted caught. === Subject: Re: Surprising Pattern of Florida's Election Results :> This application of the word easily in a new one to me. The people :> supervising the vote are community members from different parties, and :> the process is so transparent that tampering with it would be obvious, :> and have consequences just in a single voting district. : *Anyone* committing voting fraud will go to jail. That's what you don't seem to : be acknowledging. Any type of voter fraud is easy, or difficult, depending on : your point of view. But certainly just looking at ballots to see who is casting : them is not one of the more difficult tasks, relatively. I do not see that it is an easy task. In all the places I have voted the person who checks your id, the person who gives you the ballot, and the person who gives you the little 'I voted' sticker after I drop my covered ballot into the box are all different people. Also because the have several voting booths the order in which ballots are handed out to voters is not the same as the order the ballots are placed in the ballot box. I agree with others that it is quite a stretch of the word 'easily' to claim that you can 'easily' figure out how an individual voted in this system. Stephen === Subject: Re: Surprising Pattern of Florida's Election Results > I do not see that it is an easy task. In all the places I have > voted the person who checks your id, the person who gives > you the ballot, and the person who gives you the little 'I voted' > sticker after I drop my covered ballot into the box are all different > people. Most voter fraud requires more than 1 person to be in on it. > Also because the have several voting booths the order > in which ballots are handed out to voters is not the same as > the order the ballots are placed in the ballot box. Already explained in a different post. > I agree > with others that it is quite a stretch of the word 'easily' > to claim that you can 'easily' figure out how an individual > voted in this system. It's extremely easy, logistically. If you told me to do it (and it were legal), I would have no trouble at all putting faces and names with ballots. === Subject: Re: Surprising Pattern of Florida's Election Results :> I do not see that it is an easy task. In all the places I have :> voted the person who checks your id, the person who gives :> you the ballot, and the person who gives you the little 'I voted' :> sticker after I drop my covered ballot into the box are all different :> people. : Most voter fraud requires more than 1 person to be in on it. :> Also because the have several voting booths the order :> in which ballots are handed out to voters is not the same as :> the order the ballots are placed in the ballot box. : Already explained in a different post. Not really. The ballots are going to come out of the ballot box in a largely random order. :> I agree :> with others that it is quite a stretch of the word 'easily' :> to claim that you can 'easily' figure out how an individual :> voted in this system. : It's extremely easy, logistically. If you told me to do it (and it were legal), : I would have no trouble at all putting faces and names with ballots. I think you would find it a lot harder than you realize. Stephen === Subject: Re: Surprising Pattern of Florida's Election Results > :> Also because the have several voting booths the order > :> in which ballots are handed out to voters is not the same as > :> the order the ballots are placed in the ballot box. > : Already explained in a different post. > Not really. The ballots are going to come out of > the ballot box in a largely random order. Depends on the ballots and box. In my precinct, they go in in precise order and stay that way. > : It's extremely easy, logistically. If you told me to do it (and it were legal), > : I would have no trouble at all putting faces and names with ballots. > I think you would find it a lot harder than you realize. Don't think so, unless we are confusing simple with easy. === Subject: Re: Surprising Pattern of Florida's Election Results <41ab9d84_4@news1.prserv.net> <41ac8f94_1@news1.prserv.net> <41ac9eef_3@news1.prserv.net> <41acd038_1@news1.prserv.net> <41acedee_2@news1.prserv.net> <41ae000b_2@news1.prserv.net> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> :> Also because the have several voting booths the order >> :> in which ballots are handed out to voters is not the same as >> :> the order the ballots are placed in the ballot box. >> : Already explained in a different post. >> Not really. The ballots are going to come out of >> the ballot box in a largely random order. > Depends on the ballots and box. In my precinct, they go in in > precise order and stay that way. Then I'd say that you have a cross-party coalition of incompetence in your precinct. Nevertheless, it will be difficult to pay particular attention to the order of some votes (and don't forget that at some point of time absentee ballots come into play as well) without somebody else from the voting officials noticing what you are up to. When in doubt, I'd rather bet that I'd be able to steal something from a shop unnoticed. Less eyes involved there. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Surprising Pattern of Florida's Election Results > When in doubt, I'd rather bet that I'd be able to steal something from > a shop unnoticed. Less eyes involved there. Depends quite a bit on exactly what sort of shop that would be :-) === Subject: Re: Surprising Pattern of Florida's Election Results <90b9p0dtd91jl0t2nht8gt6niom4tehuaj@4ax.com> <41ab9d84_4@news1.prserv.net> <41ac8f94_1@news1.prserv.net> <41ac9eef_3@news1.prserv.net> <41acd038_1@news1.prserv.net> <41acedee_2@news1.prserv.net> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> I do not see that it is an easy task. In all the places I have >> voted the person who checks your id, the person who gives you the >> ballot, and the person who gives you the little 'I voted' sticker >> after I drop my covered ballot into the box are all different >> people. > Most voter fraud requires more than 1 person to be in on it. And will affect just a single voting precinct. The balance between probability of getting punished and the resulting effect is just much worse than being able to tamper with elections all over the States without any tangible trail (apart from negative votes, and those even _were_ present in a few precincts) in one fell swoop. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Surprising Pattern of Florida's Election Results > Most voter fraud requires more than 1 person to be in on it. > And will affect just a single voting precinct. The balance between > probability of getting punished and the resulting effect is just much > worse than being able to tamper with elections all over the States > without any tangible trail (apart from negative votes, and those even > _were_ present in a few precincts) in one fell swoop. Well, any tangible trail at the voting booth is one thing. any tangible trail at all is another. Besides, it just takes a key precinct here and there to confuse the election. Florida, obviously. Or this year, after a recount of millions of votes, one governor (Washington state?) won by something around 40 votes. === Subject: Re: Surprising Pattern of Florida's Election Results > Doesn't that open the door to someone (either a hacker or insider) being >> able to determine how every person voted? > Well gosh, isn't that true with regular paper ballots anyway? > I don't see how. Perhaps you are thinking of doing fingerprint analysis > on each and every ballot? > Fingerprints? What are you talking about? All they need to do is look at > the > votes on the paper and then look at the face of the person who just handed > them > the paper. I guess you've never voted with a paper ballot. When I have, it worked as follows: In the privacy of the voting booth I insert my ballot into a cardboard sleeve to carry it to the ballot box. Blank portions of my ballot extend about an inch above and below the sleeve. The ballot box is locked and has a slot on top. The official places the sleeve over the slot so the bottom projecting edge is in the slot, and taps the top projecting edge of my ballot, which slides into the slot without being visible at any time. This is done in front of me, and right next to the bipartisan representatives (at least one from each of the two major parties) who sign in voters. If the ballot officials remember my face out of the hundreds who voted, they'll know that I voted but not for whom. === Subject: Re: Surprising Pattern of Florida's Election Results > I don't see how. Perhaps you are thinking of doing fingerprint analysis > on each and every ballot? > Fingerprints? What are you talking about? All they need to do is look at > the > votes on the paper and then look at the face of the person who just handed > them > the paper. > I guess you've never voted with a paper ballot. When I have, it worked > as follows: In the privacy of the voting booth I insert my ballot into a > cardboard sleeve to carry it to the ballot box. Blank portions of my > ballot extend about an inch above and below the sleeve. The ballot box > is locked and has a slot on top. The official places the sleeve over > the slot so the bottom projecting edge is in the slot, and taps the top > projecting edge of my ballot, which slides into the slot without being > visible at any time. This is done in front of me, and right next to the > bipartisan representatives (at least one from each of the two major > parties) who sign in voters. Sure I have, but it was different from your way. But in any case, we're talking about fraud or corruption. If no one was corrupt, then there wouldn't be any need for any of this anyway. This just makes it more of a challenge for those who want to defraud. We make marks on a paper ballot, and simply hand that ballot to someone who puts it into an optical reader, or we put it in the reader with them watching. They could either sneak a look, or simply have access to the load of paper ballots in the reader box. They could easily match up the order of the ballots with a list of names they have in the order we checked in when we got there. Quite easy if you really wanted to do it. === Subject: Re: Surprising Pattern of Florida's Election Results >The original intent of the Founding Fathers was that state >legislators would choose the electors. They had no idea that >political parties would form. They must have had some idea, or George Washington wouldn't have found it necessary to warn us against them. He was, of course, correct in his warning, but, alas, we did not listen. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Surprising Pattern of Florida's Election Results >The original intent of the Founding Fathers was that state >legislators would choose the electors. They had no idea that >political parties would form. > They must have had some idea, or George Washington wouldn't have found > it necessary to warn us against them. He was, of course, correct in > his warning, but, alas, we did not listen. But did that warning come before or after the Constitution? I bet it was after. Besides, I don't see what can or should be done to prevent people from associating for their mutual interest. === Subject: Re: Surprising Pattern of Florida's Election Results <41aa86ab$20$fuzhry+tra$mr2ice@news.patriot.net> at 11:30 PM, frisbieinstein@yahoo.com (Patrick Powers) said: >But did that warning come before or after the Constitution? Well after. >Besides, I don't see what can or should be done to prevent people >from associating for their mutual interest. The problem isn't people associating; the problem is giving those associations special privileges. The fact that the association likes somebody should not make it easier for them to get on the ballot. Try running as an independent and you will quickly see their abuse of power. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Surprising Pattern of Florida's Election Results The election results had been forecast with great precision in an authors correctly noted the increased popularity not only of Saakashvili's party, but also of the Revival Party of Aslan Abashidze, president of the Autonomous Republic of Ajaria in Georgia's West. Meanwhile, the President's Citizens' Union, now without its former general secretary Zurab Zhvania (who had joined Mrs. Burjanadze in her brand new party project), had merged with several well-established partiesÖIrina Sarishvili-Chanturia's National Democratic Union, and Vakhtang Rcheulishvili's Socialist PartyÖinto the Union for New Georgia. On the eve of the elections, the President's bloc was gaining additional support from ethnic minorities, who fear Saakashvili's nationalist banners. And even the remains of Zviad Gamsakhurdia's movement expressed support for Shevardnadze, though the latter was first made President in the wake of Gamsakhurdia's overthrow in 1991. Meanwhile, the Revival Party had almost unanimous support in Ajaria's main city, Batumi, as well as growing influence in Tbilisi and in the Armenian-populated district of Javakheti. But Saakashvili had proclaimed, months before, that he was organizing a velvet revolution to remove Shevardnadze. And the exit polls said the elections were a fakery. These exit polls were conducted not by the Georgians, nor by official observers from the United States, Russia, or the EU. They were provided by a Washington-based polling > They must have had some idea, or George Washington wouldn't have found > it necessary to warn us against them. He was, of course, correct in > his warning, but, alas, we did not listen. --Advice 0.05; free, if wrong, again! http://tarpley.net/bush6.htm === Subject: question about lowest sampling rate... Hi all, Suppose x(t) has bandwith bandlimited in [-B, B]... so the lowest sampling rate for x(t) is Fs=2B... Does this matter when x(t) is real or complex-valued? Moreover, for (x(t))^2, the bandwidth is [-2B, 2B], the lowest Fs=4B no matter when x(t) is real or complex-valued. Am I right? More interestingly, for (x(t))^3, the bandwidth is [-3B, 3B], but we can still use Fs=2B(the same as x(t))... if x(t) is real-valued... Am I right? Does this result hold if x(t) is complex-valued? -L === Subject: Re: question about lowest sampling rate... > Hi all, > Suppose x(t) has bandwith bandlimited in [-B, B]... so the lowest sampling > rate for x(t) is Fs=2B... > Does this matter when x(t) is real or complex-valued? > Moreover, for (x(t))^2, the bandwidth is [-2B, 2B], the lowest Fs=4B no > matter when x(t) is real or complex-valued. Am I right? > More interestingly, for (x(t))^3, the bandwidth is [-3B, 3B], but we can > still use Fs=2B(the same as x(t))... if x(t) is real-valued... Am I right? > Does this result hold if x(t) is complex-valued? > -L In the thread Nyquist rate for sampling complex-valued data? starting at 12 November this item was discussed already. You may find some answers there. Jeroen === Subject: Cohomology of Lie Groups and Lie Algebras Does anyone know for a modern source of the material contained in the groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85-124 ? nojb. === Subject: Who is pulling my leg by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAUCnMh28126; 'Linear homogeneous functions' containing nothing but political views. I did not write ever a line of them.I would like to possess a so fluent english! Would mind explaining to me how all that might happen. Very sincerely, Alain. === Subject: Re: Who is pulling my leg This is an unmoderated newsgroup. >'Linear homogeneous functions' containing nothing but political views. >I did not write ever a line of them.I would like to possess a so fluent english! >Would mind explaining to me how all that might happen. to each newsgroup using forged names and e-mail addresses of regular posters and recent subject lines. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Who is pulling my leg > 'Linear homogeneous functions' containing nothing but political views. > I did not write ever a line of them.I would like to possess a so fluent > english! > Would mind explaining to me how all that might happen. > Very sincerely, > Alain. The political posts do not appear on the news server here. Nor on mathforum, http://mathforum.org/epigone/sci.math.research But they are on Google. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Who is pulling my leg > 'Linear homogeneous functions' containing nothing but political views. > I did not write ever a line of them.I would like to possess a so fluent > english! Would mind explaining to me how all that might happen. Sci.math has no moderator, but these appeared on the moderated sci.math.research. Clearly someone hacked s.m.r. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: The flux theory of gravitation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAUCnNh28178; > Since some of you have made references to the Flux Theory of Gravitation > that I developed, >>I don't remember any of us doing that .... >Lee Rudolph I was referring to some regulars here in Sci Math of the likes of Ullrich and Israel. They seem to be mathematicians. I requested them to define NUMBER a week ago but I have not seen their names ever since. Are they undergrads or mathematicians from antiquity? E. E. Escultura === Subject: Re: The flux theory of gravitation >> Since some of you have made references to the Flux Theory of Gravitation >> that I developed, >I don't remember any of us doing that .... >>Lee Rudolph >I was referring to some regulars here in Sci Math of the likes of >Ullrich and Israel. Uh, I don't remember _me_ making reference to the Flux Theory of Gravitation either. >They seem to be mathematicians. I requested >them to define NUMBER a week ago but I have not seen their names >ever since. Are they undergrads or mathematicians from antiquity? No, we're lunatics with delusions of mathematical grandeur. I am, anyway, I suppose I shouldn't speak for Israel. >E. E. Escultura ************************ David C. Ullrich === Subject: Re: Escultura affair: publication scandal by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAUCnN228211; >>He's back! >This question of whether 1 = 0.99... which I used to call Ullrich-Israel >equation has been resolved not only in the many papers I published but >>The name does have a nice ring to it, but in all modesty I must >>point out that I don't deserve any of the credit for this equation. >Very gracious of you to say this. So the credit's mine, all mine! >>Robert Israel israel@math.ubc.ca >>Department of Mathematics http://www.math.ubc.ca/~israel >>University of British Columbia Vancouver, BC, Canada >************************ >David C. Ullrich Where are you David and Robert? You seem to be out of gas. I asked you a week ago to define a NUMBER because you seem to be experts. I hsve not seen your name since then. Am I dealing with undergrads here or mathematicians from antiquity? E. E. Escultur University of the Philippines === Subject: Re: Functional Analysis: Equivalent of Taylor series for operators by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAUCnOA28252; >>thank you foryour answer. >>I am interested by both questions: >>Let T be an operator between two Banach spaces (or Hilbert spaces), >>what are the conditions on T and on these spaces such that T admits a >>Taylor series expansion, and how to perform this expansion. >The _other_ question is the topic of something called the >functional calculus, which you can read about in many >books on functional analysis. >This question makes less sense to me offhand. A Taylor >expansion for T would look something like > T(x) = sum c_n x^n, >but if x is in a Banach space there's no such thing as >x^n. >>I would appreciate any references about this. >Hi all, >is there an equivalent of Taylor series for operators between Banach >Spaces ?- if yes, what is the name of this equivalent, and what are >the conditions on this operators ? > > I can think of at least two things the question might mean > (does an operator have a Taylor series/can we apply a Taylor > series to an operator). Try to be a little more specific about > the question... > > > > ************************ > > David C. Ullrich >************************ >David C. Ullrich I believe it will be interesting to give us a clear example of the operator you are thinking about. Sometimes there are ways to 'approach' for computing aims an operator... Alain. === Subject: Re: Functional Analysis: Equivalent of Taylor series for operators dear all, discussion. For Alain: I am not thinkg of any particular operator, I am thinking of the class of operators admitting such a decomposition. === Subject: Graph coloring problem Does anyone recognize the following graph coloring problem: A directed graph (V, E) and a set of colors. Each edge in E has a set of color pairs that are allowed for the nodes that it connects. Combinations that are not in the list are not allowed. For example: V = {1, 2, 3} E = {<1, 2>, <1, 3>, <2, 3>} allowed(<1, 2>) = {, } allowed(<1, 3>) = {, } allowed(<2, 3>) = {, } A valid coloring is color(1) = R color(2) = B color(3) = G I am looking for an algorithm that assigns a color to every node. I am also looking for an algorithm that assigns as much colors as possible to every node, for example: colors(1) = {R, B} colors(2) = {B} colors(3) = {G} Jan === Subject: Gauss formula could you give me a hand please to proove the Formula of Gauss ( for gamma function)? check this link: http://140.122.140.53/~yclin/02a/gcx/gcx20.pdf the formula is in the last page, i get a similar expresion to the one used for Legendre, but i can't solve a, b ( because i cannot evaluate the product of gamma functions). === Subject: Re: help me. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAUDsjn02097; >my problem is to solve some exercises from w. rudin's book real and >complex analysis (chapters 1,2,3,6,7 and 8). where i can find >solutions for that problems? >your faitfully >h.r.sahebi That would defeat the purpose of the assignment, eh? It is to solve the exercises, not copy the solutions. === Subject: Takers and leavers: Synthesis The way I see this, the Western society is torn between two lines of thought. One is the unconditional growth of the civilization, whatever the consequences for the planet and for the future generations. The other is a reaction against it: a return-to-the-roots hippie environmentalism that seeks to Right now, the two work together in what I consider the worst possible manner. The first group engages in unconditional plunder, while the second group, being powerless to stop the plunder, instead attacks the good things that come from the civilization and the good things that civilization can achieve. Instead of keeping the first group from driving millions of species into extinction, it instead plays upon the culturally endemic fear of thought, innovation and human intelligence to fan hysterias about scientific advances, whether they consist of genetically modified corn or cloning or stem-cell research or the Human Genome Project. I believe that both are doing grave wrong. Nature contains tremendous variety and richness, and to drive millions of species into extinction in a shortsighted pursuit of profit is to destroy what is irreplaceable to pay for temporary enrichment of the enterprise at the price of permanent destruction of riches the enterprise cannot possibly replace. Like suicide, it is a permanent solution to a temporary problem; and if it continues unchecked, the result will indeed be a permanent planetary suicide. However, to stand in the way of research that can lead to more sustainable farming, cures for genetic illnesses, cures for paralysis and cancer, and synthesis of bacteria that can break down plastics and styrofoams with which the civilization has been poisoning the planet, is to do a still graver wrong: To deny humanity the tools it's been given to solve the problems that it has created, to make its life (and the life of the planet) rich and sustainable, and to create a civilization that is an improvement on nature and not a degradation. The atavistic types shout about the scientific experimentation that seeks to create new lifeforms or resurrect ones that people have driven into extinction. They did not shout when people killed off the mammoths, nor when they used DDT and Agent Orange to poison everything around them, nor when people created enough nuclear weapons to kill the world seven times over, nor when Amazonian ranchers kill thousands of species of plants and animals every year. Why this grating, atavistic hypocrisy? Why is it OK to destroy but not to create, to pillage nature and not to improve on nature, to murder but not to heal and resurrect? It is my belief that this comes from the same place as the saying that beauty is only skin-deep, but ugliness goes down to the bone and that to err is human - from the cultural notion that evil in human beings is to be expected but good is to be suspected; that destructiveness, violence, stupidity and short-sightedness is a necessary part of the human makeup but thought, innovation and inspiration is not. It comes, I believe, from the fact that most people have never been taught to think creatively and inventively, and they can identify with ugliness and destructiveness in the human nature - the ugliness and destructiveness which they've known since conception - but cannot rise to embrace the human ability to understand, to envision, to conceptualize and to create. In dividing the human beings into the leavers who live as part of nature and takers who live in the face of nature, Daniel Quinn separated the two aspects of human being: A being that lives in nature and follows the laws of nature and a being that shapes and creates out of its will, intelligence and self-awarenes. I believe that both the leavers and the takers are a necessary part of the human makeup, and that the beneficial outcome comes neither from return-to-the-soil atavism nor from turning the world into a giant strip mall. I believe that there is good in the leaver mentality and there is good in the taker mentality, and that the two need to work together in an integrative synthesis - in a synergy - that makes the best of both. The good done in service of leaver mentality comes most starkly in form of Wangari Maathai, a Kenyan activist who risked her life many times to plant millions of trees in her country - an action that allowed Kenya to escape the fate of Haiti and other countries that had ignorantly driven their forests into extinction to pay for slash-and-burn farming and in so doing condemned themselves to sickness and poverty. To resurrect rainforest where it has been pillaged - to bring back natural richness to places in which it once has existed - is a prudent, inspired and ethical project that will preserve Planet Earth for future generations and for the life to come. The Reagan official who said, If you've seen one tree, you've seen them all can be answered quite simply, You've seen one Republican, you've seen them all. And to say out of that consideration that it is right to kill off thousands of species of trees, many containing useful medicinal qualities, is as damnable as to say that one should kill off all Republicans. The good done in service of the taker mentality comes from innovators: Internet, representative democracy, antibiotics and masterpieces such as the Sistine Chapel and the Empire State Building. It comes from people who use the unique inventive capacity in the human being to produce work that beautifies the world and improves people's lives. It comes from anyone who's ever had an original idea, anyone who's ever contributed something intelligent and creative, anyone who produced something good that has not existed before. Humanity has all it needs, not only to survive long-term, but also to manifest all the good things that live in the mind and psyche. To create a civilization that's an improvement on nature and not a degradation. What am I proposing then? I propose taking the best of the Leaver mentality and the best of the Taker mentality and making them work together in an integrative synthesis. I propose preserving the planet for the future and repeating the feat of Wangari Maathai all over the world; and I propose using the human ingenuity to create a beautiful civilization - a civilization that builds upon nature, that improves upon nature, and that draws from the endless well of human ingenity and intelligence to make our world the richest, most happy, most beautiful world it can be. It is to produce masterpieces - masterpieces artistic, scientific, medical, social and technological - that use the best in the human mind to create the best possible civilization. And, in synergy with the nature as it is kept from being mindlessly plundered, arrive at a planet that has the best of nature and the best of civilization, with nature in all its richness and color remaining alive and humanity building on top of it a civilization equal to all the richness, color and inspiration that has been imparted humanity. Ilya Shambat. === Subject: Re: Takers and leavers: Synthesis > The way I see this, the Western society is torn between two lines of > thought. One is the unconditional growth of the civilization, whatever > the consequences for the planet and for the future generations. The > other is a reaction against it: a return-to-the-roots hippie > environmentalism that seeks to What old-fashioned hippie types (and their neoLuddite descendants) fail to take into account is the sheer self-serving strategic profligacy of Nature based on the policy of unconditional growth. Plants do not produce just enough seeds to ensure zero population growth for themselves, they produce enough for any given species to fill their local biome (in some cases, blanket the planet) in one generation. Similarly, animals do not produce just enough progeny to ensure ZPG for themselves. Fortunately, the Law Of Fang And Claw (for plants, chemical warfare and rapacious competition for water and sunlight) keeps any given species from taking over completely. So, who's actually acting in accordance with Nature, hippies or Capitalists? And before you bewail the sheer mass of human flesh that will occupy the planet after we've killed everything else off and have to resort to cannibalism because there's nothing else to eat, consider what happened to the poor, unsung anaerobes that formerly exclusively ruled the Earth. Now, us aerobes have to eat each other while trying to finish off the anaerobes. Nature is change in strobing neon caps, not any kind of imaginary idyllic steady state. Environmentalism, at its root, is narrow-minded arrogance. Mark L. Fergerson === Subject: Re: Takers and leavers: Synthesis >The way I see this, the Western society is torn between two lines of >thought. One is the unconditional growth of the civilization, whatever >the consequences for the planet and for the future generations. The >other is a reaction against it: a return-to-the-roots hippie >environmentalism that seeks to You know, you're right. I spend all day thinking about those two lines of thought. I'll be driving to work, thinking about what I need to accomplish first today, and my first thought is always will this advance my agenda of promoting unconditional growth and plunder? ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === Subject: Re: Takers and leavers: Synthesis > The way I see this, the Western society is torn between two lines of > thought. One is the unconditional growth of the civilization, whatever > the consequences for the planet and for the future generations. The > other is a reaction against it: a return-to-the-roots hippie > environmentalism that seeks to [snip crap] > I propose preserving the planet > for the future and repeating the feat of Wangari Maathai all over the > world; and I propose using the human ingenuity to create a beautiful > civilization Pleasantly scented, effective, with conditioner, inexpensive and limitless shampoo came from chemists engineers, and capitalists - not from idiots. Civilization knows what it is doing: screw the poor and stupid, reward the creative, industrious, and successful. The Welfare State is an abomination. The weak deserve to be destroyed by their unimpeded own hands. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: Takers and leavers: Synthesis > The atavistic types shout about the scientific experimentation that > seeks to create new lifeforms or resurrect ones that people have > driven into extinction. They did not shout when people killed off the > mammoths Well, I do remember there being something of a public outcry at the time, but the newspapers didn't carry it so nobody got to hear about it. -- ICQ 40628243 Tel 07092057581 Fax 07092308800 === Subject: FTP DOWNLOAD! FREE 24H SUPPORT! Best Softwares!!!! 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capitalism, which claims that the benefit of the collective is achieved through people striving for their own benefit, and against altruistic ideologies such as Communism and religion that claim the pursuit of self-interest to be antithetical to the benefit of the collective. What he said was something that ought to be common sense: that the benefit of the group is achieved when people strive to benefit themselves - and the group. That is, in a social universe consisting of self and others, the social universe benefits when one strives to benefit both self and the social universe. Through this is achieved an improvement in the well-being of all. The social universe benefits, in other words, not from competition alone and not from cooperation alone, but rather from the mixture of competition and cooperation. It benefits from people seeking their own benefit as well as that of the group. This mixture of competition and cooperation, an integration of two opposite approaches to life, results in a Pareto-optimal solution, an equilibrium that achieves the best outcome for everybody. Knowing what's good for yourself and knowing what's good for the group, rather than seeing the group as out to destroy you or sacrificing yourself for the group, results in the best solution both for yourself and for the group. This is the meaning of synergy. The mechanism of this approach is as follows. Competition sharpens the tool - it challenges the person to be his best. Cooperation strives for the benefit of the whole. Competition alone produces people who are good at what they do and benefit the consumer, but who are at each other's throats and never reap the full benefit of their efforts because they spend so much time competing, hence they have no time left to enjoy life. Cooperation alone produces a lazy, low-level state of affairs that sells the consumer short. Combining competition with cooperation - challenging people to be their best, and to contribute their best to the shared outcome of which shared outcome each partakes - arrives at the best outcome for the whole and for each member. This, is the beautiful outcome; synergy; reconciliation and integration; consummation; the good of the collective and the members at the same time. The optimal outcome for the world comes from cooperation of people within small entities such as companies, within the context of competition between companies and communities, within the context of cooperation among higher levels of structure - industries, sciences, technologies - to make a better world. This state is largely (and thankfully in not least degree to the people responsible) reflected in the status quo. As Covey rightly advised his clients, companies should strive for their employees collaborating among each other rather than competing against each other, and in dealing with customers and suppliers companies should strive for win-win scenarios. The first results in people expending their energy striving to benefit the company and not in divisive intramural politicking that seeks promotion of self at all costs to other employees and regardless of benefit to the company; the second results in companies having good relations with other participants in the economy except, of course, their competitors. At a higher level, there is at this time international scientific and medical cooperation, as well as international economic policy-setting that takes business and labor interests, as well as interests of the people for air and water free of disease, into account. The latter is currently under challenge by Bush administration - a state of affairs that best be corrected by giving it the benefit of the boot. What would bring larger entities to collaborate for a shared objective? Seeing the full picture and seeing how participating within it can lead to the benefit of the whole and of one's own entity. It is believed that human nature is selfishness or at least rational self-interest, but I posit that humanity has gotten as far as it has due to the fact that it has a species consciousness as well as individual consciousness - that people possess within themselves, whether through mechanism of creation or evolution, a natural interest in the benefit of people other than themselves. In other words, that people do in general, not only as a result of liberal education but innately as a result of genetic traits that are more profoundly expressed in some than in others, wish for mankind to do well, and that this impetus has allowed many people on their own accord to pursue teaching, scholarly, charitable, monastic, scientific, creative, policing and civil service paths that did not present great monetary or ego reward but allowed satisfaction of doing something significant for other people. Few feelings are better than that of having done a good deed - a feeling that brings satisfaction even without the need to boast about it; I consider only the feeling of great achievement, the feeling of mutual love and the state of grace or spiritual ecstasy to be as gratifying. The first two are rare; the third goes away unless one applies oneself in serving mankind. Doing good is a far more sure path to gratification than taking, and it is an easier path to gratification than achievement or seeking a beautiful relationship. The reason that doing good feels good is that it touches upon the best in the human beingness - the interest in the good of the species and, in some cases, of life itself. Thus, an order that benefits the self even as it benefits the species does the most for the entirety of the human nature - both for its self-directed, or self-interested, and its species-directed, or altruistic, components, which are expressed to different extents in different people and which people ought to be free to choose the path that suits them the most. This offers man the fulfillment of the entirety of the human nature - the opportunity to do good, and do well, at the same time. Love your neighbor [defined in the Bible as all people including Samaritans, not merely the person next door] as yourself means, Love yourself, and love your neighbor [defined as all humanity]. That means, do well for yourself and help others. Do good, and do well, at once, this increasing the benefit of mankind as it is contained both in self and in others. C.S. Lewis stated that the Christian recognizes God's creation as fully perfect, and the existence within a man of a certain motive therefore predicted a natural way whereby it may be fulfilled. Indeed, this is the same statement as that made by rationalist philosophers, that the Universe is rational (and all motives within human psyche are there for a reason - because they help our survival or evolution as a species). Given that both self-interest and altruism are potent motives within human psyche that have had tremendous formative power on the history of the species, both the Christian and the rationalist view demand that there be a rational and Christian way to channel both altruism and self-interest in society. There is. It is to possess a social covenant whose values encourage people to benefit both themselves and the species - to do good and do well at the same time, with those who want to do well without doing good being subjected to laws of market competition that brings them to work for the consumer's interest and those who want to do good without caring whether or not they do well monetarily being given the honor that the social usefulness of their endeavors demands. In the latter, we find America currently lacking. The disrespect that is afforded the teaching profession is, I believe, scandalous; for it is the teachers that inform the cognitive habits of the young generation, and teachers again that have the power to either affect the children into being developed human beings and productive citizens, or else through neglect or bad instruction to resign them into the hell of addiction, depression and crime. The teachers in America aren't respected, and for this reason American primary education despite money expended upon it is lagging behind that of other industrialized countries, as the best and the brightest shun teaching careers. A good teacher is a great asset to the civilization; someone who can instill in children the love of the subject, the love of learning, the love of productive endeavor, the love of the neighbor, the love of country and the love of life. I have been fortunate to have had two such teachers in an American private school. They deserve all the respect one can give them, and the more respect is afforded great teachers the more great people go into the teaching profession and have positive effect on the formation of young minds, in schools public as well as private. Representing one's effect upon the collective benefit as the sum Si = ax+a1y1+a2y1+...+anyn, Where x is self, a is contribution to the benefit of self, and each y term is an other person that stands to be affected by one's actions, S increases with increase in the sum of the terms. The benefit of each person is the sum Bi = bx+b1y1+b2y2+...+bnyn, Where x is the self, b one's own contribution to the benefit of the self, and each y term the contribution of another person to one's benefit. Collective benefit is the sum of Bi's - a sum of actions expended by all individuals to benefit themselves and other people in the social whole. This definition quantifies everyone affected by the individual's actions, whether they take economic form or other forms. It includes - one's co-workers, one's wife and children, one's neighbors, the government, the people whose health is affected by industrial and agricultural activity that forms one's consumption, the people who produce the goods one consumes, one's friends and enemies.... The interests of all these people needs to be reflected in the society and the economy in order that one's actions work toward benefit of all whose lives one touches. In this, the main tenet of Christianity - that the main commandment is to love your neighbor as yourself, with the concept of neighbor including all human beings - becomes reified in society's system of reinforcements, and people are steered toward living the lives that will get them right with God, even as they are steered toward living lives that benefit - themselves as well as each other. The equation above is true for the objective definition of the collective benefit - the benefit of the greatest good for the greatest number according to utilitarianism or the benefit of all individuals included in the society, according to Ayn Rand's concept of society as the sum total of its individual members. But game theory maintains something else: that collective benefit is not simply a sum of its parts but an entity in its own right. That is, the group exists as a real entity; whether it be Christianity claiming that all Christians are part of the body of Christ, or pantheism claiming that we are all one, or government claiming that a nation is a real unit that retains its own character regardless of which individual citizens live in it. Therefore there is, in addition to the good of the individual members of the collective, also the benefit of the collective. Which collective can be defined, respectively, as - Christendom; humanity; or the nation. This can be understood economically, but this can also apply in other aspects of life. Quite simply, actions that benefit both self and others, whether in enhancing their state of mind or enhancing their material well-being, are actions that lead to collective benefit. A person who hurts another person poses a net drain on the collective well-being; a person who helps another person enhances it. Attitudes that are ennobling and enriching - that help people to see the beauty in each other and cherish each other - are attitudes that increase total benefit and are as such attitudes that increase the well-being both of the individual and the collective. Attitudes that are prosecutorial, degrading, shriveling and abusive are attitudes that cause misery and are as such wrong for both the whole and the individual members. Ilya Shambat For more, see: http://www.geocities.com/drr0cket/pareto.htm === Subject: Series of the inverses of primes Hello I'd like some help to analyse the convergence of series of the form Sum(n =1, oo) 1/(p_n)^k, where p_n is the n_th positive prime and k>=1. I know this series diverges for k=1 and I guess it converges for k>1, but got stuck. Amanda === Subject: Re: Series of the inverses of primes >I'd like some help to analyse the convergence of series of the form >Sum(n =1, oo) 1/(p_n)^k, where p_n is the n_th positive prime and >k>=1. I know this series diverges for k=1 and I guess it converges >for k>1, but got stuck. Try comparing the series with Sum(n=1, oo) 1/n^k. Also note that Sum(n =1, oo) 1/(p_n)^k converges iff Product(n=1, oo) (1 - 1/p_n^k) converges. Mike Guy === Subject: [Fwd: pair sums applied to trignometry sums] -------- Original Message -------- === Subject: pair sums applied to trignometry sums Inc. I had used the mechanism with Bailey type of sequences and their sums in the work on b normalness in iteratives functions. It occurred to me that by adding the variable x , I could get functiond that used the nonlinear Cantor pair {1/(n+1),n/(n+1)} to split the sine and the cosine down the middle. The result is entirely new trignometric sum functions that converge very well. (* pair sums applied to trignometry sums: {1/(n+1),n/(n+1)} modulo 2 switched sums*) (* these sums break the trignometry of a circle into four functions instead of two*) (* these are subharmonic functions of a nonlinear Rational Cantor type*) fs[x_,n_]:= If[Mod[n,2]==1,(-1)^(n)*n*x^(2*n+1)/((n+1)*(2*n+1)!),(-1)^(n)* x^(2*n+1)/((n+1)*(2*n+1)!)] gs[x_,n_]:= If[Mod[n,2]==1,(-1)^n*x^(2*n+1)/((n+1)*(2*n+1)!),(-1)^n*n* x^(2*n+1)/((n+1)*(2*n+1)!)] fc[x_,n_]:= If[Mod[n,2]==1,(-1)^(n)*n*x^(2*n)/((n+1)*(2*n)!),(-1)^(n)* x^(2*n)/((n+1)*(2*n)!)] gc[x_,n_]:= If[Mod[n,2]==1,(-1)^(n)*x^(2*n)/((n+1)*(2*n)!),(-1)^(n)*n* x^(2*n)/((n+1)*(2*n)!)] digits=100; fsin[x_]:=N[Sum[fs[x,n],{n,0,digits}]]; gsin[x_]:=N[Sum[gs[x,n],{n,0,digits}]] fcos[x_]:=N[Sum[fc[x,n],{n,0,digits}]] gcos[x_]:=N[Sum[gc[x,n],{n,0,digits}]] Plot[fsin[x],{x,-Pi,Pi}] Plot[fsin[x],{x,-Pi,Pi}] Plot[gsin[x],{x,-Pi,Pi}] Plot[fcos[x],{x,-Pi,Pi}] Plot[gcos[x],{x,-Pi,Pi}] Plot[(fsin[x]+gsin[x]),{x,-Pi,Pi},PlotRange->All] Plot[(fcos[x]+gcos[x]),{x,-Pi,Pi}] ParametricPlot[{fsin[x],gsin[x]},{x,-Pi,Pi}] ParametricPlot[{fcos[x],gcos[x]},{x,-Pi,Pi}] ParametricPlot[{fsin[x],fcos[x]},{x,-Pi,Pi}] ParametricPlot[{gsin[x],gcos[x]},{x,-Pi,Pi}] ParametricPlot[{fsin[x]+gsin[x],fcos[x]+gcos[x]},{x,-Pi,Pi},PlotRange->All] Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn@netscape.net URL : http://home.earthlink.net/~tftn -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn@netscape.net URL : http://home.earthlink.net/~tftn === Subject: [Fwd: Re: pair sums applied to trignometry sums] -------- Original Message -------- === Subject: Re: pair sums applied to trignometry sums Inc. > I had used the mechanism with Bailey type of sequences > and their sums in the work on b normalness in iteratives functions. > It occurred to me that by adding the variable x , I could get > functiond that used the nonlinear Cantor pair {1/(n+1),n/(n+1)} > to split the sine and the cosine down the middle. > The result is entirely new trignometric sum functions that converge very > well. > (* pair sums applied to trignometry sums: {1/(n+1),n/(n+1)} modulo 2 > switched sums*) > (* these sums break the trignometry of a circle into four functions > instead of two*) > (* these are subharmonic functions of a nonlinear Rational Cantor type*) > fs[x_,n_]:= > If[Mod[n,2]==1,(-1)^(n)*n*x^(2*n+1)/((n+1)*(2*n+1)!),(-1)^(n)* > x^(2*n+1)/((n+1)*(2*n+1)!)] > gs[x_,n_]:= > If[Mod[n,2]==1,(-1)^n*x^(2*n+1)/((n+1)*(2*n+1)!),(-1)^n*n* > x^(2*n+1)/((n+1)*(2*n+1)!)] > fc[x_,n_]:= > If[Mod[n,2]==1,(-1)^(n)*n*x^(2*n)/((n+1)*(2*n)!),(-1)^(n)* > x^(2*n)/((n+1)*(2*n)!)] > gc[x_,n_]:= > If[Mod[n,2]==1,(-1)^(n)*x^(2*n)/((n+1)*(2*n)!),(-1)^(n)*n* > x^(2*n)/((n+1)*(2*n)!)] > digits=100; > fsin[x_]:=N[Sum[fs[x,n],{n,0,digits}]]; > gsin[x_]:=N[Sum[gs[x,n],{n,0,digits}]] > fcos[x_]:=N[Sum[fc[x,n],{n,0,digits}]] > gcos[x_]:=N[Sum[gc[x,n],{n,0,digits}]] > Plot[fsin[x],{x,-Pi,Pi}] > Plot[fsin[x],{x,-Pi,Pi}] > Plot[gsin[x],{x,-Pi,Pi}] > Plot[fcos[x],{x,-Pi,Pi}] > Plot[gcos[x],{x,-Pi,Pi}] > Plot[(fsin[x]+gsin[x]),{x,-Pi,Pi},PlotRange->All] > Plot[(fcos[x]+gcos[x]),{x,-Pi,Pi}] > ParametricPlot[{fsin[x],gsin[x]},{x,-Pi,Pi}] > ParametricPlot[{fcos[x],gcos[x]},{x,-Pi,Pi}] > ParametricPlot[{fsin[x],fcos[x]},{x,-Pi,Pi}] > ParametricPlot[{gsin[x],gcos[x]},{x,-Pi,Pi}] > ParametricPlot[{fsin[x]+gsin[x],fcos[x]+gcos[x]},{x,-Pi,Pi},PlotRange->All] > Respectfully, Roger L. Bagula > tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : > alternative email: rlbtftn@netscape.net > URL : http://home.earthlink.net/~tftn It occurs to me that these functions might be simplified, as they are each themselves sums of pairs of functions with terms satisfying simple recurrences. For example, fs can be written as the sum of n-even + n-odd terms, and these are just the sums of terms 1/(2*k+1)*x^(4*k+1)/(4*k+1)! and (-1)*(2*k+1)/(2*k+2)*x^(4*k+3)/(4*k+3)! respectively. In more detail we get the function below. InputForm[fsin2[x_] = Together[-Sum[(2*k+1)/(2*k+2)*x^(4*k+3)/(4*k+3)!, {k,0,Infinity}] + Sum[1/(2*k+1)*x^(4*k+1)/(4*k+1)!, {k,0,Infinity}]]] Out[10]//InputForm= (-4 + 4*Cosh[x] + x*Sin[x] - x*Sinh[x])/(2*x) (Isn't it great to have a symbolic math engine at ones fingertips?) As a quick check: In[11]:= InputForm[Max[Abs[Table[fsin2[x]-fsin[x], {x,-Pi,Pi,.1}]]]] Out[11]//InputForm= 3.372302437298913*^-15 (Isn't it great to have a numeric math engine at ones fingertips?) The advantage to using the closed form is twofold. One is that numeric computations are better, and the other is that they are significantly faster. To see the latter: In[5]:= Timing[Plot[fsin[x],{x,-Pi,Pi}]] Out[5]= {0.3 Second, -Graphics-} In[6]:= Timing[Plot[fsin2[x],{x,-Pi,Pi}]] Out[6]= {0.01 Second, -Graphics-} For the former, just notice what happens when we get outside the range -Pi -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn@netscape.net URL : http://home.earthlink.net/~tftn === Subject: [Fwd: Re: pair sums applied to trignometry sums] -------- Original Message -------- === Subject: Re: pair sums applied to trignometry sums >It occurs to me that these functions might be simplified, as they are >each themselves sums of pairs of functions with terms satisfying simple >recurrences. For example, fs can be written as the sum of n-even + n-odd >terms, and these are just the sums of terms 1/(2*k+1)*x^(4*k+1)/(4*k+1)! >and (-1)*(2*k+1)/(2*k+2)*x^(4*k+3)/(4*k+3)! respectively. >In more detail we get the function below. >InputForm[fsin2[x_] = Together[-Sum[(2*k+1)/(2*k+2)*x^(4*k+3)/(4*k+3)!, >{k,0,Infinity}] + > Sum[1/(2*k+1)*x^(4*k+1)/(4*k+1)!, {k,0,Infinity}]]] >Out[10]//InputForm= (-4 + 4*Cosh[x] + x*Sin[x] - x*Sinh[x])/(2*x) >(Isn't it great to have a symbolic math engine at ones fingertips?) >As a quick check: >In[11]:= InputForm[Max[Abs[Table[fsin2[x]-fsin[x], {x,-Pi,Pi,.1}]]]] >Out[11]//InputForm= 3.372302437298913*^-15 >(Isn't it great to have a numeric math engine at ones fingertips?) >The advantage to using the closed form is twofold. One is that numeric >computations are better, and the other is that they are significantly >faster. To see the latter: >In[5]:= Timing[Plot[fsin[x],{x,-Pi,Pi}]] >Out[5]= {0.3 Second, -Graphics-} >In[6]:= Timing[Plot[fsin2[x],{x,-Pi,Pi}]] >Out[6]= {0.01 Second, -Graphics-} >For the former, just notice what happens when we get outside the range >-Pi >Daniel Lichtblau >Wolfram Research > > There is a reason for using the specific pair {1/(n+1),n/(1+n)}. It stems from group theory and functional inversion. The pair function {1/(1+x),x/(1+x)} is connected to the Farey tree functions by functional inversion: x/(1-x) if the functional inverse to x/(1+x) Solve[z-x/(1-x)==0,x] (1-x)/x if the functional inverse to 1/(1+x) Solve[z-(1-x)/x==0,x] There is a larger group that contains these Farey tree transforms too called the anharmonic group: {x, 1/x, 1/(x-1),x-1,x/(1-x),(1-x)/x} and it implies an functional inversion group of: {x,1/x,(1+x)/x,x+1,x/(1+x),1/(1+x)} I have done a lot of work in this area in the last few years. I'm glad someone realized that these functions are important besides me. There should be a number of ways to do them as is usual when dealing with fundamentals in Mathematics. I chose the ones I did because they are a torus under the inverse substitution: {1/(1+x),x/(1+x)}/. x->1/x I thank you for your work and I'll see if they work on my version of Mathematica. The anharmonic group connection makes me believe that there are further subharmonic functions yet to be discovered. Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn@netscape.net URL : http://home.earthlink.net/~tftn -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn@netscape.net URL : http://home.earthlink.net/~tftn === Subject: [Fwd: Re: pair sums applied to trignometry sums] -------- Original Message -------- === Subject: Re: pair sums applied to trignometry sums This male/ female Fibonacci is another application of the pair type functions. In Mathematica: f[n_]:=(1/(n+1))^Mod[n,2]*(n/(n+1))^(1-Mod[n,2]) but g[n_]:=(n/(n+1))^Mod[n,2]*(1/(n+1))^(1-Mod[n,2]) doesn't seem to work. I had to change it to: g[n_]:=If[Mod[n,2]==1,(n/(n+1)),(1/(n+1))] The modulo power version seems functionallt equivalent, but fails completely in the zeta function versions of these. (* (1/(n+1),n/(1+n)) pair function used to get a dual population Fibonacci *) (* if the Fibonacci is a rabbit population , thn it has male and femal components*) (* in this case the gfib ( female) population is always larger or the same*) (* natural birth rate has the female popoulation slightly larger than the male in many mammals*)*) digits=50 f[n_]:=(1/(n+1))^Mod[n,2]*(n/(n+1))^(1-Mod[n,2]) g[n_]:=If[Mod[n,2]==1,(n/(n+1)),(1/(n+1))] fib[n_Integer?Positive] :=fib[n] =fib[n-1]+fib[n-2] fib[0]=0;fib[1] = 1; ffib[n_Integer?Positive] :=ffib[n] =ffib[n-1]*f[n-1]+ffib[n-2]*f[n-2] ffib[0]=0;ffib[1] = 1; gfib[n_Integer?Positive] :=gfib[n] =gfib[n-1]*g[n-1]+gfib[n-2]*g[n-2] gfib[0]=0;gfib[1] = 1; a=Table[Floor[ffib[n]*fib[n]],{n,0,digits}] b=Table[Floor[gfib[n]*fib[n]],{n,0,digits}] {0,1,0,1,1,3,4,7,11,18,29,47,75,123,197,321,514,836,1343,2181,3508,5692,9167 , 14865,23959,38838,62635,101503,163773,265344,428291,693791,1120191,1814345, 2930173,4745365,7665395,12412755,20054413,32471888,52470417,84953526, 137291667,222271983,359249034,581585233,940082660,1521822386,2460102246, 3982297570,6438059697} {0,1,0,1,2,3,5,8,13,20,34,54,88,141,230,368,599,962,1562,2512,4077,6562,1064 4, 17149,27804,44827,72655,117201,189907,306473,496500,801528,1298303,2096510, 3395454,5484273,8881231,14347563,23232342,37537787,60778546,98216903, 159015502,256996472,416059948,672493991,1088669150,1759816751,2848763556, 4605344794,7454779663} -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn@netscape.net URL : http://home.earthlink.net/~tftn -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn@netscape.net URL : http://home.earthlink.net/~tftn === Subject: Functional derivatives for discrete equations (matrices) ? Hi all, Warning: this posting is from a mathematically ignorant engineer! I am doing some image processing work. The image, i, is the convolution of the object, o, and the impulse response, h, all of which are defined on discrete N x N arrays (matrices). Thus i = h**o (1) convolution may be represented with a block-circulant circulant-block structure, so that I = HO, (2) where I and O are the vectorized (1 x N^2) versions of i and o, and H is an N^2 x N^2 matrix. I would like to take the partial derivative of i w.r.t. o - more specifically, the partial derivative of the j'th of N^2 elements of i w.r.t. the k'th of N^2 elements of o, e.g. di(j)/do(k). This seems straightforward for (2), less so for (1). However, given that i,h, and o are sampled versions of continuous functions, I am wondering if I need to approach this using functional derivatives ? Any pointers to texts or other suggestions would be greatly appreciated! Hal === Subject: Laurent........