mm-4689 === Subject: Re: Testing the slope in a regression I am running two regressions and want to find out if the slope in both > regressions is the same. > i.e. > y1=a1 + b1*x1 > y2=a2 + b2*x2 I want to find out if I can claim b1=b2 and if possible produce some > measure of the confidence with which I can make the statement. I have > all the standard errors etc for each regression. In this particular case the data is paired (x1=x2) but I suspect the > question can be answered without this. Classic use for a dummy variable. Concatenate your data sets, and add a variable D. D=0 for set 1 and D=1 for set 2 Now, y= A1 +A2*D + B1X +B2XD If B2 is not different from zero, your slopes are the same. If A2 is not different from zero, your intercepts are the same. -- Scott Reverse name to reply === Subject: Re: The Computable Reals (alpha version) =F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Interesting. So where is this number (based on Champernowne's > constant) in your list: > .0 1 10 11 100 101 110 111 1000 1001 1010 1011 ... > It has no repeating digit groups, but it falls between .0110 and .0111. off. > *plonk* I translate that as I don't know. No, it's just that I had already been asked that question (as as all others) in all possible shapes, and I had already tried with no success to answer in any possible way, actually asking for a support that has never come. In fact, if you say so, I must take that you rather didn't bother to read the thread before asking, including the fact that at that very moment I was surely not in my best mood (and for obviuos reasons). -LV === Subject: JSH: Nifty little result on quadratic diophantines =n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.27 Safari/525.13,gzip(gfe),gzip(gfe) Oh hey, after I came up with the Quadratic Diophantine Theorem, I started looking over research on quadratic diophantine equations and it's kind of interesting because hey, looks like my research can help! Like it will give criteria on Pell's equation, and may even offer a route to generally solving a 2 variable diophantine quadratic as you can just let x=1 to go from the 3 variable expression in the primary theorem. Such a powerful little result it looks like after a little research, and I was wondering if it would be a big deal as I debated about putting it on my math blog. Oh yeah, proof of the theorem is on my math blog. Turns out it's easy to prove with tautological spaces. Cool. Ok, going to read up more on quadratic diophantine equation stuff that was already done before my research. James Harris === Subject: Re: JSH: Nifty little result on quadratic diophantines =3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs 1.7; .NET CLR 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) > Oh hey, after I came up with the Quadratic Diophantine Theorem, I > started looking over research on quadratic diophantine equations and > it's kind of interesting because hey, looks like my research can help! > Yeah, but can it fight tough baked-on food or a deep-down grease stains? Such a powerful little result it looks like after a little research, > and I was wondering if it would be a big deal as I debated about > putting it on my math blog. æOh yeah, proof of the theorem is on my > math blog. æTurns out it's easy to prove with tautological spaces. But of course it's all so trivial... Ok, going to read up more on quadratic diophantine equation > stuff that was already done before my research. Now *that* will be a first : James Harris actually reading what's been done! James Harris M === Subject: Re: JSH: Nifty little result on quadratic diophantines =n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.27 Safari/525.13,gzip(gfe),gzip(gfe) Oh hey, after I came up with the Quadratic Diophantine Theorem, I > started looking over research on quadratic diophantine equations and > it's kind of interesting because hey, looks like my research can help! Yeah, but can it fight tough baked-on food or a deep-down grease > stains? Such a powerful little result it looks like after a little research, > and I was wondering if it would be a big deal as I debated about > putting it on my math blog. æOh yeah, proof of the theorem is on my > math blog. æTurns out it's easy to prove with tautological spaces. But of course it's all so trivial... Ease of the proof is important to me in case there are arguments. Ok, going to read up more on quadratic diophantine equation > stuff that was already done before my research. Now *that* will be a first : James Harris actually reading what's been > done! And I take it, you have not. This number theory result is distinct in that the theorem gives what has never been given before. So yes, if you wish, you can stupidly insult me in a case where there cannot be years of denial of basic math, no matter how cleverly posters play politics. And there can be no denial of the value of the result either. The math wars as I've called them, are, finally, really this time, over. James Harris === Subject: Re: JSH: Nifty little result on quadratic diophantines =NwriowkAAAD2revjv4goruNS3oWFbQ_H BCWEB; .NET CLR 1.0.3705; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > The math wars as I've called them, are, finally, really this time, > over. that they were over: http://mathforum.org/kb/plaintext.jspa?messageID=5298063 you were wrong? Who would have guessed that?! Jose Carlos Santos === Subject: Re: JSH: Nifty little result on quadratic diophantines =3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs 1.7; .NET CLR 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) Ok, going to read up more on quadratic diophantine equation > stuff that was already done before my research. Now *that* will be a first : James Harris actually reading what's been > done! And I take it, you have not. > James Harris- Hide quoted text - Just so we are all clear on this matter, by read do you mean that you actually intend to open a book and sit down and learn math or are you just going to revert to your old dumpster-diving ways and browse websites? JC, M === Subject: Re: JSH: Nifty little result on quadratic diophantines > The math wars as I've called them, are, finally, really this time, > over. Again. -- In a world of ideas there should be a place for people who are not experts [...] to talk out their ideas [...] without facing personal insults. And if they are frustrated[...], why should it be a surprise if they end up contacting news organizations, or Noam Chomsky? --JSH === Subject: Re: JSH: Nifty little result on quadratic diophantines <87prninkl6.fsf@phiwumbda.org> =n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > The math wars as I've called them, are, finally, really this time, > over. Again. And you too? So you don't know anything about Diophantine equations either? Go do some research. See if a theorem like mine has ever been found before. Never been done. Modern mathematicians cannot even begin to start to find an existence proof for the general equation I handled today. They wouldn't know where to begin. I used a variant on the technique I used to prove Fermat's Last Theorem. Isn't it ironic, don't you think? James Harris === Subject: Re: JSH: Nifty little result on quadratic diophantines > The math wars as I've called them, are, finally, really this time, > over. > Again. > And you too? So you don't know anything about Diophantine equations > either? No, I don't. Why should I? All my comment requires is that I know about your posting history, not Diophantine equations. The math wars are finally over this time. Just like many times previously. But this time (like the past times) are different because mathematicians can't just lie about your work, like they could with your solution of TSP, factoring, prime counting algorithm and so on and on and on. But who could lie about Diophantine equations? I'm with you on this, just like I've been with you every time before. The math wars are finally and definitely over, just like before. > Go do some research. See if a theorem like mine has ever been found > before. Never been done. Modern mathematicians cannot even begin to start to find an existence > proof for the general equation I handled today. They wouldn't know where to begin. I used a variant on the technique I used to prove Fermat's Last > Theorem. Isn't it ironic, don't you think? What can one say? Tautological spaces must be one humdinger of a tool. -- Jesse F. Hughes With [President Bush] endorsing [Intelligent Design], at the very least it makes Americans who have that position more respectable, for lack of a better phrase. -- Gary L. Bauer, in search of a thesaurus === Subject: Re: JSH: Nifty little result on quadratic diophantines > The math wars as I've called them, are, finally, really this time, > over. Let's see: You have been posting crap and making yourself the laughing stock of this and other groups for, what? Ten years now? And we are to believe that this time you are serious? My prediction is that, as usual, you will play the net's buffoon part. === Subject: Re: JSH: Nifty little result on quadratic diophantines =n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Sep 5, 5:17æpm, Jens Stueckelberger over. æ æ æ æ Let's see: You have been posting crap and making yourself the > laughing stock of this and other groups