+ hello.....doctor~ Find the Laurent series that represents the function f(z) = 1 / {(z^2) +1)} in the domain 0 < |z-i| < 2. -------------------------------------------------- um.......... f(z) = 1 / {(z^2) +1)} = [{1/(2z)} / (z+i)] + [[{1/(2z)} / (z-i)] = 1/2*[1/{z(z+i)} + 1/{z(z-i)}] = 1/2*[1/{(z-i)+i}]*[1/{(z-i)+2i}] + 1/2*[1/{(z-i)+i}]*[1/(z-i)] and, 1) 1/{(z-i)+i} = {1/(2i)} / [(z-i)/(2i) + 1/2] = {1/(2i)}*[1/2 - 1/2*{(z-i)/(2i)} + 1/2*{(z-i)/(2i)}^2 -.....] (|(z-i)/(2i)| < 1) is this possible application ? and, 2) 1/{(z-i)+2i} = 1/[(2i)*{1+(z-i)/(2i)}] = (1/(2i))*[1 - {(z-i)/(2i)} + {(z-i)/(2i)}^2 -.....] (|(z-i)/(2i)| < 1) so, by (1),(2).....i can expand f(z) to (z-i) forms. is this right method ? thank you very much for your advice. === Subject: Need a help for my book I would like to present you one cryptograph (kind of alphametics). It's Croatian, and I believe world record - three terms with 12 letters. P I S M O S I S T O K A (Letter from east - title of novel) P R O K R S T A R I T I (to cruise) S T A R O K A T O L I K (Old Catholic) Code: 7,0,8,2,4,9,3 - 9,5,6,1,3 gives additional term. Solution: Tropska klima - tropical clime (T=7, R=0, etc.) 2 6 4 1 8 4 6 4 7 8 9 3 2 0 8 9 0 4 7 3 0 6 7 6 4 7 3 0 8 9 3 7 8 5 6 9 Can you make a work with three 13-letters terms? Zoran Radisavljevic Novi Sad, Serbia P.S. Task is really hard and I'll understand if it's beyond your power. I hope that somebody can send me few works on English with === Subject: Plane geometry quickie Consider two copies of a simply connected set in R^2. Join the two sets together by rotation and translation (no mirroring) so that the sets overlap in a set of zero measure and that the union of the two sets forms a convex set in R^2. Count the distinct ways in which the two identical sets can be thus joined to form a convex set s.t. the resulting convex sets are taken to be distinct if they can not be transformed to each other by mirroring or translation. The problem is: for each n, present a set S_n s.t. the set S_n can be thus joined with a copy of itself to form exactly n distinct convex sets. The set S_n does not need to be convex by itself. For example, the half-disk is a set that can be joined with itself in exactly one way to form the convex disk. The non-equilateral isosceles triangle is a set that can be joined with itself in exactly three distinct ways to form a convex set. -- I'm not interested in mathematics that might have anything to do with reality. -- Russell Easterly, in sci.math === Subject: Re: Plane geometry quickie >Consider two copies of a simply connected set in R^2. Join the two >sets together by rotation and translation (no mirroring) so that the >sets overlap in a set of zero measure and that the union of the two >sets forms a convex set in R^2. Count the distinct ways in which the >two identical sets can be thus joined to form a convex set s.t. the >resulting convex sets are taken to be distinct if they can not be >transformed to each other by mirroring or translation. >The problem is: for each n, present a set S_n s.t. the set S_n can be >thus joined with a copy of itself to form exactly n distinct convex >sets. The set S_n does not need to be convex by itself. >For example, the half-disk is a set that can be joined with itself in >exactly one way to form the convex disk. The non-equilateral isosceles >triangle is a set that can be joined with itself in exactly three >distinct ways to form a convex set. OK, you gave an examples for n=1 and n=3. A non-equilateral parallelogram (for example, a rectangle) could be joined with itself in 2 ways. A circle is good for n=0. -- spud_demon -at- thundermaker.net The above may not (yet) represent the opinions of my employer. === Subject: Re: Plane geometry quickie Keith A. Lewis escribi.97: > 19:04:04 +0200: >> Consider two copies of a simply connected set in R^2. Join the two >> sets together by rotation and translation (no mirroring) so that the >> sets overlap in a set of zero measure and that the union of the two >> sets forms a convex set in R^2. Count the distinct ways in which the >> two identical sets can be thus joined to form a convex set s.t. the >> resulting convex sets are taken to be distinct if they can not be >> transformed to each other by mirroring or translation. >> The problem is: for each n, present a set S_n s.t. the set S_n can be >> thus joined with a copy of itself to form exactly n distinct convex >> sets. The set S_n does not need to be convex by itself. >> For example, the half-disk is a set that can be joined with itself in >> exactly one way to form the convex disk. The non-equilateral >> isosceles triangle is a set that can be joined with itself in >> exactly three distinct ways to form a convex set. A non-equilateral non-obtuse isosceles triangle. If it is obtuse, only 2 ways. A escalen triangle is also good for n = 3. > OK, you gave an examples for n=1 and n=3. > A non-equilateral parallelogram (for example, a rectangle) could be > joined > with itself in 2 ways. A circle is good for n=0. -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Powers in goup Hi I need help to solve the following problem: Let a and b be two fix elements of group G. Show that if (ab)^3=a^3b^3 and (ab)^5=a^5b^5 then ab=ba. You are allowed to use theorems of powers in group i.e. (i) a^{-n}=(a^n)^{-1}, (ii) a^ma^n=a^{m+n}, (iii) (a^m)^n=a^{mn} (iv) (ab)^n=a^nb^n iff ab=ba. ---------------------- I've tried following: --------------------- (ab)^5=(ab)(ab)(ab)(ab)(ab)=(ab)(ababab)(ab)=(ab)(aaa)(bbb)(ab) = aa(aaabbb)bb =(aaaaa)(bbbbb) =a^5b^5 => a^{-1}(ab)(aaa)(bbb)(ab)b^{-1} =b(aaabbb)a= a(aaabbb)b= a^{-1}aa(aaabbb)bb b^{-1} => b(aaabbb)a= a(aaabbb)b Now here I'm stucked. Any help is appreciated. === Subject: Re: Powers in goup ETAuAhUAukxUcK3T5xsdweFGRmj4S9YanjECFQCNb6o0tEqy7LTghNy7chL1hwcZmQ== I would take the fifth-power equation and divide it by the third-power equation. Thus (ab)^2 = a^2b^2, meaning abab = aabb. Left multiply the last equation by a^(-1) and right multiply by b^(-1). --OL === Subject: Re: Powers in goup > I would take the fifth-power equation and divide it by the third-power > equation. What do you mean by divide it by the third-power equation. Thus (ab)^2 = a^2b^2, I don't think so!How did you get this? I don't think this is correct. meaning abab = aabb. Left multiply the > last equation by a^(-1) and right multiply by b^(-1). > --OL === Subject: Re: Powers in goup > Hi > I need help to solve the following problem: > Let a and b be two fix elements of group G. Show that if (ab)^3=a^3b^3 > and (ab)^5=a^5b^5 > then ab=ba. You are allowed to use theorems of powers in group i.e. > (i) a^{-n}=(a^n)^{-1}, (ii) a^ma^n=a^{m+n}, (iii) (a^m)^n=a^{mn} > (iv) (ab)^n=a^nb^n iff ab=ba. > ---------------------- I think this exercise is just a tricky one and one need to know the trick to solve it as Ken mentioned. So it is really not a very important exercise theoretically. === Subject: Re: Powers in goup days. My association with the Department is that of an alumnus. >I need help to solve the following problem: >Let a and b be two fix elements of group G. Show that if (ab)^3=a^3b^3 >and (ab)^5=a^5b^5 >then ab=ba. You are allowed to use theorems of powers in group i.e. >(i) a^{-n}=(a^n)^{-1}, (ii) a^ma^n=a^{m+n}, (iii) (a^m)^n=a^{mn} >(iv) (ab)^n=a^nb^n iff ab=ba. >---------------------- >I've tried following: >--------------------- >(ab)^5=(ab)(ab)(ab)(ab)(ab)=(ab)(ababab)(ab)=(ab)(aaa)(bbb)(ab) = >aa(aaabbb)bb =(aaaaa)(bbbbb) =a^5b^5 >=> a^{-1}(ab)(aaa)(bbb)(ab)b^{-1} =b(aaabbb)a= a(aaabbb)b= >a^{-1}aa(aaabbb)bb b^{-1} >=> b(aaabbb)a= a(aaabbb)b >Now here I'm stucked. Any help is appreciated. (ba)^k = a^k b^k: a a^k b^k b = a^{k+1} b^{k+1} = (ab)^{k+1} = a(ba)^k b (ba)^k = a^k b^k. Also a common exercise is: PROP. Let G be a group, and fix an integer n. If for all a,b in G (ab)^{n+i} = a^{n+i} b^{n+i} for i=0, 1, and 2, then G is abelian. (ba)^{n+1}=a^{n+1}b^{n+1}=(ab)^{n+1}. So for all x,y in G (xy)^{n+1} = (yx)^{n+1} = x^{n+1}y^{n+1}=y^{n+1}x^{n+1}. Similarly, we deduce that for all x,y in G (xy)^n = (yx)^n = x^n y^n = y^n x^n. Therefore: ab = (ab)^{n+1}(ab)^{-n} = (ba)^{n+1}[(ab)^{n}]^{-1} = (ba)^{n+1}[(ba)^n]^{-1} = (ba)^{n+1}(ba)^{-n} = ba so G is abelian. QED You have that (ab)^5 = a^5 b^5 and (ab)^3 = a^3b^3 for all a,b in G. Therefore: a^4*b^4 = (a^2)^2 (b^2)^2 = (b^2 a^2)^2 = ((ab)^2)^2 = (ab)^4. Thus, we have (ab)^5 = a^5 b^5 (ab)^4 = a^4 b^4 (ab)^3 = a^3 b^3 for all a, b in G; applying the proposition above, we deduce that ab=ba for all a,b in G. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Powers in goup >Hi >I need help to solve the following problem: >Let a and b be two fix elements of group G. Show that if (ab)^3=a^3b^3 >and (ab)^5=a^5b^5 >then ab=ba. > You have that (ab)^5 = a^5 b^5 and (ab)^3 = a^3b^3 for all a,b in G. I don't think so - it says this is true for two fixed elements, not for all elements. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Powers in goup days. My association with the Department is that of an alumnus. >>Hi >>I need help to solve the following problem: >>Let a and b be two fix elements of group G. Show that if (ab)^3=a^3b^3 >>and (ab)^5=a^5b^5 >>then ab=ba. >> You have that (ab)^5 = a^5 b^5 and (ab)^3 = a^3b^3 for all a,b in G. >I don't think so - it says this is true for two fixed elements, >not for all elements. Ah, yes. Misread it. My mistake. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Powers in goup > .... > Let a and b be two fix elements of group G. Show that if (ab)^3=a^3b^3 > and (ab)^5=a^5b^5 > then ab=ba. You are allowed to use theorems of powers in group.... O.K. You showed what you'd done, and it was certainly a suitable approach to try; but these things can be rather like puzzles which need the right trick. So although this is obviously homework, here's a broad hint of a suitable trick. Cancelling a from the left and b from the right of each equation gives (ba)^2 = (a^2)(b^2) and (ba)^4 = (a^4)(b^4). Now, (ba)^4 = ((ba)^2)^2, so ....? Ken Pledger. === Subject: Re: Dead friend's memorial webpage > For Michele W., who died in October 2002, 10 days before her 26th > birthday. > She was a Caltech graduate with a scientific degree & a broad > knowledge of many subjects. > Contains her poetry and philosophical essays. > http://www.geocities.com/mswiegandrip My condolences to you and your friend's family. She suffered from depression, no? So many bright minds and promising scientists have taken their own life. Its sad. There is help. Don't bother with professional head shrinkers or phychologists (head shrinker that proscribes pills). Chemistry has helped save my life from my own hands. Venlafaxine hydrochloride has been a miracle drug for me. Just imagine what the world would be like... If Tesla had a friend If Bridgeman had oxycotin If Eastman, Boltzmann, and Coes had Venlafaxine Hydrochoride... The path of knowledge is littered with burnt out lives and bodys of the self taken. God Bless and be safe. -aSkeptic === Subject: Re: Dead friend's memorial webpage >If Tesla had a friend Well, he did have those pigeons, you know..... I think people were somewhat hesitant to get friendly with him as he tended to keep one hand in his pocket all the time, and rumor had it that if he shook your hand with _that_ hand, you would be electrocuted...by the deadly Alternating Current, no less. :-) T. Edison Menlo Park === Subject: How to find the minimum to the function Let W := (w_1,...,w_n) D := (d_1,...,d_n) n F(W):= SUM{ (d_i - w_i)^2 } + h * #{w_i not equal to 0} i=1 where: #{w_i not equal to 0} := the number of elements in W that not equal to 0. h := real const. d_i,w_i := real for all i. How to find the vector W* that minimize F. === Subject: Vector spaces of functions My textbook in linear algebra says that for each interval [a, b] the set of continuous functions defined over [a, b] is a vector space. Is the set of continuous functions over R a vector space? === Subject: Re: Vector spaces of functions Originator: grubb@lola >My textbook in linear algebra says that for each interval [a, b] the >set of continuous functions defined over [a, b] is a vector space. >Is the set of continuous functions over R a vector space? Can you *prove* that the set of continuous functions over [a,b] is a vector space (I assume you mean real-valued functions)? If, instead, you consider functions over R, does the proof change? What is your conclusion? --Dan Grubb === Subject: Re: Vector spaces of functions >My textbook in linear algebra says that for each interval [a, b] the >set of continuous functions defined over [a, b] is a vector space. >Is the set of continuous functions over R a vector space? Do you have a reason, other than that your textbook in linear algebra says so, to believe that for each interval [a, b] the set of continuous (presumably, real-valued) functions defined over [a, b] is a vector space? If you do have some such alternative reason to the authority of the textbook, is it susceptible to (your) examination and (your) consideration? Lee Rudolph === Subject: Re: Vector spaces of functions > My textbook in linear algebra says that for each interval [a, b] the > set of continuous functions defined over [a, b] is a vector space. > Is the set of continuous functions over R a vector space? Yes. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Gravitomagnetism > Ok, you don't need to respond to this comment, > but I have no major problem with non-integer > weight if you allow non-integer dimensionality. > > Non-integer? Don't understand this.. > Well you know from SR as you move an object faster > and faster it's length contracts and it's time > slows down, and at c these dimensions vanish. > So in the intermediary, between v=0 and v=c, > can we say definitely that we have a fixed integer > dimensionality? > To put that on a more rational mathematical foundation > one can integrate, (no integration constants) > $ x dx = (1/2) x^2 (area) > $$ x dx dx = (1/6) x^3 (volume) > which shows how integration generates dimensions. Well conformal weight is a concept that is independent of dimensionality, that is, a covariant in a Weyl conformal space has properties under both coordinate and scale changes. The idea of weight embodies the latter. > GR is quite clear, G_uv = T_uv is where I'll begin. > If I understand you correctly, you state > vanishing gravity (implying G_uv =>0) and > *inhomogeneous* electrodynamics > (implying T(Maxwell)_uv =/=0 > are compatible? Yes! That is what is so surprising. Nowhere is a current posited - there *is* no RHS here. In GR, as in Maxwell-Lorentz, one has a field that is driven by a *posited* current (energy tensor, charge current resp.), and then that field acts back on the current. Here, there is *no* posited current at all, rather, the assumption of strictly local metricity requires both the metric and calibration field, and these *jointly* assume roles in a manner that appears as symmetric Ricci driven by energy-momentum, and Maxwell driven by charge-current! Of course they are really 6-d *vacuum* equations: Rmn - (2R/W) Tmn + (1/2W) (DmDn + DnDm) W = 0 (Rmn = symmetric part of CCT) 1/S d/dxm ( S R Fmn ) - 5/4 (Dn W) = 0 (S = sqrt det g) The new physics would be found in the pure geometry terms 1/2W {Dm,Dn} W and (Dn W), which are respectively, energy-momentum and charge current. These are absent in flat space > Yes, I'm trying to make the simplest possible metric > consistent with a nonorthogonal space, i.e. > g_uv = g*g_uv + A_u B_v > ^ ^ > calibration EM antisymmetry => A_u B_v = - A_v B_u > (Weyl=>gauge) (Einstein) > where det g = 1 - AB, (that's a bit crude, but close). > Ken S. Tucker -drl === Subject: Re: Gravitomagnetism >This is a wonderful picture and might even be true, but I am not >making such assumptions. I'm simply saying that the extra potentials >in general mediate the presence of Matter, and that Matter is in a >sense associated with deviation from analyticity in the 5-6 (u-v) >plane of psi + i chi regarded as a complex function of u + iv. Obviously you are way past my level of competence. I just try and see what you are saying in simple physical terms. However, begging your indulgence, if I could postulate further and perhaps get some idea of what your model says, whilst being mindful that you haven't examined it in the required depth. 1) You have a 6D model that seems to express mass as related to two extra dimensions. 2) These dimensions can be seen (if you take the appropriate viewpoint) as a complex plane attached to each point in 4D spacetime. So my questions (which you probably cannot answer) would be: 1) What is the condition for one or both of your 'mass dimensions' to become real? Of course I have an event horizon or a singularity in mind here. 2) If you were asked to include EM in this model, would you expect this to generate another two (or perhaps one) extra dimensions? 3) Your model implies some oscillating something attached to each point in spacetime. One is inevitably drawn to associating this with the and out (or rather within) your two extra dimensions. >So, for >example, we might imagine poles in that plane as somehow corresponding >to Matter, with the sign of the residue giving matter vs. antimatter. >These things need a mathematician to sort out at some point. I much regret that I can be of absolutely no help in this area. -- Oz This post is worth absolutely nothing and is probably fallacious. BTOPENWORLD address has ceased. DEMON address has ceased. === Subject: Re: Gravitomagnetism 1) You have a 6D model that seems to express mass as related to two > extra dimensions. Well, Matter = matter + antimatter. Mass comes later (Dirac eqn) > 2) These dimensions can be seen (if you take the appropriate viewpoint) > as a complex plane attached to each point in 4D spacetime. Yes. In regions where there is no Matter, and assuming infinitely weak gravity, then A6 - i A5 is analytic in (u+iv), that is dA5/du + dA6/dv = 0 dA5/dv - dA6/du = 0 > So my questions (which you probably cannot answer) would be: > 1) What is the condition for one or both of your 'mass dimensions' to > become real? Of course I have an event horizon or a singularity in mind > here. Again, they are always individually real. > 2) If you were asked to include EM in this model, would you expect this > to generate another two (or perhaps one) extra dimensions? EM is *already* in the model, coming from the calibration field A. This is the whole point, to get a set of essentially coupled eqns for g and A that are of the right order and that reduce to Einstein-Maxwell in the appropriate limit. > 3) Your model implies some oscillating something attached to each point > in spacetime. One is inevitably drawn to associating this with the > and out (or rather within) your two extra dimensions. Well, wave-like solutions will clearly play a large role (ultrahyperbolic eqns) and this may be true in some context, but it densities. -drl === Subject: Re: Gravitomagnetism This is a wonderful picture and might even be true, but I am not >making such assumptions. I'm simply saying that the extra potentials >in general mediate the presence of Matter, and that Matter is in a >sense associated with deviation from analyticity in the 5-6 (u-v) >plane of psi + i chi regarded as a complex function of u + iv. > Obviously you are way past my level of competence. > I just try and see what you are saying in simple physical terms. > However, begging your indulgence, if I could postulate further and > perhaps get some idea of what your model says, whilst being mindful that > you haven't examined it in the required depth. > 1) You have a 6D model that seems to express mass as related to two > extra dimensions. Well, I avoid the term mass at this stage, because of its implications, and stick with Matter. > 2) These dimensions can be seen (if you take the appropriate viewpoint) > as a complex plane attached to each point in 4D spacetime. Yes, but the wholeness of the 6-d conformal space should be kept in mind. > So my questions (which you probably cannot answer) would be: > 1) What is the condition for one or both of your 'mass dimensions' to > become real? Of course I have an event horizon or a singularity in mind > here. They are always real. In a certain context they may be regarded as a single complex value. > 2) If you were asked to include EM in this model, would you expect this > to generate another two (or perhaps one) extra dimensions? E.M. is alrealy in the model: A_m -> E.M. -> length curvature g_mn -> gravity -> direction curvature Connection = 1/2 { g_an,m + g_am,n - g_mn,a } + 1/2 { g_an A_m + g_am A_n - g_mn A_a } By replacing replacing F_mn by a general Yang-Mills field, one can bring in other gauge fields in an analogous way. > 3) Your model implies some oscillating something attached to each point > in spacetime. One is inevitably drawn to associating this with the > and out (or rather within) your two extra dimensions. Well this is a nice image and right in a limited way - e.g. free plane waves for F_mn in flat 6-space show up as propagating Matter waves in D_m D_m W = 0 where D_m is the conformal covariant derivative (d_m - 2 A_m) and W is the electromagnetic scalar (weight -2). Perhaps the extra cross-terms coming from A_m can be thought of as generating mass. -drl === Subject: Re: The state-of-the-art in mathematics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAUIttG31770; >>Hi Folks, >>I would like to share with you the latest findings >>sqrt(1/-1) = 1/i = -i or i = -i; dividing both sides of the last >>equation by i, I obtain 1 = -1 or 1 = 0 and the real number system >>goes down the drain. If I add i on both sides instead, I obtain 2i = 0 >>or i = 0 and the complex number system vanishes in thin air. >>E. E. Escultura >i = sqrt(-1) = sqrt(1/-1) = 1/i = -i or i = -i >... every first-semester-student of mathematics knows >that the square root of a number is 2-valued. >If the rest of your work is of the same >>depth<< ... You missed the point. This contradiction comes for the fact that i is ill-defined. The sqrt mapping or operator is well-defined only on non-negative numbers. A careful teacher reminds the student about it to avoid nonsense. E. E. Escultura === Subject: Re: The state-of-the-art in mathematics >>Hi Folks, >I would like to share with you the latest findings >>sqrt(1/-1) = 1/i = -i or i = -i; dividing both sides of the last >>equation by i, I obtain 1 = -1 or 1 = 0 and the real number system >>goes down the drain. If I add i on both sides instead, I obtain 2i = 0 >>or i = 0 and the complex number system vanishes in thin air. >>E. E. Escultura >i = sqrt(-1) = sqrt(1/-1) = 1/i = -i or i = -i >... every first-semester-student of mathematics knows >that the square root of a number is 2-valued. >If the rest of your work is of the same >>depth<< ... >H > You missed the point. This contradiction comes for the fact that i > is ill-defined. That's bad English. Shouldn't that is be am? === Subject: Re: Escultura affair: publication scandal by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAUIttv31737; >He's back! >>This question of whether 1 = 0.99... which I used to call >Ullrich-Israel >>equation has been resolved not only in the many papers I published but >The name does have a nice ring to it, but in all modesty I must >point out that I don't deserve any of the credit for this equation. >Very gracious of you to say this. So the credit's mine, all mine! >Robert Israel israel@math.ubc.ca >Department of Mathematics href=http://www.math.ubc.ca/~israel>http://www.math.ubc.ca/~israelUnive rsity of British Columbia Vancouver, BC, Canada >>************************ >David C. Ullrich >> You forgot to mention that the resolution says: 0.99... < 1 and so I now >call it the Ullrich-Israel nonsense. >You're contradicting yourself. You've (claimed to have) proven that the >so-called real numbers are a nonsensical construct full of contradictions. >Based on that, there is nothing that 0.999... can possibly mean so your >so-called statement that 0.99... < 1 is just semantic noise, not >mathematics You missed the point again. The Ullrich-Israel is nonsense in the real number system and the latter is also nonsense because two of its axioms - dichotormy and completeness axioms - are false. I threw them out and constructed the NEW real number system under new sets of axioms that well-define 0.99... This new real number system has natural ordering and says: 0.99... < 1. BTW, you have not defined REAL NUMBER. Do you know what it is? I'm beginning to sense some charlatans around here. E. E. Escultura === Subject: Counterexamples to FLT by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAUIts331729; I am pleased to summarize the resolution of some issues in mathematics such as the Ullrich-Israel dilemma: whether 1 = 0.99á and the status of Wiles.89 proof of FLT. These are some of the findings: (1) The number 0.99á is not well-defined in the real number system, therefore, the equation is nonsense. (2) The real number system is itself ill-defined because two of its axioms, completeness and the dichotomy axioms are false. Banach-Tarski and Brouwer constructed counterexamples to them. Needless to say, without fixing the real number system FLT remains nonsense and Wiles.89 ëproof.89 is nonsense. (3) The way to fix the real number system is to well-define it by simple set of axioms without these false axioms. They are as follows: Denote the new real number system by R*, +, x. Axiom 1: R* contains the basic integers 0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9. Axioms 1 and 2: The addition and multiplication tables of elementary arithmetic. The well-define addition and multiplication on the inte! gers. Then I defined 10 = 1+9. Then a terminating decimal is a polynomial in decreasing powers of 10 (including negative powers) over the basic integers. For example, abc.efgh = a10^2 + b10 + c = e/10 + f/10^2 + g/10^3 + h/10^4. A nonterminating decimal is well-defined provided every digit is known or computable. Specifically, it has the form N.abcás(k)á, where the kth decimal digit s(k) is known or computable. This number can be written as standard Cauchy sequence: N.a, N.ab, áN.abcás(k), á Thus most irrationals are ill-defined since s(k) is unknown in general. At the same time all periodic decimals are well-defined. The number sqrt(p), p is prime, is well-defined because its standard Cauchy sequence can be computed. An integer is simply the integral part of a decimal. This well-defines the integers as a subspace of the new real number system. Presently, the integers are not well-defined. This is the main problem of number theory. Brouwer.89s counterexample to the dichotomy a! xiom also shows that the irrationals are ill-defined and the real number system has no natural ordering. The new real number system has a natural ordering, namely, its lexicographic ordering. Note that 0.99á is well-defined in the new real number system since every digit is known. In the natural ordering of this space 0.99á < 1. Therefore, d* = 1 .9a 0.99á is well-defined. I call it dark number; it satisfies the following properties: with two exceptions, for any nonzero number x, xd* = d*, x + d* = x, d*^N = d* for any integer N, N.99á + d* = (N+1), d* is the smallest positive new real number and it is unique in view of the natural ordering of the new real numbers. Here.89s a sampling of the new arithmetic (Applied Mathematics and Computation, Vol. 138, 2001). Let K be an integer, M.99... and N.99... new integers. Then (1) K + M.99... = (K+M).99...; (2) K(M.99...) = K(M + 0.99...) = KM + K(0.99...) = KM + (K-1).99...; (3) M.99... + N.99... = M + N + 0.99... + 0.99 = M + N + 2(0.! 99...) = M + N + 1.99...; to verify that 2(0.99...) = 1.99..., calculate (1.99...)/2 to obtain 0.99... . (4) (M.99...)(N.99...) = (M + 0.99...)(N + 0.99...) = MN + M(0.99...) + N(0.99...) + (0.99...)2 = MN + (M - 1).99... + (N -1).99... + 0.99... = MN + (M + N - 2).99... + 0.99... = MN + (M + N - 1).99... = (MN+M+N -1).99... . These numbers, including d*, are called new integers because they are isomorphic to the intgers; therefore, they share the properties of the integers. In particular, d* behaves like 0 and N.99á behaves like N+1. The exceptions for x in xd* = d* and x+d* =x are x = 0 and x = N.99á, N = 0, 1, á, respectively. The countable counterexamples FLT are as follows: Let x = (0.99á)10^T, y = d*, z = 10^T, T = 1, 2,... Then x^n + y^n = (0.99á)10^nT + d* = 10^nT = z^n. Since x, y, z are not equal to 0, we have here countable counterexamples to FLT. Therefore, the conjecture is false. Moreover, for any integer k, (kx)^n + (ky)^n = k^n(x^n + y^n) = k^n(z^n) = (kz)^n.! Therefore, the nonzero integers, k(0.99á)10^T, d*, k10^T, k = 1, 2, á, and k = 1, 2, á, satisfy Fermat.89s equation and are also counterexamples to Fermat.89s FLT. We have here countable unbounded counterexamples to Fermat.89s last theorem. My original resolution of FLT (Nonlinear Studies, Vol. 5) uses the notion of Cauchy representation and modular equality in the sense of the new nonstandard analysis. For details, visit my websites: http://www.users.bigpond.com/pidro/home.htm, http://home.iprimus.com.au/pidro/. See also Nonlinear Studies, Vol. 5, 1998, Applied Mathematics and Computation, Vols. 130 and 138 as well as the following threads in the archives of MathForge.net: Constructivist principles á; Simpler Proof of FLT. === Subject: 3D Space Curve Curvature and Torsion 1) Can you please point to references available on internet for 3D intrinsic equations of curvature= k and torsion= t of some 3D curves in terms of arc length ? Would like to see more examples concerning arc intrinsic f(k,t),apart from helices. 2) For what curve(curves) is t^2+k^2 = 1?(1/a^2?) (t*a=cos(s/a),k*a=sin(s/a)) What could be the intrinsic equation of a Belt centerline in the following two practical/simple cases using a machine/automobile V-Belt? Any leather belt will do, but to visualize torsion and curvature a continuous square/trapezium section belt should be better. 3) The belt is marked at 4 equispaced quadrant arc points. Bring 2 opposite points together and tie both the pairs together making a 3D symmetric space curve. It may or may not be stretched further along axis of fixity. See how Frenet T,N,B move and observe the belt surface and centerline for torsion and curvature variations. Where there is no torsion, curvature is maximum and vice-versa. I feel the above relation at 2) is valid qualitatively atleast. If you do not agree,what f(t,k) would you suggest in such a case? 4) Let the V-belt diameter be 2a. Holding the belt fixed at some point and twisting it by forced rotation at an opposite point through 360 degrees (it is stable after one rotation),consolidate it into a smaller coil. It falls naturally (same topology) looped to one-third original diameter. The central line is now a circle in a plane and its centerline curvature is now 3/a. Neglecting dimensions of belt section, what function can we propose for torsion of its center line with respect to arc length? Due to rotation of bi-normal and principal normal around the belt centerline, torsion is not zero, even though it is in a plane. Shall most value your indulgences in reply. Narasimham === Subject: Real Analysis If f is a continuous function and e >0, can you find a linear combination of 1, x, x^2, x^3, ... which approximates f on [-1,1]. I guess this means that given e>0 we must find a linear combination, L, of 1, x, x^2,... s/t |f(x) - L| < e for all x in [-1,1]. This is all I got so far. Let c be in [-1,1]. Then, since f is continuous, lim as x->c of f(x) = f(c). But I have no real idea of what f (c) is. Of course I can let L (x) = [f(c)/c]x, but this won't work for other values in [-1,1]. Can I please get some guidence with this one. Steven === Subject: the beautiful TOE TOE:i. Theory of Everything:i. i is the simplest explanation there is. I will tell you the story in one verse, in several verses. I,alone know the laws. I,alone dictate the laws. I, alone am supreme. I, alone, is the only law. There are infinite laws to validate me. I, alone tell you, who you are. Dont resist my delusion. Dont resist my hallucination. Understand that everything at the fundamental level is a great illusion around me. I alone am the supreme mind, who is awake. Dont fight me. I alone tell you, you all are Gods. You all are Demons. Dont resist my ego. Dont resist me. I alone command you. I alone tell you who you are. How Gerald relates to Einstien, How Fredkin relates to Feymann. Dont resist me. Learn your place. I have only one power, of all the powers. I have the power to believe or disbelieve. I disbelieve i m human. You disbelieve you are human. I tell you. The game is to convince everyone, of your delusions, of your hallucinations. Understand and accept that, i tell you. I alone, am the supreme dictator of the laws. I alone, command you. I alone tell you, who you are. I alone make you, friends, create you, enemies. Dont fight me. It is I. I alone, am the simplest explanation there is. It is I, the believer's believer, the disbeliever's disbeliever, who is awake. Dont command me. I alone, am the center of the universe, the epicenter of everything. I alone am the arbiter of arbitary facts. Dont question me. I alone program you. Dont fight your delusions. Dont fight your hallucinations. Accept the myth of reality. Understand the macrosopic correlations. Understand that effects preceed the cause. Dont validate me. I, alone tell you. I alone command you. Dont command me. The game is to convince everyone of who you are. I tell you. I m your leader. There is no leader higher than me. I m your servant. There is no servant lower than me. I alone, battle the universe. It takes infinite laws to defeat me. It is I, who set everything in motion and the power to set it and unset it. Dont command me. It is I, who flatters you, with my human form. Dont flatter me. It is I, who defeats you, who undefeats you. Dont fight me. It takes the universe to battle me....... === Subject: Re: the beautiful TOE > TOE:i. Theory of Everything:i. i is the simplest explanation there is. No, there is more simple explanation. ;-) === Subject: Re: the beautiful TOE In sci.math, Tapio : >> TOE:i. Theory of Everything:i. i is the simplest explanation there is. > No, there is more simple explanation. ;-) Well, the simplest explanation I can think of is that someone lost his medication. :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: the beautiful TOE i take the medication to keep the world sane. that alone is the true explanation. > In sci.math, Tapio > : > TOE:i. Theory of Everything:i. i is the simplest explanation there is. >> No, there is more simple explanation. ;-) > Well, the simplest explanation I can think of is that someone lost > his medication. :-) > -- > #191, ewill3@earthlink.net > It's still legal to go .sigless. === Subject: Re: the beautiful TOE > TOE:i. Theory of Everything:i. i is the simplest explanation there is. I > will tell > you the story in one verse, in several verses. I,alone know the laws. > I,alone > dictate the laws. I, alone am supreme. I, alone, is the only law. There are > infinite > laws to validate me. I, alone tell you, who you are. Dont resist my > delusion. Dont > resist my hallucination. Understand that everything at the fundamental level > is a > great illusion around me. I alone am the supreme mind, who is awake. Dont > fight me. > I alone tell you, you all are Gods. You all are Demons. Dont resist my ego. > Dont > resist me. I alone command you. I alone tell you who you are. How Gerald > relates to > Einstien, How Fredkin relates to Feymann. Dont resist me. Learn your place. > I have > only one power, of all the powers. I have the power to believe or > disbelieve. I > disbelieve i m human. You disbelieve you are human. I tell you. The game is > to > convince everyone, of your delusions, of your hallucinations. Understand and > accept > that, i tell you. I alone, am the supreme dictator of the laws. I alone, > command > you. I alone tell you, who you are. I alone make you, friends, create you, > enemies. > Dont fight me. It is I. I alone, am the simplest explanation there is. It is > I, the > believer's believer, the disbeliever's disbeliever, who is awake. Dont > command me. I > alone, am the center of the universe, the epicenter of everything. I alone > am the > arbiter of arbitary facts. Dont question me. I alone program you. Dont fight > your > delusions. Dont fight your hallucinations. Accept the myth of reality. > Understand > the macrosopic correlations. Understand that effects preceed the cause. Dont > validate me. I, alone tell you. I alone command you. Dont command me. The > game is to > convince everyone of who you are. I tell you. I m your leader. There is no > leader > higher than me. I m your servant. There is no servant lower than me. I > alone, battle > the universe. It takes infinite laws to defeat me. It is I, who set > everything in > motion and the power to set it and unset it. Dont command me. It is I, who > flatters > you, with my human form. Dont flatter me. It is I, who defeats you, who > undefeats > you. Dont fight me. It takes the universe to battle me....... Are you trolling, do you consider this to be a good joke, or are you genuinely mad? Bye, Bjoern === Subject: Re: the beautiful TOE i m perfect. >> TOE:i. Theory of Everything:i. i is the simplest explanation there is. I >> will tell >> you the story in one verse, in several verses. I,alone know the laws. >> I,alone >> dictate the laws. I, alone am supreme. I, alone, is the only law. There >> are infinite >> laws to validate me. I, alone tell you, who you are. Dont resist my >> delusion. Dont >> resist my hallucination. Understand that everything at the fundamental >> level is a >> great illusion around me. I alone am the supreme mind, who is awake. Dont >> fight me. >> I alone tell you, you all are Gods. You all are Demons. Dont resist my >> ego. Dont >> resist me. I alone command you. I alone tell you who you are. How Gerald >> relates to >> Einstien, How Fredkin relates to Feymann. Dont resist me. Learn your >> place. I have >> only one power, of all the powers. I have the power to believe or >> disbelieve. I >> disbelieve i m human. You disbelieve you are human. I tell you. The game >> is to >> convince everyone, of your delusions, of your hallucinations. Understand >> and accept >> that, i tell you. I alone, am the supreme dictator of the laws. I alone, >> command >> you. I alone tell you, who you are. I alone make you, friends, create >> you, enemies. >> Dont fight me. It is I. I alone, am the simplest explanation there is. It >> is I, the >> believer's believer, the disbeliever's disbeliever, who is awake. Dont >> command me. I >> alone, am the center of the universe, the epicenter of everything. I >> alone am the >> arbiter of arbitary facts. Dont question me. I alone program you. Dont >> fight your >> delusions. Dont fight your hallucinations. Accept the myth of reality. >> Understand >> the macrosopic correlations. Understand that effects preceed the cause. >> Dont >> validate me. I, alone tell you. I alone command you. Dont command me. The >> game is to >> convince everyone of who you are. I tell you. I m your leader. There is >> no leader >> higher than me. I m your servant. There is no servant lower than me. I >> alone, battle >> the universe. It takes infinite laws to defeat me. It is I, who set >> everything in >> motion and the power to set it and unset it. Dont command me. It is I, >> who flatters >> you, with my human form. Dont flatter me. It is I, who defeats you, who >> undefeats >> you. Dont fight me. It takes the universe to battle me....... > Are you trolling, do you consider this to be a good joke, or are > you genuinely mad? > Bye, > Bjoern === Subject: Re: the beautiful TOE you will find what you are looking for. you think I m sane, i will sound sane. you think I m joking, i will sound joking. you think I m insane. I will sound insane. You think I sound wise, I will sound wise. I m playing with your mind. I m playing with your delusion. >i m perfect. > TOE:i. Theory of Everything:i. i is the simplest explanation there is. I > will tell > you the story in one verse, in several verses. I,alone know the laws. > I,alone > dictate the laws. I, alone am supreme. I, alone, is the only law. There > are infinite > laws to validate me. I, alone tell you, who you are. Dont resist my > delusion. Dont > resist my hallucination. Understand that everything at the fundamental > level is a > great illusion around me. I alone am the supreme mind, who is awake. > Dont fight me. > I alone tell you, you all are Gods. You all are Demons. Dont resist my > ego. Dont > resist me. I alone command you. I alone tell you who you are. How Gerald > relates to > Einstien, How Fredkin relates to Feymann. Dont resist me. Learn your > place. I have > only one power, of all the powers. I have the power to believe or > disbelieve. I > disbelieve i m human. You disbelieve you are human. I tell you. The game > is to > convince everyone, of your delusions, of your hallucinations. Understand > and accept > that, i tell you. I alone, am the supreme dictator of the laws. I alone, > command > you. I alone tell you, who you are. I alone make you, friends, create > you, enemies. > Dont fight me. It is I. I alone, am the simplest explanation there is. > It is I, the > believer's believer, the disbeliever's disbeliever, who is awake. Dont > command me. I > alone, am the center of the universe, the epicenter of everything. I > alone am the > arbiter of arbitary facts. Dont question me. I alone program you. Dont > fight your > delusions. Dont fight your hallucinations. Accept the myth of reality. > Understand > the macrosopic correlations. Understand that effects preceed the cause. > Dont > validate me. I, alone tell you. I alone command you. Dont command me. > The game is to > convince everyone of who you are. I tell you. I m your leader. There is > no leader > higher than me. I m your servant. There is no servant lower than me. I > alone, battle > the universe. It takes infinite laws to defeat me. It is I, who set > everything in > motion and the power to set it and unset it. Dont command me. It is I, > who flatters > you, with my human form. Dont flatter me. It is I, who defeats you, who > undefeats > you. Dont fight me. It takes the universe to battle me....... >> Are you trolling, do you consider this to be a good joke, or are >> you genuinely mad? >> Bye, >> Bjoern === Subject: Re: the beautiful TOE > TOE:i. Theory of Everything:i. i is the simplest explanation there is. I > will tell > you the story in one verse, in several verses. I,alone know the laws. > I,alone > dictate the laws. http://www.mazepath.com/uncleal/sunshine.jpg -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: the beautiful TOE says... > > TOE:i. Theory of Everything:i. i is the simplest explanation there is. I > will tell > you the story in one verse, in several verses. I,alone know the laws. > I,alone > dictate the laws. > http://www.mazepath.com/uncleal/sunshine.jpg CamelTOE, now THAT'S beautiful. === Subject: Re: the beautiful TOE your mom > says... > TOE:i. Theory of Everything:i. i is the simplest explanation there is. >> I >> will tell >> you the story in one verse, in several verses. I,alone know the laws. >> I,alone >> dictate the laws. >> http://www.mazepath.com/uncleal/sunshine.jpg > CamelTOE, now THAT'S beautiful. === Subject: Re: the beautiful TOE > your mom > says... > TOE:i. Theory of Everything:i. i is the simplest explanation there is. >> I >> will tell >> you the story in one verse, in several verses. I,alone know the laws. >> I,alone >> dictate the laws. > http://www.mazepath.com/uncleal/sunshine.jpg > CamelTOE, now THAT'S beautiful. Good to see you're keeping your responses to under 3 words, we've seen the disasterous results when you go over... === Subject: Re: the beautiful TOE >>your mom >says... >>TOE:i. Theory of Everything:i. i is the simplest explanation there is. >I >will tell >you the story in one verse, in several verses. I,alone know the laws. >I,alone >dictate the laws. >>http://www.mazepath.com/uncleal/sunshine.jpg >CamelTOE, now THAT'S beautiful. > Good to see you're keeping your responses to under 3 words, we've seen > the disasterous results when you go over... It's amazing that his brain can generate enough energy for him to actually press keys on a keyboard to create his juvenile posts. Oh well he is comic relief and make a good playtoy. Ogie === Subject: Re: the beautiful TOE i command you. you are defeated. go home and be merry. >your mom >>says... >>TOE:i. Theory of Everything:i. i is the simplest explanation there is. >>I >>will tell >>you the story in one verse, in several verses. I,alone know the laws. >>I,alone >>dictate the laws. http://www.mazepath.com/uncleal/sunshine.jpg >CamelTOE, now THAT'S beautiful. >> Good to see you're keeping your responses to under 3 words, we've seen >> the disasterous results when you go over... > It's amazing that his brain can generate enough energy for him to actually > press keys on a keyboard to create his juvenile posts. > Oh well he is comic relief and make a good playtoy. > Ogie === Subject: Re: the beautiful TOE i defeated you. >> TOE:i. Theory of Everything:i. i is the simplest explanation there is. I >> will tell >> you the story in one verse, in several verses. I,alone know the laws. >> I,alone >> dictate the laws. > http://www.mazepath.com/uncleal/sunshine.jpg > -- > Uncle Al > http://www.mazepath.com/uncleal/ > (Toxic URL! Unsafe for children and most mammals) > http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: the beautiful TOE In sci.math, Lord of Chaos(Suresh Devanathan) <3147v2F37du29U1@individual.net>: > TOE:i. Theory of Everything:i. i is the simplest explanation there is. I > will tell > you the story in one verse, in several verses. I,alone know the laws. > I,alone > dictate the laws. >> http://www.mazepath.com/uncleal/sunshine.jpg >> -- >> Uncle Al >> http://www.mazepath.com/uncleal/ >> (Toxic URL! Unsafe for children and most mammals) >> http://www.mazepath.com/uncleal/qz.pdf > i defeated you. Solipsism defeating Uncle Al? Somehow, I don't think so. Also, quit top-posting. Followups to sci.physics. Because it's hard to read. Why is it annoying? Because it's annoying. Why is top posting bad? Because it's bad. Why shouldn't we top post? -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: the beautiful TOE you are dumb, Uncle Al. >> TOE:i. Theory of Everything:i. i is the simplest explanation there is. I >> will tell >> you the story in one verse, in several verses. I,alone know the laws. >> I,alone >> dictate the laws. > http://www.mazepath.com/uncleal/sunshine.jpg > -- > Uncle Al > http://www.mazepath.com/uncleal/ > (Toxic URL! Unsafe for children and most mammals) > http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: the beautiful TOE 'i' is the simplest explanation there is. >> TOE:i. Theory of Everything:i. i is the simplest explanation there is. I >> will tell >> you the story in one verse, in several verses. I,alone know the laws. >> I,alone >> dictate the laws. > http://www.mazepath.com/uncleal/sunshine.jpg > -- > Uncle Al > http://www.mazepath.com/uncleal/ > (Toxic URL! Unsafe for children and most mammals) > http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: the beautiful TOE u idiot >> TOE:i. Theory of Everything:i. i is the simplest explanation there is. I >> will tell >> you the story in one verse, in several verses. I,alone know the laws. >> I,alone >> dictate the laws. > http://www.mazepath.com/uncleal/sunshine.jpg > -- > Uncle Al > http://www.mazepath.com/uncleal/ > (Toxic URL! Unsafe for children and most mammals) > http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: the beautiful TOE > u idiot You are Head-Up-Your-Ass Toe HAHAHAHAHAHAHAH Merry Christmas asshole! === Subject: Re: the beautiful TOE >> u idiot > You are Head-Up-Your-Ass Toe > HAHAHAHAHAHAHAH > Merry Christmas asshole! Better yet you are an ass toe... I couldn't resist Ogie === Subject: Re: the beautiful TOE start making sense, idiot. > u idiot >> You are Head-Up-Your-Ass Toe >> HAHAHAHAHAHAHAH >> Merry Christmas asshole! > Better yet you are an ass toe... > I couldn't resist > Ogie === Subject: Re: Unstoppable Force vs Immovable Object by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iB12Wec08043; Note: when i say plaguing i meant to say in the corner of my mind. Related: that was the first thought that came to my mind, i am only a graded school student Related: I just kinda wanted other peoples views... masturbation... ok... === Subject: distributing points to high dimensional shapes? HI all, I post a question about distributing points on some space several days ago I still have a question that I am not very sure about: How to distribute points onto a ellipsoid in N-dimensional space? There are many literature about distributing points onto a sphere... but I want to know if putting points onto an ellipsoid and maximize the minimum distance among all these points is a trivial extension of sphere or a complete different prolbem? Any analysis and proof? Using an example, how to put 3 points onto an ellipse in a 2D plane to maximize the minimum distance among the 3 points? Can anybody give me a proof? For circle, it should be equal-lateral triangle, that's easy... Now what if the high dimensional shape is not sphere or ellipsoid? How about any arbitrary shape? What can be said about it? === Subject: Re: distributing points to high dimensional shapes? >How to distribute points onto a ellipsoid in N-dimensional space? There are >many literature about distributing points onto a sphere... but I want to >know if putting points onto an ellipsoid and maximize the minimum distance >among all these points is a trivial extension of sphere or a complete >different prolbem? Any analysis and proof? Is it a requirement that all points be on the surface of the ellipsoid? >Using an example, how to put 3 points onto an ellipse in a 2D plane to >maximize the minimum distance among the 3 points? Can anybody give me a >proof? For circle, it should be equal-lateral triangle, that's easy... The solution for an ellipse is an isosoles triangle with the apex on one end of the minor axis and the other 2 vertices at the farthest points on the ellipse from that point. For proof, partition the ellipse into 3 pieces -- the minor axis and the 2 pieces on each side of it. Look at any 2 points in the same piece, and they will be closer than the minimum distance with the isosoles solution, so there must be 1 point in each piece. It's obvious that the isosoles solution uses the longest distance from a side to the axis, then all you have to do is prove that the base of the triangle is the longest side. >Now what if the high dimensional shape is not sphere or ellipsoid? How about >any arbitrary shape? What can be said about it? There are a lot of potential situations where one axis will dominate, so the packing will be almost linear. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: how to prove that lamda_max(A) <= ||A||? Hi all, I want to ask how to prove that lamda_max(A) <= ||A||? where lamda_max(A) is the maximum eigenvalue of A and ||A|| is the 2-norm of the matrix A and ||A|| = sigma_max(A) where sigma_max(A) is the maximal singular value of A... Generally what should be the relationship between lamda(A) and sigma(A)? === Subject: Re: how to prove that lamda_max(A) <= ||A||? >Hi all, >I want to ask how to prove that lamda_max(A) <= ||A||? What is the definition of ||A||? And what is the definition of eigenvalue? If you look at both those definitions the proof is easy. >where lamda_max(A) is the maximum eigenvalue of A and ||A|| is the 2-norm of >the matrix A and ||A|| = sigma_max(A) where sigma_max(A) is the maximal >singular value of A... >Generally what should be the relationship between >lamda(A) and sigma(A)? ************************ David C. Ullrich === Subject: please I need an answer. Consider the sequence b0=1 and bn= for all n 1 Find the formula for b n and prove it by induction. (So by definition b1= b0=1 b2= b0+b1 =1+1=2) === Subject: Re: please I need an answer. > Consider the sequence b0=1 and bn= for all n 1 > Find the formula for b n and prove it by induction. > (So by definition b1= b0=1 > b2= b0+b1 =1+1=2) Your formula for bn didn't come through clearly. -- Will Twentyman email: wtwentyman at copper dot net === Subject: proving Consider the sequence b0=1 and bn= summation for all n 1 Find the formula for b n and prove it by induction. (So by definition b1= b0=1 b2= b0+b1 =1+1=2) === Subject: Re: proving > Consider the sequence b0=1 and bn= summation for all n 1 > Find the formula for b n and prove it by induction. > (So by definition b1= b0=1 > b2= b0+b1 =1+1=2) I'm guessing that you are trying to type something in using a fancy editor. Don't. Type in your question in notepad, then copy/paste it. We can only read ASCII, no special fonts or graphics. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: proving > Consider the sequence b0=1 and bn= summation for all n 1 > Find the formula for b n and prove it by induction. > (So by definition b1= b0=1 > b2= b0+b1 =1+1=2) Why don't you work out by hand what b1, b2, b3, b4, b5, b6, b7 and b8 are, and post them here? Does this sequence suggest what the formula for bn is? After you have done that, then we can discuss the proof by induction bit. === Subject: Re: proving >Consider the sequence b0=1 and bn= summation for all n 1 This doesn't make any more sense than the first version you posted. You need to say what the problem is first... >Find the formula for b n and prove it by induction. >(So by definition b1= b0=1 > b2= b0+b1 =1+1=2) ************************ David C. Ullrich === Subject: proving 1) use the definition of continuity to prove that the function f(x)= (x-1)/x is continuous at x0=2. 