for, what? Ten years now? And we > are to believe that this time you are serious? My prediction is that, as > usual, you will play the net's buffoon part. But that's not true. Posters have routinely lied about my results for years now. The difference with the Quadratic Diophantine Theorem is that the lies are not so easy. My other pure math results were either complicated or traveled over previous ground just enough that posters would just claim there wasn't anything important about them, like my prime counting function. But with this result in an area where there is over a thousand years of research going back to ancient Asia, and the Greeks, and including Fermat and Gauss, there is no other result like this one in terms of reach. Remarkably though in the initial replies I see an indication of those among you who know very little if any number theory or you'd have some inkling that it's better to wait on this one, as otherwise you look real stupid, if it's correct. And I can assure you the theorem IS correct. Those wishing to see the proof can just check my math blog and see how easily it is derived, though it was over a thousand years in the finding. History being made. Fun!!! James Harris === Subject: Re: JSH: Nifty little result on quadratic diophantines On Sep 5, 5:17 pm, Jens Stueckelberger The math wars as I've called them, are, finally, really this time, > over. > Let's see: You have been posting crap and making yourself the > laughing stock of this and other groups for, what? Ten years now? And we > are to believe that this time you are serious? My prediction is that, as > usual, you will play the net's buffoon part. >But that's not true. Posters have routinely lied about my results for >years now. Eh, you started it. By lying when you post results. >The difference with the Quadratic Diophantine Theorem is that the lies >are not so easy. it is trivial, already solved; http://www.alpertron.com.ar/QUAD.HTM >My other pure math results were either complicated or traveled over >previous ground just enough that posters would just claim there wasn't >anything important about them, like my prime counting function. agreed, not important, trivial >But with this result in an area where there is over a thousand years >of research going back to ancient Asia, and the Greeks, and including >Fermat and Gauss, there is no other result like this one in terms of >reach. Who in Asia? >Remarkably though in the initial replies I see an indication of those >among you who know very little if any number theory or you'd have some >inkling that it's better to wait on this one, as otherwise you look >real stupid, if it's correct. trivial, go read a book, http://www.alpertron.com.ar/QUAD.HTM >And I can assure you the theorem IS correct. you lie. >Those wishing to see the proof can just check my math blog and see how >easily it is derived, though it was over a thousand years in the >finding. >History being made. Fun!!! Trolling Fun!!! >James Harris === Subject: Re: JSH: Nifty little result on quadratic diophantines =_k7REQoAAACDJL2M6OKCfSBj5_wTcvrO Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Oh hey, after I came up with the Quadratic Diophantine Theorem, I > started looking over research on quadratic diophantine equations and > it's kind of interesting because hey, looks like my research can help! Like it will give criteria on Pell's equation, and may even offer a > route to generally solving a 2 variable diophantine quadratic as you > can just let x=1 to go from the 3 variable expression in the primary > theorem. Such a powerful little result it looks like after a little research, > and I was wondering if it would be a big deal as I debated about > putting it on my math blog. æOh yeah, proof of the theorem is on my > math blog. æTurns out it's easy to prove with tautological spaces. Cool. æOk, going to read up more on quadratic diophantine equation > stuff that was already done before my research. James Harris Cool, if this finds it's way into a 'peer-reviewed' paper you'll be sure to post the reviewers' comments here won't you? That way we can't go around thinking nobody reviewed it. === Subject: Re: Very basic mistakes > This is not a physics newsgroup. I see physics affected by mistakes in basics of mathematrics. Why should I hate or blame any person? Ebbinghaus quoted Lessing as to express distance from naive set theory. Why to go on hiding the obvious lack of foundation for mathematics? Why not look for the very cause of problems affecting physics? > Charles Francis is a mathematician, Who? I found no evidence of a mathematician Charles Francis on > the web, and no evidence of any mathematical publications > of anyone of that name. He himself revealed being a mathematician. Most likely he was trained at university as a mathematician. Archimedes was rather rambling in positive sense as also were Galilei, Leibniz, Newton, Weyl, Hilbert, v. Neumann and many others. Fermat was a jurist, Einstein allegedly was a poor mathematician who skipped Minkowski's lessons. Why do you intend to make mathematics esoteric? > http://www.teleconnection.info/rqg/MainIndex Ah, a reference! A web page concerned with *physics* > (albeit with some gentle introductions to some mathematical topics > as used by physicists)! Yes, some mathematics there, but featured in > a purely instrumental role as a language for the author's > physical theories, not for its own end. This website > demonstrates that its author is a physicist and not a mathematician. I see it demonstrating that physics depends on possibly questionable mathematics, not the other way round. Do finitism and deductionism belong to physics? Didn't misuse of unscrupulously fabricated axioms spread from mathematics to physics? > Well, I am resuming what has almost been forgotten. > Well, I refuse to exclusively accept modern terminology. > Well, I am trying to judge the missing success of aleph_2, etc. > Well, I admire the rigorosity in Galilei's thinking. Well, you refuse to post to relevant newsgroups. Aleph_2 is not physical nonsense but mathematical one. So called fundamental crisis of m. has almost been forgotten. Modern terminology has been structured as to hide moot points. Galilei was correct in mathematics too. > I contempt those who more or less in vain take naive or axiomatic > set theory a gospel, including all deduced physical structure. Contempt isn't a verb. # > But you may (but probably will not) wish > to remedy your ignorance: there is no deduced physical structure > in set theory. Set theory is not a physical theory. Set theory lacks any tangible basis. Dedekind admitted that he did not have any evidence for his guess that the entity of all rational numbers can be split into larger and smaller ones wrt his cut. While all science including physics applies mathematics, the very basics of mathematics are nonetheless obliged to obey selfconsistency not just within a limited paradise but in a more comprehensive manner that respects the mutual logical exclusion between countable discreta and uncountable continua. For instance, Buridan's donkey has primarily to do with poorly understood logical fundamentals rather than with physics. > John v. Neuman was honest when he in 1935 admitted to Birkhoff that he > did no longer believe in Cantor space. > Do you understand that? I guess you don't. I certainly don't, since I have no idea what Cantor space is. > Is it like Hilbert space? Certainly John von Neumann did important > work with that. Of course I meant Hilbert space. Cantor space was just a typo I have to apologize for. However, the problem with Hilbert space actually goes back to Cantor's naive ideas. Still in 1925 Hilbert tried to defend them as einfaches Hinueberzaehlen (from finite to transfinite numbers). Fraenkel admitted in 1923 that Cantor's naive definition of a set (including an infinite one) was untenable. Until now there is no corrected definition. By the way, if you must descend to argument by name-dropping > you might at least have the decency to spell the name of your > uncritically admired hero correctly. I do not understand this hint. I do not uncritically admire Galileo Galilei. I decided to borrow his synonym Salviati because I would like to stress that his ultimate conclusion is fully convincing to me and not made out-dated by Cantor. > Of course, it's stupid trying to bolster your adoption > of a controversial position by claiming that a well-known figure > supported it, since there will be many figures of equal eminence who will > have opposed it. Having read some criticism concerning Galilei, I see his outstanding eminence in his consequent factual fight against mistakes. Ascribing the Aristotelian position of the pope to Simplicius, he was anything but compromizing. In contrast to Cantor, Galilei was perhaps never guided by mere intentions, speculations and belief. He provided compelling arguments. Who can be called a figure of eminence comparable to Galilei? Who opposed Galilei's clear distinction between finite and infinite, countable and uncountable, discrete and continuous? When Berkeley criticised Leibniz and Newton, I consider this justified to some extent because Leibniz used his first level of infinity in the sense of being infinitesimal = infinite relative to something much smaller. Leibniz equated