2) prove that if p is an odd integer then the equation x^4 + x^2-p=0 has no integer solutions. Dos the equation have rational solutions? Justify your answer. === Subject: Re: proving > 1) use the definition of continuity to prove that the function > f(x)= (x-1)/x is continuous at x0=2. f(x) = 1 - 1/x and 1/x is continuous at x = 2. > 2) prove that if p is an odd integer then the equation x^4 + x^2-p=0 > has no integer solutions. x^4 + x^2 is always even for n in Z. > Dos the equation have rational solutions? Justify your answer. No. Let q = n/m with coprime n,m. q^4 + q^2 = p n^4 + n^2 m^2 = m^4 p n odd, m odd not possible n odd, m even not possible n even, m odd not possible Thus n and m are even, but n,m are coprime. === Subject: Groups of order 28-errors at Mathworld and web I found the following error for groups of order 28 on the web page for finite groups (I post it after I find it again) and at Mathworld. I recall a post about errors to Mathworld here a while back. Anyway, I'm right about this, am I not? The error I think, is saying that the 2 non-Abelian groups are D_14 and C_2 x D_7, which are in fact the same. Here is the letter I sent. I was looking over your excellent web page on finite groups, and found an error for n = 28. I just did a problem showing that when n = 2m and m is odd, D_2m = C_2 x D_m ; thus D_14 and C_2 x D_7 are the same group. (Consider n = 12, D_6 and C_2 x S_3 = C_2 x D_3 are the same). The other non-Ablelian group for n = 28 = 4.7 is like the one for n = 20 = 4.5 with G = , x^7 = y^4 = 1, and yxy^(-1) = x^(-1) You should correct this. I will also write to Mathworld, as they have repeated this error. Sylvan Jacques === Subject: Re: Groups of order 28-errors at Mathworld and web Here is the link http://www.math.usf.edu/~eclark/algctlg/small_groups.html The mail bounced from the original author, perhaps the eclark is the Ed Clark who has some notes on intro algebra and elementary number theory. Van === Subject: Re: Groups of order 28-errors at Mathworld and web Here is the link http://www.math.usf.edu/~eclark/algctlg/small_groups.html The mail bounced from the original author, perhaps the eclark is the Ed Clark who has some notes on intro algebra and elementary number theory. Van === Subject: A rather complicated pseudorandom number generator In my explorations of FTL I have discovered that associated with each solution to a^n + b^n = c^n there is a sum (a^n + b^n) mod c + (c^n - a^n) mod b + (c^n - b^n) mod a = 0 and that for each sum S = 0, there may be a solution, but there may be merely a false hit. I am now wondering if S.n might be a useful pseudorandom number generator. It probably isn't. Look at all the computations. Also, it seeds with three values, not one. But I will try. Marsaglia gives some tests which I have, and Knuth gives more which I do not have. A triple a,b,c with aIn my explorations of FTL I have discovered that associated with each solution Looks like a homework problem to me... >a^n + b^n = c^n there is a sum >(a^n + b^n) mod c + >(c^n - a^n) mod b + >(c^n - b^n) mod a = 0 Of course. Consider the following: a^n + b^n = c^n therefore (a^n +c^n) mod c = c^n mod c = 0 The others should follow logically. I hope this has made it easier for you to figure out where you are going with this... === Subject: Re: A rather complicated pseudorandom number generator > I am now wondering if S.n might be a useful pseudorandom number generator. It > probably isn't. There is a NIST test suite available for testing. You might want to run it. M. K. Shen === Subject: Advanced Mathematics books by Perkins & Perkins When I studied for my 'A'-Level in Maths we used the two Advanced Mathematics books by Perkins & Perkins (the first book was blue, and the second was green). I left school 6 years ago and have just found my old maths work, which includes my own workings to many of the problems in the two books. Does anyone know if any official book of solutions to the problems in the books exists? Ideally I would like to get rid of my old schoolwork to make some space in our house, but if the solutions to the problems were never published I might think twice about doing this. Incidentally, how popular (if at all) are the Perkins & Perkins books? I was able to purchase new copies of both books a few years ago, for my own reference, but the original versions were published in 1982. I was wondering if they are used by teachers or lecturers these days. Rich === Subject: Solving Solvable Sextics Using Polynomial Decomposition Hello all, For those who like solvable equations and groups, here's something that might be interesting: Solving Solvable Sextics Using Polynomial Decomposition ABSTRACT: Using basic results established by Etienne Bezout (1730-1783) and Niels Henrik Abel (1802-1829), we devise a general method to solve the solvable sextic in radicals by deriving two kinds of resolvents, one kind of the 15th degree and the other of the 10th degree, that factors when the sextic is solvable and thus enabling us to decompose the solvable sextic either as: a) three quadratics whose coefficients are determined by a cubic or b) two cubics whose coefficients are determined by a quadratic. Mathematics Subject Classification: Primary: 12E12. http://www.geocities.com/titus_piezas/sextics.html Just click at the link to the .pdf file. --Titus === Subject: Re: final proof Einstein 1905 is crap. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iB1DHxV29498; >> Let's face it, it is beyond the intellectual capacity of moortel to >> understand >> how the sheer genius of Einstein hoodwinked the community for 100 >> years. >> Recall that the light leaves A and reflects at B in >> 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in >> empty >> space. It's not just this which is a crap. The whole theory theory of relativity is. Both its two postules and Lorentz transformation have been disputed by reports from the Hubble. For example, the claim that the velocity of light is the upper limit of velocity is false. The galaxies are speeding outwards at the rate of 10^20 km/sec. The basis of the postulate of relativity, that the Cosmos is essentially empty E. E. Escultura === Subject: Re: JSH: Simple proof by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iB1DHxY29540; Why don't you simply and briefly state the >>problematic<< property of the ring of algebraic integers and/or the statement that you want to prove. Using standard mathematical terms and notions (for example from commutative algebra) this should be possible in a few lines instead of making a long story. Why do we have to discuss things like >>what is a polynomial?<< here? The notion of a polynomial is defined since a long time and can be found in every introductory book on algebra. If we consistently use the common definitions of mathematical objects like polynomials we should rather quickly be able to clarify the situation and avoid all the frustration that frequently seems to culminate in personal attacks. H === Subject: Re: JSH: Simple proof > Why don't you simply and briefly state the >>problematic<< property > of the ring of algebraic integers and/or the statement that you > want to prove. The ring of algebraic integers is determined by roots of *monic* polynomials with integer coefficients. It is possible to show with basic algebra that there are numbers which are properly units but because their multiplicative inverse is not the root of some monic polynomial with integer coefficients they are not units in the ring of algebraic integers. To see how it works consider that in rationals you can have (3x + 1)(x + 1) = 3x^2 + 4x + 1 where, of course, one of the roots is a unit in the ring of algebraic integers. But now consider (3x + u_1)(x + u_2) = 3x^2 + kx + 1, where u_1 u_2 =1, and k is an integer. You find that if the u's are irrational, then u_1, while an algebraic integer is not a unit in the ring, while u_2 cannot then even be an algebraic integer. My research shows though that both u_1 and u_2 can be units in a ring where -1 and 1 are the only rational units, and no non-unit member of the ring is a factor of any two integers that are coprime in the ring of integers. You see, I abstracted out two key properties of rings like the ring of integers and the ring of algebraic integers. > Using standard mathematical terms and notions (for example from > commutative algebra) this should be possible in a few lines instead > of making a long story. It's not complicated. Basically you can't just rely on whether or not some number is in the ring of algebraic integers when considering factors of roots of a polynomial. The mathematics is mostly REALLY simple. > Why do we have to discuss things like >>what is a polynomial?<< I'm not discussing that, other posters made a big deal out of it. > here? The notion of a polynomial is defined since a long time and > can be found in every introductory book on algebra. So? > If we consistently use the common definitions of mathematical > objects like polynomials we should rather quickly be able to > clarify the situation and avoid all the frustration that frequently > seems to culminate in personal attacks. I've seen posters come and go, and every once in a while there's a poster like you who claims to care about working things out. When it turns out that I'm right, you go over to the other side, and either run away, or turn to bizarre behavior. Psychologists call it cognitive dissonance. Basically, deep down you believe that I must be wrong, so your post is not really in good faith. But simply *saying* certain things that indicate objectivity or willingness to be objective sets you up psychologically. That is, you feel a need to be consistent with what you said. But later, when you run into the rigid mathematics, which goes against what you wish to believe, you basically kind of break. Your mind breaks, and you run away or behave weird. I've seen it lots of times. Do yourself a favor, and just walk away now. James Harris === Subject: Re: JSH: Simple proof >>Why don't you simply and briefly state the >>problematic<< property >>of the ring of algebraic integers and/or the statement that you >>want to prove. > The ring of algebraic integers is determined by roots of *monic* > polynomials with integer coefficients. That's the definition. It is not a property of a ring. What is the problem with the ring? What prevents it from being everything that's been claimed? > It is possible to show with basic algebra that there are numbers which > are properly units but because their multiplicative inverse is not the > root of some monic polynomial with integer coefficients they are not > units in the ring of algebraic integers. No, it is not possible to show this, unless the term properly a unit is defined. You have not done this. An element of a ring is or is not a unit. If it *is* a unit, then it has a multiplicative inverse in the ring. If it isn't a unit, then its multiplicative inverse fails to be in the ring. There's no ambiguity here. How about providing a definition for your bogus terminology? > To see how it works consider that in rationals you can have > (3x + 1)(x + 1) = 3x^2 + 4x + 1 > where, of course, one of the roots is a unit in the ring of algebraic > integers. And the polynomial is of course reducible over the integers. > But now consider > (3x + u_1)(x + u_2) = 3x^2 + kx + 1, where u_1 u_2 =1, and k is an > integer. > You find that if the u's are irrational, then u_1, while an algebraic > integer is not a unit in the ring, while u_2 cannot then even be an > algebraic integer. So what? Here's what you get from the linear term of the product: u_1 + 3 u_2 = k Letting u = u_2 (since we already know that u_1 is an algebraic integer): 1/u + 3 u = k 3 u^2 - k u + 1 = 0 We solve for u, via the quadratic formula u = (k +/- sqrt(k^2 - 12))/6 Now, let's look at a specific example: k = 5 25 - 12 = 13 u = (5 +/- sqrt(13))/6 I'll let the root with + be u, and the root with - be ubar. Note that you can't have *both* solutions in your ring: u ubar = (25 - 13)/36 = 12/36 = 1/3 Considering your recent rant about how you can't get rid of the evil ambiguity of root extractions, how do you determine *which* of these roots is in your ring? Note that the fields Q(u) and Q(ubar) are isomorphic: any arithmetic statement you make in Q(u) can be translated automatically (by mechanically replacing each occurrence of u with ubar) into an equivalent arithmetic statement in Q(ubar), so that if one is true then so is the other one. That means that you cannot use algebraic means (e.g., any algebraic formula involving the integers and the desired root) to determine which root to select. If memory serves me correctly, the issue becomes even more problematic once you start trying to throw in roots of more and more non-monic polynomials. > My research shows though that both u_1 and u_2 can be units in a ring > where -1 and 1 are the only rational units, and no non-unit member of > the ring is a factor of any two integers that are coprime in the ring > of integers. Which u_2 are you going to choose? What I've called u, or what I've called ubar? You can't have both, and you can't tell them apart using algebra. You have not constructed such a ring. You have not even shown how to establish whether a given complex number is an element of the ring (such as: which of u,ubar is in the ring). Many here have agreed that such a ring can exist, but it is not unique. Note the above example. You may in fact be able to admit one of every candidate quadratic, one or two from every candidate cubic, and so forth, but you *cannot* admit all roots of *any* non-monic polynomial, without admitting a non-integral rational number. Once that happens, you'll get invertible integers other than +/- 1. BTW, if you have two coprime elements, u and v, of a ring R, then they remain coprime in any ring R' that contains R. It is also not difficult to show that any common factor of coprime elements of a ring must be a unit. > You see, I abstracted out two key properties of rings like the ring of > integers and the ring of algebraic integers. Your second property: no non-unit member of the ring is a factor of any two integers that are coprime in the ring of integers. is a corollary of the remarks I made above. This has been known for far longer than I have been alive; it was most likely obvious to Dedekind. It has nothing to do with key properties like the ring of integers. It holds for arbitrary commutative rings. Hooray for you for having abstracted it. >>Using standard mathematical terms and notions (for example from >>commutative algebra) this should be possible in a few lines instead >>of making a long story. > It's not complicated. Basically you can't just rely on whether or not > some number is in the ring of algebraic integers when considering > factors of roots of a polynomial. That's what you say. However, every specific example you've proposed that is supposed to highlight the flaws of the ring of algebraic integers (such as your polynomial factorization in your ill-fated paper Advanced Polynomial Factorization) has been proven to have been incorrect. > The mathematics is mostly REALLY simple. >>Why do we have to discuss things like >>what is a polynomial?<< > I'm not discussing that, other posters made a big deal out of it. >>here? The notion of a polynomial is defined since a long time and >>can be found in every introductory book on algebra. > So? >>If we consistently use the common definitions of mathematical >>objects like polynomials we should rather quickly be able to >>clarify the situation and avoid all the frustration that frequently >>seems to culminate in personal attacks. > I've seen posters come and go, and every once in a while there's a > poster like you who claims to care about working things out. > When it turns out that I'm right, you go over to the other side, and > either run away, or turn to bizarre behavior. It has not turned out you're right. What has invariably happened is that you have turned to some form of antisocial behavior, such as what you are doing right now. > Psychologists call it cognitive dissonance. > Basically, deep down you believe that I must be wrong, so your post is > not really in good faith. But simply *saying* certain things that > indicate objectivity or willingness to be objective sets you up > psychologically. This must be part of your morning Stuart Smalley chant. > That is, you feel a need to be consistent with what you said. > But later, when you run into the rigid mathematics, which goes against > what you wish to believe, you basically kind of break. Your mind > breaks, and you run away or behave weird. > I've seen it lots of times. Do yourself a favor, and just walk away > now. I think you're trying to censor the poster. Shame on you. > James Harris Dale. === Subject: Re: JSH: Simple proof !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> It is possible to show with basic algebra that there are numbers >> which are properly units but because their multiplicative inverse >> is not the root of some monic polynomial with integer coefficients >> they are not units in the ring of algebraic integers. > No, it is not possible to show this, unless the term > properly a unit > is defined. > You have not done this. An element of a ring is or is not > a unit. If it *is* a unit, then it has a multiplicative > inverse in the ring. If it isn't a unit, then its multiplicative > inverse fails to be in the ring. > There's no ambiguity here. > How about providing a definition for your bogus terminology? How about an example? When considering the ring of ordinary integers, 1/2 is properly a unit(TM), while its inverse (which is 2) is not a unit in the ordinary integers. And this is a problem with the ordinary integers. Or something. Why you'd be considering 1/2 in the first place when talking about the ring of integers, is a different question. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: JSH: Simple proof Discussion, linux) >> Why don't you simply and briefly state the >>problematic<< property >> of the ring of algebraic integers and/or the statement that you >> want to prove. > The ring of algebraic integers is determined by roots of *monic* > polynomials with integer coefficients. > It is possible to show with basic algebra that there are numbers which > are properly units but because their multiplicative inverse is not the > root of some monic polynomial with integer coefficients they are not > units in the ring of algebraic integers. What is the meaning of properly units? Can you give an example of a non-zero complex number which is not properly a unit? (Note: Answering the second question does not make the first question irrelevant.) -- Come on people!!! The US just blew up a lot of people in Iraq, don't you realize that a person with my exposure might just end up dead, by mysterious circumstances? --James Harris, on the dangers of proving Fermat's last theorem === Subject: Re: JSH: Simple proof !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> Why don't you simply and briefly state the >>problematic<< property >> of the ring of algebraic integers and/or the statement that you >> want to prove. > The ring of algebraic integers is determined by roots of *monic* > polynomials with integer coefficients. Well, that's the definition. That's what makes them interesting in the first place. If you don't like the particular class of numbers defined by this property, you are free to define a different class of numbers defined by different properties. You just can't expect it to obey the same laws then. > It is possible to show with basic algebra that there are numbers > which are properly units This is nonsense. Units are always units in a specified ring. > but because their multiplicative inverse is not the root of some > monic polynomial with integer coefficients they are not units in the > ring of algebraic integers. You have no clue what units means. In analogy to your argument, I could say that the ring of plain integers has a problem because 1/2 is a proper unit, but its multiplicative inverse 2 is not a unit in the ring of integers. This is complete folly, because _of course_ 1/2 is not even subject to discussion when I am talking about integers. > But later, when you run into the rigid mathematics, which goes > against what you wish to believe, you basically kind of break. Your > mind breaks, and you run away or behave weird. Good self-description. > I've seen it lots of times. Do yourself a favor, and just walk away > now. Good self-advice. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: JSH: Simple proof >>Why don't you simply and briefly state the >>problematic<< property >>of the ring of algebraic integers and/or the statement that you >>want to prove. > The ring of algebraic integers is determined by roots of *monic* > polynomials with integer coefficients. > It is possible to show with basic algebra that there are numbers which > are properly units but because their multiplicative inverse is not the > root of some monic polynomial with integer coefficients they are not > units in the ring of algebraic integers. Has it occurred to you that the problem might be with your undefined notion of properly units? What you just said is that something can be properly a unit, but not actually a unit. To me it sounds like non-sense. > To see how it works consider that in rationals you can have > (3x + 1)(x + 1) = 3x^2 + 4x + 1 > where, of course, one of the roots is a unit in the ring of algebraic > integers. Which is obvious from the fact that x+1 is a monic polynomial. > But now consider > (3x + u_1)(x + u_2) = 3x^2 + kx + 1, where u_1 u_2 =1, and k is an > integer. > You find that if the u's are irrational, then u_1, while an algebraic > integer is not a unit in the ring, while u_2 cannot then even be an > algebraic integer. Not surprising. > My research shows though that both u_1 and u_2 can be units in a ring > where -1 and 1 are the only rational units, and no non-unit member of > the ring is a factor of any two integers that are coprime in the ring > of integers. True. But that ring is likely to vary based on the values of the u's. You are probably talking about a set of rings, rather than a single ring. > You see, I abstracted out two key properties of rings like the ring of > integers and the ring of algebraic integers. Now, is there a maximal ring in the set? No. So it seems unlikely you have anything ground-breaking, nor do you have a flaw in the algebraic integers. What made you think you did? >>Using standard mathematical terms and notions (for example from >>commutative algebra) this should be possible in a few lines instead >>of making a long story. > It's not complicated. Basically you can't just rely on whether or not > some number is in the ring of algebraic integers when considering > factors of roots of a polynomial. > The mathematics is mostly REALLY simple. Here's the problem as I see it: You would rather force something to be a unit, thus going to a ring extension, rather than discuss the properties of a number in the algebraic integers. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: JSH: Simple proof > It is possible to show with basic algebra that there are numbers which > are properly units but because their multiplicative inverse is not the > root of some monic polynomial with integer coefficients they are not > units in the ring of algebraic integers. What does properly units mean? === Subject: Re: JSH: Simple proof >>It is possible to show with basic algebra that there are numbers which >>are properly units but because their multiplicative inverse is not the >>root of some monic polynomial with integer coefficients they are not >>units in the ring of algebraic integers. > What does properly units mean? It appears to mean things I want to be units, because they make my argument work. The fact that there are such things that are not algebraic integers leads James to conclude that there is some fundamental problem with the algebraic integers. It's sort of like complaining that there's no solution to 3x - 1 = 0 in the ring of integers and then going on to conclude that there's a fundamental problem with the integers. Rick === Subject: Re: JSH: Simple proof !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >It is possible to show with basic algebra that there are numbers which >are properly units but because their multiplicative inverse is not the >root of some monic polynomial with integer coefficients they are not >units in the ring of algebraic integers. >> What does properly units mean? > It appears to mean things I want to be units, because they make > my argument work. The fact that there are such things that are > not algebraic integers leads James to conclude that there is some > fundamental problem with the algebraic integers. > It's sort of like complaining that there's no solution to 3x - 1 = 0 > in the ring of integers and then going on to conclude that there's a > fundamental problem with the integers. Oh, but there is. That's why we have rationals in the first place. In a similar vein, we have algebraic numbers as a superset of algebraic integers. What James does not realize is that the shortcomings and their structures are what actually makes study of those fields (uh, pun gone bad) interesting in the first place. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: JSH: Simple proof > Why don't you simply and briefly state the >>problematic<< property > of the ring of algebraic integers and/or the statement that you > want to prove. > Using standard mathematical terms and notions (for example from > commutative algebra) this should be possible in a few lines instead > of making a long story. > Why do we have to discuss things like >>what is a polynomial?<< > here? The notion of a polynomial is defined since a long time and > can be found in every introductory book on algebra. > If we consistently use the common definitions of mathematical > objects like polynomials we should rather quickly be able to > clarify the situation and avoid all the frustration that frequently > seems to culminate in personal attacks. But if he cleared away the obfuscation and removed the personal stuff, what would he have left? === Subject: Putting FLT to Rest by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iB1DI0q29566; PUTTING FERMAT.89S LAST THEOREM TO REST Now that counterexamples to FLT have been constructed it is appropriate to put this false conjecture in perspective. Andrew Wiles made the most earnest attempt to prove it. His strategy was quite simple: consider elliptic curves on the surface, the latter supposedly defined by Fermat.89s equation for n > 2, namely, x^n + y^n = z^n, and prove that they contain no point with integer coordinates. A problem arises: curves and surfaces are topics in analysis. This is minor, however, since Wiles could have devised suitable analytical tools. He didn.89t. Instead, as number theorist, he resorted to classical algebraic tools such as Galois theory and Hecke rings which are vintage 19th Century. They are inadequate and uninformed by recent developments. The more serious problem, however, as L. C. Young pointed out in the thirties, is: all classical theories of curves and surfaces including elliptic curves are flawed because they are not well-defined by their functional representations. A c! urve is well-defined by a pair of functions .9a its functional representation and derivative. He developed this idea into the theory of generalized curves in a series of papers from 1933 to 1937. Wiles was unaware of this development, a fatal error of omission that shut down his ëproof..89 Moreover, elliptic curves were already known. He merely tried to use them but without success. Therefore, Wiles did not open up any area and I am not aware of any major paper that sprung up from his work. For the benefit of the reader, we may define an elliptic curve as the intersection of a surface and a plane and, for Wiles, a suitable family of elliptic curves lie on planes perpendicular to and passing through integral values on the z-axis. Every point on an elliptic curve satisfies both the equation of the surface and the plane. It would have sufficed to prove that no point on this curve has integral x- and y-coordinates and that this is so for the other elliptic curves on Fermat.89s surface! . However, this scheme failed because Wiles.89 tools lacked validity. Like a curve, a surface is not well-defined by its functional representation alone. Young fixed this error by well-defining a surface not only by a system of partial differential equations with suitable conditions along its boundary but also by associating with every element of area on the surface a directed line segment perpendicular to it. This is called the jacobian. Young developed this idea into the theory of generalized surfaces in a series of papers from 1938 to 1954. Again, Wiles was not aware of it .9a another fatal error since one cannot have a valid proof unless it is justified by a valid theory. The most fundamental error Wiles committed, however, is: he tried to tackle FLT without assessing the status of the underlying fields .9a foundations, number theory and analysis. Present mathematical reasoning is flawed. Number theory has no valid axiomatization; therefore, no valid proof exists. Two of the a! xioms of the real number system .9a the completeness and dichotomy axioms .9a are false. Therefore, the real number system, the base space of analysis, is ill-defined. Consequently, FLT as posed by Wiles in the context of curves and surfaces, which belong to analysis, is ill-defined. The remedy is the reconstruction of the real number system without these axioms as well as upgrading of foundations and number theory that I undertook in 1998 (Nonlinear Studies, Vol. 5). The reconstruction upgrades number theory by embedding the integers in the new real number system. After discarding the nonsense of the real number system the new real number system becomes finite but unbounded, free from contradictions, enriched by the dark number d* and unbounded number u* and has natural ordering, namely, the lexicographic ordering. Wiles was also unaware of recent results especially my discovery of ambiguous sets (Nonlinear Analysis, Vol. 35, 2001) such as the set of points in an elliptic curve! . Scientific standards require that when an error is committed it is not sufficient to present a correct alternative. The error must be analyzed and criticized, which I did in a series of papers from 1996 to the present, and the one who committed it must either refute the criticism and defend his work or accept it by default. Wiles did the latter. Finally, new mathematics sprung up from my resolution of FLT among which are: the new real number system, new nonstandard calculus, new arithmetic, dynamic modeling and characterization of undecidable propositions, published in renowned journals. (For the interested reader visit the websites: http://home.iprimus.com.au/pidro/ and http://www.users.bigpond.com/pidro/home.htm) === Subject: Re: Putting FLT to Rest Wow... You must be so bloody stupid... === Subject: Re: Putting FLT to Rest So give us ONE counter example! >PUTTING FERMAT.89S LAST THEOREM TO REST >Now that counterexamples to FLT have been constructed it is appropriate to put this false conjecture in perspective. Andrew Wiles made the most earnest attempt to prove it. His strategy was quite simple: consider elliptic curves on the surface, the latter supposedly defined by Fermat.89s equation for n > 2, namely, x^n + y^n = z^n, and prove that they contain no point with integer coordinates. A problem arises: curves and surfaces are topics in analysis. This is minor, however, since Wiles could have devised suitable analytical tools. He didn.89t. Instead, as number theorist, he resorted to classical algebraic tools such as Galois theory and Hecke rings which are vintage 19th Century. They are inadequate and uninformed by recent developments. The more serious problem, however, as L. C. Young pointed out in the thirties, is: all classical theories of curves and surfaces including elliptic curves are flawed because they are not well-defined by their functional representations. A c! >urve is well-defined by a pair of functions .9a its functional representation and derivative. He developed this idea into the theory of generalized curves in a series of papers from 1933 to 1937. Wiles was unaware of this development, a fatal error of omission that shut down his ëproof..89 Moreover, elliptic curves were already known. He merely tried to use them but without success. Therefore, Wiles did not open up any area and I am not aware of any major paper that sprung up from his work. For the benefit of the reader, we may define an elliptic curve as the intersection of a surface and a plane and, for Wiles, a suitable family of elliptic curves lie on planes perpendicular to and passing through integral values on the z-axis. Every point on an elliptic curve satisfies both the equation of the surface and the plane. It would have sufficed to prove that no point on this curve has integral x- and y-coordinates and that this is so for the other elliptic curves on Fermat.89s surface! >. However, this scheme failed because Wiles.89 tools lacked validity. Like a curve, a surface is not well-defined by its functional representation alone. Young fixed this error by well-defining a surface not only by a system of partial differential equations with suitable conditions along its boundary but also by associating with every element of area on the surface a directed line segment perpendicular to it. This is called the jacobian. Young developed this idea into the theory of generalized surfaces in a series of papers from 1938 to 1954. Again, Wiles was not aware of it .9a another fatal error since one cannot have a valid proof unless it is justified by a valid theory. The most fundamental error Wiles committed, however, is: he tried to tackle FLT without assessing the status of the underlying fields .9a foundations, number theory and analysis. Present mathematical reasoning is flawed. Number theory has no valid axiomatization; therefore, no valid proof exists. Two of the a! >xioms of the real number system .9a the completeness and dichotomy axioms .9a are false. Therefore, the real number system, the base space of analysis, is ill-defined. Consequently, FLT as posed by Wiles in the context of curves and surfaces, which belong to analysis, is ill-defined. The remedy is the reconstruction of the real number system without these axioms as well as upgrading of foundations and number theory that I undertook in 1998 (Nonlinear Studies, Vol. 5). The reconstruction upgrades number theory by embedding the integers in the new real number system. After discarding the nonsense of the real number system the new real number system becomes finite but unbounded, free from contradictions, enriched by the dark number d* and unbounded number u* and has natural ordering, namely, the lexicographic ordering. Wiles was also unaware of recent results especially my discovery of ambiguous sets (Nonlinear Analysis, Vol. 35, 2001) such as the set of points in an elliptic curve! >. Scientific standards require that when an error is committed it is not sufficient to present a correct alternative. The error must be analyzed and criticized, which I did in a series of papers from 1996 to the present, and the one who committed it must either refute the criticism and defend his work or accept it by default. Wiles did the latter. Finally, new mathematics sprung up from my resolution of FLT among which are: the new real number system, new nonstandard calculus, new arithmetic, dynamic modeling and characterization of undecidable propositions, published in renowned journals. (For the interested reader visit the websites: http://home.iprimus.com.au/pidro/ and http://www.users.bigpond.com/pidro/home.htm) === Subject: Re: Putting FLT to Rest ... > So give us ONE counter example! That is easy. E. Escultura is in the clan of people that think that the set of integers also contains infinite integers, and are in fact 10-adics. Finding counter-examples in the 10-adics is simple (and I think he has posted them already long ago; at least I did). Consider the 10-adics as the direct sum of the 2-adics and the 5-adics. We find two idempotents in the 10-adics and their sum is 1. So this is a counterexample for each power you may wish. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Putting FLT to Rest > So give us ONE counter example! Easy: nonsense^handwave + absurdity^handwave = crank^handwave what could be clearer? === Subject: Re: Putting FLT to Rest In sci.math, John Coleman > So give us ONE counter example! > Easy: nonsense^handwave + absurdity^handwave = crank^handwave > what could be clearer? You forgot to divide by the incomprehensibility factor. That would have made it a *lot* clearer. There's also the fudge factor, which is the amount of chocolate one eats during a certain timeframe... -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Putting FLT to Rest > So give us ONE counter example! ...And if You are able to parametrize the counter example by using or applying finite integers, then You have right to put FLT to rest. ;-) === Subject: Re: The flux theory of gravitation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iB1DHw929472; >> Since some of you have made references to the Flux Theory of Gravitation >> that I developed, >They seem to be mathematicians. I requested >them to define NUMBER a week ago but I have not seen their names >ever since. Are they undergrads or mathematicians from antiquity? >Some of my students might agree with the latter... >Robert Israel israel@math.ubc.ca >Department of Mathematics http://www.math.ubc.ca/~israel >University of British Columbia Vancouver, BC, Canada E. E. Escultura === Subject: Re: The flux theory of gravitation >to define NUMBER. That was two weeks ago and neither of you is able to >do it. That is why I wondered if I was dealing undergrads >mathematicians from antiquity or, perhaps, even charlatans. >E. E. Escultura Actually what you said was: > Now, let us put our money where our mouth is and make this forum > interesting. I make this challenge: DEFINE A REAL NUMBER The definition > must be original and different from mine. If you post a correct > definition, no flaw, etc. I.89ll send you a check for $1000, indicate your > address. However, if your definition is wrong or not original, you > should send me a check for the same amount. I.89ll post my address when needed. I'm not interested in this challenge, for the following reasons: 1) This is not interesting. Rigourous definitions of the real numbers have been available for well over 100 years. Look it up in a book. 2) Any correct definition I might make would likely be the same as that in some book, and therefore not original. 3) I don't anticipate I'd have much chance of collecting your $1000 in any case, especially if you are the judge of what is correct, no flaw, etc. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Cantor's diagonal proof wrong? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iB1DHw529447; >In sci.math, Shmuel (Seymour J.) Metz >But it is still curious that it happens at 2^oo and not oo^oo. >> Why? Is it curious that 3*3 is smaller than 3^3, or that 16*16 is >> smaller than 2^16? >I suspect that, if a set S has cardinality of at least card(N), >then a bijective mapping can be found between the elements of 2^S >and the elements of n^S, where n > 1 is an integer. >However, I'd have to look. Sure, that's correct; in fact, n can have cardinality anywhere from 2 to 2^S. There's an injection 2^S --> n^S, and there's an injection n --> 2^S (and so an injection 2^S --> (2^S)^S ~ 2^S); now apply Schroeder-Bernstein. Todd Trimble === Subject: Vector invariants I have 3 vectors A,B,C in the plane (with some origin O) and like to construct invariants which are symmetric (or antisymmetric) in A,B,C and independent of O, by piling up cross and scalar products. (Example: [AxB]*C. Not very helpful :-) Same with 4 vectors. Any suggestions? -- Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn === Subject: Re: Vector invariants >I have 3 vectors A,B,C in the plane (with some origin O) >and like to construct invariants which are symmetric >(or antisymmetric) in A,B,C and independent of O, First of all, a function F(A,B,C) which is antisymmetric has the property that G(A,B,C) = ( F(A,B,C) )^2 is symmetric, so in some sense it's sufficient to find all possible symmetric functions. Try this: Writing A = (a1,a2), etc., it appears that you are looking for polynomials in six variables which are invariant under the diagonal action of S_3 (the symmetric group on three letters). These form a ring which is a finitely-generated module over a polynomial ring. Now, I was working on a similar problem just three years ago and asked in sci.math.symbolic about this sort of thing. The computations -- at least for n=3 ! -- are not too bad. Indeed, the invariant ring R can be viewed as containing the polynomial ring P on Z1=a1+b1+c1, Y1=a2+b2+c2, Z2=a1^2+b1^2+c1^2, Y2=a2^2+b2^2+c2^2, Z3=a1^3+b1^3+c1^3, Y3=a2^3+b2^3+c2^3 ; the whole invariant ring R is generated, as a ring, by these and three more invariants, W1 = a1*a2 + b1*b2 + c1*c2 , W2 = a1^2*a2 + b1^2*b2 + c1^2*c2 , W3 = a1*a2^2 + b1*b2^2 + c1*c2^2 , which are subject to certain relations (attached below). If you like, the whole invariant ring may be presented as a free module over P on six generators: 1, W1, W2, W3, and W4 = W1^2, W5 = W2*W3 ; the five relations show (respectively) how to express W1 W2, W1 W3, W1^3, W2^2, and W3^2 in terms of these generators. Now, there is a homomorphism phi from this invariant ring R to R[u,v] defined by sending a1 -> a1+u, a2 -> a2+v and likewise b1 -> b1+u, etc. When you say you want expressions which are independent of O, I take it that means you want the subring of invariants which are fixed by phi. I would have told you it was easy to compute this, except that in a thread several weeks ago I realized I didn't know how to compute invariants of just this type (e.g. phi(x,y) = (2x, 3y) on F[x,y]. The thread was called Curves invariant under a rational map.) But in this case life is not so complicated because we easily find phi(Z1)=Z1 + 3u, phi(Y1) = Y1 + 3v ; so it's not too hard to use phi(Z2) = Z2 + 2u Z1 + 3 u^2 and so we compute that Z2' = Z2 - Z1^2/3 is phi-invariant. Likewise Z3' = Z3 - Z1^2*Z2 + (2/9)*Z1^3 and similar expressions for the Y's. Therefore our invariant subring R may be written F[ Z1, Y1, Z2', Y2', Z3', Y3' ] on which it seems pretty clear that the portion preserved by phi is just F[ Z2', Y2', Z3', Y3' ]. Likewise we may replace our module generators by phi-invariant ones: W1' = W1 - Z1*Y1/3, W2' = W2 - (Y1*Z2/3)-2*(Z1*W1)/3 + 2*(Y1*Z1^2)/9, W3' = W3 - (Y2*Z1/3)-2*(Y1*W1)/3 + 2*(Z1*Y1^2)/9, so that the whole S_3 - invariant ring R is now described as: the rank-6 free module over F[ Z1, Y1, Z2', Y2', Z3', Y3'] spanned by 1, W1', W2', W3', W4' = (W1')^2, and W5=(W2')(W3'); the action of phi is easily given since all but two generators are phi-invariant. It's obvious (isn't it?) that if S is any ring and phi : S[Z1,Y1] --> S[Z1,Y1,u,v] is the map which is the identity on S but sends phi(Z1) = Z1+u, phi(Y1)=Y1+v, then the subring fixed by phi is precisely S. In our case, that means (drumroll...) The invariants you are looking for are the polynomials in : Z2= (a1^2+b1^2+c1^2) - (b1*a1+c1*a1+b1*c1) Y2= (a2^2+b2^2+c2^2) - (b2*a2+c2*a2+b2*c2) Z3= 2*(a1^3+b1^3+c1^3) + 12*a1*b1*c1 - 3*(c1*b1^2+c1^2*b1+c1*a1^2+b1*a1^2+a1*c1^2+a1*b1^2) Y3= 2*(a2^3+b2^3+c2^3) + 12*a2*b2*c2 - 3*(c2*b2^2+c2^2*b2+c2*a2^2+b2*a2^2+a2*c2^2+a2*b2^2) W1= 2*(a1*a2+b1*b2+c1*c2) - (a1*b2+a1*c2+b2*c1+b1*c2+a2*c1+a2*b1) W2= 2*(a1^2*a2+b1^2*b2+c1^2*c2) + 4*(b1*a1*c2+c1*a1*b2+b1*c1*a2) -(a2*c1^2+a2*b1^2+c2*b1^2+c1^2*b2+a1^2*b2+a1^2*c2) -2*(b1*a1*b2+c1*a1*a2+c1*a1*c2+b1*c1*b2+b1*c1*c2+b1*a1*a2) W3= 2*(a1*a2^2+b1*b2^2+c1*c2^2) + 4*(c1*a2*b2+b1*a2*c2+a1*b2*c2) -( c2^2*a1+b2^2*a1+b2^2*c1+c2^2*b1+a2^2*b1+a2^2*c1) -2*( a2*b2*b1+c2*a1*a2+a2*c2*c1+c2*b2*b1+b2*c1*c2+b2*a1*a2) (I scaled the generators to get rid of fractions.) Just an example, one of the obvious invariants is the square of the area of the triangle formed by the endpoints of the vectors, a quantity which is known (Heron) to be expressible as a polynomial in the coordinates. It turns out to be (1/3)( 4 Z2 Y2 - (W1)^2 ) . Something related to the perimeter ought to be in there too. I will let you find another five or so geometric quantities which are clearly both polynomial and invariant so that there is a completely geometric description of all the invariants. (Note that we don't expect most of them to be _rotation_ - invariant.) You said something about taking cross products but I'm going to ignore that because if you REALLY meant to do that, then I'd have to repeat all these computations within the ring F[a1,a2,a3,b1...,c3]. That will be uglier. You go on to ask, >Same with 4 vectors. Any suggestions? which would force me to work out Sym(4) invariants in a polynomial ring on 12 generators; if you really want that, it's going to cost you... dave Here are the five generators among the Zs,Ys, and Ws: -6*W1*W2 + Z1^3*Y2-2*Z1^2*Y1*W1-Z1^2*W3+Z1*Y1^2*Z2+2*Z1*Y1*W2 -4*Z1*Z2*Y2+4*Z1*W1^2-Y1^2*Z3+3*Z2*W3+3*Y2*Z3, -6*W1*W3 + Z1^2*Y1*Y2-Z1^2*Y3-2*Z1*Y1^2*W1+2*Z1*Y1*W3+Y1^3*Z2 -Y1^2*W2-4*Y1*Z2*Y2+4*Y1*W1^2+3*Z2*Y3+3*Y2*W2, -3*W1^3 + Z1^3*Y1*Y2+Z1^3*Y3-2*Z1^2*Y1^2*W1-Z1^2*Y1*W3-3*Z1^2*Y2*W1 +Z1*Y1^3*Z2-Z1*Y1^2*W2-Z1*Y1*Z2*Y2+7*Z1*Y1*W1^2-6*Z1*Z2*Y3 +6*Z1*Y2*W2+Y1^3*Z3-3*Y1^2*Z2*W1+6*Y1*Z2*W3-6*Y1*Y2*Z3 +3*Z2*Y2*W1+9*Z3*Y3-9*W2*W3, -6*W2^2 + Z1^4*Y2-2*Z1^3*Y1*W1+Z1^2*Y1^2*Z2-4*Z1^2*Z2*Y2+3*Z1^2*W1^2 +2*Z1*Y1*Z2*W1-2*Z1*Z2*W3+2*Z1*Y2*Z3-Y1^2*Z2^2+4*Y1*Z2*W2 -4*Y1*Z3*W1+Z2^2*Y2-Z2*W1^2+6*Z3*W3, -6*W3^2 + Z1^2*Y1^2*Y2-Z1^2*Y2^2-2*Z1*Y1^3*W1+2*Z1*Y1*Y2*W1+4*Z1*Y2*W3 -4*Z1*Y3*W1+Y1^4*Z2-4*Y1^2*Z2*Y2+3*Y1^2*W1^2+2*Y1*Z2*Y3 -2*Y1*Y2*W2+Z2*Y2^2-Y2*W1^2+6*Y3*W2 === Subject: Re: Vector invariants a sentence. (Well, at least I do understand almost every term :-) Yes, I think I do need rotational invariance too. -- Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn === Subject: Re: Vector invariants >Yes, I think I do need rotational invariance too. Oh, then the problem is trivial! There are natural one-to-one correspondences between these categories of thing: * lists of three vectors, up to permutations, translations, & rotations * sets of three points, up to translations, & rotations * congruence classes of triangles * sets of three positive lengths (subject to triangle inequality) * solutions to cubic polynomials (of some restricted type) In other words, all the invariants of the type you're looking for will be naturally built from the three symmetric functions of the three distances between the ends of the vectors. You can use the squares of the distances just as well; they can be expressed as polynomials in the coordinates. In other words, try using just these three invariants (and polynomials in them): -2*a1^2+2*a1*b1-2*b1^2-2*a2^2+2*a2*b2-2*b2^2+2*a1*c1-2*c1^2 +2*a2*c2-2*c2^2+2*c1*b1+2*c2*b2 (-a2*b1+a1*b2+a2*c1-b2*c1-a1*c2+b1*c2)^2 (a1^2-2*a1*b1+b1^2+a2^2-2*a2*b2+b2^2)*(a1^2-2*a1*c1+c1^2+a2^2-2*a2*c2+c2^2) *(c1^2-2*c1*b1+b1^2+c2^2-2*c2*b2+b2^2) > a sentence. dave === Subject: Re: Uncountable many reals without Cantor > there are often threads in this group concerning > the cardinality of the set of real numbers. Some > persons seem to have strong objections against the > Cantor Proof of the fact that the set of real > numbers is not denumerable by the naturals. > Cantor's Proof uses diagonalization. But there is > a mesaure theoretic argument for the uncountability > of the reals due to Borel which does not use this > technique. There are many alternative proofs, all of which presuppose the consistency and truth of the definitions, in particular the abstraction of actual infinity is present in any proof. Borel's proof or any of the covering arguments is exactly that, showing that the measure of computable reals is 0, based on the actual infinite divisibility of the continuum. (Technically, that sounds such a nice and acceptable thing, but *philosophically* I don't accept that the extent of anything should be 0 if it exists. I will not argue that, I don't believe that there is anybody here to argue over that point.) So, I will say that none of this formalism solves the problems pertinent in this implausible theory of the transfinite. The problems are in the premises and concepts, not in the proofs per se. But yes, we can fully expect several nonsensical consequences of such faulty premises, which is indeed the case for continuum. -- Eray Ozkural === Subject: Re: Uncountable many reals without Cantor > there are often threads in this group concerning > the cardinality of the set of real numbers. Some > persons seem to have strong objections against the > Cantor Proof of the fact that the set of real > numbers is not denumerable by the naturals. > > Cantor's Proof uses diagonalization. But there is > a mesaure theoretic argument for the uncountability > of the reals due to Borel which does not use this > technique. > There are many alternative proofs, all of which presuppose the > consistency and truth of the definitions, in particular the > abstraction of actual infinity is present in any proof. It should be noted that Cantor's diagonal proof goes through even without supposing the actual infinite (or what you probably mean by that). Consider an axiomatization for arithmetic without the successor axiom. I could state specific axioms, but what is important to take is that models for the axiomatization will be the natural numbers as well as all initial segments of the natural numbers. That is, {0} is a model, as is {0,1}, {0,1,2}, etc. So the actual infinite is not assumed; the universe could be finite. Define a real number to be a function f from the natural numbers to the natural numbers, where, if n is non-zero, f(n) is either 0 or 1. Then you can consider f(0).f(1)f(2)... to be the binary expansion. In the model where the natural numbers are {0,1,2}, the reals are {n + i/2 + i/4 : n = 0, 1, or 2 and i = 0 or 1}. In the standard model the reals would be the standard reals. Now Cantor's diagonal argument still goes through. Suppose there is a 1-1 function f from the set of natural numbers onto the set of reals. Consider the function g from the set of natural numbers to the set of reals where g(i) is set to be different than (f(i))(i). Then f must be different from every g, hence there is no 1-1 function from the naturals onto the reals. In the case where the natural numbers are {0,1,2}, there are 3 naturals and 12 reals. So more reals than naturals. In brief, Cantor's diagonal proof is not about the actual infinite; it's about lower and higher types, and how there are no 1-1 functions from lower types onto higher types. (Caveats: (1) Actually, our reals are only the non-negative reals. (2) You have to do something to ensure numbers with expansions with infinitely repeating 1s are equated with the equivalent rational number. (3) In the infinitary case you have to manually take care of the case when (f(0))(0) = 0 and (f(1))(i) = 1 for all i <> 0. All simplifications are for expository purposes. ) === Subject: Re: Uncountable many reals without Cantor > In the model where the natural numbers are {0,1,2}, the reals are {n + > i/2 + i/4 : n = 0, 1, or 2 and i = 0 or 1}. Sorry this should read the reals are {n + i/2 + j/4 : n = 0, 1, or 2 and i,j = 0 or 1} Anyway, I wanted to add one more point, perhaps a little more tangential. Where Cantor uses what you would probably call actual infinity, is in the proofs that the natural numbers can be put into 1-1 correspondence with other sets, e.g. the rationals. For instance, define a rational to be a number where, when you multiply its expansion by a natural number, is a natural number. Then in the model where the natural numbers are {0,1,2}, the rationals are {n + i/2 : n = 0, 1, or 2 and i = 0 or 1}. There are therefore 3 natural numbers but 6 rationals. Actual infinity compresses rather than distinguishes so-called equinumerosity. === Subject: Re: Uncountable many reals without Cantor >there are often threads in this group concerning >the cardinality of the set of real numbers. Some >persons seem to have strong objections against the >Cantor Proof of the fact that the set of real >numbers is not denumerable by the naturals. People may have strong objections, but nobody has any _coherent_ objections - the people who object seem to be unable to follow very simple reasoning. Hence I doubt that they're going to be able to follow complicated chains of reasoning... >Cantor's Proof uses diagonalization. But there is >a mesaure theoretic argument for the uncountability >of the reals due to Borel which does not use this >technique. >Let (a_i), i e {1,2,3,...