the fiction of infinity in the sense of a quality that can absolutely not be exhausted with his third and highest level, i.e., with god. Hilbert was a rather extroverted brute who got support from many people who admired him and appreciated his interest in physics, even from Woldemar Vogt. However, Hilbert's successor Weyl called Hilbert a piper whom all experts were following, and obviously Weyl was not sure whether or not Hilbert's way was the correct one. Once again: Do you understand why v. Neumann in 1935 did no longer believe in Hilbert space? His famous book was published just a few years earlier in 1932. The reason for v. Neumann to abandon his belief was most likely a paper by EPR. You will certainly agree that Hilbert space is pure mathematics. By the way, Hilbert was still alife in Germany until he died in 1943, and after 1935, v. Neumann did not find back to his former hoity-toity attitude of understanding all. Salviati: ... in ultima conclusione, gli attributi di eguale maggiore e minore non aver luogo ne gl'infiniti, ma solo nelle quantit.88 terminate. IR>|>IR+>|>IR === Subject: Blumschien's very basic mistakes <48c19ea6$0$9318$9b4e6d93@newsspool2.arcor-online.net> =IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) > This is not a physics newsgroup. I see physics affected by mistakes in basics of mathematrics. Again, you are posting off-topic. This is not a physics newsgroups, and you have not isolated any mistakes in basics of mathematics that can be profitably discussed by mathematicians. > Charles Francis is a mathematician, Who? I found no evidence of a mathematician Charles Francis on > the web, and no evidence of any mathematical publications > of anyone of that name. He himself revealed being a mathematician. Most likely he was trained > at university as a mathematician. So maybe he calls himself a mathematician (or maybe not), but his website indicates an interest in physics rather than mathematics, and his contribution to mathematics per se seems invisible. > Archimedes was rather rambling in positive sense as also were Galilei, > Leibniz, Newton, Weyl, Hilbert, v. Neumann and many others. Aha! Blumshein rambles, but so did all these other great names, so Blumscheit is their intellectual equal!!! > Fermat was a jurist, Einstein allegedly was a poor mathematician who > skipped Minkowski's lessons. Why do you intend to make mathematics esoteric? Have you stopped beating your wife? I see it demonstrating that physics depends on possibly questionable > mathematics, not the other way round. Do finitism and deductionism > belong to physics? Didn't misuse of unscrupulously fabricated axioms > spread from mathematics to physics? Again you can't help yourself. You are talking about physics again. But the answer to your rhetorical question is no. I neither know nor care whether physicists use (or misuse) unscrupulously fabricated axioms but since mathematicians do not, the use can't have spread from mathematics. > Aleph_2 is not physical nonsense but mathematical one. What aleph_2 actually is, is the second uncountable cardinal. > Set theory lacks any tangible basis. I conjecture that what you mean by tangible is physical, that is subordinate to the interests of physicists. If so then it is all to the good that mathematics has no such basis! > Dedekind admitted that he did not have any evidence for his guess > that the entity of all rational numbers can be split into larger and > smaller ones wrt his cut. Reference? Do you know what a Dedekind cut is? They are trivial to construct. > For instance, Buridan's donkey has primarily to do with poorly understood > logical fundamentals rather than with physics. Buridan's ass has nothing to do with mathematics. By the way, if you must descend to argument by name-dropping > you might at least have the decency to spell the name of your > uncritically admired hero correctly. I do not understand this hint. I do not uncritically admire Galileo Galilei. > I decided to borrow his synonym Salviati because I would like to stress > that his ultimate conclusion is fully convincing to me and not made > out-dated by Cantor. Not because all sci.math readers had worked out what an arrogant, conceited, self-satisfied, ineducable ignoramus Eckard Blumschein is? > Once again: Do you understand why v. Neumann in 1935 > did no longer believe in Hilbert space? His famous book > was published just a few years earlier in 1932. > The reason for v. Neumann to abandon his belief was most likely > a paper by EPR. > You will certainly agree that Hilbert space is pure mathematics. Hilbert space is a topic in mathematics, and a very important and fertile