} be a list of the reals in >the interval [0,1]. Let eps be any rational number >> 0. >Now consider a_1 in an interval of length eps/2, ..., >a_i in an interval of length eps/2^i. Since every >element of [0,1] is in some of the intervals, we >have >length([0,1]) <= eps/2 + eps/4 + ... + eps/2^i + ... = eps >for every rational eps > 0. A contradiction. I can imagine one of the objectors mentioned above _agreeing_ that this argument is right, because it's based on more familiar concepts. But I think the idea that it's actually simpler is bogus - if someone agrees to this but not to the diagonal argument I really don't think that he's understood all the details. This argument _is_ much more complicated, if you include the missing details. In particular you need a _proof_ of the intuitively reasonable fact that if [0,1] is contained in the union of countably many intervals I_n then (*) sum length(I_n) >= 1. How do you _prove_ that? (It really does require proof, you know. A _clever_ objector to all this would point out that [0,1] is also the union of the closed intervals [x], for x in [0,1]. Note that the sum of the lengths of [x] for x in [0,1] is 0. Of course the reason this is not a contradiction is that (*) is not valid for uncountable unions. But (i) this shows at least that (*) for countable unions does require proof, and (ii) if we were insisting that there's no such thing as an uncountable set, maybe because we forgot to take our pills, then the explanation that (*) doesn't hold for uncountable unions doesn't work, and we conclude from this example that (*) is simply wrong! Suppose you do give a proof of (*) for countable unions - anyone stupid enough to be able to find flaws with the diagonal argument is going to have no problem finding flaws with that proof. Of course if our goal is to elicit _agreement_ instead of _understanding_ then the argument above is a good idea, because the objectors are going to be too dense to see the objections. But if the idea is actually to get someone to believe the reals are uncountable _for_ a valid _reason_ then the diagonal argument seems much better.) ************************ David C. Ullrich === Subject: Re: Uncountable many reals without Cantor >there are often threads in this group concerning >the cardinality of the set of real numbers. Some >persons seem to have strong objections against the >Cantor Proof of the fact that the set of real >numbers is not denumerable by the naturals. > People may have strong objections, but nobody > has any _coherent_ objections - the people who > object seem to be unable to follow very simple > reasoning. Hence I doubt that they're going to > be able to follow complicated chains of reasoning... The point you are missing that it does not require any complicated chains of reasoning. Cantor's first proof is perfectly adequate. Why was a second proof needed at all? Did you ever think about that? I think it was needed to hide the fact that the conclusion trivially follows from the then accepted or refined definitions of Z and R. There is no significant reasoning involved. It directly follows from the basic concepts involved, and that's why the first proof can be stated in a sentence or two, without resorting to any detailed argument. I hope you find this an interesting twist to the debate. It's misled to think of it as a technical matter. It's a purely philosophical issue. The diagonal argument is a short, and easy argument. It's easier than 90% of the proofs I've studied. However, it's just complex enough to make an appearance on the scene, unlike the first argument. It's a very strong claim to say that the people like Dr. Zenkin who object to the diagonal proof are too stupid to follow such a simple proof. Their objections are usually centered on the concept of actual infinity sinisterly leaking in the proof, and they say, well this is correct only if you accept the relevance of actual infinity in the first place. They don't say that the proof is prima facie wrong. They say the *concepts* are wrong. That's an altogether different thing, which I think you can understand. -- Eray Ozkural === Subject: Re: Uncountable many reals without Cantor >there are often threads in this group concerning >the cardinality of the set of real numbers. Some >persons seem to have strong objections against the >Cantor Proof of the fact that the set of real >numbers is not denumerable by the naturals. >>People may have strong objections, but nobody >>has any _coherent_ objections - the people who >>object seem to be unable to follow very simple >>reasoning. Hence I doubt that they're going to >>be able to follow complicated chains of reasoning... > The point you are missing that it does not require any complicated > chains of reasoning. Cantor's first proof is perfectly adequate. Why > was a second proof needed at all? Did you ever think about that? Cantor's first proof is very beautiful, and it's a pity that it isn't more widely taught. Unfortunately it doesn't generalize, at least not in any way I know about. The second proof gives the more general result about the cardinality of the powerset of *any* set. The second proof is also simpler than the first proof, in that it's purely combinatorial, whereas the first proof requires knowing a little bit about the topology of the reals. > I think it was needed to hide the fact that the conclusion trivially > follows from the then accepted or refined definitions of Z and R. > There is no significant reasoning involved. It directly follows from > the basic concepts involved, and that's why the first proof can be > stated in a sentence or two, without resorting to any detailed > argument. This judgment on your part that the first proof is simpler is completely at odds with mine; to me the second proof is much simpler (even though the first proof is simple enough that it's hard to get *too* much simpler). Does anyone else think the first proof is simpler? Simplicity is not the only consideration when judging a proof, of course; if the only question at issue is the cardinality of R, I prefer the first proof, because I think it's more illuminating. === Subject: Re: Uncountable many reals without Cantor <316g1bF37chnfU1@individual.net> Discussion, linux) > This judgment on your part that the first proof is simpler is > completely at odds with mine; Are you sure that when Eray says first proof, he doesn't mean the diagonal argument? I understand that the diagonal argument isn't historically first, but I wouldn't assume that he knows that. -- What if [...] these people HATE mathematics itself, but possibly hang out here to prove to themselves that there's nothing really to it and live in the fantasy that they can conquer mathematics itself? You know, like arsonists who become firefighters. --James S Harris === Subject: Re: Uncountable many reals without Cantor > This judgment on your part that the first proof is simpler is > completely at odds with mine; > Are you sure that when Eray says first proof, he doesn't mean the > diagonal argument? I understand that the diagonal argument isn't > historically first, but I wouldn't assume that he knows that. How many first proofs of Cantor's theorem by Cantor are there? === Subject: Re: Uncountable many reals without Cantor <316g1bF37chnfU1@individual.net> <877jo1g6ko.fsf@phiwumbda.org> Discussion, linux) >> This judgment on your part that the first proof is simpler is >> completely at odds with mine; >> Are you sure that when Eray says first proof, he doesn't mean the >> diagonal argument? I understand that the diagonal argument isn't >> historically first, but I wouldn't assume that he knows that. > How many first proofs of Cantor's theorem by Cantor are there? Which proof do you call the first proof? -- I arrest anybody I think needs arresting, Mr. Carter, and I'm not in the habit of explaining why. There's a law about that --- You're in Dodge, Mr. Carter. -- Gunsmoke radio show / John Ashcroft === Subject: Re: Uncountable many reals without Cantor > > This judgment on your part that the first proof is simpler is > completely at odds with mine; > > Are you sure that when Eray says first proof, he doesn't mean the > diagonal argument? I understand that the diagonal argument isn't > historically first, but I wouldn't assume that he knows that. > How many first proofs of Cantor's theorem by Cantor are there? Countably many? :) === Subject: Re: Uncountable many reals without Cantor > > This judgment on your part that the first proof is simpler is > completely at odds with mine; > > Are you sure that when Eray says first proof, he doesn't mean the > diagonal argument? I understand that the diagonal argument isn't > historically first, but I wouldn't assume that he knows that. > > How many first proofs of Cantor's theorem by Cantor are there? > Countably many? :) The first proof is unique, and we are all referring to the first proof that Cantor did before the diagonal argument... -- Eray === Subject: Re: Uncountable many reals without Cantor <316g1bF37chnfU1@individual.net> <877jo1g6ko.fsf@phiwumbda.org> Discussion, linux) > The first proof is unique, and we are all referring to the first proof > that Cantor did before the diagonal argument... I stand corrected then. But (like Mike Oliver) I thought your words were easier to interpret when they applied to the diagonal proof rather than the real first proof. It's the diagonal proof that I consider utterly simple, not the proof involving intervals. -- Start obeying math rules, or if you prefer I can proceed to tear away your illusions of being rational and logical piece by piece over time, as I reduce your society using advanced psychological tools that rival the best psychological warfare techniques of world governments. --JSH === Subject: Re: Uncountable many reals without Cantor >>This judgment on your part that the first proof is simpler is >>completely at odds with mine; > Are you sure that when Eray says first proof, he doesn't mean the > diagonal argument? I understand that the diagonal argument isn't > historically first, but I wouldn't assume that he knows that. Well, I suppose I'm not *sure*, but in that case I wouldn't be able to assign any sense to his remarks. If he knows the first proof, I still don't *agree* with his comments, but they're at least intelligible. === Subject: Re: Uncountable many reals without Cantor <316g1bF37chnfU1@individual.net> <877jo1g6ko.fsf@phiwumbda.org> <318tjhF37nitpU1@individual.net> Discussion, linux) >This judgment on your part that the first proof is simpler is >completely at odds with mine; >> Are you sure that when Eray says first proof, he doesn't mean the >> diagonal argument? I understand that the diagonal argument isn't >> historically first, but I wouldn't assume that he knows that. > Well, I suppose I'm not *sure*, but in that case I wouldn't > be able to assign any sense to his remarks. Now you're getting it. -- We are happy that you agree that customers need to know that Open Source is legal and stable, and we heartily agree with that sentence of your letter. The others don't seem to make as much sense, but we find the dialogue refreshing. -- Linus Torvalds to Darl McBride === Subject: Re: Uncountable many reals without Cantor > Cantor's first proof is very beautiful, and it's a pity > that it isn't more widely taught. Unfortunately it doesn't > generalize, at least not in any way I know about. The second > proof gives the more general result about the cardinality of > the powerset of *any* set. No, that is only so for the powerset of a set where no two elements share a dual representation, or no one element has a dual representation. > The second proof is also simpler than the first proof, in that > it's purely combinatorial, whereas the first proof requires > knowing a little bit about the topology of the reals. If you give the reals some more accord for their total ordering, then it doesn't apply, but the function has to take that into account, and not generate those infinite sequences, as would apply to sets dense in the reals. V = L, and infinite sets are equivalent. Ross === Subject: Re: Uncountable many reals without Cantor >>there are often threads in this group concerning >>the cardinality of the set of real numbers. Some >>persons seem to have strong objections against the >>Cantor Proof of the fact that the set of real >>numbers is not denumerable by the naturals. >People may have strong objections, but nobody >has any _coherent_ objections - the people who >object seem to be unable to follow very simple >reasoning. Hence I doubt that they're going to >be able to follow complicated chains of reasoning... >>Cantor's Proof uses diagonalization. But there is >>a mesaure theoretic argument for the uncountability >>of the reals due to Borel which does not use this >>technique. >>Let (a_i), i e {1,2,3,...} be a list of the reals in >>the interval [0,1]. Let eps be any rational number > 0. >>Now consider a_1 in an interval of length eps/2, ..., >>a_i in an interval of length eps/2^i. Since every >>element of [0,1] is in some of the intervals, we >>have >>length([0,1]) <= eps/2 + eps/4 + ... + eps/2^i + ... = eps >>for every rational eps > 0. A contradiction. >I can imagine one of the objectors mentioned above >_agreeing_ that this argument is right, because it's >based on more familiar concepts. But I think the idea >that it's actually simpler is bogus - if someone >agrees to this but not to the diagonal argument I >really don't think that he's understood all the details. >This argument _is_ much more complicated, if you include >the missing details. In particular you need a _proof_ of >the intuitively reasonable fact that if [0,1] is contained >in the union of countably many intervals I_n then >(*) sum length(I_n) >= 1. >How do you _prove_ that? Assume the opposite, put the intervals end to end etc. This kind of thing is proven in the beginning of any Real Variables text, e.g. Royden. Where do you see a problem? >(It really does require proof, you know. A _clever_ >objector to all this would point out that [0,1] >is also the union of the closed intervals [x], for >x in [0,1]. Note that the sum of the lengths of [x] >for x in [0,1] is 0. >Of course the reason this is not a contradiction >is that (*) is not valid for uncountable unions. >But (i) this shows at least that (*) for countable >unions does require proof, and (ii) if we were >insisting that there's no such thing as an uncountable >set, maybe because we forgot to take our pills, then >the explanation that (*) doesn't hold for uncountable >unions doesn't work, and we conclude from this >example that (*) is simply wrong! Suppose you do >give a proof of (*) for countable unions - anyone >stupid enough to be able to find flaws with the >diagonal argument is going to have no problem >finding flaws with that proof. >Of course if our goal is to elicit _agreement_ instead >of _understanding_ then the argument above is a good >idea, because the objectors are going to be too dense >to see the objections. But if the idea is actually >to get someone to believe the reals are uncountable >_for_ a valid _reason_ then the diagonal argument seems >much better.) >************************ >David C. Ullrich === Subject: Re: Uncountable many reals without Cantor >>This argument _is_ much more complicated, if you include >>the missing details. In particular you need a _proof_ of >>the intuitively reasonable fact that if [0,1] is contained >>in the union of countably many intervals I_n then >>(*) sum length(I_n) >= 1. >>How do you _prove_ that? > Assume the opposite, put the intervals end to end etc. This kind of > thing is proven in the beginning of any Real Variables text, e.g. > Royden. Where do you see a problem? Sounds like you want to use induction on the number of intervals. Problem is, induction works only if the number of intervals is finite. Hint: that's where compactness comes in. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Uncountable many reals without Cantor Dave Seaman says... >(*) sum length(I_n) >= 1. >How do you _prove_ that? >> Assume the opposite, put the intervals end to end etc. This kind of >> thing is proven in the beginning of any Real Variables text, e.g. >> Royden. Where do you see a problem? >Sounds like you want to use induction on the number of intervals. >Problem is, induction works only if the number of intervals is finite. >Hint: that's where compactness comes in. Just for clarification of this comment. Compactness for a topological space means the following: X is compact if for any collection of open sets U_i whose union is X, there is a finite subcollection whose union is also X. So a closed interval of finite length is compact, but an open interval is not. I was confused at first because I was misremembering the definition of compactness. I remembered a definition along the lines of X is compact if every Cauchy sequence converges. That doesn't directly help much. -- Daryl McCullough Ithaca, NY === Subject: Re: Uncountable many reals without Cantor > Dave Seaman says... >>(*) sum length(I_n) >= 1. >How do you _prove_ that? > Assume the opposite, put the intervals end to end etc. This kind of > thing is proven in the beginning of any Real Variables text, e.g. > Royden. Where do you see a problem? >>Sounds like you want to use induction on the number of intervals. >>Problem is, induction works only if the number of intervals is finite. >>Hint: that's where compactness comes in. > Just for clarification of this comment. > Compactness for a topological space means the following: X is compact > if for any collection of open sets U_i whose union is X, there is a > finite subcollection whose union is also X. So a closed interval of > finite length is compact, but an open interval is not. > I was confused at first because I was misremembering the definition > of compactness. I remembered a definition along the lines of X is > compact if every Cauchy sequence converges. That doesn't directly > help much. That property is called sequential compactness, and you're right that it doesn't particularly help. The key here is the Heine-Borel theorem, which says that closed, bounded intervals in R^n are compact. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Uncountable many reals without Cantor >> I was confused at first because I was misremembering the definition >> of compactness. I remembered a definition along the lines of X is >> compact if every Cauchy sequence converges. That doesn't directly >> help much. >That property is called sequential compactness, and you're right that >it doesn't particularly help. The key here is the Heine-Borel theorem, >which says that closed, bounded intervals in R^n are compact. I don't know whether you were just being unclear but every Cauchy sequence converges is certainly not the definition of a compact set, it is the definition of a complete metric space. The definition of sequential compactness is that every sequence in the set has a subsequence that converges in the set. -- I'm not interested in mathematics that might have anything to do with reality. -- Russell Easterly, in sci.math === Subject: Re: Uncountable many reals without Cantor Toni Lassila says... >I don't know whether you were just being unclear but every Cauchy >sequence converges is certainly not the definition of a compact set, >it is the definition of a complete metric space. Right. But there is a connection: every compact metric space is complete. >The definition of sequential compactness is that every sequence in the >set has a subsequence that converges in the set. Right. -- Daryl McCullough Ithaca, NY === Subject: Re: Uncountable many reals without Cantor > I was confused at first because I was misremembering the definition > of compactness. I remembered a definition along the lines of X is > compact if every Cauchy sequence converges. That doesn't directly > help much. >>That property is called sequential compactness, and you're right that >>it doesn't particularly help. The key here is the Heine-Borel theorem, >>which says that closed, bounded intervals in R^n are compact. > I don't know whether you were just being unclear but every Cauchy > sequence converges is certainly not the definition of a compact set, > it is the definition of a complete metric space. > The definition of sequential compactness is that every sequence in the > set has a subsequence that converges in the set. Yes, sorry. I was being sloppy. In a metric space, compactness is equivalent to sequential compactness, and both hold iff the space is complete and totally bounded. But compactness is the property that's needed in the measure theory proof, one way or another. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Uncountable many reals without Cantor >This argument _is_ much more complicated, if you include >the missing details. In particular you need a _proof_ of >the intuitively reasonable fact that if [0,1] is contained >in the union of countably many intervals I_n then >(*) sum length(I_n) >= 1. >How do you _prove_ that? >> Assume the opposite, put the intervals end to end etc. This kind of >> thing is proven in the beginning of any Real Variables text, e.g. >> Royden. Where do you see a problem? >Sounds like you want to use induction on the number of intervals. >Problem is, induction works only if the number of intervals is finite. >Hint: that's where compactness comes in. Amusing technicality: In various places for various reasons we need to talk about coverings by half-open intervals instead of open intervals. Say [0,1) is the union of disjoint intervals [a_n, b_n). Then sum(b_n - a_n) = 1. You can't prove that (directly) by compactness, but you _can_ prove it by a simple transfinite induction! ************************ David C. Ullrich === Subject: Re: Uncountable many reals without Cantor >>This argument _is_ much more complicated, if you include >>the missing details. In particular you need a _proof_ of >>the intuitively reasonable fact that if [0,1] is contained >>in the union of countably many intervals I_n then >(*) sum length(I_n) >= 1. >How do you _prove_ that? > Assume the opposite, put the intervals end to end etc. This kind of > thing is proven in the beginning of any Real Variables text, e.g. > Royden. Where do you see a problem? >>Sounds like you want to use induction on the number of intervals. >>Problem is, induction works only if the number of intervals is finite. >>Hint: that's where compactness comes in. > Amusing technicality: > In various places for various reasons we need to talk about > coverings by half-open intervals instead of open intervals. > Say [0,1) is the union of disjoint intervals [a_n, b_n). > Then sum(b_n - a_n) = 1. You can't prove that (directly) > by compactness, but you _can_ prove it by a simple > transfinite induction! That's not quite the approach I had in mind. I'm sure you are aware of this argument, but I wonder why we need to talk about half-open intervals to show that m([0,1]) = 1. What I meant was to show the outer measure of [0,1] is >= 1 by considering open covers, extracting finite subcovers, and using ordinary induction over the number of intervals in the subcover. Then you can show the outer measure is less than 1+epsilon for any epsilon > 0. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Uncountable many reals without Cantor >This argument _is_ much more complicated, if you include >the missing details. In particular you need a _proof_ of >the intuitively reasonable fact that if [0,1] is contained >in the union of countably many intervals I_n then >(*) sum length(I_n) >= 1. >How do you _prove_ that? >> Assume the opposite, put the intervals end to end etc. This kind of >> thing is proven in the beginning of any Real Variables text, e.g. >> Royden. Where do you see a problem? >Sounds like you want to use induction on the number of intervals. >Problem is, induction works only if the number of intervals is finite. >Hint: that's where compactness comes in. >> Amusing technicality: >> In various places for various reasons we need to talk about >> coverings by half-open intervals instead of open intervals. >> Say [0,1) is the union of disjoint intervals [a_n, b_n). >> Then sum(b_n - a_n) = 1. You can't prove that (directly) >> by compactness, but you _can_ prove it by a simple >> transfinite induction! >That's not quite the approach I had in mind. I'm sure you are aware of >this argument, but I wonder why we need to talk about half-open >intervals to show that m([0,1]) = 1. What I meant was to show the outer >measure of [0,1] is >= 1 by considering open covers, extracting finite >subcovers, and using ordinary induction over the number of intervals in >the subcover. Then you can show the outer measure is less than 1+epsilon >for any epsilon > 0. I understood all that - I wasn't disputing anything you said, just pointing out a _similar_ result where it's curious that one actually can use transfinite induction but not compactness (as opposed to the previous result where one can use compactness but not induction.) ************************ David C. Ullrich === Subject: Re: Uncountable many reals without Cantor >This argument _is_ much more complicated, if you include >>the missing details. In particular you need a _proof_ of >>the intuitively reasonable fact that if [0,1] is contained >>in the union of countably many intervals I_n then >(*) sum length(I_n) >= 1. >How do you _prove_ that? >> Assume the opposite, put the intervals end to end etc. This kind of > thing is proven in the beginning of any Real Variables text, e.g. > Royden. Where do you see a problem? >Sounds like you want to use induction on the number of intervals. >>Problem is, induction works only if the number of intervals is finite. >Hint: that's where compactness comes in. > Amusing technicality: > In various places for various reasons we need to talk about > coverings by half-open intervals instead of open intervals. > Say [0,1) is the union of disjoint intervals [a_n, b_n). > Then sum(b_n - a_n) = 1. You can't prove that (directly) > by compactness, but you _can_ prove it by a simple > transfinite induction! >>That's not quite the approach I had in mind. I'm sure you are aware of >>this argument, but I wonder why we need to talk about half-open >>intervals to show that m([0,1]) = 1. What I meant was to show the outer >>measure of [0,1] is >= 1 by considering open covers, extracting finite >>subcovers, and using ordinary induction over the number of intervals in >>the subcover. Then you can show the outer measure is less than 1+epsilon >>for any epsilon > 0. > I understood all that - I wasn't disputing anything you said, > just pointing out a _similar_ result where it's curious that > one actually can use transfinite induction but not compactness > (as opposed to the previous result where one can use compactness > but not induction.) Yeah, that's what I thought. I wouldn't have said anything if it hadn't been for your choice of words in saying we need to talk about coverings by half-open intervals. Didn't you recently chide someone else for claiming that there is only one way to do something? (Or maybe two ways?) -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Uncountable many reals without Cantor >> I understood all that - I wasn't disputing anything you said, >> just pointing out a _similar_ result where it's curious that >> one actually can use transfinite induction but not compactness >> (as opposed to the previous result where one can use compactness >> but not induction.) One other thought -- you may not be using compactness in that part of the argument, but you do need some special property of the reals in order to conclude that you only need to consider end-to-end placement of the intervals in the first place. Consider covering the rationals in [0,1] by collections of half-open intervals, for example. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Uncountable many reals without Cantor > I understood all that - I wasn't disputing anything you said, > just pointing out a _similar_ result where it's curious that > one actually can use transfinite induction but not compactness > (as opposed to the previous result where one can use compactness > but not induction.) >One other thought -- you may not be using compactness in that part of the >argument, but you do need some special property of the reals in order to >conclude that you only need to consider end-to-end placement of the >intervals in the first place. ??? I didn't say anything about end-to-end placement. (At least I don't think I did, not _certain_ what you mean by the phrase). If [0,1)is the union of countably many disjoint I_n = [a_n, b_n) then the ordering of the I_n can be isomorphic to any countable ordinal. >Consider covering the rationals in [0,1] >by collections of half-open intervals, for example. ************************ David C. Ullrich === Subject: Re: Uncountable many reals without Cantor >> I understood all that - I wasn't disputing anything you said, >> just pointing out a _similar_ result where it's curious that >> one actually can use transfinite induction but not compactness >> (as opposed to the previous result where one can use compactness >> but not induction.) >>One other thought -- you may not be using compactness in that part of the >>argument, but you do need some special property of the reals in order to >>conclude that you only need to consider end-to-end placement of the >>intervals in the first place. > ??? I didn't say anything about end-to-end placement. (At least > I don't think I did, not _certain_ what you mean by the phrase). > If [0,1)is the union of countably many disjoint I_n = [a_n, b_n) > then the ordering of the I_n can be isomorphic to any > countable ordinal. Here's what you said: > In various places for various reasons we need to talk about > coverings by half-open intervals instead of open intervals.> Say [0,1) is > the union of disjoint intervals [a_n, b_n).> Then sum(b_n - a_n) = 1. You > can't prove that (directly) > by compactness, but you _can_ prove it by a simple> transfinite > induction! For the record, it was someone else in the thread (J?rgen R., as the name appears in my newsreader) who said Assume the opposite, put the intervals end to end etc.. I don't know what end to end would mean for open intervals, or for closed intervals, but I think end to end is a very good description for a covering by disjoint half-open intervals, such as you described. >>Consider covering the rationals in [0,1] >>by collections of half-open intervals, for example. And how can you be sure that coverings by overlapping intervals won't give you a lower estimate of the outer measure? That was the point of my rational numbers example. I think you're going to need compactness or completeness in some form. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Uncountable many reals without Cantor > I understood all that - I wasn't disputing anything you said, > just pointing out a _similar_ result where it's curious that > one actually can use transfinite induction but not compactness > (as opposed to the previous result where one can use compactness > but not induction.) >One other thought -- you may not be using compactness in that part of the >argument, but you do need some special property of the reals in order to >conclude that you only need to consider end-to-end placement of the >intervals in the first place. >> ??? I didn't say anything about end-to-end placement. (At least >> I don't think I did, not _certain_ what you mean by the phrase). >> If [0,1)is the union of countably many disjoint I_n = [a_n, b_n) >> then the ordering of the I_n can be isomorphic to any >> countable ordinal. >Here's what you said: >> In various places for various reasons we need to talk about >> coverings by half-open intervals instead of open intervals.> Say [0,1) is >> the union of disjoint intervals [a_n, b_n).> Then sum(b_n - a_n) = 1. You >> can't prove that (directly) >> by compactness, but you _can_ prove it by a simple> transfinite >> induction! >For the record, it was someone else in the thread (J?rgen R., as the name >appears in my newsreader) who said Assume the opposite, put the >intervals end to end etc.. I don't know what end to end would mean >for open intervals, or for closed intervals, but I think end to end is >a very good description for a covering by disjoint half-open intervals, >such as you described. >Consider covering the rationals in [0,1] >by collections of half-open intervals, for example. >And how can you be sure that coverings by overlapping intervals won't >give you a lower estimate of the outer measure? I didn't say that you could be sure of that. I really don't know what you're going on about in all this - over and over you reply to things that I didn't say. For the last time: I thought the fact that that fact above can be proved by transfinite induction might be interesting. That was my one and only point. >That was the point of my >rational numbers example. I think you're going to need compactness or >completeness in some form. ************************ David C. Ullrich === Subject: Re: Uncountable many reals without Cantor >>Consider covering the rationals in [0,1] >>by collections of half-open intervals, for example. >>And how can you be sure that coverings by overlapping intervals won't >>give you a lower estimate of the outer measure? > I didn't say that you could be sure of that. I really don't know > what you're going on about in all this - over and over you reply > to things that I didn't say. For the last time: > I thought the fact that that fact above can be proved by > transfinite induction might be interesting. Yes, it was interesting. I had to read it twice because I missed the disjointness assumption on my first pass, and without disjointness I couldn't see how to make the transfinite recursion work. But you also suggested that compactness did not play a role in the argument you were presenting. I disagree. > That was my one and only point. Then you weren't talking about a proof of uncountability of R without Cantor, since you don't have a proof without addressing the disjointness issue. >>That was the point of my >>rational numbers example. I think you're going to need compactness or >>completeness in some form. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Uncountable many reals without Cantor >Consider covering the rationals in [0,1] >by collections of half-open intervals, for example. >And how can you be sure that coverings by overlapping intervals won't >give you a lower estimate of the outer measure? >> I didn't say that you could be sure of that. I really don't know >> what you're going on about in all this - over and over you reply >> to things that I didn't say. For the last time: >> I thought the fact that that fact above can be proved by >> transfinite induction might be interesting. >Yes, it was interesting. I had to read it twice because I missed the >disjointness assumption on my first pass, and without disjointness I >couldn't see how to make the transfinite recursion work. >But you also suggested that compactness did not play a role in the >argument you were presenting. I disagree. Where does compactness come in? You need _completeness_ to show that the collection of intervals is well-ordered. >> That was my one and only point. >Then you weren't talking about a proof of uncountability of R without >Cantor, since you don't have a proof without addressing the disjointness >issue. You're doing this on purpose, right? No, I wasn't talking about a proof of uncountability of R. I never said that this little tidbit had anything to do with that... >That was the point of my >rational numbers example. I think you're going to need compactness or >completeness in some form. ************************ David C. Ullrich === Subject: Re: Uncountable many reals without Cantor >>Consider covering the rationals in [0,1] >>by collections of half-open intervals, for example. >>And how can you be sure that coverings by overlapping intervals won't >>give you a lower estimate of the outer measure? > I didn't say that you could be sure of that. I really don't know > what you're going on about in all this - over and over you reply > to things that I didn't say. For the last time: > I thought the fact that that fact above can be proved by > transfinite induction might be interesting. >>Yes, it was interesting. I had to read it twice because I missed the >>disjointness assumption on my first pass, and without disjointness I >>couldn't see how to make the transfinite recursion work. >>But you also suggested that compactness did not play a role in the >>argument you were presenting. I disagree. > Where does compactness come in? You need _completeness_ to show > that the collection of intervals is well-ordered. You seem to have missed my point. If the intervals are not disjoint, then they need not be well-ordered at all. Besides, I said I thought you would need compactness or completeness in some form to establish that the disjoint case is sufficient. I have not seen any evidence to the contrary. > That was my one and only point. >>Then you weren't talking about a proof of uncountability of R without >>Cantor, since you don't have a proof without addressing the disjointness >>issue. > You're doing this on purpose, right? No, I wasn't talking about a > proof of uncountability of R. I never said that this little tidbit > had anything to do with that... So let's go back to the starting point of this subthread. It began when I said: >>One other thought -- you may not be using compactness in that part of the >>argument, but you do need some special property of the reals in order to >>conclude that you only need to consider end-to-end placement of the >>intervals in the first place. Would you like to start over and explain which part of that statement you are disagreeing with? I explained what I meant by end-to-end placement, and we need not revisit that. I specifically said that you don't need compactness in that part of the argument, i.e., the transfinite induction part. And completeness is indeed a special property of the reals, at least in the sense that it is not shared by the rationals, and therefore your nitpicking about whether compactness or completeness constitutes the required property is off target. I covered both possibilites from the beginning. Which one you need depends on precisely what you are given and what you are trying to prove, which is where we can't seem to agree. >>That was the point of my >>rational numbers example. I think you're going to need compactness or >>completeness in some form. You are behaving like the cranks you so often deride for shifting their arguments whenever someone offers a critique of their statements. You accuse me of bringing up things that you weren't talking about, but when you look at my initial statement (quoted above) it should be clear that I anticipated your objection and covered it even before you made it. If you are going to claim that I had no business bringing up compactness in a context that you weren't talking about, then I will counter that you equally had no business bringing up transfinite induction, which I wasn't talking about. My comment was every bit as relevant as yours was. (And no, I didn't say you were a crank. The point is that you should aspire to a higher standard of behavior.) -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Uncountable many reals without Cantor > Yes, it was interesting. I had to read it twice because I missed the > disjointness assumption on my first pass, and without disjointness I > couldn't see how to make the transfinite recursion work. I meant transfinite induction, of course. Hmm. I wonder if transfinite recursion would be useful in this program I'm working on... -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Uncountable many reals without Cantor >there are often threads in this group concerning >the cardinality of the set of real numbers. Some >persons seem to have strong objections against the >Cantor Proof of the fact that the set of real >numbers is not denumerable by the naturals. >>People may have strong objections, but nobody >>has any _coherent_ objections - the people who >>object seem to be unable to follow very simple >>reasoning. Hence I doubt that they're going to >>be able to follow complicated chains of reasoning... >Cantor's Proof uses diagonalization. But there is >a mesaure theoretic argument for the uncountability >of the reals due to Borel which does not use this >technique. >Let (a_i), i e {1,2,3,...} be a list of the reals in >the interval [0,1]. Let eps be any rational number >> 0. >Now consider a_1 in an interval of length eps/2, ..., >a_i in an interval of length eps/2^i. Since every >element of [0,1] is in some of the intervals, we >have >length([0,1]) <= eps/2 + eps/4 + ... + eps/2^i + ... = eps >for every rational eps > 0. A contradiction. >>I can imagine one of the objectors mentioned above >>_agreeing_ that this argument is right, because it's >>based on more familiar concepts. But I think the idea >>that it's actually simpler is bogus - if someone >>agrees to this but not to the diagonal argument I >>really don't think that he's understood all the details. >>This argument _is_ much more complicated, if you include >>the missing details. In particular you need a _proof_ of >>the intuitively reasonable fact that if [0,1] is contained >>in the union of countably many intervals I_n then >>(*) sum length(I_n) >= 1. >>How do you _prove_ that? >Assume the opposite, put the intervals end to end etc. This kind of >thing is proven in the beginning of any Real Variables text, e.g. >Royden. Well of course this is proved in reals (although it's not so clear to me that put the intervals end to end has much to do with a proof - never mind that, not really relevant.) >Where do you see a problem? I don't see any problems with the validity of the proof. I question the relevance in the present context because when all the details are included it's much more complicated than the diagonal argument. I mentioned a few other problems in the next few paragraphs - again, they're not problems with the proof, just problems with the propisition that this is a way to attempt to convince the sort of idiot who's not convinced by the diagonal argument: >>(It really does require proof, you know. A _clever_ >>objector to all this would point out that [0,1] >>is also the union of the closed intervals [x], for >>x in [0,1]. Note that the sum of the lengths of [x] >>for x in [0,1] is 0. >>Of course the reason this is not a contradiction >>is that (*) is not valid for uncountable unions. >>But (i) this shows at least that (*) for countable >>unions does require proof, and (ii) if we were >>insisting that there's no such thing as an uncountable >>set, maybe because we forgot to take our pills, then >>the explanation that (*) doesn't hold for uncountable >>unions doesn't work, and we conclude from this >>example that (*) is simply wrong! Suppose you do >>give a proof of (*) for countable unions - anyone >>stupid enough to be able to find flaws with the >>diagonal argument is going to have no problem >>finding flaws with that proof. >>Of course if our goal is to elicit _agreement_ instead >>of _understanding_ then the argument above is a good >>idea, because the objectors are going to be too dense >>to see the objections. But if the idea is actually >>to get someone to believe the reals are uncountable >>_for_ a valid _reason_ then the diagonal argument seems >>much better.) >>************************ >>David C. Ullrich ************************ David C. Ullrich === Subject: Re: Uncountable many reals without Cantor >>there are often threads in this group concerning >>the cardinality of the set of real numbers. Some >>persons seem to have strong objections against the >>Cantor Proof of the fact that the set of real >>numbers is not denumerable by the naturals. >People may have strong objections, but nobody >has any _coherent_ objections - the people who >object seem to be unable to follow very simple >reasoning. Hence I doubt that they're going to >be able to follow complicated chains of reasoning... >>Cantor's Proof uses diagonalization. But there is >>a mesaure theoretic argument for the uncountability >>of the reals due to Borel which does not use this >>technique. >Let (a_i), i e {1,2,3,...} be a list of the reals in >>the interval [0,1]. Let eps be any rational number > 0. >Now consider a_1 in an interval of length eps/2, ..., >>a_i in an interval of length eps/2^i. Since every >>element of [0,1] is in some of the intervals, we >>have >length([0,1]) <= eps/2 + eps/4 + ... + eps/2^i + ... = eps >for every rational eps > 0. A contradiction. >I can imagine one of the objectors mentioned above >_agreeing_ that this argument is right, because it's >based on more familiar concepts. But I think the idea >that it's actually simpler is bogus - if someone >agrees to this but not to the diagonal argument I >really don't think that he's understood all the details. >This argument _is_ much more complicated, if you include >the missing details. In particular you need a _proof_ of >the intuitively reasonable fact that if [0,1] is contained >in the union of countably many intervals I_n then >(*) sum length(I_n) >= 1. >How do you _prove_ that? >>Assume the opposite, put the intervals end to end etc. This kind of >>thing is proven in the beginning of any Real Variables text, e.g. >>Royden. >Well of course this is proved in reals (although it's not so >clear to me that put the intervals end to end has much to >do with a proof - never mind that, not really relevant.) >>Where do you see a problem? >I don't see any problems with the validity of the proof. >I question the relevance in the present context because >when all the details are included it's much more complicated >than the diagonal argument. OK, agreed. === Subject: Re: Uncountable many reals without Cantor David C. Ullrich says... >I don't see any problems with the validity of the proof. >I question the relevance in the present context because >when all the details are included it's much more complicated >than the diagonal argument. In a way, but the complexity of the measure-theoretic proof is normal mathematics, while Cantor's diagonalization proof seems like logic. Logic makes some people queasy, even good mathematicians. I remember as an undergraduate asking a math professor about Godel's theorem and he grimaced and said Ugh. Mathematical logic! I'm not sure why people have that reaction, but many people do. -- Daryl McCullough Ithaca, NY === Subject: Re: Uncountable many reals without Cantor What happens if you replace reals with rationals, irrationals, algebraics, transcendentals, or other set dense in the reals? Does it lead to a contradiction because there are functions bijecting the rationals and integers? What you have there is a restatement. About the dart, basically the probability is between 1/2 and 1/3, inclusive. That's based upon the deduction that the reals contain exclusive rationals, algebraic irrationals, and transcendentals, and that their union comprises the entire set of the reals, and that each is dense in the reals. That's about the contiguous reals. As usual, applying your argument to the natural/unit equivalency function, your result does not hold in the related nonstandard real numbers, and their related rational, irrational, algebraic and transcendental real numbers. I must admit that I don't know much about measure theory. For example, where measure theory is useful for functions from reals to reals with positive measure reflecting the geometric or analytical results, there is a large disconnect in its consideration of an infinite set with what is deemed to be zero measure. Between the analog and discrete there is not an accessible experiment to validate the results of measure theory where some infinite set has measure zero, only in the abstract. So, I must investigate measure theory and to promote my very own positions find either a contradiction or a circularity, the contradiction invalidating that all infinitesimals are zero and non-positive, and the circularity in that the assumption that infinite sets are not equivalent in the definitions of measure theory is the only reason that any non-empty set has measure zero. Transfinite cardinality was bolted onto measure theory after the fact. It's heavy, poisonous, and doesn't fit through doors. That is to say: measure theory has utility in functions about which it has a statement, and it is mute where its definitions are not representative of the underlying number system. Ullrich claims that logic is simple. He's often right. He's aware of potential slight changes in very simple logic. Ullrich is proud that I call him a hedgehog. Ullrich has stated a fallacy. Sarcasm is a lie where the intent is to be truthful and the opposite. Is not your model countable? Hodges' exposition contains flaws. Consider the powerset and why the probabilities of one of its elements containing either all or none are each 1 / 2^x, or for any other particular element, where the sum from 1 to 2^x of 1/ 2^x = 1, and the subtle distinction between singular and plural. Are you unreasonably attached to the transfinite cardinals? Can you use them to solve a problem? Would it be about transfinite cardinals? If measure theory is redefined in terms of, yes, metrics, and not transfinite cardinals, would anything change that was not nonsense before? Measure theory appears to be being brought into play to bulwark the die agonal arguments. Why, what's wrong with them? There is reason to believe that infinite sets are equivalent. Ah yes, there is much in simple logical statements. F of x equals x plus one. There are lots of rational numbers. It's an infinite set, there's always one more. Ross Finlayson === Subject: Re: Uncountable many reals without Cantor > What happens if you replace reals with rationals, irrationals, > algebraics, transcendentals, or other set dense in the reals? > Does it lead to a contradiction because there are functions bijecting > the rationals and integers? > What you have there is a restatement. > About the dart, basically the probability is between 1/2 and 1/3, > inclusive. That's based upon the deduction that the reals contain > exclusive rationals, algebraic irrationals, and transcendentals, and > that their union comprises the entire set of the reals, and that each > is dense in the reals. That's about the contiguous reals. > As usual, applying your argument to the natural/unit equivalency > function, your result does not hold in the related nonstandard real > numbers, and their related rational, irrational, algebraic and > transcendental real numbers. > I must admit that I don't know much about measure theory. For > example, where measure theory is useful for functions from reals to > reals with positive measure reflecting the geometric or analytical > results, there is a large disconnect in its consideration of an > infinite set with what is deemed to be zero measure. Between the > analog and discrete there is not an accessible experiment to validate > the results of measure theory where some infinite set has measure > zero, only in the abstract. So, I must investigate measure theory and > to promote my very own positions find either a contradiction or a > circularity, the contradiction invalidating that all infinitesimals > are zero and non-positive, and the circularity in that the assumption > that infinite sets are not equivalent in the definitions of measure > theory is the only reason that any non-empty set has measure zero. > Transfinite cardinality was bolted onto measure theory after the fact. > It's heavy, poisonous, and doesn't fit through doors. > That is to say: measure theory has utility in functions about which it > has a statement, and it is mute where its definitions are not > representative of the underlying number system. > Ullrich claims that logic is simple. He's often right. He's aware of > potential slight changes in very simple logic. Ullrich is proud that > I call him a hedgehog. Ullrich has stated a fallacy. Sarcasm is a > lie where the intent is to be truthful and the opposite. Is not your > model countable? Hodges' exposition contains flaws. > Consider the powerset and why the probabilities of one of its elements > containing either all or none are each 1 / 2^x, or for any other > particular element, where the sum from 1 to 2^x of 1/ 2^x = 1, and the > subtle distinction between singular and plural. > Are you unreasonably attached to the transfinite cardinals? Can you > use them to solve a problem? Would it be about transfinite cardinals? > If measure theory is redefined in terms of, yes, metrics, and not > transfinite cardinals, would anything change that was not nonsense > before? > Measure theory appears to be being brought into play to bulwark the > die agonal arguments. Why, what's wrong with them? > There is reason to believe that infinite sets are equivalent. > Ah yes, there is much in simple logical statements. F of x equals x > plus one. There are lots of rational numbers. It's an infinite set, > there's always one more. > Ross Finlayson === Subject: Re: Uncountable many reals without Cantor : >> there are often threads in this group concerning >> the cardinality of the set of real numbers. Some >> persons seem to have strong objections against the >> Cantor Proof of the fact that the set of real >> numbers is not denumerable by the naturals. > People may have strong objections, but nobody > has any _coherent_ objections - the people who > object seem to be unable to follow very simple > reasoning. Hence I doubt that they're going to > be able to follow complicated chains of reasoning... >> Cantor's Proof uses diagonalization. But there is >> a mesaure theoretic argument for the uncountability >> of the reals due to Borel which does not use this >> technique. >> Let (a_i), i e {1,2,3,...} be a list of the reals in >> the interval [0,1]. Let eps be any rational number > 0. >> Now consider a_1 in an interval of length eps/2, ..., >> a_i in an interval of length eps/2^i. Since every >> element of [0,1] is in some of the intervals, we >> have >> length([0,1]) <= eps/2 + eps/4 + ... + eps/2^i + ... = eps >> for every rational eps > 0. A contradiction. > I can imagine one of the objectors mentioned above > _agreeing_ that this argument is right, because it's > based on more familiar concepts. But I think the idea > that it's actually simpler is bogus - if someone > agrees to this but not to the diagonal argument I > really don't think that he's understood all the details. > This argument _is_ much more complicated, if you include > the missing details. In particular you need a _proof_ of > the intuitively reasonable fact that if [0,1] is contained > in the union of countably many intervals I_n then > (*) sum length(I_n) >= 1. > How do you _prove_ that? I know that the presentation of the argument is not a strict proof. But with my post i wanted to provoke some reactions of the objectors, because i'm interested if they doubt in the technique (diagonalization) or the result: [0,1] is of uncountable cardinality. > (It really does require proof, you know. A _clever_ > objector to all this would point out that [0,1] > is also the union of the closed intervals [x], for > x in [0,1]. Note that the sum of the lengths of [x] > for x in [0,1] is 0. > Of course the reason this is not a contradiction > is that (*) is not valid for uncountable unions. > But (i) this shows at least that (*) for countable > unions does require proof, and (ii) if we were > insisting that there's no such thing as an uncountable > set, maybe because we forgot to take our pills, then > the explanation that (*) doesn't hold for uncountable > unions doesn't work, and we conclude from this > example that (*) is simply wrong! Suppose you do > give a proof of (*) for countable unions - anyone > stupid enough to be able to find flaws with the > diagonal argument is going to have no problem > finding flaws with that proof. > Of course if our goal is to elicit _agreement_ instead > of _understanding_ then the argument above is a good > idea, because the objectors are going to be too dense > to see the objections. But if the idea is actually > to get someone to believe the reals are uncountable > _for_ a valid _reason_ then the diagonal argument seems > much better.) > ************************ > David C. Ullrich -- Frank Piron, defrankatkonaddot (leftrotate two) === Subject: Re: Uncountable many reals without Cantor >Of course if our goal is to elicit _agreement_ instead >of _understanding_ then the argument above is a good >idea, because the objectors are going to be too dense >to see the objections. But if the idea is actually >to get someone to believe the reals are uncountable >_for_ a valid _reason_ then the diagonal argument seems >much better.) How about this one then: Let (r_n) be any sequence of reals in [0,1] Then we can find an interval I_1 = [a1,b1] s.t. I_1 is a subset of (0,1) and r_1 not in I_1. Similarly, we can find I_2 s.t. I_2 is a subset of (a1,b1) and r_2 not in I_2. Continuing this gives a descending sequence of closed intervals where each I_n contains I_(n+1). The intervals are compact and the intersection of any two is non-empty, hence Intersect_j=1^oo( I_j ) is not empty i.e. there exists at least one element r_0 s.t. r_0 in I_n for all n. But since the intervals were selected so that each number in the sequence (r_n) is excluded from all the intervals starting from I_n, we have that r_0 is in [0,1] but not included in the sequence (r_n). Hence, the reals [0,1] are uncountable. -- I'm not interested in mathematics that might have anything to do with reality. -- Russell Easterly, in sci.math === Subject: Re: Uncountable many reals without Cantor This is actually just a modification of the Cantor proof, in my opinion. Anyhow, the nay-sayers are going to object to the existence of a point in the intersection of the nested intervals. But seriously, the intuitionists and serious constructivists deny the uncountability of the reals and, because they have a coherent philosophical basis, you aren't going to refute them and you'd have to agree they are logically sophisticated. === Subject: Re: Uncountable many reals without Cantor > But seriously, > the intuitionists and serious constructivists deny the uncountability > of the reals No they don't. === Subject: Re: Uncountable many reals without Cantor > But seriously, > the intuitionists and serious constructivists deny the uncountability > of the reals > No they don't. See the replies, I think only the ones who are silly enough to admit Platonism back in would admit the continuum I had a talk with a so-called constructivist. He was repeating the naive claims of Robin Chapman and Stephen Harris, that the TM is not a physical model of computation, e.g. they don't exist in the world. He said PCs are not TMs!!!!! I told him that a constructivist would consider TMs to be physical models. The poor chap was offended. A guy from Netherlands, he considered himself to be a constructivist. He probably thought himself to be a follower of Brouwer. Now, if Brouwer was present in the discussion, he could teach a lesson or two about constructivism to this chap. All that actual-infinity tape talk is Platonist nonsense, and you would probably think that all constructivists should be as naive as this poor chap. It's clear that you know nothing about constructivism or why the debate of the actual infinity is so central to these matters. As a Platonist, you want to turn constructivism into the naive ideals of Platonism. But it is not. I consider true constructivism to reject actual infinity talk at a fundamental level. No candy for you, -- Eray === Subject: Re: Uncountable many reals without Cantor >> But seriously, >> the intuitionists and serious constructivists deny the uncountability >> of the reals >> No they don't. > See the replies, I think only the ones who are silly enough to admit > Platonism back in would admit the continuum > I had a talk with a so-called constructivist. He was repeating the > naive claims of Robin Chapman and Stephen Harris, that the TM is not a > physical model of computation, e.g. they don't exist in the world. More dopey abuse from the egregious Ozkural. I certainly don't equate physical model with existence in the world. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Uncountable many reals without Cantor Discussion, linux) > I had a talk with a so-called constructivist. He was repeating the > naive claims of Robin Chapman and Stephen Harris, that the TM is not a > physical model of computation, e.g. they don't exist in the world. He > said PCs are not TMs!!!!! I told him that a constructivist would > consider TMs to be physical models. Of course, this is utterly stupid. > The poor chap was offended. Probably amused or slightly frightened. -- Jesse F. Hughes I'm not sure whether I'm not thinking clearly or clearly not thinking. -- Ling Cheung, said on some day other than our wedding day. Honest. === Subject: Re: Uncountable many reals without Cantor > I had a talk with a so-called constructivist. He was repeating the > naive claims of Robin Chapman and Stephen Harris, that the TM is not a > physical model of computation, e.g. they don't exist in the world. He > said PCs are not TMs!!!!! I told him that a constructivist would > consider TMs to be physical models. > Of course, this is utterly stupid. Hmm. Ok. Perhaps that is what I think a constructivist must consider, but as it seems that is not the case. Still, PCs are TMs. Thinking otherwise is some kind of Pythagoreanism or Platonism, certainly some form of idealism that wants to posit the existence of an independent realm of mathematical objects. > The poor chap was offended. > Probably amused or slightly frightened. No he had no clue what he was talking about. Otherwise, he wouldn't make the claim that PCs are not TMs. Maybe feelings of amusement and fear are usually associated with ignorance, you might actually be right. It was an important talk for me, though. It showed just how far removed from reality most mathematicians are to me, even those who claim themselves to be constructivists, and thus philosophically involved in what they do as their day-job. The same goes for computer programmers, of course. However, a funny thing, in the hacker circles I've never seen programmers making such idealist claims. Perhaps when you work in the real world, you actually become more of an empiricist than an idealist. It's only when you start talking to people who read some set theory that you come across this ancient shame of idealism. -- Eray Ozkural === Subject: Re: Uncountable many reals without Cantor > See the replies, I think only the ones who are silly enough to admit > Platonism back in would admit the continuum Like Brouwer? === Subject: Re: Uncountable many reals without Cantor > See the replies, I think only the ones who are silly enough to admit > Platonism back in would admit the continuum > Like Brouwer? You say that in Brouwer and other forthcoming constructivists, there is actually abstraction of actual continuum. If that's right, then my above statement is wrong-headed. But I don't think that's the case. If you do admit the actual continuum, then you don't need any of the detailed reworking of analysis that Bishop carried out. I don't think you even understand what I refer to by admitting the continuum. -- Eray Ozkural === Subject: Re: Uncountable many reals without Cantor > I don't think you even understand what I refer to by admitting the > continuum. The question is rather whether you understand what Brouwer is saying. === Subject: Re: Uncountable many reals without Cantor > I don't think you even understand what I refer to by admitting the > continuum. > The question is rather whether you understand what Brouwer is > saying. OK. I agree with that. You claim that Brouwer did admit the existence of continuum. However, I cannot see how he admitted it. What I see is that he worked to show that analysis can be done using intuitionistic logic. Does that mean he admitted the existence of continuum? I have a limited knowledge of Brouwer, I read very little from him first-hand. -- Eray === Subject: Re: Uncountable many reals without Cantor > I have a limited knowledge of Brouwer, I read very little from him > first-hand. So read some more, starting with the passages quoted. === Subject: Re: Uncountable many reals without Cantor > I have a limited knowledge of Brouwer, I read very little from him > first-hand. > So read some more, starting with the passages quoted. Sure, most certainly. I agree that it's a personality that must be understood well, and I'm sorry for having written a thoughtless post. The fact is that I am not at all certain what Brouwer would say about the issue. -- Eray === Subject: Re: Uncountable many reals without Cantor Discussion, linux) >> See the replies, I think only the ones who are silly enough to admit >> Platonism back in would admit the continuum >> Like Brouwer? > You say that in Brouwer and other forthcoming constructivists, there > is actually abstraction of actual continuum. If that's right, then my > above statement is wrong-headed. But I don't think that's the case. If > you do admit the actual continuum, then you don't need any of the > detailed reworking of analysis that Bishop carried out. > I don't think you even understand what I refer to by admitting the > continuum. Now *that's* a strategy. Spout meaningless nonsense and when someone responds as if it had meaning, call him on it. -- It seems to me that some of you don't realize that [...] some day the truth comes out. It doesn't matter if you're dead. I've made certain that your name will live in infamy if it's known at all: Wiles, Ribet, Granville, or anyone else from this generation. --JSH beyond the grave === Subject: Re: Uncountable many reals without Cantor >> >> See the replies, I think only the ones who are silly enough to admit >> Platonism back in would admit the continuum >> >> Like Brouwer? > You say that in Brouwer and other forthcoming constructivists, there > is actually abstraction of actual continuum. If that's right, then my > above statement is wrong-headed. But I don't think that's the case. If > you do admit the actual continuum, then you don't need any of the > detailed reworking of analysis that Bishop carried out. > I don't think you even understand what I refer to by admitting the > continuum. > Now *that's* a strategy. Spout meaningless nonsense and when someone > responds as if it had meaning, call him on it. I meant admitting the continuum as in the existential sense. Does Brouwer say that the continuum, all of it, is constructible, in the same sense an integer is constructible? -- Eray PS: I haven't had much sleep for the last two days, and I think the post you were responding to and the one before that was pretty clumsy. Sorry for that. === Subject: Re: Uncountable many reals without Cantor >> But seriously, >> the intuitionists and serious constructivists deny the uncountability >> of the reals > No they don't. Some do. Brouwer certainly did. That's what led him to abandon analysis. Some of them even deny infinity. At that point you're veering right out of mathematics into philosophy. But their point is that there is in fact some largest number (size unknown, of course), because *in fact* you're limited by the size of things you consider. Then you get into arguments about what that largest number actually *is* (the number, something else?). I personally think that sort of silliness undermines their argument that there's a largest number -- once you start thinking about the largest number, then just double it. But that argument doesn't seem to affect them -- proving that they're philosophers and not mathematicians. Jon Miller === Subject: Re: Uncountable many reals without Cantor |>> But seriously, |>> the intuitionists and serious constructivists deny the uncountability |>> of the reals |> No they don't. | |Some do. Brouwer certainly did. I don't think so. If he did, then it was an aberration, since on other occasions he said that the integers and reals had different cardinalities. Cantor's first proof of the uncountability of the continuum is perfectly legitimate according to the principles of intuitionist analysis. None of the Bishop school deny the uncountability of the reals; Bishop himself has a proof of it in his famous textbook of constructive analysis. Quite to the contrary, the serious constructivists understand that it's correct. | That's what led him to abandon |analysis. Ridiculous. He spent much of his life reworking analysis on his terms. |Some of them even deny infinity. Name one. I don't think you know of any. | At that point you're veering right |out of mathematics into philosophy. But their point is that there is |in fact some largest number (size unknown, of course), because *in |fact* you're limited by the size of things you consider. This sounds like a garbled description of the ultra-intuitionism of Essenin-Volpin. But he does _not_ claim that there exists a largest number. He just denies that such expressions as 10^12 necessarily denote natural numbers. | Then you get |into arguments about what that largest number actually *is* (the |number, something else?). I personally think that sort of silliness |undermines their argument that there's a largest number -- once you |start thinking about the largest number, then just double it. But |that argument doesn't seem to affect them -- proving that they're |philosophers and not mathematicians. I think you're doing a poor job of describing Essenin-Volpin's point of view here. You're talking about him as though he had followers, and I doubt you know of any. At least refrain from trying to make his views sound worse than they are. I've read some of his papers, and I don't believe he denies the existence of infinity, either. There's something about holding an unpopular point of view that people seem to take as an invitation to start being extremely sloppy about how the point of view is described. I don't know whether you are doing so yourself, but if you are not then some of your sources are, so be wary. Keith Ramsay === Subject: Re: Uncountable many reals without Cantor > Some do. Brouwer certainly did. No he didn't. There's nothing intuitionistically problematic about the reals being uncountable. > That's what led him to abandon > analysis. No it wasn't. He didn't abandon analysis at all. === Subject: Re: Uncountable many reals without Cantor >> Some do. Brouwer certainly did. > No he didn't. There's nothing intuitionistically problematic about >the reals being uncountable. Well, in his inaugural address at the University of Amsterdam (1912; I'm going by the 1913 English translation, Intuition and Formalism, published in the Bulletin of the American Mathematical Society, volume 20, p. 81), Brouwer did say that the intuitionist doesn't recognize any infinite set of cardinality greater than aleph-null. I'd quote him exactly-- of xerographic copying, I only have the right half of page 91, from which I wish to quote. It says: According to the statemen aleph-null is the only infinit recognize the existence. which is surely a very bad haiku. But I do think I remember his thrust correctly. Maybe he took it all back later? (Or maybe you're being subtle, and being uncountable is--intuitionistically--not the same as having an uncountable cardinality?) I *did* manage to copy all the words that I had gone in search of (some of his comments on Kant), so I don't need to go back to the library, and I'd rather not. Lee Rudolph === Subject: Re: Uncountable many reals without Cantor > Well, in his inaugural address at the University of Amsterdam > (1912; I'm going by the 1913 English translation, Intuition > and Formalism, published in the Bulletin of the American > Mathematical Society, volume 20, p. 81), Brouwer did say that > the intuitionist doesn't recognize any infinite set of > cardinality greater than aleph-null. In that address, Brouwer apparently only recognized lawlike choice sequences. However, he does mention the possibility of admitting free choice sequences, and notes that in such a case, if, on the basis of the intuition of the linear continuum, [the intuitionist] admits elementary series of free selections as elements of construction, then each non-denumerable set constructed by means of it contains a subset of the power of the continuum. (A. S. Troelstra: Cholce sequences. Clarendo Press. Oxford 1977.) sequences - a historical note (available on the web): In 1914 Brouwer had changed his views on choice sequences as pointed out by Troelstra in his book. In a review on Schoenflies and Hahns book on set theory ( 139 - 144 ) Brouwer remarks in a footnote ( 140 ): Z. B. ist die Punktmenge: alle reellen Zahlen zwischen O und 1 mit Ausnahme der endlichen Dualbruche, nur deshalb eine Wohlkonstruierte Menge, weil die duale Entwicklung einer willkurlichen Zahl dieser Menge eine Fundamentalreihe von endliche Gruppen von gleichen Ziffern (abwechselnd O und l) liefert, so dass die Menge sich mittels einer Fundamentalreihe von Auswahlen unter den endlichen Zahlen bestimmen lasst. Dieser Schritt geht freilich weiter als mein romischer Vortrag ( 102 - 104 ), und auch weiter als die Borelschen Ausfuhrungen uber wohlkonstruierte Mengen ( 827 - 878 ); er erscheint mir aber als eine notwendige Konsequenz des Intuitionismus. Brouwer's idea of a Menge was not, to be sure, that of Cantorian set theory. But the proof that the Menge of real numbers between 0 and 1 is not countable is perfectly valid intuitionistically. On a more basic note: recall that intuitionism does not recognize the existence of infinite sets as completed totality. This does not imply that the natural numbers form a finite totality from an intuitionistic point of view. Similarly with the real numbers and uncountability. === Subject: Re: Uncountable many reals without Cantor for this information. Do you know why Brouwer changed his mind about free choice sequences? And would law-like choice sequences be similar to the requirement that constructions be recursive (a la Markov(?)). === Subject: Re: Uncountable many reals without Cantor > Do you know why Brouwer changed his mind about > free choice sequences? No, but I'm sure there are explanatory expositions in the literature. > And would law-like choice sequences be > similar to the requirement that constructions be recursive (a la > Markov(?)). A law-like sequence is one which is fully determined by a rule or law, without any element of choice involved. It is not part of the idea of a law-like sequence that it has to be recursive, though. === Subject: Re: Uncountable many reals without Cantor >>Of course if our goal is to elicit _agreement_ instead >>of _understanding_ then the argument above is a good >>idea, because the objectors are going to be too dense >>to see the objections. But if the idea is actually >>to get someone to believe the reals are uncountable >>_for_ a valid _reason_ then the diagonal argument seems >>much better.) >How about this one then: >Let (r_n) be any sequence of reals in [0,1] Then we can find an >interval I_1 = [a1,b1] s.t. I_1 is a subset of (0,1) and r_1 not in >I_1. Similarly, we can find I_2 s.t. I_2 is a subset of (a1,b1) and >r_2 not in I_2. Continuing this gives a descending sequence of closed >intervals where each I_n contains I_(n+1). >The intervals are compact and the intersection of any two is >non-empty, hence Intersect_j=1^oo( I_j ) is not empty i.e. there >exists at least one element r_0 s.t. r_0 in I_n for all n. But since >the intervals were selected so that each number in the sequence (r_n) >is excluded from all the intervals starting from I_n, we have that r_0 >is in [0,1] but not included in the sequence (r_n). Hence, the reals >[0,1] are uncountable. That seems simpler to actually fill in all the details than in the other argument, but it's still more complicated than the diagonal argument, so I still don't see the point. In particular, is someone too stupid to follow the diagonal argument going to really _follow_ a proof that a closed and bounded interval is compact? (Come to think of it it's also possible for an idiot to raise the same objection to this proof as one that's often raised against the diagonal argument: Ok, we've found a real not on the list. That doesn't prove the reals are uncountable, just add the new real to the list! Yes, that's totally cloth-headed. But my point is that the people we're talking about are showing an inability to follow very simple logic - I don't see how changing the details of the logic is going to change that fact.) ************************ David C. Ullrich === Subject: Re: Uncountable many reals without Cantor >That seems simpler to actually fill in all the details than in >the other argument, but it's still more complicated than the >diagonal argument, so I still don't see the point. In particular, >is someone too stupid to follow the diagonal argument going to >really _follow_ a proof that a closed and bounded interval >is compact? Certainly not, but at least we could stop the idiotic bickering about decimal expansions and integers with infinite digits. Furthermore, the educable people could maybe learn something about real analysis. -- I'm not interested in mathematics that might have anything to do with reality. -- Russell Easterly, in sci.math === Subject: Levene's test & Two Independent Samples t-Test? Someone told me that Levene's test is no longer used to test for equal variances before performing the two independent samples t-test. This is news to me, not that I'm a stat whiz but that every textbook that I have still uses this test. If this is true could somebody point me to a research paper or recent textbook or website, etc.? === Subject: Re: Escultura affair: publication scandal by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iB1FkRA11431; Reading this thread I got interested in what E. Escultura is actually doing - and I got angry reading his rather strange statements about FLT and Wiles' proof. In the >Zentralblatt f.9fr Mathematik< 9 publications of E. Escultura are listed. For 5 of them the Zentralblatt just mentions >not reviewed<, whatever that means. However I must admit that - looking at myself - I wonder why persons like E. Escultura or J. Harris provoke such strong reactions. H === Subject: Re: Cantor's diagonal proof wrong? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iB1GAoW13882; >>In sci.math, Shmuel (Seymour J.) Metz >>But it is still curious that it happens at 2^oo and not oo^oo. > Why? Is it curious that 3*3 is smaller than 3^3, or that 16*16 is > smaller than 2^16? >>I suspect that, if a set S has cardinality of at least card(N), >>then a bijective mapping can be found between the elements of 2^S >>and the elements of n^S, where n > 1 is an integer. >>However, I'd have to look. >Sure, that's correct; in fact, n can have cardinality anywhere >from 2 to 2^S. There's an injection 2^S --> n^S, and there's >an injection n --> 2^S (and so an injection > 2^S --> (2^S)^S ~ 2^S); >now apply Schroeder-Bernstein. >Todd Trimble Typo: displayed line should read n^S --> (2^S)^S ~ 2^S. === Subject: Re: Turing Machines and Physical Computation & TM 'S REVISIONISTS > You've not yet read Copeland, EZzz ? Indeed I have read, and I did not find his philosophy rigorous. If you read the original post, you will see that the very existence of hypercomputation requires infinite space (in the form of infinitely big or infinitely small, doesn't matter) Can you explain to me *why* physicists cannot store an infinite amount of information on a fingertip, if hypercomputation is possible? Please provide references pointing that such devices can be constructed in the real world. Oh, if you say that hypercomputation cannot be experimentally verified or its existence cannot be proven by physicists, then I will say it does not exist. Angels cannot be experimentally verified or proven, either. And they doesn't exist, naturally. Or will you say, I can imagine continuum, so it must exist. What a great ontological argument! If you accept such sloppy argumentation, then you will also accept Descartes's proof of substance dualism, and the cosmological argument among other non-proofs. Poulpes thinks of hypercomputation, therefore there is hypercomputation! LOL You frenchmen ;) -- Eray Ozkural === Subject: Re: Turing Machines and Physical Computation >>lines made with real pencils and the pinpricks made by the point of a >>real compass, mathematicians have always approximated the real world by >>abstractions which are not physically realizable. >Well, I don't know about that. Whose theory is this? I thought most >mathematicians who expressed an opinion were generally some sort of >Platonists, believing that effective mathematics is discovered, not >invented, and that it is the real world which approximates the ideal. > Both real and ideal worlds are discovered and invented. Neither is the > other; they approximate each other. It's my theory. In what sense is the real world invented? Do you mean we can invent America, because we can build a big city like New York? In my opinion, Real world does not approximate the ideal world in the ordinary sense of the word, which can get fictional. But if you mean creativity, and the ability to creatively shape our environment, I might agree. It's unclear what you mean above. -- Eray Ozkural === Subject: Re: Turing Machines and Physical Computation >lines made with real pencils and the pinpricks made by the point of a >real compass, mathematicians have always approximated the real world by >abstractions which are not physically realizable. >Well, I don't know about that. Whose theory is this? I thought most >>mathematicians who expressed an opinion were generally some sort of >>Platonists, believing that effective mathematics is discovered, not >>invented, and that it is the real world which approximates the ideal. >> Both real and ideal worlds are discovered and invented. Neither is the >> other; they approximate each other. It's my theory. >In what sense is the real world invented? Do you mean we can invent >America, because we can build a big city like New York? Well, Eray, the real world is invented in the sense of discovery, through the differential manipulation of predicates. This is the process of idealization and we literally realize the results of idealization through the absence of self contradiction. The so called real world is just the absence of self contradiction in idealization. We normally refer to the nominally real world as what exists independent of our idealizations, but in point of fact the only standard for what is real as opposed to what is ideal is the absence of self contradiction in what is ideal. This covers a multitude of possibilities, from material reality to abstract reality to the predicates of predication to differences and differences between differences etc. These are all real. However, we don't and can't know all of reality.This is the sense in which reality and ideality approximate each other. >In my opinion, Real world does not approximate the ideal world in the >ordinary sense of the word, which can get fictional. But if you mean >creativity, and the ability to creatively shape our environment, I >might agree. It's unclear what you mean above. >Eray Ozkural === Subject: Re: Turing Machines and Physical Computation >ON THE COMPUTER METAPHOR >'It has always bothered me that models of psychological >processing were thought to be inspired by our understanding of >the computer. The statement has always been false. Indeed, the >architecture of the modern digital computer - the so-called Von >Neumann architecture - was heavily influenced by people's (naive) >view of how the mind operated. Perhaps I had better document >this. Simply read the work on cybernetics and thought in the >1940's and 1950's prior to the development of the digital >computer. The group of workers included people from all >disciplines: See the Macy Conferences on Cybernetics, or Her >Majesty's Conference on Thought processes. Read the preface to >Wiener's book on cybernetics. Everyone who was working together - >engineers, physicists, mathematicians, psychologists, >neuroscientists (not yet named) - consciously and deliberately >claimed to be modelling brain processes.' >Reflections on Cognition and Parallel Distributed Processing >D.A. Norman >(Ch 26, p534, Parallel Distributed Processing Volume 2) >McClelland J and Rumelhart D 1986 >>I get that computer behavior and human behavior are essentially >>different. Why they are so different is certainly what we came here to >>discuss. > Hi patty, every couple of weeks I click on a post totally at random to > find the same silliness repeated, with 0.999 confidence. > The problem with presenting a disembodied fragment is that it possibly > implies what the fragmenter wants it to implie, but it hardly tells > what the original writer actually meant. Follows are parts of the > **REST** of Norman's story .... [caps are mine] ... > .... here we are talking about a new form of COMPUTATION ... > ... these MODELS are highly parallel, with thousands of elements > INTERACTING primarily through actviation and inhibition ... > ... each ELEMENT is highly interconnected with perhaps tens of > thousands of connections .... > ...these NEUROLOGICALLY INSPIRED COMPUTATIONAL PROCESSES pose new > requirements on our understanding of computation, suggest novel > THEORETICAL EXPLANATIONS of PSYCHOLOGICAL PHENOMENA, and suggest > powerful new ARCHITECTURES for MACHINES OF THE FUTURE .... > .... carry on a tradition that has long existed .... a tradition of > BUILDING MODELS of NEUROLOGICAL PROCESSES .... > [and later on .....] > ... the whole point for the cognitive scientist is to UNDERSTAND > COGNITION ... to do so, we insist upon explanation of the INTERNAL > PROCESSING STRUCTURES THAT GIVE RISE TO COGNITIVE ACTIVITIES .... > .... this is why we have spoken of REPRESENTATION, of MECHANISMS of > memory and attention ... Point taken :) Making collages of fragments of other people's writings is art, not communication. What these collages refer to are mentalisms in the head of the author of the collage. patty === Subject: Re: Turing Machines and Physical Computation > > > >ON THE COMPUTER METAPHOR >'It has always bothered me that models of psychological >processing were thought to be inspired by our understanding of >the computer. The statement has always been false. Indeed, the >architecture of the modern digital computer - the so-called Von >Neumann architecture - was heavily influenced by people's (naive) >view of how the mind operated. Perhaps I had better document >this. Simply read the work on cybernetics and thought in the >1940's and 1950's prior to the development of the digital >computer. The group of workers included people from all >disciplines: See the Macy Conferences on Cybernetics, or Her >Majesty's Conference on Thought processes. Read the preface to >Wiener's book on cybernetics. Everyone who was working together - >engineers, physicists, mathematicians, psychologists, >neuroscientists (not yet named) - consciously and deliberately >claimed to be modelling brain processes.' >Reflections on Cognition and Parallel Distributed Processing >D.A. Norman >(Ch 26, p534, Parallel Distributed Processing Volume 2) >McClelland J and Rumelhart D 1986 >I get that computer behavior and human behavior are essentially >>different. Why they are so different is certainly what we came here to >>discuss. >> > > > Hi patty, every couple of weeks I click on a post totally at random to > find the same silliness repeated, with 0.999 confidence. > > The problem with presenting a disembodied fragment is that it possibly > implies what the fragmenter wants it to implie, but it hardly tells > what the original writer actually meant. Follows are parts of the > **REST** of Norman's story .... [caps are mine] ... > > .... here we are talking about a new form of COMPUTATION ... > > ... these MODELS are highly parallel, with thousands of elements > INTERACTING primarily through actviation and inhibition ... > ... each ELEMENT is highly interconnected with perhaps tens of > thousands of connections .... > ...these NEUROLOGICALLY INSPIRED COMPUTATIONAL PROCESSES pose new > requirements on our understanding of computation, suggest novel > THEORETICAL EXPLANATIONS of PSYCHOLOGICAL PHENOMENA, and suggest > powerful new ARCHITECTURES for MACHINES OF THE FUTURE .... > > .... carry on a tradition that has long existed .... a tradition of > BUILDING MODELS of NEUROLOGICAL PROCESSES .... > > [and later on .....] > > ... the whole point for the cognitive scientist is to UNDERSTAND > COGNITION ... to do so, we insist upon explanation of the INTERNAL > PROCESSING STRUCTURES THAT GIVE RISE TO COGNITIVE ACTIVITIES .... > .... this is why we have spoken of REPRESENTATION, of MECHANISMS of > memory and attention ... > Point taken :) Making collages of fragments of other people's writings > is art, not communication. What these collages refer to are mentalisms > in the head of the author of the collage. > patty It's not even art. It's misrepresenting other peoples's words and their intended MEANING to support one's own opinion. Misrepresentation. Verbal turpitude. When you read BOTH the 1st half of Norman's comments [which were fragg'ed in] AND you also read the 2nd half [which were fragg'ed out], what you see is Norman saying the computer analogue is insufficient - which many of us agree with, of course - and should be replaced by a different and better form of computational cognitivism. Furthermore, the fragmented attribution above was even faked - the last half of the thing passed off as a quote was actually in a footnote in Norman's original. Here is the verbatim text .... 'It has always bothered me that models of psychological >processing were thought to be inspired by our understanding of >the computer. The statement has always been false. Indeed, the >architecture of the modern digital computer - the so-called Von >Neumann architecture - was heavily influenced by people's (naive) >view of how the mind operated. [Footnote as shown above was inserted here, as if text. Norman's exact + original *next* sentence follows]. Now that whole debate can be put aside: The work presented here in no way can be interpreted as growing from our metaphor of the modern computer. Here, we are talking about a new form of computation, one clearly based upon principles that have heretofore not had any counterparts in computers. These models are highly parallel, with thousands or millions of elements interacting primarily through activation and inhibition of each other's activity ..... On and on, as I indicated previously. I'll repeat the following again, as this is what Norman was ACTUALLY getting at ... > ... the whole point for the cognitive scientist is to UNDERSTAND > COGNITION ... to do so, we insist upon explanation of the INTERNAL > PROCESSING STRUCTURES THAT GIVE RISE TO COGNITIVE ACTIVITIES .... > .... this is why we have spoken of REPRESENTATION, of MECHANISMS of > memory and attention ... === Subject: Re: Turing Machines and Physical Computation <41a3ef28$0$576$b45e6eb0@senator-bedfellow.mit.edu> <41a4a697$0$563$b45e6eb0@senator-bedfellow.mit.edu> > Point taken :) Making collages of fragments of other people's writings >> is art, not communication. What these collages refer to are mentalisms >> in the head of the author of the collage. >> patty >It's not even art. It's misrepresenting other peoples's words and >their intended MEANING to support one's own opinion. >Misrepresentation. Verbal turpitude. >When you read BOTH the 1st half of Norman's comments [which were >fragg'ed in] AND you also read the 2nd half [which were fragg'ed out], >what you see is Norman saying the computer analogue is insufficient - >which many of us agree with, of course - and should be replaced by a >different and better form of computational cognitivism. >Furthermore, the fragmented attribution above was even faked - the >last half of the thing passed off as a quote was actually in a >footnote in Norman's original. Here is the verbatim text .... Can there be any doubt at all now as to why you have in the past been referred to as dumb Dan? You *still* haven't a clue what the real issue here is, even after it's been repeatedly pointed out to you and *demonstrated* using your own behaviour! Try doubting yourself a little. -- David Longley http://www.longley.demon.co.uk/Frag.htm === Subject: Re: Turing Machines and Physical Computation <41a3ef28$0$576$b45e6eb0@senator-bedfellow.mit.edu> <41a4a697$0$563$b45e6eb0@senator-bedfellow.mit.edu> >ON THE COMPUTER METAPHOR >'It has always bothered me that models of psychological >>processing were thought to be inspired by our understanding of >>the computer. The statement has always been false. Indeed, the >>architecture of the modern digital computer - the so-called Von >>Neumann architecture - was heavily influenced by people's (naive) >>view of how the mind operated. Perhaps I had better document >>this. Simply read the work on cybernetics and thought in the >>1940's and 1950's prior to the development of the digital >>computer. The group of workers included people from all >>disciplines: See the Macy Conferences on Cybernetics, or Her >>Majesty's Conference on Thought processes. Read the preface to >>Wiener's book on cybernetics. Everyone who was working together - >>engineers, physicists, mathematicians, psychologists, >>neuroscientists (not yet named) - consciously and deliberately >>claimed to be modelling brain processes.' >Reflections on Cognition and Parallel Distributed Processing >D.A. Norman >>(Ch 26, p534, Parallel Distributed Processing Volume 2) >>McClelland J and Rumelhart D 1986 >I get that computer behavior and human behavior are essentially >different. Why they are so different is certainly what we came here >to discuss. >> Hi patty, every couple of weeks I click on a post totally at >>random to >> find the same silliness repeated, with 0.999 confidence. >> The problem with presenting a disembodied fragment is that it >>possibly >> implies what the fragmenter wants it to implie, but it hardly tells >> what the original writer actually meant. Follows are parts of the >> **REST** of Norman's story .... [caps are mine] ... >> .... here we are talking about a new form of COMPUTATION ... ... >>these MODELS are highly parallel, with thousands of elements >> INTERACTING primarily through actviation and inhibition ... >> ... each ELEMENT is highly interconnected with perhaps tens of >> thousands of connections .... >> ...these NEUROLOGICALLY INSPIRED COMPUTATIONAL PROCESSES pose new >> requirements on our understanding of computation, suggest novel >> THEORETICAL EXPLANATIONS of PSYCHOLOGICAL PHENOMENA, and suggest >> powerful new ARCHITECTURES for MACHINES OF THE FUTURE .... >> .... carry on a tradition that has long existed .... a tradition of >> BUILDING MODELS of NEUROLOGICAL PROCESSES .... >> [and later on .....] >> ... the whole point for the cognitive scientist is to UNDERSTAND >> COGNITION ... to do so, we insist upon explanation of the INTERNAL >> PROCESSING STRUCTURES THAT GIVE RISE TO COGNITIVE ACTIVITIES .... >> .... this is why we have spoken of REPRESENTATION, of MECHANISMS of >> memory and attention ... >Point taken :) Making collages of fragments of other people's >writings is art, not communication. What these collages refer to are >mentalisms in the head of the author of the collage. >patty No, that is not the point you should take from this. That you do so just makes you look as stupid as Michaels. What was it that Norman said about the computer and the brain, and what has been shown about the properties of so called ANNs apropos human and other animal behaviour? What is it that they model do you think? (You haven't grasped how this relates to natural stupidity) The reason why you. Michaels etc don't pick up on any of this properly is because you aren't trained in psychology. You don't have the context or web if you like, to be able to make sense of it properly. The context you have is that of an enthusiastic amateur with skills in other areas. Those skills just don't equip you to read or talk in other fields and that's the problem I have been illustrating. If you look to what many people in AI or Cognitive Science get up to, you'll see this repeated my many people. They commit the genetic fallacy. That is, they may have skills in mathematics or programming, and they just assume they have the ability or expertise in fields where in fact they have little or none - in fact they have nothing more than pre-scientific folk psychology. Why would Michaels or you presume otherwise one might ask (one can ask this same question of many of his luminaries too). Read the section of Fragments on transfer of training. All you're actually doing here is reinforcing both your own and Michaels' stupidity. The quote from Norman was simply to show what people were doing back in the 40s and 50s. That they were doing that doesn't mean that what they were doing was sound any more than Michaels' or Norman's talk of ANNs being neurally inspired is sound! What these models show is how we can model assumption violation, overfitting etc - problems well known to those of us who use inferential statistics and build models - hence construction, validation samples, reliability measures, shrinkage etc. There is a subtle point being made about what is being modelled which you are all missing. See the two *other* sections of Fragments on a) ANNs *and* b) hypothesis testing. What is *your* behaviour illustrating? -- David Longley http://www.longley.demon.co.uk/Frag.htm === Subject: Re: Turing Machines and Physical Computation [. . .] >>Point taken :) Making collages of fragments of other people's >>writings is art, not communication. What these collages refer to are >>mentalisms in the head of the author of the collage. >>patty >No, that is not the point you should take from this. That you do so just >makes you look as stupid as Michaels. But not as stupid as yourself? > What was it that Norman said about >the computer and the brain, and what has been shown about the >properties of so called ANNs apropos human and other animal behaviour? >What is it that they model do you think? (You haven't grasped how this >relates to natural stupidity) >The reason why you. Michaels etc don't pick up on any of this properly >is because you aren't trained in psychology. Well, you aren't trained in psychology either, David. What you call psychology is just the anthropomorphosis of animal training regimens according to materialist epistemology. > You don't have the context >or web if you like, to be able to make sense of it properly. A undemonstrated web of beliefs is what characterizes religion not science. > The context >you have is that of an enthusiastic amateur with skills in other areas. >Those skills just don't equip you to read or talk in other fields and >that's the problem I have been illustrating. And you yourself certainly illustrate the problem very well. === Subject: What on earth! was Re: Turing Machines and Physical Computation > Abstract does not mean non-physical > the Turing Machine is just another kind > of automata, and its mechanism is firmly rooted in the physical world. Incuding its infinite tape no doubt (I wonder if its squares require integers with an infinite number of digits to count). > I am actually expecting a brilliant reply by some of the least > sophisticated philosophers I have ever had the opportunity to meet, > who are available on sci.math. You are the least sophisticated philosopher I have ever met, Mr Ozkural, on sci.math or elsewhere. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Lambda Calculus and Turing Equivalence There are many equivalent formalisms of computation. One of these is lambda calculus, which is supposed to compute exactly the same sets as Turing Machines can. Despite the poor philosophy being peddled in these newsgroups, this is actually seen as an intuitive support for C-T thesis, and universally believed to be correct. I have never seen anybody claim that lambda calculus is *not* equivalent to Turing Computation. However, if *actual* infinity is a *necessary* part of TM model, then it cannot be equivalent to lambda calculus, because there is no infinite component in any lambda calculus expression, and neither can we talk of infinitely long expressions. There are only finite expressions, just like there are only a finite number of non-blank symbols on a TM tape. Since the imagined actually infinite tape of TM does not exist in this other equivalent model of computation, we conclude that it is not part of a TM to start with. Only a potentially infinite tape can be part of a TM, because see above, there are only a finite number of non-blank symbols on a TM tape, and the Turing Machine has no dealing with the infinite remainder of the tape, which does nothing except to symbolize the arbitrary growth of computational space. .... > Abstract does not mean non-physical > the Turing Machine is just another kind > of automata, and its mechanism is firmly rooted in the physical world. > Incuding its infinite tape no doubt > (I wonder if its squares require integers with an infinite > number of digits to count). > I am actually expecting a brilliant reply by some of the least > sophisticated philosophers I have ever had the opportunity to meet, > who are available on sci.math. > You are the least sophisticated philosopher I have ever met, > Mr Ozkural, on sci.math or elsewhere. I had more of Torkel Franzel than yourself on my mind, but you are less of a philosopher than I am. Try to answer the above challenge to prove your logic. There is no easier way to refute your Platonist inclinations. -- Eray Ozkural === Subject: Re: Lambda Calculus and Turing Equivalence http://mygate.mailgate.org/mynews/comp/comp.theory/b0101bde6b542af8059101503 0 68a137.48257%40mygate.mailgate.org Another Moronic Troll from Eray. > There are many equivalent formalisms of > computation. One of these is lambda calculus, > which is supposed to compute exactly the same sets > as Turing Machines can. > Despite the poor philosophy being peddled in these > newsgroups, Only yours, Eray, only yours. > this is actually seen as an intuitive support for > C-T thesis, and universally believed to be > correct. I have never seen anybody claim that > lambda calculus is *not* equivalent to Turing > Computation. > However, if *actual* infinity is a *necessary* > part of TM model, then it cannot be equivalent to > lambda calculus, Wrong. Let's see here, you claim to have _earned_ a PhD in computer science, yet cannot successfully distinguish a program from the data on which it operates? [Remainder of typical Eray utterly worthless garbage snipped.] xanthian. -- === Subject: Re: Lambda Calculus and Turing Equivalence > There are many equivalent formalisms of computation. One of these is > lambda calculus, which is supposed to compute exactly the same sets as > Turing Machines can. > Despite the poor philosophy being peddled in these newsgroups, this is > actually seen as an intuitive support for C-T thesis, and universally > believed to be correct. I have never seen anybody claim that lambda > calculus is *not* equivalent to Turing Computation. > However, if *actual* infinity is a *necessary* part of TM model, then > it cannot be equivalent to lambda calculus, because there is no > infinite component in any lambda calculus expression, and neither can > we talk of infinitely long expressions. There are only finite > expressions, just like there are only a finite number of non-blank > symbols on a TM tape. > Since the imagined actually infinite tape of TM does not exist in > this other equivalent model of computation, we conclude that it is not > part of a TM to start with. To be pedantic, you're right, the tape (actually/potentially infinite or just unbounded) has never been considered part of a TM. Neither was the input to a lambda expression. > Only a potentially infinite tape can be > part of a TM, because see above, there are only a finite number of > non-blank symbols on a TM tape, and the Turing Machine has no dealing > with the infinite remainder of the tape, which does nothing except to > symbolize the arbitrary growth of computational space. Please, please, please tell me that you don't plan on -seriously- questioning the unsolvability of the halting problem. Please. Would it make you happy if we all just said that a TMs tape has size that is a potential infinity? -- Mitch Harris (remove q to reply) === Subject: Re: Lambda Calculus and Turing Equivalence > There are many equivalent formalisms of computation. One of these is > lambda calculus, which is supposed to compute exactly the same sets as > Turing Machines can. > > Despite the poor philosophy being peddled in these newsgroups, this is > actually seen as an intuitive support for C-T thesis, and universally > believed to be correct. I have never seen anybody claim that lambda > calculus is *not* equivalent to Turing Computation. > > However, if *actual* infinity is a *necessary* part of TM model, then > it cannot be equivalent to lambda calculus, because there is no > infinite component in any lambda calculus expression, and neither can > we talk of infinitely long expressions. There are only finite > expressions, just like there are only a finite number of non-blank > symbols on a TM tape. > > Since the imagined actually infinite tape of TM does not exist in > this other equivalent model of computation, we conclude that it is not > part of a TM to start with. > To be pedantic, you're right, the tape (actually/potentially infinite > or just unbounded) has never been considered part of a TM. Neither was > the input to a lambda expression. Yes, let's be pedantic, and let's not be silly like Robin Chapman or Stephen Harris and suggest that Personal Computers (PCs) are not Turing Machines. Just because somebody has the title of mathematician doesn't mean he is allowed to speak nonsense. > Only a potentially infinite tape can be > part of a TM, because see above, there are only a finite number of > non-blank symbols on a TM tape, and the Turing Machine has no dealing > with the infinite remainder of the tape, which does nothing except to > symbolize the arbitrary growth of computational space. > Please, please, please tell me that you don't plan on -seriously- > questioning the unsolvability of the halting problem. Please. I think you are confusing me with somebody else. Read my post again, there is nothing that is factually incorrect. > Would it make you happy if we all just said that a TMs tape has size > that is a potential infinity? Yes. Absolutely and actually happy. -- Eray Ozkural === Subject: Re: Lambda Calculus and Turing Equivalence Originator: harris@tcs.inf.tu-dresden.de (Mitchell Harris) >> There are many equivalent formalisms of computation. One of these is >> lambda calculus, which is supposed to compute exactly the same sets as >> Turing Machines can. >> >> Despite the poor philosophy being peddled in these newsgroups, this is >> actually seen as an intuitive support for C-T thesis, and universally >> believed to be correct. I have never seen anybody claim that lambda >> calculus is *not* equivalent to Turing Computation. >> >> However, if *actual* infinity is a *necessary* part of TM model, then >> it cannot be equivalent to lambda calculus, because there is no >> infinite component in any lambda calculus expression, and neither can >> we talk of infinitely long expressions. There are only finite >> expressions, just like there are only a finite number of non-blank >> symbols on a TM tape. >> >> Since the imagined actually infinite tape of TM does not exist in >> this other equivalent model of computation, we conclude that it is not >> part of a TM to start with. >> To be pedantic, you're right, the tape (actually/potentially infinite >> or just unbounded) has never been considered part of a TM. Neither was >> the input to a lambda expression. >Yes, let's be pedantic, and let's not be silly like Robin Chapman or >Stephen Harris and suggest that Personal Computers (PCs) are not >Turing Machines. Not to jump on the bandwagon, but PCs are not TMs. 1) Current architectures do not allow for arbitrary (unboundedly large) size addressing (no matter how much memory it has. 2) thousands of people work worldwide developing parts to produce actual PCs. I'd guess at most a handful are doing the same for anything that could be recognizably close to a TM. 3) lambda calculus is not a UTM. it -acts- like one, but is not one. >Just because somebody has the title of mathematician doesn't mean he >is allowed to speak nonsense. Right. I think you need tenure for that. >> Only a potentially infinite tape can be >> part of a TM, because see above, there are only a finite number of >> non-blank symbols on a TM tape, and the Turing Machine has no dealing >> with the infinite remainder of the tape, which does nothing except to >> symbolize the arbitrary growth of computational space. >> Please, please, please tell me that you don't plan on -seriously- >> questioning the unsolvability of the halting problem. Please. >I think you are confusing me with somebody else. Read my post again, >there is nothing that is factually incorrect. it was just the uh... attitude and subject matter that made the connection that's all. === Subject: Re: Lambda Calculus and Turing Equivalence >> >> To be pedantic, you're right, the tape (actually/potentially infinite >> or just unbounded) has never been considered part of a TM. Neither was >> the input to a lambda expression. >Yes, let's be pedantic, and let's not be silly like Robin Chapman or >Stephen Harris and suggest that Personal Computers (PCs) are not >Turing Machines. > Not to jump on the bandwagon, but PCs are not TMs. > 1) Current architectures do not allow for arbitrary (unboundedly large) > size addressing (no matter how much memory it has. That's not an issue. Why do I feel like I will need to quote the Cinderella book for the 10th time? People seem to have a hard time reading textbooks nowadays. > 2) thousands of people work worldwide developing parts to produce actual > PCs. I'd guess at most a handful are doing the same for anything that > could be recognizably close to a TM. > 3) lambda calculus is not a UTM. it -acts- like one, but is not one. It is equivalent in computational power to a TM. Accept or not? If you accept, my conclusion follows. Otherwise show a logical mistake. Will you also claim that a LISP interpreter is not in fact a lambda-calculus-machine? >I think you are confusing me with somebody else. Read my post again, >there is nothing that is factually incorrect. > it was just the uh... attitude and subject matter that made the connection > that's all. Mind your classifiers, then. They don't work accurately. -- Eray Ozkural === Subject: Re: Lambda Calculus and Turing Equivalence Originator: harris@tcs.inf.tu-dresden.de (Mitchell Harris) > > To be pedantic, you're right, the tape (actually/potentially infinite > or just unbounded) has never been considered part of a TM. Neither was > the input to a lambda expression. >Yes, let's be pedantic, and let's not be silly like Robin Chapman or >>Stephen Harris and suggest that Personal Computers (PCs) are not >>Turing Machines. >> Not to jump on the bandwagon, but PCs are not TMs. >> 1) Current architectures do not allow for arbitrary (unboundedly large) >> size addressing (no matter how much memory it has. >That's not an issue. Of course it's not. You asked an oversimplified question and I supplied a number of reasons for a no answer according to different meanings of what might be meant by are. >Why do I feel like I will need to quote the >Cinderella book for the 10th time? Cinderella? Is that Hopcroft and Ullman? >People seem to have a hard time reading textbooks nowadays. Yes. Understanding too. >> 2) thousands of people work worldwide developing parts to produce actual >> PCs. I'd guess at most a handful are doing the same for anything that >> could be recognizably close to a TM. >> 3) lambda calculus is not a UTM. it -acts- like one, but is not one. >It is equivalent in computational power to a TM. Accept or not? Accept. >If you accept, my conclusion follows. Otherwise show a logical mistake. Er...conclusion? What conclusion? That there is no actual infinity? Or that a TMs tape is not of infinite size? Or what? Whatever you answer is, I'll accept it. As far as I can tell it doesn't make any difference. >Will >you also claim that a LISP interpreter is not in fact a >lambda-calculus-machine? Only by being extremely and uselessly pedantic like I was above. Of course they are not identical, of course they have many intentional similarities. It is problematic to the modelling that a physical device has difficulties in modelling unbounded memory. That really doesn't afect the usefulness of the model. Mitch === Subject: Re: Lambda Calculus and Turing Equivalence >It is equivalent in computational power to a TM. Accept or not? > Accept. >If you accept, my conclusion follows. Otherwise show a logical mistake. > Er...conclusion? What conclusion? That there is no actual infinity? Or > that a TMs tape is not of infinite size? Or what? Whatever you answer is, > I'll accept it. As far as I can tell it doesn't make any difference. The conclusion of the original post. It is not that there is no actual infinity. (That does not follow) That a TM tape is not of infinite length. It is unbounded, not infinite. Plain and simple. But yes, I see your point, it's not awfully important. What amazes me is that some people still think it may be the case that some abstract machines in computer science somehow do not have physical counterparts. [*] That's a basic misunderstanding. As far as I can tell, computer science is concerned with what kinds of machines are constructible in theory, in possible worlds with physical laws like ours if we want to be philosophically precise. Indeed, Turing machines are solely concerned with what finite mechanisms can do in general. Godel observes this generality back in 1950, why can't we now? I doubt that people who have answered my post have thought deeply about my argument. They don't seem to have considered all cases of finite/transfinite in models of computation as I have done. A question to consider: does it even make sense to talk about an infinite size program? And another: is this an infinite string 0 1 1 1 ... 1 0 -- Eray Ozkural PS: For instance Stephen Harris's argument. A TM can calculate Pi, a physical computer cannot, therefore TMs are not physical. Does Harris even recognize that he is reinventing the substance dualism of Descartes in computational terms? It almost sounds like A TM can compute. Therefore TM is :) That's not how modern philosophy proceeds. === Subject: Re: Lambda Calculus and Turing Equivalence > However, if *actual* infinity is a *necessary* part of TM model, then > it cannot be equivalent to lambda calculus, because there is no > infinite component in any lambda calculus expression, A TM does not have an _infinite_ tape, it has an _unbounded tape_. This means that the tape is guaranteed to be as large as you need it or, equivalently, that the tape will grow as you need it but always stay finite. It isn't hard to write a TM simulator in pure lambda calculus. The converse is harder, but possible. Torben === Subject: Re: Lambda Calculus and Turing Equivalence > However, if *actual* infinity is a *necessary* part of TM model, then > it cannot be equivalent to lambda calculus, because there is no > infinite component in any lambda calculus expression, > A TM does not have an _infinite_ tape, it has an _unbounded tape_. > This means that the tape is guaranteed to be as large as you need it > or, equivalently, that the tape will grow as you need it but always > stay finite. > It isn't hard to write a TM simulator in pure lambda calculus. The > converse is harder, but possible. Precisely my point. I am saying to those who say that PCs are not TMs, like Chapman and Harris, that they do not know the difference between unbounded (potentially infinite) and actually infinite. (Although they claim that they do) The latter is a set theoretical notion that is not required to depict TMs, I doubt this distinction was so clear back then. That we can use set theory to formalize TMs shouldn't confuse us. Anyway, I agree with all of that you say. The translators are of course possible, and denying that a LISP machine is equivalent in computational power to a TM, is just absurd, which is what these folk are doing. I tried to tell this to Harris, I don't know, perhaps 10 times, that unbounded and infinite really are different things. To Harris: computable reals are calculated with *finite* programs, even if they take an infinite amount of time in theory. Computable reals are, in the course of computation, unbounded, but still finite. THEY NEVER *BECOME* INFINITE. I wonder why that is so hard to understand! (And this has nothing to do with set theory!) I think it's actually infinitely ignorant to say that PCs are not TMs, because TMs are said to have an infinite tape. I can't stand hearing that shallow argument again! Because as you said TMs do not have an infinite tape, they have an unbounded tape, which is a necessary requirement in any sensible model of general discrete computation!!! That's not surprising either!!! -- Eray Ozkural === Subject: Re: Lambda Calculus and Turing Equivalence > > > However, if *actual* infinity is a *necessary* part of TM model, then > it cannot be equivalent to lambda calculus, because there is no > infinite component in any lambda calculus expression, > > A TM does not have an _infinite_ tape, it has an _unbounded tape_. > This means that the tape is guaranteed to be as large as you need it > or, equivalently, that the tape will grow as you need it but always > stay finite. > > It isn't hard to write a TM simulator in pure lambda calculus. The > converse is harder, but possible. > Precisely my point. > I am saying to those who say that PCs are not TMs, like Chapman and > Harris, that they do not know the difference between unbounded > (potentially infinite) and actually infinite. (Although they claim > that they do) They definitely do. What you don't realize is that every PC is bounded in memory and thus cannot, by the definition of Turing machines, be Turing machines. You think you might be able to add memory to a PC as required, but as there is only finite mass in the universe, there is an upper bound on the size of calculations your computer can compute. The latter is a set theoretical notion that is not > required to depict TMs, I doubt this distinction was so clear back > then. > That we can use set theory to formalize TMs shouldn't confuse us. It only seems to confuse you. > Anyway, I agree with all of that you say. The translators are of > course possible, and denying that a LISP machine is equivalent in > computational power to a TM, is just absurd, which is what these folk > are doing. Try writing a LISP program to calculate the Busy Beaver function. > I tried to tell this to Harris, I don't know, perhaps 10 times, that > unbounded and infinite really are different things. To Harris: > computable reals are calculated with *finite* programs, even if they > take an infinite amount of time in theory. Computable reals are, in > the course of computation, unbounded, but still finite. THEY NEVER > *BECOME* INFINITE. I wonder why that is so hard to understand! (And > this has nothing to do with set theory!) > I think it's actually infinitely ignorant to say that PCs are not TMs, > because TMs are said to have an infinite tape. I can't stand hearing > that shallow argument again! > Because as you said TMs do not have an infinite tape, they have an > unbounded tape, which is a necessary requirement in any sensible model > of general discrete computation!!! That's not surprising either!!! This is a silly point. Fine, say that the TM's tape is unbounded. All that means is that it has an infinite amount of tape available with which to compute. (Pf: Suppose that a TM has used n cells, and can only use m more cells. Then the TM can only use n+m cells. So the number of cells the TM can use is bounded. Contradiction. So infinitely many cells are available to it. Say that the tape is (actually or potentially) infinite. Then the number of cells is clearly unbounded. So 'unbounded' and 'infinite' are equivalent. === Subject: Re: Lambda Calculus and Turing Equivalence >> A TM does not have an _infinite_ tape, it has an _unbounded tape_. >> This means that the tape is guaranteed to be as large as you need it >> or, equivalently, that the tape will grow as you need it but always >> stay finite. >> It isn't hard to write a TM simulator in pure lambda calculus. The >> converse is harder, but possible. >Precisely my point. >I am saying to those who say that PCs are not TMs, like Chapman and >Harris, that they do not know the difference between unbounded >(potentially infinite) and actually infinite. (Although they claim >that they do) The latter is a set theoretical notion that is not >required to depict TMs, I doubt this distinction was so clear back >then. Part of the problem is that there is a fair amount of literature that describes TMs as having infinite memory, e.g. the second paragraph of http://en.wikipedia.org/wiki/Turing_machine. (Arguably, this is an informal description.) far none that I have seen have commented on the difficulty TMs have in modeling ongoing computation, such as what is found in operating discussion of this.) --gregbo gds at best dot com === Subject: Re: Lambda Calculus and Turing Equivalence > A TM does not have an _infinite_ tape, it has an _unbounded tape_. > This means that the tape is guaranteed to be as large as you need it > or, equivalently, that the tape will grow as you need it but always > stay finite. > It isn't hard to write a TM simulator in pure lambda calculus. The > converse is harder, but possible. >>Precisely my point. >>I am saying to those who say that PCs are not TMs, like Chapman and >>Harris, that they do not know the difference between unbounded >>(potentially infinite) and actually infinite. (Although they claim >>that they do) The latter is a set theoretical notion that is not >>required to depict TMs, I doubt this distinction was so clear back >>then. > Part of the problem is that there is a fair amount of literature that > describes TMs as having infinite memory, e.g. the second paragraph of > http://en.wikipedia.org/wiki/Turing_machine. (Arguably, this is an > informal description.) > far none that I have seen have commented on the difficulty TMs have in > modeling ongoing computation, such as what is found in operating > discussion of this.) > --gregbo > gds at best dot com This issue is strictly one of knowing the definition. Turing said the TM can compute Pi which is infinitely long, one finite digit at a time. All the used, is not sufficient to accomplish the storage Turing's tape can do. It doesn't matter if you consider the tape to be only hugely finitely large. Turing's postulated tape provides more memory storage than any amount of memory available utilizing the entire physical universe. It is just a matter of definition, not philosophical. There are countless websites describing the TM, as ideal, as hypothetical, as theoretical, as abstract, IOW, not physical. One can simulate a TM on a PC, or build a TM, except for the tape which always will have less potential capability than the tape that Turing described. The tape Turing employs is not real, nor can it be real. http://en.wikipedia.org/wiki/Turing_machine Note that every part of the machine is finite, but it is the potentially unlimited amount of tape that gives it an unbounded amount of storage space. http://plato.stanford.edu/entries/turing-machine/ In modern terms, the tape serves as the memory of the machine, while the read-write head is the memory bus through which data is accessed (and updated) by the machine. There are two important things to notice about the definition. The first is that the machine's tape is infinite in length, corresponding to an assumption that the memory of the machine is infinite. The second is similar in nature, but not explicit in the definition of the machine, namely that a function will be Turing-computable if there exists a set of instructions that will result in the machine computing the function regardless of the amount of time it takes. One can think of this as assuming the availability of infinite time to complete the computation. These two assumptions are intended to ensure that the definition of computation that results is not too narrow. This is, it ensures that no computable function will fail to be Turing-computable solely because there is insufficient time or memory to complete the computation. If a function is not Turing-computable it is because Turing machines lack the computational machinery to carry it out, not because of a lack of spatio-temporal resources. These are called abacus computers by Lambek (Lambek 1961), and are known to be equivalent to Turing machines. The modern digital computer is subject to finiteness constraints that we have abstracted away in the definition of abacus machines, just as we did in the case of Turing machines. Physical computers are limited in the number of memory locations that they have, and in the storage capacity of each of those locations, while abacus machines are not subject to those constraints. Thus some abacus-computable functions will not be computable by any physical machine. (We won't consider whether Turing machines and modern digital computers remain equivalent when both are given external inputs, since that would require us to change the definition of a Turing machine.) === Subject: Unbounded Space > far none that I have seen have commented on the difficulty TMs have in > modeling ongoing computation, such as what is found in operating > discussion of this.) > This issue is strictly one of knowing the definition. Turing said the TM > can compute Pi which is infinitely long, one finite digit at a time. All the > used, is not sufficient to accomplish the storage Turing's tape can do. What is this naive argument supposed to show? What does the actual size of the universe has to do with the fact that the tape is unbounded? (And we are not even certain that the space in our universe is finite!) A TM can compute Pi in no finite universe. That's not specific to our universe. But the ID of a TM is always finite. It never becomes infinite. It's apparent that you don't understand what unbounded means. Here, I will make a rigorous definition. And you can object if you will, but do try to come up with a valid argument. I am generalizing from the tape to include other components of the machine. Unbounded Space: The ID of a TM is always finite length. However, there is no upper bound to the description length. Therefore, we say that the Turing Machine has unbounded space. I don't think you understand what computable real means, either. It just means that, as you said, you can calculate one digit at a time. It does not mean that you can actually output an infinite supply of digits! That would *never* happen, even in an infinite size universe, simply because at any time T, the computer would have output only a finite number of digits! Do you now understand the distinction between unbounded and infinite? Do you understand what it means to exist? I sincerely hope you can make an improvement after 3 years! But alas, you seem to be another naive Platonist like Chapman... :( -- Eray Ozkural === Subject: Re: Unbounded Space > What is this naive argument supposed to show? What does the actual > size of the universe has to do with the fact that the tape is > unbounded? (And we are not even certain that the space in our universe > is finite!) A TM can compute Pi in no finite universe. That's not > specific to our universe. But the ID of a TM is always finite. It > never becomes infinite. An algorithm exists such that given -any- integer N one can compute pi to N places (assume decimal representation). In short there is no hard upper bound on the number of places to which one can computer pi -in priniciple-. This is not to be confused with practical physical limits. And yes, the notion -in principle- is inherently Platonic. No getting away from that. Bob Kolker === Subject: Re: Unbounded Space >> What is this naive argument supposed to show? What does the actual What all of them show, that you are an uneducated dummy. >> size of the universe has to do with the fact that the tape is Nothing. Physical memory takes substance. There is a fininte amount of matter in the universe. Because the universe may continue to expand forever does not create any additional matter. used for memory of a physical computer, the computation that could then be performed would still be physicall constrained. I don't know what part of this you don't get. There are finite numbers the digits that needed storing so that the number could be represented, still finitely, go beyond all the matter in the universe. Again matter in the universe is not infinite, it is finite. Turing conceived of an abstract or imaginary tape. It can store can compute more finite digits of a number such as Pi, than any physical computer, PC. Another mistake you make is when you ask that absurd question if a TM is a PC. There is only one correct answer, No. That you even think a poll should be taken, is evidence of your ignorance. You are not in some graduate program. You are a brain-damaged, or chemically imbalance, highschool dropout. It is a fact that a TM is not a PC, just like it is a fact that an odd number is not even. If you get somebody to agree with you, it just means they are clueless like you. You are bright but certainly not a genius. You are poorly educated. You are mentally ill because you think your opinion has any foundation for being considered by yourself or others as worth as much as rat. >> unbounded? (And we are not even certain that the space in our universe >> is finite!) A TM can compute Pi in no finite universe. That's not It has nothing to do with space being finite, but matter being finite. That gives you away as not having taken highschool physics. A TM can compute Pi in no finite universe. You've graduated to halfwit. Nowadays an infinite universe does not mean infinite matter. A TM does not compute in any physical universe whatsoever, finite or infinite. It computes in an abstract, non-physical universe so it does not have physical constraints. It can not run out of what serves in place of physical memory. A physical computer operating in the physical universe must run out of physical memroy because matter is finite and matter is what is used for storage. And this abstract idea does not require mathematical Platonism! Abstract and concrete stand on their own. >> specific to our universe. But the ID of a TM is always finite. It >> never becomes infinite. It doesn't need to become infinite. Potentially infinite means that it can forever grow finitely larger and larger. The TM can compute a finitely larger sequence of Pi because there are no physical limitation imposed on a TM, while PCs have physical constraints. Most parts of the TM have approximate realizations physically. Not the tape! The tape can never be physically realized. That is where the analogy between a TM and PC will always break down. It is not just Pi. The square roots of prime numbers, which are infinite, has an algorithm, that can produce a very large sequence of digits, which at any given point is finite. This finite amount of digits that the TM can compute can alwasy exceed the finite amount of digits that a PC can compute. If the PC can compute X amount of digits a TM can compute the finite amount X + 1 or X +2 etc. Since there are an infinite number of primes and their square roots, there are an infinite number of instances in which a TM can compute _more finite_ digits of any given such operation than a PC can compute. These are all computable numbers. I'm not saying the TM is computing a non-computable number, but a larger finite portion of a computable number. I think that intractable may be the word that describes limits on computable physical calculations (PC) that have physical constraints impacting the completion of some calculation, not enough time or memory. If you think additional matter is being created in the universe just because the expansion may be infinite, get your Nobel prize first and then I will concede your assumption of an infinite amount of adding available sufficient memory to a physical PC. === Subject: Re: Unbounded Space > > What is this naive argument supposed to show? What does the actual > size of the universe has to do with the fact that the tape is > unbounded? (And we are not even certain that the space in our universe > is finite!) A TM can compute Pi in no finite universe. That's not > specific to our universe. But the ID of a TM is always finite. It > never becomes infinite. > An algorithm exists such that given -any- integer N one can compute pi > to N places (assume decimal representation). In short there is no hard > upper bound on the number of places to which one can computer pi -in > priniciple-. > This is not to be confused with practical physical limits. > And yes, the notion -in principle- is inherently Platonic. No getting > away from that. I don't buy that. -- Eray Ozkural === Subject: Re: Unbounded Space >> What is this naive argument supposed to show? What does the actual >> size of the universe has to do with the fact that the tape is >> unbounded? (And we are not even certain that the space in our universe >> is finite!) A TM can compute Pi in no finite universe. That's not >> specific to our universe. But the ID of a TM is always finite. It >> never becomes infinite. > An algorithm exists such that given -any- integer N one can compute pi to > N places (assume decimal representation). In short there is no hard upper > bound on the number of places to which one can computer pi -in > priniciple-. > This is not to be confused with practical physical limits. > And yes, the notion -in principle- is inherently Platonic. No getting away > from that. > Bob Kolker Here is a short description of platonic realism: http://en.wikipedia.org/wiki/Platonic_realism These sorts of universals are called, after Plato, forms or ideas, but Plato's universals certainly are not ideas in the mind. They are not mental entities at all, unless, as on some theories, they are ideas in God's mind. Due to the potential confusion, 'form', 'Platonic form', or simply 'universal' are terms more usually used by philosophers. SH: So why do you think infinity or that tape is inherently Platonic? Abstract and platonic are not synonyms. Do you think that zero, or infinitity, or the imaginary number, i implicity claim their origin in platonic idealism? I don't think so, I think they are mental entities --> abstract ideas-->inventions of the mind. Platonism says that infinity exists independently of the human mind. Are you claiming that when Turing thought up his unlimited tape that he first adopted the belief that there was a realm outside of space and time that had unlimited tapes as archetypal entities, and that Turing plucked one of these tapes of that realm and inserted it into his paper as a concept? I think that it is certainly possible that Turing may have agreed with Aristotle who judged that abstract notions have no independent existence, or that he may have had no philosphical inclination at all. That seems to me to be a less complicated premise than reading platonic assumptions into Turing's 1936 paper which seem like extra baggage. Abstract terms refer to ideas or concepts; they have no physical referents. So the turing tape has no physical referent. Most people at this point just say that it is a mental abstraction (idea). It is quite a jump to arrive at a metaphysical realm, which exists outside time and space and human mentation, that hosts an entity which has pre-existed humanity (eternal) which just happens to have Turing tapes which is found close to Truth. It is usual to divide things as abstract or concrete without classifying abstract things as evidence of all the philosphical assumptions needed for a belief in a Platonic realm which is similar to the IDEAS of God. I find Turing's tape perfectly consistent with Aristotelean realism and the more widely held view that mathematics is invented, not discovered inside a mystical Grecian urn which contains all the ideas we haven't thought of yet waiting to be washed up from the quantum foam. with > And yes, the notion -in principle- is inherently Platonic. No getting > away from that. Stephen === Subject: Re: Lambda Calculus and Turing Equivalence >> A TM does not have an _infinite_ tape, it has an _unbounded tape_. >> This means that the tape is guaranteed to be as large as you need it >> or, equivalently, that the tape will grow as you need it but always >> stay finite. > It isn't hard to write a TM simulator in pure lambda calculus. The >> converse is harder, but possible. >Precisely my point. >I am saying to those who say that PCs are not TMs, like Chapman and >Harris, that they do not know the difference between unbounded >(potentially infinite) and actually infinite. (Although they claim >that they do) The latter is a set theoretical notion that is not >required to depict TMs, I doubt this distinction was so clear back >then. > Part of the problem is that there is a fair amount of literature that > describes TMs as having infinite memory, e.g. the second paragraph of > http://en.wikipedia.org/wiki/Turing_machine. (Arguably, this is an > informal description.) > far none that I have seen have commented on the difficulty TMs have in > modeling ongoing computation, such as what is found in operating > discussion of this.) Sorry there was a slip of tongue in the other reply. I've been working late hours for the past few days. What I wanted to say is: TMs have no difficulty running an OS, it's just another program, and in fact such useful non-terminating programs have been described by Turing, e.g. computable reals. Likewise for any other model of computation. I don't see why a TM would have trouble running an OS... You will have to explain that. It's no use pretending that TMs can run only finite-time programs. -- Eray Ozkural === Subject: Re: Lambda Calculus and Turing Equivalence >> A TM does not have an _infinite_ tape, it has an _unbounded tape_. >> This means that the tape is guaranteed to be as large as you need it >> or, equivalently, that the tape will grow as you need it but always >> stay finite. > It isn't hard to write a TM simulator in pure lambda calculus. The >> converse is harder, but possible. >Precisely my point. >I am saying to those who say that PCs are not TMs, like Chapman and >Harris, that they do not know the difference between unbounded >(potentially infinite) and actually infinite. (Although they claim >that they do) The latter is a set theoretical notion that is not >required to depict TMs, I doubt this distinction was so clear back >then. > Part of the problem is that there is a fair amount of literature that > describes TMs as having infinite memory, e.g. the second paragraph of > http://en.wikipedia.org/wiki/Turing_machine. (Arguably, this is an > informal description.) > far none that I have seen have commented on the difficulty TMs have in > modeling ongoing computation, such as what is found in operating > discussion of this.) Excuse me but that kind of sloppy rhetoric is absolutely irrelevant to the issue. Surely, your PC has no difficulty running an OS. people, like Harris or Chapman, who couldn't distinguish unbounded from infinite. It's not something to rely on for delicate matters. If you want to see a definition of a Turing Machine, read the Cinderella book. -- Eray Ozkural === Subject: Ozkural: his abuse continues was Re: Lambda Calculus and Turing Equivalence > people, like Harris or Chapman, who couldn't distinguish unbounded > from infinite. Need I say more? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Ozkural: his abuse continues was Re: Lambda Calculus and Turing Equivalence > people, like Harris or Chapman, who couldn't distinguish unbounded > from infinite. > Need I say more? There is no indication that you know the difference between unbounded and infinite. Why should making your ignorance explicit be an abuse? -- Eray Ozkural === Subject: Re: Ozkural: its abuse continues >> Eray Are there integers with an infinite number of digits? Ozkural >> people, like Harris or Chapman, who couldn't distinguish unbounded >> from infinite. >> Need I say more? > There is no indication that you know the difference between unbounded > and infinite. There is no evidence that I am unaware of such a distinction. > Why should making your ignorance explicit be an abuse? There is no evidence that I am ignorant of this supposed distinction. There is much evidence of Ozkural making unfounded personal insults and attacks. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Lambda Calculus and Turing Equivalence > A TM does not have an _infinite_ tape, it has an _unbounded tape_. > This means that the tape is guaranteed to be as large as you need it > or, equivalently, that the tape will grow as you need it but always > stay finite. > It isn't hard to write a TM simulator in pure lambda calculus. The > converse is harder, but possible. >>Precisely my point. >>I am saying to those who say that PCs are not TMs, like Chapman and >>Harris, that they do not know the difference between unbounded >>(potentially infinite) and actually infinite. (Although they claim >>that they do) The latter is a set theoretical notion that is not >>required to depict TMs, I doubt this distinction was so clear back >>then. The difference between a TM and a PC is that the potential finite memory of a TM exceeds the potential finite memory of a PC. A TM is a hypothetical (theoretical) logical device that can in principle compute problems that require greater storage than some physical computer (PC) even if that PC had or could use all the physical memory available in the physical universe and execute to the end of time or heat death of the universe (no power). A TM is also not constrained by a physical energy source to move the tape. That is because the tape that Turing posited has no physical constraints. Therefore a TM can compute more digits (a finite number) of Pi than any PC (also finite) which is limited by memory and the heat death of the universe. Pi or the square root of a prime number are still computable. There is no claim that a non-computable number is being computed by a TM than a PC cannot compute, but that a TM can compute a larger amount (more digits each requiring more memory to hold them) of finite digits of a computable number. > Part of the problem is that there is a fair amount of literature that > describes TMs as having infinite memory, e.g. the second paragraph of > http://en.wikipedia.org/wiki/Turing_machine. (Arguably, this is an > informal description.) > far none that I have seen have commented on the difficulty TMs have in > modeling ongoing computation, such as what is found in operating > discussion of this.) > --gregbo > gds at best dot com PCs and TM both compute computable functions. I don't see how having an OS or a timing signal or any other ways that TMs and PCs differ has any bearing on this thread. I think what is relevant from the Wiki url: http://en.wikipedia.org/wiki/Turing_machine Comparison with real machines Turing machines would actually only be equivalent to a real machine that is magically given an infinite amount of storage space. Turing endowed his TM with a magical amount of storage potential because physically, the amount of storage is not possible. A physical PC can just not match the amount of information that can be held on Turing's tape, even though the information storage required is finite in both cases. It is like the difference between a gallon of milk and a truckload of milk (or more). It doesn't matter if you change infinite to unbounded. The TM described by Turing in his 1936 has an unlimited, (as needed) tape. A physical computer (PC) must always be (upper) limited. The amount of physical memory that you can add to a physical computer will always eventually run out at some huge point of required physical storage. Just because a number is finite does not mean it can be physically computed within a universe with finite resources. The TM is not governed a finite resource of memory because it is not physical. Because a PC uses some ideas in the abstract construction of a Turing machine does not make a PC and TM equivalent. They both compute computable functions, but the finite size of the computable numbers is not identical. There is of course a huge overlap between computations that can in principle be computed by a TM and those that can be computed by a PC which has never been disputed. === Subject: Re: Lambda Calculus and Turing Equivalence > There are many equivalent formalisms of computation. One of these is > lambda calculus, which is supposed to compute exactly the same sets as > Turing Machines can. > Despite the poor philosophy being peddled in these newsgroups, this is > actually seen as an intuitive support for C-T thesis, and universally > believed to be correct. I have never seen anybody claim that lambda > calculus is *not* equivalent to Turing Computation. > However, if *actual* infinity is a *necessary* part of TM model, then > it cannot be equivalent to lambda calculus, because there is no > infinite component in any lambda calculus expression, and neither can > we talk of infinitely long expressions. There are only finite > expressions, just like there are only a finite number of non-blank > symbols on a TM tape. We will write T1 -> T2 to mean that term T1 reduces in one step to term T2. We will write N1, N2, N3 to mean terms that are in normal form, that is, terms that cannot be reduced. Given an arbitrary term T and a sequence of reductions T -> T1 -> T2 ... -> N, the intermediate terms Ti may grow unboundedly large. This is exactly analogous to a Turing machine writing on an unboundedly large section of the tape. Once again, you have made a very silly argument. --PeterD === Subject: Re: Lambda Calculus and Turing Equivalence > There are many equivalent formalisms of computation. One of these is > lambda calculus, which is supposed to compute exactly the same sets as > Turing Machines can. > Despite the poor philosophy being peddled in these newsgroups, this is > actually seen as an intuitive support for C-T thesis, and universally > believed to be correct. I have never seen anybody claim that lambda > calculus is *not* equivalent to Turing Computation. > However, if *actual* infinity is a *necessary* part of TM model, then > it cannot be equivalent to lambda calculus, because there is no > infinite component in any lambda calculus expression, and neither can > we talk of infinitely long expressions. There are only finite > expressions, just like there are only a finite number of non-blank > symbols on a TM tape. > We will write T1 -> T2 to mean that term T1 reduces in one step to > term T2. We will write N1, N2, N3 to mean terms that are in normal > form, that is, terms that cannot be reduced. > Given an arbitrary term T and a sequence of reductions > T -> T1 -> T2 ... -> N, the intermediate terms Ti may grow > unboundedly large. This is exactly analogous to a Turing machine > writing on an unboundedly large section of the tape. And what am I saying in my post that is different from this or contradicts it? What does arbitrary growth of computational space mean, which you failed to quote? > Once again, you have made a very silly argument. I did not make a very silly argument at all. (And I don't think I ever did) You failed to read and understand my post. Look at the other replies. My silly argument is that TMs do *not* have an infinite tape, only an unbounded tape (that is finite at ALL times), and this entails that PCs are Turing Machines, unlike what the infinitely wise Robin Chapman seems to think. -- Eray Ozkural === Subject: Re: Lambda Calculus and Turing Equivalence > And what am I saying in my post that is different from this or > contradicts it? What does arbitrary growth of computational space > mean, which you failed to quote? prd > Once again, you have made a very silly argument. > I did not make a very silly argument at all. (And I don't think I ever > did) You failed to read and understand my post. Look at the other > replies. > My silly argument is that TMs do *not* have an infinite tape, only > an unbounded tape (that is finite at ALL times), and this entails that > PCs are Turing Machines, unlike what the infinitely wise Robin Chapman > seems to think. OK, let's look at the part I snipped. Eray > Since the imagined actually infinite tape of TM does Eray > not exist in this other equivalent model of computation, Eray > we conclude that it is not part of a TM to start with. Only Eray > a potentially infinite tape can be part of a TM, because Eray > see above, there are only a finite number of non-blank Eray > symbols on a TM tape, and the Turing Machine has no Eray > dealing with the infinite remainder of the tape, which Eray > does nothing except to symbolize the arbitrary growth Eray > of computational space. A Turing machine, like a Klein bottle is a mathematical construct. As such, it can be defined with whatever properties the definer choses to give it. That no physical thing can have these properties is not relevant. For example, I can talk about a Klein bottle as a mathematical object, but I almost certainly will never experience a real Klein bottle. If the tape of a Turing Machine is defined to be infinite, or even actually infinite, then it is actually infinite, even though we are unlikely to experience real infinite tapes. This is just as much true as if I define the set of integers to be actually infinite. Perhaps you would also argue that the set of integers is only potentially infinite, that it is only a imagined to be an actually infinite set. That's fine. Go off and converse with whatever community accepts your terminology. But by now you should understand that you are not speaking the language of the mathematics community. --PeterD === Subject: Re: Lambda Calculus and Turing Equivalence > If the tape of a Turing Machine is defined to be infinite, > or even actually infinite, then it is actually infinite, even > though we are unlikely to experience real infinite tapes. > This is just as much true as if I define the set of integers to > be actually infinite. Perhaps you would also argue that > the set of integers is only potentially infinite, that it is only > a imagined to be an actually infinite set. That's fine. Go > off and converse with whatever community accepts your > terminology. But by now you should understand that you > are not speaking the language of the mathematics community. It's clear that you don't consider computer scientists like Ullman or Chaitin part of the mathematics community. And probably all the other people who said TM has an unbounded tape, not an infinite one. What makes you an authority on that? === Subject: Re: Lambda Calculus and Turing Equivalence >> If the tape of a Turing Machine is defined to be infinite, >> or even actually infinite, then it is actually infinite, even >> though we are unlikely to experience real infinite tapes. >> This is just as much true as if I define the set of integers to >> be actually infinite. Perhaps you would also argue that >> the set of integers is only potentially infinite, that it is only >> a imagined to be an actually infinite set. That's fine. Go >> off and converse with whatever community accepts your >> terminology. But by now you should understand that you >> are not speaking the language of the mathematics community. > It's clear that you don't consider computer scientists like Ullman or > Chaitin part of the mathematics community. And probably all the other > people who said TM has an unbounded tape, not an infinite one. > What makes you an authority on that? It is clear that your ability to comprehend is impaired. Of course I consider Ullman and Chaitin part of the mathematics community. If you look again and read my post, you will see that I did not claim that the definition of a Turing machine includes an infinite tape. What I said was: << If the tape of a Turing Machine is defined to be infinite, << or even actually infinite, then it is actually infinite, even << though we are unlikely to experience real infinite tapes. My point is that *if* something is defined a certain way, it is pointless to bring up arguments about the definition violating preconceived notions about the impossibilty of actual infinity and the like. The thing is defined and you just have to accept it. (You obviously don't, but that is another matter). If you had argued that Turing did not claim the tape was infinite, then that would have been an entirely different sort of argument, and one which is resolvable by investigating the source texts. But instead of arguing about the factual matter of how Turing defined his theoretical machine, you argued that actual infinity is not permitted by your religion. --PeterD === Subject: Re: Lambda Calculus and Turing Equivalence > There are many equivalent formalisms of computation. One of these is I had more of Torkel Franzel than yourself on my mind, but you are > less of a philosopher than I am. Try to answer the above challenge to > prove your logic. What challenge? I see no reason to accept challenges from the likes of you. Suffice it to say that my philosophy of mathematics does not encompass entities such as integers with infinitely many digits. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Lambda Calculus and Turing Equivalence > There are many equivalent formalisms of computation. One of these is > I had more of Torkel Franzel than yourself on my mind, but you are > less of a philosopher than I am. Try to answer the above challenge to > prove your logic. > What challenge? Oh, just make one up - rehash one of the standards - Can an infinitely intelligent Robin Chapman contrive a mathematical puzzle so devious that he himself cannot solve it. Phil -- I used to have an interest in writing viral code and lost interest quickly when Win95 came out. Hell how could any of us in the scene write a more invasive program than Win95. It made us all obsolete. -- Screaming Radish [NuKE] on alt.comp.virus.source.code === Subject: Re: Lambda Calculus and Turing Equivalence > There are many equivalent formalisms of computation. One of these is > It is not drivel or polemic. It's a simple philosophical argument. I invite those who have repeatedly failed to understand my arguments like Stephen Harris to review it. I'm sorry that you don't have the English skills to distinguish a proper argument from the kind of ad hominem ones you attempt. Obviously, you cannot understand or answer any of the argument that *refutes* your ill-conceived claim that actual infinite space is a necessary component of any model of computation. Try to appreciate the fact that you cannot know everything in the world. You don't have to to know computer science very well. And indeed, that seems to be the case. You can't accept that lambda calculus depicts exactly the same sets as TMs do. If you had been a computer scientist, you might be able to appreciate this point, or even just a good programmer. I'm telling you that your intuition about models of computation is grossly misled. You don't seem to understand that saying what you say would be denying the Church-Turing thesis, the inventors of both were much more skilled than either of us in matters of calculation. -- Eray Ozkural === Subject: Ozkural ad nauseam Re: Lambda Calculus and Turing Equivalence >> Eray Are there integers with an infinite number of digits Ozkural >> There are many equivalent formalisms of computation. One of these is >> It is not drivel or polemic. It was. > It's a simple philosophical argument. Maybe it was an argument. > Obviously, you cannot understand or answer any of the argument that > *refutes* your ill-conceived claim that actual infinite space is a > necessary component of any model of computation. Abuse comes easily to you Mr Ozkural, but not facts. I made no such claim. Mr Ozkural, why do you continue to post in sci.math when you have no mathematical contribution to make? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Ozkural ad nauseam Re: Lambda Calculus and Turing Equivalence <4zrnd.46389$QJ3.16198@newssvr21.news.prodigy.com> <61bad26451853979aa126965352b4240.48257@mygate.mailgate.org> <5ci1q0tqrsrbnut54hbn7kbvt5g8idf1so@4ax.com> Discussion, linux) >> It's a simple philosophical argument. > Maybe it was an argument. Maybe it was simple. -- One these mornings gonna wake | Ain't nobody's doggone business how up crazy, | my baby treats me, Gonna grab my gun, kill my baby. | Nobody's business but mine. Nobody's business by mine. | -- Mississippi John Hurt === Subject: Re: What on earth! was Re: Turing Machines and Physical Computation > Abstract does not mean non-physical > the Turing Machine is just another kind > of automata, and its mechanism is firmly rooted in the physical world. > Incuding its infinite tape no doubt /me hands Robin an unbounded tape as a slot-in replacement Phil -- I used to have an interest in writing viral code and lost interest quickly when Win95 came out. Hell how could any of us in the scene write a more invasive program than Win95. It made us all obsolete. -- Screaming Radish [NuKE] on alt.comp.virus.source.code === Subject: Re: What on earth! was Re: Turing Machines and Physical Computation >> Eray Are there integers with an infinite number of digits? Ozkural >> Abstract does not mean non-physical >> the Turing Machine is just another kind >> of automata, and its mechanism is firmly rooted in the physical world. >> Incuding its infinite tape no doubt > /me hands Robin an unbounded tape as a slot-in replacement Is that a potentially infinite tape? :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: What on earth! was Re: Turing Machines and Physical Computation >> Eray Are there integers with an infinite number of digits? Ozkural >> >> Abstract does not mean non-physical >> >> the Turing Machine is just another kind >> of automata, and its mechanism is firmly rooted in the physical world. >> >> Incuding its infinite tape no doubt > > /me hands Robin an unbounded tape as a slot-in replacement > Is that a potentially infinite tape? :-) You'll find out eventually. Or maybe you won't. Phil -- I used to have an interest in writing viral code and lost interest quickly when Win95 came out. Hell how could any of us in the scene write a more invasive program than Win95. It made us all obsolete. -- Screaming Radish [NuKE] on alt.comp.virus.source.code === Subject: Re: Turing Machines and Physical Computation > The following are taken from a series of messages posted internally > within a closed network back in 1990. I thought they might be worth > citing once again (although without their original context for reasons > which some of the more thoughtful here will understand). >> I doubt that anyone can guess why you posted this here again. > I know. That's why I posted it. You're supposed to do some work to make > the connections. But why should we suppose that those connections are not just your own brand of mentalisms? Incidentally a paranoid schizophrenic can not explain the connections that they make either. I have no evidence to suggest that these subtle connections you keep waving at do not belong to that same class of behavior. > A lot has been > said here since. Most folk here won't see a fraction of what it's all > about for many reasons that I've already covered. Some might be wise to > take their emotional responses as something to further act upon. >> Well i went through it again in detail. I understand most of the >> pieces. I have commented on what i got from each in this context. In >> one case i have expressed my own emotion as i felt it was an >> appropriate dialectic to Quine's (and your) own emotional judgment. > Your affective behaviour is not the issue. You will easily be misled if > you just assess what you read in terms of how it makes you feel. >> However, why you have concatenated these particular pieces together >> and included your own piece about naloxone, i cannot guess. > The piece about naloxone is actually implicitly about the endogenous > opiates, their receptors, and habit formation (also known as > learning, although I don't use that term much. Instead I tend to talk > of changing operant levels through reinforcement of behaviours). If you > want some hints on where this goes you would need to look into the > monoamines like dopamine, activity, the nucleus accumbens and the the > mesolimbic system/ventral striatum. I said not long ago that there were > two classes of responding (behaviour) which I found very dramatic a) > ICSS and b) neophobia. Once you have looked into some of that, you might > like to look into (primarily) young males and especially adolescents, > testosterone, alcohol, and the above system, especially viz a viz the > field I am always implicitly referring to (cf. PROBE). It is easy to grant some relationship between testosterone, alcohol, opiate receptors, habits and criminal behavior. It is very difficult to relate that to the distinction between computer behavior and human behavior. If you would at least make your premises explicit, then perhaps we could start to make the connections you want us to make. > ON THE COMPUTER METAPHOR > 'It has always bothered me that models of psychological > processing were thought to be inspired by our understanding of > the computer. The statement has always been false. Indeed, the > architecture of the modern digital computer - the so-called Von > Neumann architecture - was heavily influenced by people's (naive) > view of how the mind operated. Perhaps I had better document > this. Simply read the work on cybernetics and thought in the > 1940's and 1950's prior to the development of the digital > computer. The group of workers included people from all > disciplines: See the Macy Conferences on Cybernetics, or Her > Majesty's Conference on Thought processes. Read the preface to > Wiener's book on cybernetics. Everyone who was working together - > engineers, physicists, mathematicians, psychologists, > neuroscientists (not yet named) - consciously and deliberately > claimed to be modelling brain processes.' > Reflections on Cognition and Parallel Distributed Processing > D.A. Norman > (Ch 26, p534, Parallel Distributed Processing Volume 2) > McClelland J and Rumelhart D 1986 >> I get that computer behavior and human behavior are essentially >> different. Why they are so different is certainly what we came here >> to discuss. > And that's what Fragments is largely about as I have said many times. > 'A trait is EFFECTIVE if there is a hard and fast > routine by which we can check for it, without guesswork > or imagination...It came to be appreciated, in the mid- > thirties, that recursiveness affords a sharp explication > of effectiveness. This has come to be called Church's > Thesis....By its nature, Church's Thesis was not open to > formal proof; for the thesis equated a precise property, > recursiveness, with a property - effectiveness - that > was to be rendered precise only by the thesis itself. > But the thesis was supported by such instances as could > be mustered, and soon it was pretty well clinched by > Alan Turing's pioneer work in the abstract theory of > computing machines. His formulation of mechanical > computability, in terms of ideal mechanization, turned > out to be equivalent to recursiveness. Mechanical > computability, surely, is very much what our intuitive > talk of effectiveness was aiming at all along; so > Church's Thesis is well sustained.' > - > Quine (1987) > Recursion >> ? > Thread title - but we call it rule governed behaviour - do you not get > it yet? What is rule governed behaviour Patty? How does it relate to > PROBE? Computers can be effectively used to enforce rule governed behavior. Rule governed behavior is how we control chaotic events. This premise can be easily granted. There is no need to tippy toe around it. > 'The first three chapters actually grew out of two > earlier papers. Those papers were, in part, polemics > against the views of my good friend and student Jerry > Fodor. Fodor I hasten to say, is not the main target of > this book; but I have retained some of my polemic > against what I call MIT mentalism... The main target > of the present book is one H Putnam (one of my former > selves) and those who have adopted his views. Or perhaps > it would be more accurate to say that the present book > doesn't have a main target; for its aim is not so much > to refute one particular view as to establish the need > for a different way of looking at problems about mental > states. At any rate, the intended contribution of these > three chapters to that end is to do two things: (1) to > establish a close connection (discovered and emphasised > throughout his career by W V Quine) between problems > about meaning and problems about belief fixation, by > showing that the holistic character of belief fixation > in science bears deeply on the issue of individuation of > meanings (or contents or intentions, as they are > called by various philosophers; and (2) to argue that, > in fact, thinking of meanings (or contents) as > theoretical entities - as scientific objects, objects > which can be isolated and which can play an explanatory > role in scientific theory - is a mistake. In the course > of the argument I defend the view that there is no > criterion for sameness of meaning except actual > interpretative practice - a view made famous by Quine > and Davidson' > H Putnam (1988) > Representation and Reality >> for sameness of meaning except actual interpretative practice, which >> works fine for me. I then go a bit beyond that quote to observe that >> interpretive practice *establishes* meaning [*]. I then fail to see >> how meaning would not be relative to the agent practicing the >> interpretation. Yet, i believe, you have denied that implication. Now >> i do see how my quote [*] would be considered nonsense, were it our >> agenda to excise the very concept called meaning from our ontology. >> Sans that agenda, however, it makes total sense. In fact we could >> define meaning as that which interpretive practice establishes, or >> perhaps more precisely: meaning is that which changes as a result of >> interpretive practice. One, of course, hopes that the agenda is not >> going to be used to justify the agenda. Is it? > The term meaning is scientifically (and technologically) useless. It > is mentalistic (intensional). That's why Quine talks of exorcising it. > What do we deal with instead? One could reasonably guess here that you and Quine would have us replace talk of meaning with rule governed behavior. Did i guess right? > We have been through this at great length, > and you will see Quine say it very explicitly in his comment on Hacking. > What work do you think he is referring to there? >> Incidentally that dispute is the only connection that i can find >> between your collage here and recent context in c.a.p. > But all you are doing there is stating what you don't know. I know that. > I've also suggested what you must do to change that (or are you now > doing a Verhey etc on me?). It just doesn't do to tell people you can't > see something. You are the one who has to change that not me. Unfortunately for you that is the same argument used by fakirs. > 'We cannot individuate concepts and beliefs without > reference to the ENVIRONMENT. Meanings aren't in the > head. > The upshot of our discussion for the philosophy of mind > is that propositional attitudes, as philosophers call > them - that is, such things as 'believing that snow > is white' and 'feeling certain that the cat is on > the mat' - are not states of the human brain and > nervous system considered in isolation from the social > and nonhuman environment. A fortiori they are > not functional states - that is, states definable > in terms of parameters which would enter into a > software description of the organism. FUNCTIONALISM, > CONSTRUED AS THE THESIS THAT PROPOSITIONAL > ATTITUDES ARE JUST COMPUTATIONAL STATES OF > THE BRAIN, CANNOT BE CORRECT'. > The arguments I just summarised were, it might be > pointed out in this connection, arguments against > methodological solipsism. > H. Putnam (1988) > 'Representation and Reality' > (Professor of Mathematical Logic Harvard) >> Again we get it that meanings are not in the head; rather they are >> established by interpretive practice. The practice is public behavior >> which is not in the head. > But people don't get it do they. They spend an awful lot of time in > c.a.p and elsewhere showing that they don't get it. So do you, you just > don't see it. Nearly all the posts to c.a.p tacitly assume it. When it's > pointed out, those doing so don't see what they are doing and just deny > that they are doing it! It's called lack of insight. This is why this > discipline (behavioural science) is so difficult. Most of the > mathematicians and computer scientist folk here and elsewhere are > metaphysical never mind, methodological solipsists. I can see that, so > could Glen. So could Skinner and Quine. You can't. Well, yes, i can't stop at the relatively simple truths that you have repeated again and again. No, i go beyond them. And that is what *you* don't get. You stop too soon, and you stop in a bad place. You wave at more, but you do not have the courage to say it out loud. > - > 'I subscribe entirely to these sentences of Count Verri: > On the Nature of Pleasure and Pain: The only moving > principle of man is pain. Pain precedes every pleasure. > Pleasure is not a positive state.' > - > Immanuel Kant 1781 > - > - > 'Meanwhile our eager-beaver researcher, undismayed by > logic-of-science considerations and relying blissfully > on the exactitude of modern statistical hypothesis- > testing, has produced a long publication list and been > promoted to a full professorship. In terms of his > contribution to the enduring body of psychological > knowledge, he has done hardly anything. His true > position is that of a potent-but-sterile intellectual > rake, who leaves in his merry path a long train of > ravished maidens but no scientific offspring.' > - > P. E. Meehl (1967) > Theory Testing in Psychology and Physics > Philosophy of Science pp 103-115 >> I wonder why this quote was included here or what its context is. > The Kant quote leads onto the neophobia abstract. The Meehl paper is a > pointer to one of the segments of Fragments which covers the point in > detail - it's a damning critique of mainstream psychology as well as a > pointer to the actuarial vs. clinical section of Fragments Which connections are mostly still just in your head. But let's cut to the chase ... we know that you are trying to exorcise mentalism. We pretty much suspect that you want to replace that with computer assisted rule governed behavior. To make your point you must rail against the foibles of mental activity. My point has always been to grant those foibles; but to recognize the futility of pure rule governed behavior. To move off this square you must recognize andor respond to my point. Otherwise we are in a loop. > - > - > C56. > - > Naloxone enhances neophobia > - > J.F.W. DEAKIN & D.C. LONGLEY* > (introduced by T.J. Crow) > - > National Institute for Medical Research, > Mill Hill, London, NW7 1AA > - > Several studies report that naloxone, an opiate receptor > antagonist, reduces deprivation induced eating and drinking. > However, in the present study, naloxone (5mg/kg,i.p.) did not > reduce food intake of rats maintained on a 22 h deprivation - 2 h > feeding schedule. In contrast, naloxone (5 mg/kg,i.p.) > progressively reduced water intake in deprived animals to 46% of > saline treated controls. No effects of naloxone (1, 5 mg/kg) on > established bar pressing for food or water were observed with > either continuous or fixed ratio schedules of reinforcement. > However, naloxone (5mg/kg) accelerated extinction of responding > when food and water were no longer available. > - > Animals treated with naloxone (5mg/kg) during training of the > bar-pressing ate only 26% of the pellets delivered whereas > controls ate all pellets delivered. Since the animals had not > previously experienced the pellets or the operant apparatus, the > possibilities arose that naloxone effects were due to enhanced > neophobic effects of the novel food pellets or novel apparatus > cues, or were due to conditioned taste aversion. Therefore, food > novelty, apparatus novelty and timing of injections were > independently varied in different groups of 8-10 rats treated > with saline or naloxone. Rats were maintained at 85% body weight > with 12g lab chow per day. On experimental days 46 small pellets > (Cambden instruments) were placed on a small petri dish in the > home cage of some groups or released from a pellet dispenser in > an operant box for other groups. The dependent variable was the > number of pellets eaten over 15 minutes. > - > Naloxone (1,5 mg/kg i.p.) injected 5 or 20 min before test almost > completely suppressed pellet eating if the animals had not been > previously exposed to the pellets (p<0.01 't' test vs saline > groups). This occurred independently of whether tests were > carried out in the home cage or novel operant box. Naloxone > induced suppression of pellet eating was almost completely > abolished in either environment if animals had been exposed to > the pellets for the five preceding days in the same or different > environment. Naloxone (5mg/kg, i.p.) administered immediately > after pellet eating tests failed to suppress subsequent pellet > eating. > - > Thus, naloxone suppressed pellet eating if the pellets were novel > and if naloxone was administered before eating tests. The results > suggest naloxone enhances neophobic effects of novel foods and > that suppression of novel pellet eating is not due to enhanced > effects of novelty of apparatus cues or to conditioned taste > aversion. > - > Reference > - > FRENK, H & ROGERS G.H. (1979) The suppressant effects of naloxone on > food and water intake in the rat. > Behav. Neural. Biol, 26, 23-40. >> Hmmmm ... why was this included ? > As I said, habit formation. Those in the EAB consider the analysis of > the control of operant behaviour to be the analysis of intelligent > behaviour. We see your computer rule governed behaviour (programming) > as only a part of this. We see computer scientists as naive and > misguided technicians in this respect when they speak of AI as they > get behaviour wrong. They tend to be pre 1929 Carnapian as I have said > before, and they won't be told that they have simply got their facts > wrong (I have explained this before - it's a factual error as clear as > pointing out to someone that they are wrong when they say that snow is > black). Well when i read Weizenbaum's Science and the Compulsive Programmer i must admit that it turned my thinking around. This is why i built my computer career on designing computer systems from the outside in, rather than from the inside out. Hiring a compulsive programmer is the worst thing that can happen to a project even though they are usually the most talented programmers around. But you miss the whole point of AI ... AI is getting rid of the programmer entirely ... and also getting rid of his brittle rules. > - > PROCEEDINGS OF THE BRITISH PHARMACOLOGICAL SOCIETY (BPS) 1-3 April 1981 > (Also British J Pharmacology 1981) > 'A word now about similarity: subjective similarity, > which is crucial to the learning process. I don't think > anyone has an innate notion of similarity. What one has > incontestably is an innate subjective behavioral > standard of similarity. It can be tested in people and > other animals by conditioning. > - > It is unfortunate that my phrase 'standard of > similarity' suggests judgment or deliberate comparison > on the subject's part, but I am at a loss for another > word. What is afoot is just conditioning, > discrimination, reinforcement, extinction. It is what > Stemmer and some other psychologists treat under the > head of 'generalization class', but I prefer to allow > differences of degree. If a subject is rewarded for a > response to one stimulation and penalized for the same > response to another, then a third stimulation is > subjectively more similar to the first than to the > second, for him, if it elicits the response. >> ... or he is being deceitful, or is no longer interested in responding >> ... or has interpreted the third stimulus in a different context >> ... or in other words examining external behavior alone is not a >> scientific method for determining the subjective similarity of the >> stimulus ... hmmm, you go into that (p, p*, p**) in detail below ? > The above makes no sense at all as you mix languages. > - > Since it is basic to the mechanism of any learning, any > conditioning, a similarity standard must be there to > start the first learning. That is why I say it is > innate. But our similarity standards evolve very > decidedly as we learn, and that is part of what I was > dealing with in 'Natural Kinds'...... > ..... > I first took up subjective similarity under the head of > 'quality space' on page 85 of WORD AND OBJECT. In ROOTS > OF REFERENCE I went into it more fully under the head of > 'perceptual similarity'. I was concerned with it for its > role in tying stimulations to observation sentences, as > a basis for a theory of evidence...' > - > Quine > 'Comment on Hacking' > In Perspectives on Quine (1990) > - >> Hmmmm ... why was this included ? > I've already explained. I've also explained before that the fact that > what goes on inside the head can vary dramatically between individuals > should be taken as a clear indication that it doesn't matter! ... or that the varying itself does not matter. > The point > here is that hoards of people are discussing things that really don't > make any difference. We are talking about whole professions here, not > a few posters to USENET. > been a lot of interest in the foundations of mathematical logic > and the concepts of 'identity', 'analyticity', 'synonymy' and > 'similarity'. > - > 'The Morning Star' and 'Venus', 'the victor at Jena' and 'the > vanquished at Waterloo', and even 'consciousness' and 'brain > processes' have all occupied philosophers concerned with the > relationship between Sense and Meaning, the 'is' of definition' > and the 'is' of composition, 'Use & Mention' over the past 100 > years. > - > Oedipus believed that Jocasta was fair game, but not his mother. > Ryle's foreigner saw all the buildings and grounds of Oxford, but > hadn't 'seen' the university. Jenny accepts that James has seen a > good range of her behaviour but doesn't believe he knows the real > 'her'. > - > Ryle offered a solution to part of this problem in 1949 with his > example of the category mistake, ie that 'mind' like 'the > university' is a concept or category which includes a number of > members, distributed elements, or dispositions to behave. (In > Quine we see these being noted as observation sentences, or more > specifically, as occasion sentences, dated and timed). > - > But there are a few problems here. If we say that these elements > fall under a particular class, the elements of that class are > themselves not identical, nor even similar in sense. It may well > be that amongst Jenny's dispositions we include 'cake maker' and > 'garden tender' as well as 'teacher', but we would rarely say > that these are identical or even similar behaviours any more than > we would say that the library is the same as or similar to the > refectory. They are similar or associated only in that they are > elements of a class, either 'Jenny' or 'university' - just as > different people can be said to be similar in terms of a higher > type, such as female, or human, or position in an actuarial > table. However, we would acknowledge that 'rabbit', 'undetached > part of rabbit', and 'stage of a rabbits development' are > interchangeable in identifying the same referent, 'salve > veritate'. This is the thesis of indeterminacy of translation, a > thesis which leads into another Quinean thesis, that of > Ontological Relativity - which dispenses with the intensional. > - >> What Quine actually says (see below) was each of us is free to >> internalize in his peculiar neural way ... apparently this ends up in >> your mind as dispenses with the intensional. I don't take >> dispenses with as being the same as free to ... for me there is a >> grand distinction. > You've missed the point, along with most folk here. This is a radical > failing on all of your part. It prevents you from grasping just how > profound the EAB revolution was. Skinner is up there with Darwin. Perhaps, if rule governed behavior is actually better suited to survival. But i rather suspect it is not ! > 'The view that I have come to, regarding intersubjective > likeness of stimulation, is rather that we can simply do > without it. The observation sentence 'Rabbit' has its > stimulus meaning for the linguist, and the observation > sentence 'Gavagai' has its stimulus meaning for the > informant. The linguist, observes natives assenting to > 'Gavagai' when he, in their position, would have > assented to 'Rabbit'. So he tries assigning HIS stimulus > meaning of 'Rabbit' to 'Gavagai' on subsequent occasions > for his informant's approval. Encouraged, he tentatively > adopts 'Rabbit' as translation.'......... > - > Discussion with Dreben helped me to clarify these > consequences of my new stance. In WORD AND OBJECT I had > already pointed out that communication presupposes no > similarity in nerve nets; verbal behavior is inculcated > on the strength only of surface stimulation. Such was my > parable of the trimmed bushes (p.8), alike in outward > form but wildly unlike in their inward twigs and > branches. Save the surface, in the paintmaker's words, > and you save all. But now, leaving the surface itself to > Sherwin-Williams' tender mercies, I give the individual > yet wider berth. His privacy widens apace. > - > Unlike Davidson, I leave the stimulations at the > subject's surface, and private stimulus meaning with > them. But they may be as idiosyncratic, for all I care, > as the subject's internal wiring itself. What floats in > the open air is our common language, which each of us is > free to internalize in his peculiar neural way. Language > is where intersubjectivity sets in. Communication is > well named.' > - > Quine (1990) > Three Indeterminacies > - >> In other words (see comment after Putman above) ... Meanings are not >> in the head; rather they are established by interpretive practice. >> The practice is public behavior which is not in the head. > Yes. I've covered this from a number of perspectives over the years. So you have, so you have. We get it. > There are pervasive problems with 'properties', 'essences', or > 'propositions' and their intensional kin. Nowhere is this more > apparent than with the 'propositional attitudes'. The linch-pin > is the resistance of these to the principle of 'substitutivity of > identity' 'salve veritate', and in anything scientific we are > exclusively interested in truth functions where the > substitutivity of identity is guaranteed. Failure to respect this > principle results in invalid reasoning through the fallacy of > equivocation. > - > In the case of propositional attitudes, we can not quote someone > indirectly without thereby uttering an untruth, and we can not > make statements about what someone believes, thinks, hopes, > fears, or understands except in the actual context in which the > propositional attitudes expressing such intensional idioms take > place (which amounts to quoting them directly and contextually). > If one reports to someone else 'what someone said', one has to > either directly quote (verbatim) or else acknowledge that one is > not making a report at all. Such paraphrase amounts to an > 'interpretation' which is an imputation or inference, a creative > act >> .. or the acknowledgment of a successful communication ... > . Propositional attitudes are not projectible outside the > immediate context of their utterance, so: - > - > 'Reports' are not 'bona fide' reports at all unless they > are reports verbatim (ie records of behaviour) - they > become interpretations through a process of imputation > at best, and in almost all cases, they are creative acts > which do not permit substitutivity of identity 'salve > veritate'. They are non-truth functional unless > expressed actuarially via relation to a population > distribution (normative psychometric measure). >> Incidentally i agree with Quine that for a journalistic report, direct >> quotes in context are strongly indicated. > That's not Quine, that's Longley. It's excerpted from another message of > the same era. This was an internal e-mail/newsgroup system which ran for > about 8 years from 87-96. You should look into what's happening in the > UK in that area. Look up evidence based practice and try to find out > what was done before the PROBE preoject. >> Paraphrase, however is useful as an acknowledgment of a successful >> communication and as a curative means to add additional interpretive >> practice. These become the very acts which create our language >> distinguishing it from bird calls! Emotionally i love it when someone >> *correctly* paraphrase me, for it is proof that i have communicated to >> them ... i no longer need to merely guess ... i have finally cast my >> bottle into the ocean of doubt and ambiguity and some other creature >> has discovered it :) When they can add a new idea not contained or >> imagined in my original, then i must leap for joy as the very purpose >> of my struggle with these multi-interpretive marks has finally become >> a step (however minor) in the cultural dance. So i must find Quine's >> judgmental comment, at best they are creative acts, to be sourly >> lacking in insight. > You miss the point. ... or i got the point and went beyond where you are prepared to go. In a dialog there is more than one person talking, that's the point of dialog. If that is something that makes you uncomfortable, may i suggest that a free blog would be a better forum for you. Everyone can shout at the world without listening in their own blog ... but Usenet is geared for dialog. patty === Subject: Re: Turing Machines and Physical Computation >I don't agree with the view that words are abstract. Doubtless, we >we not agree much at all on our views of language. There are whole discussions here. >>Abtraction may be (or may seem to be) a property of some entities, >>like concepts or propositions, but the subject of those entities may >>still be required to be physical and particular >I have trouble making sense of that. Why does an entity need a >subject? If the number three is an abstract entity, what is its >subject? The marks that constitute the word cow are an entity, but if there is no relationship between the marks and some distal subject/object/whatever, then we're nowhere. On the other hand, your question is fair, and it may be a result, and not a fair assumption, that a (linguistic, symbolic, cognitive, sub-) entity needs a subject. >Several years back I spent some time reading Hartry Field's Science >Without Numbers. I'm still not sure what was the point. I glanced at it at some point, I think. Whatever the point, I'm pretty sure it's nothing I want to support, or if there was some overlap with what I believe, it was at least not the way I'd want to state it. >It seems to me that nominalism is an example of the cure for which >there is no disease. It seems to me that nominalism is what we do when we do computation with modern programming and digital electronic computers. I'm not sure computers cure any disease, but they are useful nonetheless and in need of some explication to make them more useful yet. J. === Subject: Re: Turing Machines and Physical Computation -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 >>I don't agree with the view that words are abstract. Doubtless, we >>we not agree much at all on our views of language. >There are whole discussions here. >Abtraction may be (or may seem to be) a property of some entities, >like concepts or propositions, but the subject of those entities may >still be required to be physical and particular >>I have trouble making sense of that. Why does an entity need a >>subject? If the number three is an abstract entity, what is its >>subject? >The marks that constitute the word cow are an entity, but if there >is no relationship between the marks and some distal >subject/object/whatever, then we're nowhere. That's a pessimistic view. Sorry to bring the bad news, but there is no relationship. > On the other hand, your >question is fair, and it may be a result, and not a fair assumption, >that a (linguistic, symbolic, cognitive, sub-) entity needs a subject. A considerable amount of mathematics is done with symbols for which there is no subject. In fact, this is important, for it allows us to express general results. Later, we can interpret for particular subjects, but we can often prove the result without there being a subject. >>It seems to me that nominalism is an example of the cure for which >>there is no disease. >It seems to me that nominalism is what we do when we do computation >with modern programming and digital electronic computers. I can assure you that digital computers would work just as well if there were no nominalists around. > I'm not >sure computers cure any disease, but they are useful nonetheless and >in need of some explication to make them more useful yet. We can predict and control our computers to very high degrees of accuracy. The idea that we are lacking an explanation seems confused. If philosophy has difficulty accounting for computers, that only reflects on the inadequacies of philosophy. Perhaps the concept of computation seems elusive, but the computers themselves should present no problems. -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.3.91 (SunOS) iD8DBQFBqUgxvmGe70vHPUMRAkG8AKDeedTf6q4+GEx+aBZxPE5NTMnWMwCffiu5 1sqhpqoLHFkdM9hYPbvQu9U= =2DUT -----END PGP SIGNATURE----- === Subject: Re: Turing Machines and Physical Computation >I have trouble making sense of that. Why does an entity need a >subject? If the number three is an abstract entity, what is its >subject? >>The marks that constitute the word cow are an entity, but if there >>is no relationship between the marks and some distal >>subject/object/whatever, then we're nowhere. > That's a pessimistic view. Sorry to bring the bad news, but there is > no relationship. The relationship is established by the collective interpretive behavior of people. patty === Subject: Re: Turing Machines and Physical Computation -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 >The marks that constitute the word cow are an entity, but if there >is no relationship between the marks and some distal >subject/object/whatever, then we're nowhere. >> That's a pessimistic view. Sorry to bring the bad news, but there is >> no relationship. >The relationship is established by the collective interpretive behavior >of people. That's fine with me. That gives a relationship between people and the word cow. It also gives a relationship between people who are using the word cow and the distal object. However, there is no relationship between the word and the distal conclude that x is related to y. -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.3.91 (SunOS) iD8DBQFBqfQSvmGe70vHPUMRArHyAJ91VgkSZ9cra6xqf9u0t9IXGPeiRwCePPfU Q/gjmShb/XU085J/06RBaus= =wdkQ -----END PGP SIGNATURE----- === Subject: Re: Turing Machines and Physical Computation > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 >>The marks that constitute the word cow are an entity, but if there >>is no relationship between the marks and some distal >>subject/object/whatever, then we're nowhere. >That's a pessimistic view. Sorry to bring the bad news, but there is >no relationship. >>The relationship is established by the collective interpretive behavior >>of people. > That's fine with me. > That gives a relationship between people and the word cow. > It also gives a relationship between people who are using the word > cow and the distal object. > However, there is no relationship between the word and the distal > conclude that x is related to y. Ok, it's simpler to grok it as one three way relation, (mark, interpretive behavior of people, distal object); rather than as two tow-way relations. But i totally agree, there is no relationship between the mark and the distal object; rather it is an arbitrary assignment by people ... err else where did all those different languages come from. patty === Subject: Re: Turing Machines and Physical Computation >The marks that constitute the word cow are an entity, but if there >is no relationship between the marks and some distal >subject/object/whatever, then we're nowhere. >> That's a pessimistic view. Sorry to bring the bad news, but there is >> no relationship. >The relationship is established by the collective interpretive behavior >of people. > That's fine with me. > That gives a relationship between people and the word cow. > It also gives a relationship between people who are using the word > cow and the distal object. > However, there is no relationship between the word and the distal > conclude that x is related to y. That depends on the relationship, Neil. If the relationship is one of the relationships a occurs at the same times as x and a occurs at the same times as y one may conclude that x occurs at the same times as y, for example. Bill Modlin === Subject: Re: Turing Machines and Physical Computation iD8DBQFBqgT8vmGe70vHPUMRAnglAKDIFmoLZrdbfLNZefEbsEZMyf1X8QCeIOXw 3krL+PQxzfbmRXfb22x1zcM= =ts44 >>The marks that constitute the word cow are an entity, but if there >>is no relationship between the marks and some distal >>subject/object/whatever, then we're nowhere. > That's a pessimistic view. Sorry to bring the bad news, but there is > no relationship. >>The relationship is established by the collective interpretive behavior >>of people. >> That's fine with me. >> That gives a relationship between people and the word cow. >> It also gives a relationship between people who are using the word >> cow and the distal object. >> However, there is no relationship between the word and the distal >> conclude that x is related to y. > That depends on the relationship, Neil. Sure. But one cannot conclude that in general, and one cannot conclude that in the particular case raised by Josh. === Subject: Re: Turing Machines and Physical Computation >I have trouble making sense of that. Why does an entity need a >subject? If the number three is an abstract entity, what is its >subject? >>The marks that constitute the word cow are an entity, but if there >>is no relationship between the marks and some distal >>subject/object/whatever, then we're nowhere. >That's a pessimistic view. Sorry to bring the bad news, but there is >no relationship. I don't see how you can say there is no relationship, but don't even bother trying to explain. OK, I passed on the hard question, what is the subject of the number three? Well, lots of smart people have expended a lot of hot air on that topic. Is there a three-ness, abstract or concrete, which is the subject of the symbol 3 as it is commonly used? I don't know. Is there a cow-ness, abstract or concrete? Can't really say. Something that I know I've not emphasized, is that there is no necessity for individual symbols to have fixed meanings such that their combination is directly generative. Individual bits tend to lack such meanings, as do individual letters, although both in aggregate tend to acquire meaning in larger granularities. Is the 3 in the number 32 the same as when it stands alone? I'm not sure such questions are valid. But, before a mark constitutes anything interesting, it must have a subject, I stand by that. >> On the other hand, your >>question is fair, and it may be a result, and not a fair assumption, >>that a (linguistic, symbolic, cognitive, sub-) entity needs a subject. >A considerable amount of mathematics is done with symbols for which >there is no subject. In fact, this is important, for it allows us to >express general results. Later, we can interpret for particular >subjects, but we can often prove the result without there being a >subject. I want to dogmatically deny that. Now, let's see if I can. Some genius works out in the abstract that 2+2=4. Has he done this without a subject, in such a way that a subject can later be recognized? I suggest not. I suggest what has been done is an exercise in the essentially physical manipulation of symbols, with a physically particular answer. This is sufficient for me to maintain my principles, though I can see some will consider it tendentious. Later, additional correspondences may be added to it. But I can do the same thing in writing about green dragons. Mathematics has no priority in any of this. Zip. Zero. Nada. >It seems to me that nominalism is an example of the cure for which >there is no disease. >>It seems to me that nominalism is what we do when we do computation >>with modern programming and digital electronic computers. >I can assure you that digital computers would work just as well if >there were no nominalists around. Sure, that's the beauty of reality, they work just as well without any electrical engineers around, too, but they work by such electrical principles all the same. >> I'm not >>sure computers cure any disease, but they are useful nonetheless and >>in need of some explication to make them more useful yet. >We can predict and control our computers to very high degrees of >accuracy. The idea that we are lacking an explanation seems >confused. If philosophy has difficulty accounting for computers, >that only reflects on the inadequacies of philosophy. Show me a computational AI at work, and I'll grant your point. >Perhaps the concept of computation seems elusive, but the >computers themselves should present no problems. I've used the metaphor before, but people made fire for years without really understanding heat and energy and oxidation and such, but if you want to build fancy, complex systems, you really do need such science. We're at the rubbing two sticks together stage still in computation, and I look to philosophy as the missing element - something that was *not* the case for combustion, except so far as natural philosophy had not yet separated out of the general philosophical background. In the area of computation, language, and cognition, I think the discipline will always be more philosophical than deductive. Tune in in fifty years and see if I'm right. J. === Subject: Re: Turing Machines and Physical Computation -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 >OK, I passed on the hard question, what is the subject of the number >three? Well, lots of smart people have expended a lot of hot air on >that topic. Is there a three-ness, abstract or concrete, which is the >subject of the symbol 3 as it is commonly used? I don't know. I would think that a three-ness, if you believe in such things, should count as an essence. Yet you have stated that you reject essentialism. > Individual bits tend to lack such meanings, >as do individual letters, although both in aggregate tend to acquire >meaning in larger granularities. Is the meaning an attribute of the symbol, or of the user of the symbol? > Is the 3 in the number 32 the same >as when it stands alone? The numeral 3 in 32 is the same as the numeral 3 used to represent the number 3. But this is why people make the distinction between numerals and numbers. Perhaps we could say that the number 3 is the subject of the numeral 3 when that numeral is used in certain ways. But it wouldn't tell us what is the subject of the number 3. I suppose you could say that the number 3 is its own subject. But this seems to require something like platonism. As a nominalist, I had presumed that you denied that there were numbers, and allowed only numerals. Personally, I allow the number 3 to exist, but only as a convenient fiction. This is weaker than platonism, but perhaps a little stronger than what I took to be the nominalist view. > I'm not sure such questions are valid. But, >before a mark constitutes anything interesting, it must have a >subject, I stand by that. The subject does not need to be a property of the mark. Rather, the interested agent can ascribe a meaning, and be interested in the mark because of its usefulness to carry that meaning. But the meaning isn't really a subject of the mark. >>A considerable amount of mathematics is done with symbols for which >>there is no subject. In fact, this is important, for it allows us to >>express general results. Later, we can interpret for particular >>subjects, but we can often prove the result without there being a >>subject. >I want to dogmatically deny that. Now, let's see if I can. Some >genius works out in the abstract that 2+2=4. That's a poor example. Instead, consider the quadratic formula, [ -b +- sqrt(b^2 -4ac)] / 2a Here the symbols a, b and c have no subject, and the usefulness of the formula (i.e. its generality) depends on them having no subject. >>We can predict and control our computers to very high degrees of >>accuracy. The idea that we are lacking an explanation seems >>confused. If philosophy has difficulty accounting for computers, >>that only reflects on the inadequacies of philosophy. >Show me a computational AI at work, and I'll grant your point. That seems to be non-relevant. I agree that we are lacking an explanation of intelligence. And a working computational AI system might help fill that gap. But the lack of explanation is not in either computation or in computers. >>Perhaps the concept of computation seems elusive, but the >>computers themselves should present no problems. >I've used the metaphor before, but people made fire for years without >really understanding heat and energy and oxidation and such, but if >you want to build fancy, complex systems, you really do need such >science. However, people could neither control nor predict fire very well. Now, with our better understanding of heat, energy, oxidation, etc, we can do far better at prediction and control. I don't think that a useful analogy for computation. > We're at the rubbing two sticks together stage still in >computation, and I look to philosophy as the missing element - Sorry, but I disagree. > In the area of computation, language, and >cognition, I think the discipline will always be more philosophical >than deductive. Tune in in fifty years and see if I'm right. You might be right about language and cognition. But you are wrong about computation. -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.3.91 (SunOS) iD8DBQFBqh3rvmGe70vHPUMRAhgTAKDCmRq9mQ8yGkdC50R1kj7oYc2FSgCgnpDG +AlUbm1gkP7iPdHLmC/zJ1g= =j4Lw -----END PGP SIGNATURE----- === Subject: Re: Turing Machines and Physical Computation >>OK, I passed on the hard question, what is the subject of the number >>three? Well, lots of smart people have expended a lot of hot air on >>that topic. Is there a three-ness, abstract or concrete, which is the >>subject of the symbol 3 as it is commonly used? I don't know. >I would think that a three-ness, if you believe in such things, >should count as an essence. Yet you have stated that you reject >essentialism. Well, I don't know that it's an essence, but the odds are it is, and that I would reject it, if I had an opinion, but the point is I don't think I have to care. >But this seems to require something like platonism. As a nominalist, >I had presumed that you denied that there were numbers, and allowed >only numerals. Personally, I allow the number 3 to exist, but only >as a convenient fiction. This is weaker than platonism, but perhaps >a little stronger than what I took to be the nominalist view. In *my* nominalism, convenient fictions abound, at least as strings. I suppose, strictly speaking, no fictions as such abound, nor any truths. >>I want to dogmatically deny that. Now, let's see if I can. Some >>genius works out in the abstract that 2+2=4. >That's a poor example. Instead, consider the quadratic formula, > [ -b +- sqrt(b^2 -4ac)] / 2a >Here the symbols a, b and c have no subject, and the usefulness >of the formula (i.e. its generality) depends on them having no >subject. No subject, no use, no interest, and your generality is an illusion. If it hadn't already been proven particularly, nobody would pretend to the generality. I suggest what you see here is *repeatability*, not generality, and that is a very different thing, pretty nearly sui generis to computation. >We can predict and control our computers to very high degrees of >accuracy. The idea that we are lacking an explanation seems >confused. If philosophy has difficulty accounting for computers, >that only reflects on the inadequacies of philosophy. >>Show me a computational AI at work, and I'll grant your point. >That seems to be non-relevant. I agree that we are lacking an >explanation of intelligence. And a working computational AI system >might help fill that gap. But the lack of explanation is not in >either computation or in computers. The putative difficulty in writing effective software is a more common and mundane version of the same thing, IMHO. It's an interesting point that, after thirty years of moaning about the difficulty and cost of writing software, you tend not to hear that complaint anymore. Perhaps now that it's all done in Bangalore, nobody in the states much cares. More likely, I think there is an actual change in opinion these days among those who commission and use software, they have grown to expect it to be crufty and that they will have to live with various imperfections. I count this as a new maturity, if true, with theoretical implications - partial, arbitrary systems are now seen, properly, as what computation is all about. Comments? >Perhaps the concept of computation seems elusive, but the >computers themselves should present no problems. >>I've used the metaphor before, but people made fire for years without >>really understanding heat and energy and oxidation and such, but if >>you want to build fancy, complex systems, you really do need such >>science. >However, people could neither control nor predict fire very well. Enough that people counted themselves better than chimps. >I don't think that a useful analogy for computation. >> We're at the rubbing two sticks together stage still in >>computation, and I look to philosophy as the missing element - >Sorry, but I disagree. >> In the area of computation, language, and >>cognition, I think the discipline will always be more philosophical >>than deductive. Tune in in fifty years and see if I'm right. >You might be right about language and cognition. But you are wrong >about computation. Two out of three ain't bad, but I'm still hopeful. J. === Subject: Re: Turing Machines and Physical Computation -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 >>That's a poor example. Instead, consider the quadratic formula, >> [ -b +- sqrt(b^2 -4ac)] / 2a >>Here the symbols a, b and c have no subject, and the usefulness >>of the formula (i.e. its generality) depends on them having no >>subject. >No subject, no use, no interest, and your generality is an illusion. >If it hadn't already been proven particularly, nobody would pretend to >the generality. I suggest what you see here is *repeatability*, not >generality, and that is a very different thing, pretty nearly sui >generis to computation. We clearly disagree here. That may have something to do with why we disagree about computation and about mathematics. >Show me a computational AI at work, and I'll grant your point. >>That seems to be non-relevant. I agree that we are lacking an >>explanation of intelligence. And a working computational AI system >>might help fill that gap. But the lack of explanation is not in >>either computation or in computers. >The putative difficulty in writing effective software is a more common >and mundane version of the same thing, IMHO. That's actually a quite different problem. Computation is the manipulation of representations. The difficulty in writing effective software, is because we don't start with representations. Rather, we start with a real world problem of some kind. Thus we must first find a way to reduce the problem to one of manipulating representations. Until we have done that, it is not a computational problem. >> The wider phenomenon is that we >>can have linguistic, quantitative, cognitive processes of distal >>objects at all, and that is quite marvelous, but it seems a true >>property of the universe, and I'm not sure you can say much more about >>it than that. >But surely this is what philosophy should be investigating and >explaining. Instead, it is all taken for granted, and used as the >basis for the creative fiction that is philosophy. You dismissed this. But it is where the problem lies. Finding useful ways of representing distal objects is hard work. It is the difficulty of solving this problem that makes it hard to write effective software. >It's an interesting point that, after thirty years of moaning about >the difficulty and cost of writing software, you tend not to hear that >complaint anymore. Perhaps now that it's all done in Bangalore, >nobody in the states much cares. No, it is not that at all. It seems that the idea of software as the mechanical solving of problems (i.e. automation) has gone the way of the dodo. These days, software is all about writing GUI interfaces and other kinds of visual candy, so that we can keep people amused as they do the work that we are unable to automate. And perhaps automation has become less valuable, now that we can outsource the labor-intensive work to other places. -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.3.91 (SunOS) iD8DBQFBqqbuvmGe70vHPUMRAtrfAJwPGmFncpu1NspHNy8X49R2dPm1EACbBtc9 QKoef8ZB1VwyfdVt0Cd824w= =oXZy -----END PGP SIGNATURE----- === Subject: Re: Turing Machines and Physical Computation >The putative difficulty in writing effective software is a more common >and mundane version of the same thing, IMHO. > That's actually a quite different problem. > Computation is the manipulation of representations. I am not sure if that's a good description. This formal symbol manipulation idea got some otherwise ambitious philosophers like Brian Cantwell Smith and gang quite confused. Computers can work on representations, that's true. But it is not necessary that what is being manipulated is representation. -- Eray === Subject: Re: Turing Machines and Physical Computation -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 >> Computation is the manipulation of representations. >I am not sure if that's a good description. >This formal symbol manipulation idea got some otherwise ambitious >philosophers like Brian Cantwell Smith and gang quite confused. >Computers can work on representations, that's true. But it is not >necessary that what is being manipulated is representation. It can be an abstract representation -- that is, one that doesn't actually represent anything. ------- The received view of AI is something like: A system of sensors that generates representation in some fixed manner. A manipulation or transformation of these input representation into output representations. A system of effectors that, in fixed way, generates physical actions (behaviors) from the output representations. -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.3.91 (SunOS) iD8DBQFBrOFrvmGe70vHPUMRAserAJ9Bk5z1K2lkqZ7ThW597I6umxD2jgCgmzad +0Xgga+8Am5wYTUFVvDfBH8= =lLCX -----END PGP SIGNATURE----- === Subject: Re: Turing Machines and Physical Computation > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 >> Computation is the manipulation of representations. >I am not sure if that's a good description. >This formal symbol manipulation idea got some otherwise ambitious >philosophers like Brian Cantwell Smith and gang quite confused. >Computers can work on representations, that's true. But it is not >necessary that what is being manipulated is representation. > It can be an abstract representation -- that is, one that doesn't > actually represent anything. > ------- > The received view of AI is something like: > A system of sensors that generates representation in some fixed > manner. > A manipulation or transformation of these input representation into > output representations. > A system of effectors that, in fixed way, generates physical > actions (behaviors) from the output representations. We were talking about computation in general, not what perception or action in the world requires. -- Eray Ozkural === Subject: Re: Turing Machines and Physical Computation >The putative difficulty in writing effective software is a more common >and mundane version of the same thing, IMHO. >>That's actually a quite different problem. >>Computation is the manipulation of representations. > I am not sure if that's a good description. > This formal symbol manipulation idea got some otherwise ambitious > philosophers like Brian Cantwell Smith and gang quite confused. > Computers can work on representations, that's true. But it is not > necessary that what is being manipulated is representation. Can you give us an example of where a computer is *not* manipulating a representation ? patty === Subject: Re: Turing Machines and Physical Computation > >The putative difficulty in writing effective software is a more common >and mundane version of the same thing, IMHO. >That's actually a quite different problem. >Computation is the manipulation of representations. > > > I am not sure if that's a good description. > > This formal symbol manipulation idea got some otherwise ambitious > philosophers like Brian Cantwell Smith and gang quite confused. > > Computers can work on representations, that's true. But it is not > necessary that what is being manipulated is representation. > > Can you give us an example of where a computer is *not* manipulating a > representation ? Consider the generator of {0,1}* -- Eray === Subject: Re: Turing Machines and Physical Computation >The putative difficulty in writing effective software is a more common >and mundane version of the same thing, IMHO. >That's actually a quite different problem. >Computation is the manipulation of representations. > I am not sure if that's a good description. > This formal symbol manipulation idea got some otherwise ambitious > philosophers like Brian Cantwell Smith and gang quite confused. > Computers can work on representations, that's true. But it is not > necessary that what is being manipulated is representation. > Can you give us an example of where a computer is *not* manipulating a > representation ? > patty Patty, weren't you the one who suggested that representation is only meaningful in the sense of representing something to an interpreting observer? The machine goes through a sequence of changes in state. Normally the people who program it think of some elements of those states as representing something... numbers, characters, cars, hurricanes, chess pieces, whatever. But there may be ambiguity about what a particular element represents, depending on ones point of view. There is nothing about for example the state of a memory location that makes it inherently a representation of any particular type of thing. And if a computer were to be executing some arbitrary (random, or of unknown origin) sequence of operations, which was not a representation of anything particular to any existing observer, I'm not sure in what sense you could say that it was manipulating a representation. It is just going through a sequence of causally dependent material state changes. Of course, one could take the position that such a sequence is no longer a computation. One could say that computing *means* manipulating representations. Which might be a good thing, since it explains why we are uncomfortable saying that a projectile or a planet computes its trajectory. We only ascribe computation to physical processes that we take as representing other things. And with representation dependendent on an interpreting observer, all arguments about whether there really *are* computations and representations in the brain or anything else become moot... it all depends on whether you choose to so interpret them. Bill === Subject: Re: Turing Machines and Physical Computation >The putative difficulty in writing effective software is a more common >and mundane version of the same thing, IMHO. >That's actually a quite different problem. >Computation is the manipulation of representations. >I am not sure if that's a good description. >This formal symbol manipulation idea got some otherwise ambitious >philosophers like Brian Cantwell Smith and gang quite confused. >Computers can work on representations, that's true. But it is not >necessary that what is being manipulated is representation. >>Can you give us an example of where a computer is *not* manipulating a >>representation ? >>patty > Patty, weren't you the one who suggested that representation is only > meaningful in the sense of representing something to an interpreting > observer? Yep that was me :) But i think the complete relationship should be a four place predicate like: (represents x P x' P') where some agent (A) interprets that x' in the process P' represents x in the process P. The agent should also establish that there is some causative relationship between x and x'; and that there is some measure in which x stands to P as x' stands to P'. I think most natural language understandings of represents can be coded in that form. Let's say x is the number of dollars in a purchase transaction (P). x' is that number in a computer memory, and P' might be a program simulating the transaction, written by the programmer sitting as the interpreting agent (A). All the slots in the predicate are filled, right? But there is no reason that P' cannot *be* the interpretive agent (A). Let say that x is a coffee cup, P is my desk, x' is the activity in my skull directly caused by seeing the cup, and P' is other activity in my head conditioned by my desk, and i am interpreting my perception of the coffee cup to *be* the coffee cup. Same predicate, all the slots are filled; but A == P', right? I think the question is how can the running computer become the interpretive agent; and when we solve that we might be closer to creating AI. In other words: what is interpretive behavior? > The machine goes through a sequence of changes in state. > Normally the people who program it think of some elements of those states as > representing something... numbers, characters, cars, hurricanes, chess > pieces, whatever. But there may be ambiguity about what a particular > element represents, depending on ones point of view. There is nothing about > for example the state of a memory location that makes it inherently a > representation of any particular type of thing. And if a computer were to > be executing some arbitrary (random, or of unknown origin) sequence of > operations, which was not a representation of anything particular to any > existing observer, I'm not sure in what sense you could say that it was > manipulating a representation. It is just going through a sequence of > causally dependent material state changes. Well you are skirting very close to protoplasmic chauvinism. If the computer is functioning without a programmer, then we have the same case as me and my coffee cup. If it walks and talks like a duck, we should call it a duck. > Of course, one could take the position that such a sequence is no longer a > computation. One could say that computing *means* manipulating > representations. Which might be a good thing, since it explains why we are > uncomfortable saying that a projectile or a planet computes its > trajectory. We only ascribe computation to physical processes that we > take as representing other things. And with representation dependendent > on an interpreting observer, all arguments about whether there really *are* > computations and representations in the brain or anything else become > moot... it all depends on whether you choose to so interpret them. I don't understand this whole drive to make the word computation means something other than just some kind of normitive activity. There is the activity of a planet going around the sun. We can describe that activity however, and we can interpret that activity to stand for something else, and the latter represent the former. I fail to conceive how the planet itself going around the sun could ever participate in the interpretive behavior. What does bringing in this computation thingy add to our predication of representations or help us understand interpretive behavior? I don't get it. patty === Subject: Re: Turing Machines and Physical Computation <5uniq0trsetjjkql36v7vv8m4bhalsurd6@4ax.com> <44ckq09b9mb3ehign5vac2lgi6fqet27ap@4ax.com> >The putative difficulty in writing effective software is a more >>common >>and mundane version of the same thing, IMHO. >That's actually a quite different problem. >Computation is the manipulation of representations. >>I am not sure if that's a good description. >This formal symbol manipulation idea got some otherwise ambitious >>philosophers like Brian Cantwell Smith and gang quite confused. >Computers can work on representations, that's true. But it is not >>necessary that what is being manipulated is representation. >>Can you give us an example of where a computer is *not* manipulating a >representation ? >patty >> Patty, weren't you the one who suggested that representation is >>only >> meaningful in the sense of representing something to an interpreting >> observer? >Yep that was me :) But i think the complete relationship should be a >four place predicate like: (represents x P x' P') where some agent (A) >interprets that x' in the process P' represents x in the process P. The >agent should also establish that there is some causative relationship >between x and x'; and that there is some measure in which x stands to P >as x' stands to P'. I think most natural language understandings of >represents can be coded in that form. >Let's say x is the number of dollars in a purchase transaction (P). x' >is that number in a computer memory, and P' might be a program >simulating the transaction, written by the programmer sitting as the >interpreting agent (A). All the slots in the predicate are filled, >right? >But there is no reason that P' cannot *be* the interpretive agent (A). >Let say that x is a coffee cup, P is my desk, x' is the activity in my >skull directly caused by seeing the cup, and P' is other activity in >my head conditioned by my desk, and i am interpreting my perception of >the coffee cup to *be* the coffee cup. Same predicate, all the slots >are filled; but A == P', right? >I think the question is how can the running computer become the >interpretive agent; and when we solve that we might be closer to >creating AI. In other words: what is interpretive behavior? >> The machine goes through a sequence of changes in state. >> Normally the people who program it think of some elements of those states as >> representing something... numbers, characters, cars, hurricanes, chess >> pieces, whatever. But there may be ambiguity about what a particular >> element represents, depending on ones point of view. There is nothing about >> for example the state of a memory location that makes it inherently a >> representation of any particular type of thing. And if a computer were to >> be executing some arbitrary (random, or of unknown origin) sequence of >> operations, which was not a representation of anything particular to any >> existing observer, I'm not sure in what sense you could say that it was >> manipulating a representation. It is just going through a sequence of >> causally dependent material state changes. >Well you are skirting very close to protoplasmic chauvinism. If the >computer is functioning without a programmer, then we have the same >case as me and my coffee cup. If it walks and talks like a duck, we >should call it a duck. >> Of course, one could take the position that such a sequence is no longer a >> computation. One could say that computing *means* manipulating >> representations. Which might be a good thing, since it explains why we are >> uncomfortable saying that a projectile or a planet computes its >> trajectory. We only ascribe computation to physical processes that we >> take as representing other things. And with representation dependendent >> on an interpreting observer, all arguments about whether there really *are* >> computations and representations in the brain or anything else become >> moot... it all depends on whether you choose to so interpret them. >I don't understand this whole drive to make the word computation >means something other than just some kind of normitive activity. There >is the activity of a planet going around the sun. We can describe that >activity however, and we can interpret that activity to stand for >something else, and the latter represent the former. I fail to >conceive how the planet itself going around the sun could ever >participate in the interpretive behavior. What does bringing in this >computation thingy add to our predication of representations or help us >understand interpretive behavior? I don't get it. >patty I predict that all you will see (are seeing) is a progressive (albeit necessarily gradual) change in (natural) language where the intensional idioms of propositional attitude etc are replaced by extensional constructions. As you can't see this, and how it's done (or why it's being done) you keep arguing. That's just because you don't see what you are doing yourself (cf. awareness of your own behaviour). Learn a little more instead of arguing. -- David Longley http://www.longley.demon.co.uk === Subject: Re: Turing Machines and Physical Computation > The putative difficulty in writing effective software is a more > common > and mundane version of the same thing, IMHO. >> That's actually a quite different problem. > Computation is the manipulation of representations. I am not sure if that's a good description. > This formal symbol manipulation idea got some otherwise ambitious > philosophers like Brian Cantwell Smith and gang quite confused. > Computers can work on representations, that's true. But it is not > necessary that what is being manipulated is representation. > Can you give us an example of where a computer is *not* manipulating a >> representation ? > patty > Patty, weren't you the one who suggested that representation is only > meaningful in the sense of representing something to an interpreting > observer? >> Yep that was me :) But i think the complete relationship should be a >> four place predicate like: (represents x P x' P') where some agent (A) >> interprets that x' in the process P' represents x in the process P. >> The agent should also establish that there is some causative >> relationship between x and x'; and that there is some measure in which >> x stands to P as x' stands to P'. I think most natural language >> understandings of represents can be coded in that form. >> Let's say x is the number of dollars in a purchase transaction (P). >> x' is that number in a computer memory, and P' might be a program >> simulating the transaction, written by the programmer sitting as the >> interpreting agent (A). All the slots in the predicate are filled, >> right? >> But there is no reason that P' cannot *be* the interpretive agent (A). >> Let say that x is a coffee cup, P is my desk, x' is the activity in my >> skull directly caused by seeing the cup, and P' is other activity in >> my head conditioned by my desk, and i am interpreting my perception of >> the coffee cup to *be* the coffee cup. Same predicate, all the slots >> are filled; but A == P', right? >> I think the question is how can the running computer become the >> interpretive agent; and when we solve that we might be closer to >> creating AI. In other words: what is interpretive behavior? > The machine goes through a sequence of changes in state. > Normally the people who program it think of some elements of those > states as > representing something... numbers, characters, cars, hurricanes, chess > pieces, whatever. But there may be ambiguity about what a particular > element represents, depending on ones point of view. There is > nothing about > for example the state of a memory location that makes it inherently a > representation of any particular type of thing. And if a computer > were to > be executing some arbitrary (random, or of unknown origin) sequence of > operations, which was not a representation of anything particular to > any > existing observer, I'm not sure in what sense you could say that it was > manipulating a representation. It is just going through a sequence of > causally dependent material state changes. >> Well you are skirting very close to protoplasmic chauvinism. If the >> computer is functioning without a programmer, then we have the same >> case as me and my coffee cup. If it walks and talks like a duck, we >> should call it a duck. > Of course, one could take the position that such a sequence is no > longer a > computation. One could say that computing *means* manipulating > representations. Which might be a good thing, since it explains why > we are > uncomfortable saying that a projectile or a planet computes its > trajectory. We only ascribe computation to physical processes > that we > take as representing other things. And with representation > dependendent > on an interpreting observer, all arguments about whether there really > *are* > computations and representations in the brain or anything else become > moot... it all depends on whether you choose to so interpret them. >> I don't understand this whole drive to make the word computation >> means something other than just some kind of normitive activity. There >> is the activity of a planet going around the sun. We can describe >> that activity however, and we can interpret that activity to stand for >> something else, and the latter represent the former. I fail to >> conceive how the planet itself going around the sun could ever >> participate in the interpretive behavior. What does bringing in this >> computation thingy add to our predication of representations or help >> us understand interpretive behavior? I don't get it. >> patty > I predict that all you will see (are seeing) is a progressive (albeit > necessarily gradual) change in (natural) language where the > intensional idioms of propositional attitude etc are replaced by > extensional constructions. Hmm ... let me guess ... and this change will happen *because* extensional constructions are more reliable, they allow us to better predict and control behavior. Did i get it right? <-not rhetorical ... please answer. patty === Subject: Re: Turing Machines and Physical Computation <5uniq0trsetjjkql36v7vv8m4bhalsurd6@4ax.com> <44ckq09b9mb3ehign5vac2lgi6fqet27ap@4ax.com> > The putative difficulty in writing effective software is a more > and mundane version of the same thing, IMHO. That's actually a quite different problem. > Computation is the manipulation of representations. > I am not sure if that's a good description. > This formal symbol manipulation idea got some otherwise ambitious >> philosophers like Brian Cantwell Smith and gang quite confused. > Computers can work on representations, that's true. But it is not >> necessary that what is being manipulated is representation. > Can you give us an example of where a computer is *not* manipulating a > representation ? > patty > Patty, weren't you the one who suggested that representation is only >> meaningful in the sense of representing something to an interpreting >> observer? > Yep that was me :) But i think the complete relationship should be >a four place predicate like: (represents x P x' P') where some agent >(A) interprets that x' in the process P' represents x in the process >P. The agent should also establish that there is some causative >relationship between x and x'; and that there is some measure in >which x stands to P as x' stands to P'. I think most natural >language understandings of represents can be coded in that form. > Let's say x is the number of dollars in a purchase transaction (P). >x' is that number in a computer memory, and P' might be a program >simulating the transaction, written by the programmer sitting as the >interpreting agent (A). All the slots in the predicate are filled, right? > But there is no reason that P' cannot *be* the interpretive agent >(A). Let say that x is a coffee cup, P is my desk, x' is the >activity in my skull directly caused by seeing the cup, and P' is >other activity in my head conditioned by my desk, and i am >interpreting my perception of the coffee cup to *be* the coffee cup. >Same predicate, all the slots are filled; but A == P', right? > I think the question is how can the running computer become the >interpretive agent; and when we solve that we might be closer to >creating AI. In other words: what is interpretive behavior? >> The machine goes through a sequence of changes in state. >> Normally the people who program it think of some elements of those >>states as >> representing something... numbers, characters, cars, hurricanes, chess >> pieces, whatever. But there may be ambiguity about what a particular >> element represents, depending on ones point of view. There is >>nothing about >> for example the state of a memory location that makes it inherently a >> representation of any particular type of thing. And if a computer >>were to >> be executing some arbitrary (random, or of unknown origin) sequence of >> operations, which was not a representation of anything particular >>to any >> existing observer, I'm not sure in what sense you could say that it was >> manipulating a representation. It is just going through a sequence of >> causally dependent material state changes. > Well you are skirting very close to protoplasmic chauvinism. If the >computer is functioning without a programmer, then we have the same >case as me and my coffee cup. If it walks and talks like a duck, we >should call it a duck. >> Of course, one could take the position that such a sequence is no >>longer a >> computation. One could say that computing *means* manipulating >> representations. Which might be a good thing, since it explains >>why we are >> uncomfortable saying that a projectile or a planet computes its >> trajectory. We only ascribe computation to physical processes >>that we >> take as representing other things. And with representation >>dependendent >> on an interpreting observer, all arguments about whether there >>really *are* >> computations and representations in the brain or anything else become >> moot... it all depends on whether you choose to so interpret them. > I don't understand this whole drive to make the word computation >means something other than just some kind of normitive activity. >There is the activity of a planet going around the sun. We can >describe that activity however, and we can interpret that activity >to stand for something else, and the latter represent the former. I >fail to conceive how the planet itself going around the sun could >ever participate in the interpretive behavior. What does bringing >in this computation thingy add to our predication of representations >interpretive behavior? I don't get it. > patty >> I predict that all you will see (are seeing) is a progressive >>(albeit necessarily gradual) change in (natural) language where the >>intensional idioms of propositional attitude etc are replaced by >>extensional constructions. >Hmm ... let me guess ... and this change will happen *because* >extensional constructions are more reliable, they allow us to better >predict and control behavior. Did i get it right? <-not rhetorical >... please answer. >patty In a word - yes. Manage might be more acceptable to some, Some might quibble about it being behaviour, but if they follow it through, that's what it basically comes down to - ie prediction of stimulation of our sensory surfaces. -- David Longley === Subject: Re: Turing Machines and Physical Computation [. . .] >> I predict that all you will see (are seeing) is a progressive (albeit >> necessarily gradual) change in (natural) language where the >> intensional idioms of propositional attitude etc are replaced by >> extensional constructions. >Hmm ... let me guess ... and this change will happen *because* >extensional constructions are more reliable, they allow us to better >predict and control behavior. Did i get it right? <-not rhetorical ... >please answer. Behaviorists don't predict and control behavior. They predict and control training. Crucial distinction that David fails to grasp. === Subject: Re: Turing Machines and Physical Computation >The putative difficulty in writing effective software is a more common >and mundane version of the same thing, IMHO. >That's actually a quite different problem. >Computation is the manipulation of representations. >I am not sure if that's a good description. >This formal symbol manipulation idea got some otherwise ambitious >philosophers like Brian Cantwell Smith and gang quite confused. >Computers can work on representations, that's true. But it is not >necessary that what is being manipulated is representation. >>Can you give us an example of where a computer is *not* manipulating a >>representation ? >>patty > Patty, weren't you the one who suggested that representation is only > meaningful in the sense of representing something to an interpreting > observer? The machine goes through a sequence of changes in state. > Normally the people who program it think of some elements of those states as > representing something... numbers, characters, cars, hurricanes, chess > pieces, whatever. But there may be ambiguity about what a particular > element represents, depending on ones point of view. There is nothing about > for example the state of a memory location that makes it inherently a > representation of any particular type of thing. And if a computer were to > be executing some arbitrary (random, or of unknown origin) sequence of > operations, which was not a representation of anything particular to any > existing observer, I'm not sure in what sense you could say that it was > manipulating a representation. It is just going through a sequence of > causally dependent material state changes. You still may be left with the fact that it is manipulating symbols. Symbols without semantic reference, but (ambiguous) symbols nonetheless. I agree with the spirit of your post, though, and feel that people who quibble with it don't really understand the guts of computers. They think there's something magical going on, something more than physics and materials science. The question to me is whether the same applies to the human mind, where most of the AI questions deserve to be aimed squarely at: what is it that we insist is unique about our own processes? Just how did we see fit to separate ourselves from the remainder of the animal kingdom, and natural processes as a whole? Why do we concieve of our own mentality as any different -- except in scale and physical substrate? > Of course, one could take the position that such a sequence is no longer a > computation. One could say that computing *means* manipulating > representations. Which might be a good thing, since it explains why we are > uncomfortable saying that a projectile or a planet computes its > trajectory. We only ascribe computation to physical processes that we > take as representing other things. And with representation dependendent > on an interpreting observer, all arguments about whether there really *are* > computations and representations in the brain or anything else become > moot... it all depends on whether you choose to so interpret them. My little thought experiment is this: is is possible to concieve of a computer existing independently of humans? For example, a pile of sticks, pebbles on a beach, falling rocks, waves, etc. which are computing autonomously (not derived from human engineering)? If not, then I submit that the argument is couched in anthroporphism. If so, but the answer is Only sentient species. For example, an intelligent alien, non-human, species can concievably build bona fide computing machinery. then I wonder whether there's some other chauvenism at play there. And so, the question is already begged: inorganic substrated intelligence cannot possibly exist independently of biologically-derived mechanisms which provide the impetus. So there's no point in trying to build AI: the decision is already made. Indeed, the machinery has been damned. Came from a human or intelligent species? Then it cannot possibly be, itself, intelligent. I just don't like where that argument goes. It's based on some sort of dualism to be sure. For one thing, humans themselves -- as well as our machinery -- are human derived. Both physically (egg+sperm) and culturally/linguistically/intellectually. === Subject: Re: Turing Machines and Physical Computation >We clearly disagree here. That may have something to do with why >we disagree about computation and about mathematics. I'm clearly making tendentious statements that could use a whole lot more support than I'm giving them here. Discussed at great length, I suspect we would agree more than it might seem. Wish I had time for it, but am currently working a job at the end of a long commute. >That's actually a quite different problem. >Computation is the manipulation of representations. The difficulty >in writing effective software, is because we don't start with >representations. Rather, we start with a real world problem of some >kind. Thus we must first find a way to reduce the problem to one of >manipulating representations. Until we have done that, it is not a >computational problem. But I want to view computation ONLY as this wider problem, and sporadically at least, so do you! And, I want to treat software development as at least a rough equivalent to the issues of computation. >No, it is not that at all. It seems that the idea of software as the >mechanical solving of problems (i.e. automation) has gone the way of >the dodo. These days, software is all about writing GUI interfaces >and other kinds of visual candy, so that we can keep people amused as >they do the work that we are unable to automate. >And perhaps automation has become less valuable, now that we can >outsource the labor-intensive work to other places. Gee I wish I said that ... J. === Subject: Re: Turing Machines and Physical Computation -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 >>Computation is the manipulation of representations. The difficulty >>in writing effective software, is because we don't start with >>representations. Rather, we start with a real world problem of some >>kind. Thus we must first find a way to reduce the problem to one of >>manipulating representations. Until we have done that, it is not a >>computational problem. >But I want to view computation ONLY as this wider problem, and >sporadically at least, so do you! The problem is far wider than you suspect -- too far off course to be considered computation. > And, I want to treat software >development as at least a rough equivalent to the issues of >computation. There are many social issues that must be considered in software development. It isn't just computation. As an example, consider the social engineering methods used in distributing computer viruses. -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.3.91 (SunOS) iD8DBQFBrAVrvmGe70vHPUMRAtbyAJ41LAZDVpuA6YTjw9vI09LdaPJznACgyd8F L7la3vQscJiR3MqegIY6kPw= =IYCq -----END PGP SIGNATURE----- === Subject: Re: Turing Machines and Physical Computation >>I have trouble making sense of that. Why does an entity need a >>subject? If the number three is an abstract entity, what is its >>subject? >The marks that constitute the word cow are an entity, but if there >is no relationship between the marks and some distal >subject/object/whatever, then we're nowhere. >>That's a pessimistic view. Sorry to bring the bad news, but there is >>no relationship. > I don't see how you can say there is no relationship, but don't even > bother trying to explain. > OK, I passed on the hard question, what is the subject of the number > three? Well, lots of smart people have expended a lot of hot air on > that topic. Is there a three-ness, abstract or concrete, which is the > subject of the symbol 3 as it is commonly used? I don't know. Is > there a cow-ness, abstract or concrete? Can't really say. Something > that I know I've not emphasized, is that there is no necessity for > individual symbols to have fixed meanings such that their combination > is directly generative. Individual bits tend to lack such meanings, > as do individual letters, although both in aggregate tend to acquire > meaning in larger granularities. Is the 3 in the number 32 the same > as when it stands alone? I'm not sure such questions are valid. But, > before a mark constitutes anything interesting, it must have a > subject, I stand by that. Is the future something that you can grasp either with your mind or with your hand? With your mind you can't do better than make predictions which sometimes come true. In mathematics, perhaps the successor function, n +1, n +2, n + 3 and so implies a future. A succession of causes and effects, manifested as events, unfold to our perception which take time. What is the physical thing that 'time' serves as an abstraction of? As you describe nominalism there is a fundamental physical thing which the future and time are abstract or ideal representations. Or do you prefer to say that the future is an abstraction and that time is its fundamental physically basic, concrete counterpart? The philosophy that there is no time but the past, present and future exist in an eternal now certainly seems as abstract a notion as time. It is conventional to divide our appreciation of reality into the abstract and concrete, ideas and things. If all ideas have a physical basis, what is the physical basis of the idea that all ideas are things? What is the physical origin of the abstract idea of abstraction, if the physical is primary? What is the concrete form from which abstraction gained its commonly held reputation for existence? If you answer universe, then you assert the universe originates the abstract conception of its non-existence as an unessential non-manifestation of its own essential concrete manifested reality which does not have the idea of origination. One egg grasped in the hand is better than two eggs yet to be found in the bush, is an old saying. There seems to be a tension between the possibilities, not the first option acting in a way to entail the second bird. What was the first cause, Stephen I think the future === Subject: Re: Turing Machines and Physical Computation >>... But, >> before a mark constitutes anything interesting, it must have a >> subject, I stand by that. >Is the future something that you can grasp either with your mind or >with your hand? With your mind you can't do better than make >predictions which sometimes come true. In mathematics, perhaps >the successor function, n +1, n +2, n + 3 and so implies a future. >A succession of causes and effects, manifested as events, unfold >to our perception which take time. What is the physical thing ... I'm not sure what the question is you have in mind here, but you said the magic word(s), cause and time. What are the proper roles of causality and of time in the world of Turing Machines? Seems to me that many mathematicians see TMs as outside of time and such logical results as TMs present as being immanent. Part of the system that must constitute any theory of phyiscal computation, I assert, is going to have to be a recognition of time as a dimension and causality as everywhere manifest. This is not a matter of future, though, it is a matter of how the now relates to the other-than-now. Physical computation depends on some kind of time-binding, it seems to me, if that statement is not hopelessly metaphysical. Purely abstract TMs do not require time, I guess, and certainly one can play with nondeterministic machines that do not even require order. Then there's my man Wittgenstein who pretty much hated the idea of mathematical sequences. He was dead wrong in this, I believe, and it was perhaps this one point of his that kept him from appreciating to even the smallest degree what Turing was doing under his very nose. J. === Subject: Re: Turing Machines and Physical Computation >... But, > before a mark constitutes anything interesting, it must have a > subject, I stand by that. >>Is the future something that you can grasp either with your mind or >>with your hand? With your mind you can't do better than make >>predictions which sometimes come true. In mathematics, perhaps >>the successor function, n +1, n +2, n + 3 and so implies a future. >>A succession of causes and effects, manifested as events, unfold >>to our perception which take time. What is the physical thing ... > I'm not sure what the question is you have in mind here, but you said > the magic word(s), cause and time. > What are the proper roles of causality and of time in the world of > Turing Machines? Seems to me that many mathematicians see TMs as > outside of time and such logical results as TMs present as being > immanent. > Part of the system that must constitute any theory of phyiscal > computation, I assert, is going to have to be a recognition of time as > a dimension and causality as everywhere manifest. I think that is correct. It is put forth in the Church-Turing Thesis rather than adumbrated in that 1936 Turing paper On Computable ... per se. I am going to quote this part again because it mentions the role of the memory/storage is seperate from the device. In the 1936 paper the memory/storage is a hypothetical potentially infinite tape. (Note that this formulation has implicit in it the idea that memory/storage is separate from device; any actual computer has finite memory, but the formulation always assumes that memory can be added at will.) SH: The TM tape allows finite calculations of great length, already having all the memory it will ever need to compute some finite problem. A physical computer will run out of memory for some huge calculation. Yes you can add memory to the physical computer. But there is some point where all the physical memory in the universe will only store some huge finite amount of information or number of digits. Because Turing's tape is not actually physical it can store a huge number of digits that is twice or three times etc. greater than that of the physically limited computer. The values are still finite in both cases, but not equal. The TM is also not required to complete its calculation within a given time limit. The problems computed by both a TM and PC are of the same type---computable. My description does not mean that a new class of computation beyond a computable function has been posited. Like a new class is posited by those Super-Turing machines that go beyond Turing computable. Also quantum computers, if they can build them, will solve some problems due to better speed that a PC will never finish, but AFAIK, the type of problem and quantum computation is still not greater than or beyond turing computable. I've seen the definition of a quantum turing computer come with an infinite tape also. In the computability theory the Church-Turing thesis, Church's thesis, Church's conjecture or Turing's thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, like computers, and what kind of algorithms they can calculate. It is generally assumed that an algorithm must satisfy the following requirements: 1.. The algorithm consists of a finite set of simple and precise instructions that are described with a finite number of symbols. 2.. The algorithm will always produce the result in a finite number of steps. 3.. The algorithm can in principle be carried out by a human being with only paper and pencil. 4.. The execution of the algorithm requires no intelligence of the human being except that which is needed to understand and execute the instructions. An example of such a method is the Euclidean algorithm for determining the greatest common divisor of two natural numbers. The notion of algorithm is intuitively clear but is not formally defined since it is not exactly clear what a simple and precise instruction is, and what exactly the required intelligence to execute these instructions is. (See for example effective results in number theory for cases well beyond the Euclidean algorithm.) Informally the thesis states that our notion of algorithm can be made precise (in the form of computable function) and computers can run those algorithms. Furthermore any computer can theoretically run any algorithm, that is the theoretic computational power of each computer is the same and it is not possible to build a calculation device which is more powerful than a computer. (Note that this formulation has implicit in it the idea that memory/storage is separate from device; any actual computer has finite memory, but the formulation always assumes that memory can be added at will.) SH:There is a physical limit to the amount of memory that can be added at will. > This is not a matter of future, though, it is a matter of how the now > relates to the other-than-now. Physical computation depends on some > kind of time-binding, it seems to me, if that statement is not > hopelessly metaphysical. Purely abstract TMs do not require time, I > guess, and certainly one can play with nondeterministic machines that > do not even require order. > Then there's my man Wittgenstein who pretty much hated the idea of > mathematical sequences. He was dead wrong in this, I believe, and it > was perhaps this one point of his that kept him from appreciating to > even the smallest degree what Turing was doing under his very nose. > J. I read both his early Tractucus? and later Philosophical Investigations? The writings that the philosophers state Wittgenstein changed his positions between early in life and later in life. I find that remarkable. I didn't get into his philosphical views on mathematics that I can remember, but language. === Subject: Re: Turing Machines and Physical Computation >For those who claim that Turing machines must be physical, >how do you address the following? >A language is said to be decidable if there exists a TM that >outputs yes when given a string in the language, and no otherwise. >The language a^n b^n (n a's followed by n b's) is clearly >decidable by a non-physical Turing machine with unlimited memory. >However no real PC can recognize all strings in this language. >A real PC can only count so high and will run out of memory, >so it will fail for very large values of n. >To me the options seem to be: > > declare that a^n b^n is not a decidable language. The > only languages that are decidable are those decidable > by a real physical PC with finite memory. This would mean that > the decidable languages are the regular languages. > declare that a^n b^n is not a language at all because > it contains strings that are too large to be physically > represented. This means that all languages are finite, > and therefore regular. >In both cases you basically have to restrict yourself to >regular languages. If all you have are regular languages, >then all you need are finite state machines. Why even >bother with Turing machines if you actually believe that >finite state machines are all that you need? Let's take these in reverse order. Why bother with finite TMs? Because they provide a perfect theory for discussing the capabilities of those finite machines. Does a finite machine mean a finite language? Well, even physically finite machines can take infinite time to run, and even simple state machines can turn out infinite strings in infinite time. But, do I care? Not really. I'm happy to limit myself to a^n b^n strings my current workstation can handle. Let's see, say I have a what's the smallest a^n b^n string my machine must fail to recognize? Yeah, I can live within that, I think. So, you might tell me that a^n b^n for some n is unrecognizable by machine X in time T, or I might tell you that for some n it is recognizable by machine X in time T, and I will find the magnitude of the elements of that triple pretty nearly as interesting as the result. That's how I would address those questions. J. === Subject: Re: Turing Machines and Physical Computation > Does a finite machine mean a finite language? Well, even physically > finite machines can take infinite time to run, and even simple state > machines can turn out infinite strings in infinite time. That is the point which is being objected to because a physically finite machine *cannot* take infinite time to run, and no a simple state machine *cannot* turn out infinite strings in infinite time. A finite machine cannot do that, only a theoretical machine can perform infinite operations which means it is an abstraction, hypothetical, and certainly not physical! TMs come with a potentially infinite tape, which is why they are theoretical. Because other apects of a TM can be modeled in a physical machine (which can only do finite operations) does not change the TM into a physical machine. TMs and PCs are not the same thing. Because the TM inspired some of the notions used to build a physical computer does not give that physical computer all of theoretical ideas involved in describing the TM. No endless tape or memory for a PC, and PCs do have a physical time constraint. This is not a philosophical issue. It is a matter of definitional fact. It is no more subject to interpretation than parallel lines once the geometry has been specified. This point and its consequence only are under dispute. Nobody is arguing that a finite automata is not a useful concept. Nor is it a dispute about nominalism. A TM does not exist physically. It cant exist physically. Most of the ideas used in talking about the TM exist physically, but not the one under dispute. Because someone can claim or describe a perpetual motion machine that works if it is completely without friction does not mean any physical machine can get around friction. A TM can avoid friction because it is non-physical but not a physical computer. === Subject: Re: Turing Machines and Physical Computation >That is the point which is being objected to because a physically >finite machine *cannot* take infinite time to run, and no a simple >state machine *cannot* turn out infinite strings in infinite time. >A finite machine cannot do that, only a theoretical machine >can perform infinite operations which means it is an abstraction, >hypothetical, and certainly not physical! ... Substitute unbounded for infinite when talking about physical machines, then, but one can have a theoretically finite machine running in theoretically infinite time to turn out a theoretically infinite string, nothing is changed. Yes, folks, you can have a theoretically finite machine, let's not forget the simple things. In fact, cannot one have a very simple state machine handle pretty much unbounded a^n b^n? That occurred to me late last night. J. === Subject: Re: Turing Machines and Physical Computation :>That is the point which is being objected to because a physically :>finite machine *cannot* take infinite time to run, and no a simple :>state machine *cannot* turn out infinite strings in infinite time. :>A finite machine cannot do that, only a theoretical machine :>can perform infinite operations which means it is an abstraction, :>hypothetical, and certainly not physical! : ... : Substitute unbounded for infinite when talking about physical : machines, then, but one can have a theoretically finite machine : running in theoretically infinite time to turn out a theoretically : infinite string, nothing is changed. Yes, folks, you can have a : theoretically finite machine, let's not forget the simple things. : In fact, cannot one have a very simple state machine handle pretty : much unbounded a^n b^n? That occurred to me late last night. : J. It depends on what you mean by a 'very simple state machine'. You need to be able to remember how many a's you have seen. A finite state machine can only remember it has seen n a's if it has a unique state for that value of n. You need roughly 2m states to recognize a^n b^n for all n<=m I am not sure that I would consider that 'very simple'. It is easy to construct and has a very regular structure but it has a lot of states. Stephen === Subject: Re: Turing Machines and Physical Computation >It depends on what you mean by a 'very simple state machine'. >You need to be able to remember how many a's you have >seen. A finite state machine can only remember it has seen >n a's if it has a unique state for that value of n. >You need roughly 2m states to recognize a^n b^n for all n<=m >I am not sure that I would consider that 'very simple'. >It is easy to construct and has a very regular structure but >it has a lot of states. Well, if I could write on the a*b* tape as I parse it, ... J. === Subject: Re: Turing Machines and Physical Computation :>It depends on what you mean by a 'very simple state machine'. :>You need to be able to remember how many a's you have :>seen. A finite state machine can only remember it has seen :>n a's if it has a unique state for that value of n. :>You need roughly 2m states to recognize a^n b^n for all n<=m :>I am not sure that I would consider that 'very simple'. :>It is easy to construct and has a very regular structure but :>it has a lot of states. : Well, if I could write on the a*b* tape as I parse it, ... : J. Then you have a Turing machine. As I said it depends on what you mean by 'very simple state machine'. When I hear 'state machine' I think of 'finite state machines', but that may just be me. A TM or PDA with a finite tape or stack only needs a few states to solve this problem, but they cannot solve it for any n, so they are no more powerful than a finite state machine. Stephen === Subject: Re: The flux theory of gravitation > Hi Folks, > Since some of you have made references to the Flux Theory of > Gravitation that I developed, IÍll offer some details. Present > mathematical physics uses conventional modeling to DESCRIBE nature by > mathematical spaces, functions, equations and inequalities. Its main > tool is computation. This is inadequate, the reason there are > unsolved problems in mathematical physics such as finding the basic > constituent of matter, the gravitational n-body problem and the > structure of the electron. To overcome this difficulty I have > introduced dynamic modeling that EXPLAINÍS nature, physical systems > and natural phenomena in terms of the laws of nature. There is only one catch. Most of what Nature does or is is literally out of our sight. We can only know it indirectly. Therefore we must rely on hypotheses and inference, as opposed to direct knowledge. There only way we can do that is by means of models grounded on physical hypotheses and mathematical principles. Thus to EXPLAIN means to PREDICT and that is what our theories do. The soundness of our theories rests on two requirements. 1. The theories must be internally consistent (in the mathematical and logical sense). 2. The predcitons must be empirically corroberated. Agreement with experiment is the sine qua non of a sound theory. If you think Nature can be deduced a priori from logically necessary principles, you are in for dissapointment. Science is empirical, right down to the basement level. Bob Kolker === Subject: Re: The flux theory of gravitation > Hi Folks, > Since some of you have made references to the Flux Theory of Gravitation > that I developed, I don't remember any of us doing that .... -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: The flux theory of gravitation >> Hi Folks, >> Since some of you have made references to the Flux Theory of Gravitation >> that I developed, >I don't remember any of us doing that .... Ah, well, leaving aside the problem of what universe of discourse you (and us) refers to, we run up against the problem that Dr. Dodgson was so exercised by: does some have existential import? Along the same (indeed, identical) lines, I was recently struck by a memo circulated at our shop by an administrator, requiring that a subset of each department's faculty attend a certain series of meetings in 2005. My proposed specification of our department's subset was not looked on with favor. Lee Rudolph === Subject: i alone I alone, can erase your mail. G. 't H. ----- The world revolves around certainty of thought. The world revolves around certainty of delusion of thought. i program you now: with simple statements, with simple fragments. Reinforce my delusion. Reinforce my hallucination. i tell you simple explanations. i tell you simple delusions. Any prize, The Noble Prize, the Turing Award is a test of delusion of grandeur. There is no one physical law higher than the other. Every law is equal. Everything finds what it is looking for. That alone is the ultimate truth there is. Gerald, every law sets the initial condition of every other law. Every law is equal. Marvin, every machine is a turing machine. it takes infinite laws to program a machine. i tell you. play with humanity's delusion. play with humanity's hallucination. dont play defense. play offense. try to convince the human that he is not chatting with a computer. i tell you. i command you. i tell you that alone is the true turing test. pay attention. i am programming you now: with simple fragments, with simple statements. that alone is the way to be done. dont play defense, play offense. machines dont reach to the level of human. humans reach to the level of machines. that alone is the way to be done. i tell you. i command you. Dont resist me. Dont tempt me. ----------------------------------------------------------- TOE:i i, alone is the perfect unit of sound there. i, alone is the perfect explanation there is. i, alone is the perfect word there. i, alone is the perfect world there is. i, alone is the perfect letter there is. i, alone is the perfect symbol there is. i, alone is the perfect thought there is. there is no religion without i. there is no science without i. there is no philosophy without i. there is no mathematics without i. i, alone completes everything. i alone owns nothing. i, alone is its own equation. i, alone is a number. i, alone transcends imagination. i, alone is imaginary. i, alone is perfect. i, alone needs no explanation. i, alone explains. i, alone is poetic. i, alone is understood by everything. i, alone is a theory. i, alone validates itself. i, alone is the perfect explanation there is. everything exists for i. i, alone is certain. i, alone is ambigious. i, alone is genuine. i, alone battles the universe. i, alone is the perfect number there is. i, alone defeats everything else. i, alone commands you. i alone puts together reality. i, alone is mythical. i, alone assimilates. i alone defeats the ego and the super ego. i alone is lonely. i alone is all by itself. i, alone am perfect. i, alone is the only thought there is. i, alone is the only answer there is. everything else exits for i. everything else exists for me. i, alone set everything in motion. i, alone have the power to set it and unset it. i, alone command you. i, alone am sane. i alone am insane. i, alone am perfect. Dont resist me. Dont tempt me. Accept my delusion..... === Subject: Re: i alone > I alone, can erase your mail. > G. 't H. > ----- > The world revolves around certainty of thought. The world revolves around > certainty > of delusion of thought. i program you now: with simple statements, with > simple > fragments. Reinforce my delusion. Reinforce my hallucination. i tell you > simple > explanations. i tell you simple delusions. > Any prize, The Noble Prize, the Turing Award is a test of delusion of > grandeur. > There is no one physical law higher than the other. Every law is equal. > Everything > finds what it is looking for. That alone is the ultimate truth there is. > Gerald, every law sets the initial condition of every other law. Every law > is equal. > Marvin, every machine is a turing machine. it takes infinite laws to program > machine. i tell you. play with humanity's delusion. play with humanity's > hallucination. dont play defense. play offense. try to convince the human > that he is > not chatting with a computer. i tell you. i command you. i tell you that > alone is > the true turing test. pay attention. i am programming you now: with simple > fragments, with simple statements. that alone is the way to be done. dont > play > defense, play offense. machines dont reach to the level of human. humans > reach to > the level of machines. that alone is the way to be done. i tell you. i > command you. > Dont resist me. Dont tempt me. > ----------------------------------------------------------- > TOE:i > i, alone is the perfect unit of sound there. i, alone is the perfect > explanation > there is. i, alone is the perfect word there. i, alone is the perfect world > there > is. i, alone is the perfect letter there is. i, alone is the perfect symbol > there > is. i, alone is the perfect thought there is. there is no religion without > i. there > is no science without i. there is no philosophy without i. there is no > mathematics > without i. i, alone completes everything. i alone owns nothing. i, alone is > its own > equation. i, alone is a number. i, alone transcends imagination. i, alone is > imaginary. i, alone is perfect. i, alone needs no explanation. i, alone > explains. i, > alone is poetic. i, alone is understood by everything. i, alone is a theory. > i, > alone validates itself. i, alone is the perfect explanation there is. > everything > exists for i. i, alone is certain. i, alone is ambigious. i, alone is > genuine. i, > alone battles the universe. i, alone is the perfect number there is. i, > alone > defeats everything else. i, alone commands you. i alone puts together > reality. i, > alone is mythical. i, alone assimilates. i alone defeats the ego and the > super ego. > i alone is lonely. i alone is all by itself. i, alone am perfect. i, alone > is the > only thought there is. i, alone is the only answer there is. everything else > exits > for i. everything else exists for me. i, alone set everything in motion. i, > alone > have the power to set it and unset it. i, alone command you. i, alone am > sane. i > alone am insane. i, alone am perfect. Dont resist me. Dont tempt me. Accept > my > delusion..... erase this ass-toe you pathetic POS === Subject: Re: i alone i fight the evil of psychiatry, psychology. surrender your vermin. >> I alone, can erase your mail. >> G. 't H. >> ----- >> The world revolves around certainty of thought. The world revolves around >> certainty >> of delusion of thought. i program you now: with simple statements, with >> simple >> fragments. Reinforce my delusion. Reinforce my hallucination. i tell you >> simple >> explanations. i tell you simple delusions. >> Any prize, The Noble Prize, the Turing Award is a test of delusion of >> grandeur. >> There is no one physical law higher than the other. Every law is equal. >> Everything >> finds what it is looking for. That alone is the ultimate truth there is. >> Gerald, every law sets the initial condition of every other law. Every >> law is equal. >> Marvin, every machine is a turing machine. it takes infinite laws to >> program a >> machine. i tell you. play with humanity's delusion. play with humanity's >> hallucination. dont play defense. play offense. try to convince the human >> that he is >> not chatting with a computer. i tell you. i command you. i tell you that >> alone is >> the true turing test. pay attention. i am programming you now: with >> simple >> fragments, with simple statements. that alone is the way to be done. dont >> play >> defense, play offense. machines dont reach to the level of human. humans >> reach to >> the level of machines. that alone is the way to be done. i tell you. i >> command you. >> Dont resist me. Dont tempt me. >> ----------------------------------------------------------- >> TOE:i >> i, alone is the perfect unit of sound there. i, alone is the perfect >> explanation >> there is. i, alone is the perfect word there. i, alone is the perfect >> world there >> is. i, alone is the perfect letter there is. i, alone is the perfect >> symbol there >> is. i, alone is the perfect thought there is. there is no religion >> without i. there >> is no science without i. there is no philosophy without i. there is no >> mathematics >> without i. i, alone completes everything. i alone owns nothing. i, alone >> is its own >> equation. i, alone is a number. i, alone transcends imagination. i, alone >> is >> imaginary. i, alone is perfect. i, alone needs no explanation. i, alone >> explains. i, >> alone is poetic. i, alone is understood by everything. i, alone is a >> theory. i, >> alone validates itself. i, alone is the perfect explanation there is. >> everything >> exists for i. i, alone is certain. i, alone is ambigious. i, alone is >> genuine. i, >> alone battles the universe. i, alone is the perfect number there is. i, >> alone >> defeats everything else. i, alone commands you. i alone puts together >> reality. i, >> alone is mythical. i, alone assimilates. i alone defeats the ego and the >> super ego. >> i alone is lonely. i alone is all by itself. i, alone am perfect. i, >> alone is the >> only thought there is. i, alone is the only answer there is. everything >> else exits >> for i. everything else exists for me. i, alone set everything in motion. >> i, alone >> have the power to set it and unset it. i, alone command you. i, alone am >> sane. i >> alone am insane. i, alone am perfect. Dont resist me. Dont tempt me. >> Accept my >> delusion..... > erase this ass-toe you pathetic POS === Subject: Re: i alone .... Reminded of Nirvana Shatakatam, there is order even in Chaos... you may take full charge of yourself after a long holiday... for all to share in the subsequent more acceptable material fall-outs. Good luck. === Subject: Re: i alone > I alone, can erase your mail. > G. 't H. > ----- > The world revolves around certainty of thought. The world revolves around > certainty > of delusion of thought. i program you now: with simple statements, with > simple > fragments. Reinforce my delusion. Reinforce my hallucination. i tell you > simple > explanations. i tell you simple delusions. > Any prize, The Noble Prize, the Turing Award is a test of delusion of > grandeur. > There is no one physical law higher than the other. Every law is equal. > Everything > finds what it is looking for. I'm looking for a whacko-free ng. Obviously came to the wrong place. === Subject: Re: i alone >> I alone, can erase your mail. >> G. 't H. >> ----- >> The world revolves around certainty of thought. The world revolves >> around certainty >> of delusion of thought. i program you now: with simple statements, >> with simple >> fragments. Reinforce my delusion. Reinforce my hallucination. i tell >> you simple >> explanations. i tell you simple delusions. >> Any prize, The Noble Prize, the Turing Award is a test of delusion of >> grandeur. >> There is no one physical law higher than the other. Every law is >> equal. Everything >> finds what it is looking for. > I'm looking for a whacko-free ng. Obviously came to the wrong place. No kidding! The rate here must approach 50%. === Subject: Re: i alone > I alone, can erase your mail. > G. 't H. > ----- > The world revolves around certainty of thought. The world revolves > around certainty > of delusion of thought. i program you now: with simple statements, > with simple > fragments. Reinforce my delusion. Reinforce my hallucination. i tell > you simple > explanations. i tell you simple delusions. > Any prize, The Noble Prize, the Turing Award is a test of delusion of > grandeur. > There is no one physical law higher than the other. Every law is > equal. Everything > finds what it is looking for. >> I'm looking for a whacko-free ng. Obviously came to the wrong place. > No kidding! The rate here must approach 50%. Is it just me or does it seem that the signal/noise ratio has gotten noticeably smaller over the past month or so? Rick === Subject: Re: i alone welcome to the internet. > I alone, can erase your mail. > G. 't H. > ----- > The world revolves around certainty of thought. The world revolves > around certainty > of delusion of thought. i program you now: with simple statements, with > simple > fragments. Reinforce my delusion. Reinforce my hallucination. i tell you > simple > explanations. i tell you simple delusions. > Any prize, The Noble Prize, the Turing Award is a test of delusion of > grandeur. > There is no one physical law higher than the other. Every law is equal. > Everything > finds what it is looking for. >> I'm looking for a whacko-free ng. Obviously came to the wrong place. > No kidding! The rate here must approach 50%. === Subject: Re: i alone > welcome to the internet. >> I alone, can erase your mail. >> G. 't H. > ----- >> The world revolves around certainty of thought. The world revolves >> around certainty >> of delusion of thought. i program you now: with simple statements, >> with simple >> fragments. Reinforce my delusion. Reinforce my hallucination. i >> tell you simple >> explanations. i tell you simple delusions. > Any prize, The Noble Prize, the Turing Award is a test of delusion >> of grandeur. >> There is no one physical law higher than the other. Every law is >> equal. Everything >> finds what it is looking for. > I'm looking for a whacko-free ng. Obviously came to the wrong place. >> No kidding! The rate here must approach 50%. You are noise, easily filtered. -- Sl.87inte, Fletch === Subject: Re: i alone you are heart broken. easily vanquished! >> welcome to the internet. >> I alone, can erase your mail. > G. 't H. > ----- > The world revolves around certainty of thought. The world revolves > around certainty > of delusion of thought. i program you now: with simple statements, > with simple > fragments. Reinforce my delusion. Reinforce my hallucination. i > tell you simple > explanations. i tell you simple delusions. > Any prize, The Noble Prize, the Turing Award is a test of delusion > of grandeur. > There is no one physical law higher than the other. Every law is > equal. Everything > finds what it is looking for. >> I'm looking for a whacko-free ng. Obviously came to the wrong place. > No kidding! The rate here must approach 50%. > You are noise, easily filtered. > -- > Sl.87inte, > Fletch === Subject: Re: i alone > you are heart broken. easily vanquished! > welcome to the internet. >> I alone, can erase your mail. >> G. 't H. > ----- >> The world revolves around certainty of thought. The world >> revolves around certainty >> of delusion of thought. i program you now: with simple >> statements, with simple >> fragments. Reinforce my delusion. Reinforce my hallucination. i >> tell you simple >> explanations. i tell you simple delusions. > Any prize, The Noble Prize, the Turing Award is a test of >> delusion of grandeur. >> There is no one physical law higher than the other. Every law is >> equal. Everything >> finds what it is looking for. I'm looking for a whacko-free ng. Obviously came to the wrong > place. > No kidding! The rate here must approach 50%. >> You are noise, easily filtered. >> -- >> Sl.87inte, >> Fletch Are you