contribution. But why is it of interest whether or not one particular mathematician may (or may not) have decided not to believe in it? Again, as with all your anti-mathematical arguments, you do not isolate any contradiction in the mathematics but list names of prominent figures who you insinuate (but do not prove) support your position. > By the way, Hilbert was still alife in Germany until he died in 1943, > and after 1935, v. Neumann did not find back to his former > hoity-toity attitude of understanding all. And Eckard Bumschein has found his way back to his comfortable hoity-toity attitide of understanding nothing. Victor Meldrew I don't believe it! === Subject: Re: Very basic mistakes > So maybe he calls himself a mathematician (or maybe not), but his > website indicates an interest in physics rather than mathematics, He is a moderator of sci.physics.foundations, and mathematics is a most important foundation of physics. If there are basic mistakes in mathematics then they may affect physics. > Aha! Blumshein ... Blumscheit ... Eckhard... Blumshien's... !!!! > Why do you intend to make mathematics esoteric? Have you stopped beating your wife? ? Something is esoteric if it is only understood by a small number of people who have special knowledge of it. > I see it demonstrating that physics depends on possibly questionable > mathematics, not the other way round. Do finitism and deductionism > belong to physics? Didn't misuse of unscrupulously fabricated axioms > spread from mathematics to physics? Again you can't help yourself. You are talking about physics again. > But the answer to your rhetorical question is no. I neither know > nor care whether physicists use (or misuse) unscrupulously fabricated > axioms but since mathematicians do not, Are you familiar with the jungle of ZF, ZFC, NF, NBG, ... ? Do you know why mathematics suddenly demanded the pointless quarrel about for instance AC? > the use can't have spread from mathematics. Hilbert intended to also create an axiomatic physics and metaphysics. > Aleph_2 is not physical nonsense but mathematical one. What aleph_2 actually is, is the second uncountable cardinal. While aleph_0 and aleph_1 can be interpreted as the infiniteness of the natural numbers, and the fiction of perfect infinity, respectively, any aleph in excess is a phantasmagoria without any reasonable application since more than a hundred years. Cantor's interpretation of the second diagonal argument is based on assumed comparability. Salviati already understood that for infinite quantities the relations equal, larger or smaller do not apply. Therefore the idea of cardinality lacks any tangible basis. basis. > Set theory lacks any tangible basis. I conjecture that what you mean by tangible is physical, No. I mean logical. Fraenkel in 1923 admitted that there is a so called 4th logical possibility besides =, >, and <: uncomparable. I add: Even the rational number 4 is incomparabel with the real number 4. Berkeley was not so far from the truth when he asked: May we not call them the ghost of departed quantities? A rational number and a (genuine) real number are within quite different categories. > that is subordinate to the interests of physicists. > If so then it is all to the good that mathematics has no > such basis! Application and comprehensive selfconsistency are valuable touchstones for mathematics. > Dedekind admitted that he did not have any evidence for his guess > that the entity of all rational numbers can be split into larger and > smaller ones wrt his cut. Reference? Stertigkeit und irrationale Zahlen, Vieweg, Braunschweig (1872). > For instance, Buridan's donkey has primarily to do with poorly understood > logical fundamentals rather than with physics. Buridan's ass has nothing to do with mathematics. Perhaps you are too ignorant. > By the way, if you must descend to argument by name-dropping > you might at least have the decency to spell the name of your > uncritically admired hero correctly. > I do not understand this hint. I do not uncritically admire Galileo > Galilei. > I decided to borrow his synonym Salviati because I would like to stress > that his ultimate conclusion is fully convincing to me and not made > out-dated by Cantor. Not because all sci.math readers had worked out what an arrogant, > conceited, self-satisfied, ineducable ignoramus Eckard Blumschein > is? If you are unable to reply factually, I am sorry.... > Once again: Do you understand why v. Neumann in 1935 > did no longer believe in Hilbert space? His famous book > was published just a few years earlier in 1932. > The reason for v. Neumann to abandon his belief was most likely > a paper by EPR. > You will certainly agree that Hilbert space is pure mathematics. Hilbert space is a topic in mathematics, and a very important > and fertile contribution. But why is it of interest whether or > not one particular mathematician may (or may not) have decided > not to believe in it? He was not just one mathematician. It was the same Janos, alias Johann alias John v. Neumann who was perhaps the most intelligent pupil of Hilbert and who introduced Hilbert space. Springer, Berlin 1932. > Again, as with all your anti-mathematical > arguments, you do not isolate any contradiction in the mathematics Pointing to a putative very basic mistake is not anti-mathematical. I do not intend to destroy anything but what might be an illusion for millenia. Was the so called fundamental crisis of mathematics actually settled when Hilbert, Fraenkel, and Zermelo managed to ban the contradiction between countable and uncountable out of the very axioms into their interpretation between the lines? In the end, application decides. Let's wait for the outcome of the search for Higgs bosons and the promised performance of quantum computers? Again, do you know why v. Neumann did no longer believe in his own creation Hilbert space? It has to do with completeness as introduced by Dedekind. Salviati === Subject: Blumshein's very basic mistakes <48c273b7$0$7488$9b4e6d93@newsspool4.arcor-online.net> =IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) > So maybe he calls himself a mathematician (or maybe not), but his > website indicates an interest in physics rather than mathematics, He is a moderator of sci.physics.foundations, and mathematics is a most > important foundation of physics. If there are basic mistakes in mathematics > then they may affect physics. Then that is a matter for physics, and a physics newsgroup is more appropriate for such discussion. > æWhy do you intend to make mathematics esoteric? Have you stopped beating your wife? ? Have you? Again you can't help yourself. You are talking about physics again. > But the answer to your rhetorical question is no. I neither know > nor care whether physicists use (or misuse) unscrupulously fabricated > axioms but since mathematicians do not, Are you familiar with the jungle of ZF, ZFC, NF, NBG, ... ? Hardly a (weasel word) jungle. > Do you know why mathematics suddenly demanded the pointless > quarrel about for instance AC? pointless quarrel: loading your terms again, Herr Blumsheit. > the use can't have spread from mathematics. Hilbert intended to also create an axiomatic physics and metaphysics. Then discuss Hilbert's ideas on physics in sci.physics (or worse). > What aleph 2 actually is, is the second uncountable cardinal. While aleph 0 and aleph 1 can be interpreted as the infiniteness > of the natural numbers, and the fiction of perfect infinity, respectively, What aleph 0 and aleph 1 actually are (as opposed to being interpreted as (maybe by no one other than Bumschien)) are the first infinite cardinal and the first uncountable infinite cardinal. > Salviati already understood that for infinite quantities the relations > equal, larger or smaller do not apply. Therefore the idea of cardinality > lacks any tangible basis. > basis. Again arguing from authority, Herr Blumschien. > Set theory lacks any tangible basis. I conjecture that what you mean by tangible is physical, No. I mean logical. Then you are wrong. ZF set theory is formalizable in first-order logic. > Fraenkel in 1923 admitted that there is a > so called 4th logical possibility besides =, >, and <: > uncomparable. Inequality signs are not logical symbols, they are predicate symbols. Incidentally there are partially ordered sets where pairs of elements may be incomparable. A partially ordered set where there are no pairs of incomparable elements is called totally ordered. Examples of totally ordered sets include N, Z, Q and R with the standard ordering. > I add: Even the rational number 4 is incomparabel with the > real number 4. The image of 4 under the natural embedding of Q into R is *equal* to (not incomparable in any sense) to the real number 4. > A rational number and a (genuine) real number are > within quite different categories. There is a natural embedding of the field Q of rationals into the complete ordered field R of reals. In many (but not all contexts) it causes no harm to identify Q with its image in R. > that is subordinate to the interests of physicists. > If so then it is all to the good that mathematics has no > such basis! Application and comprehensive selfconsistency are valuable > touchstones for mathematics. I presume this means that you want to subjugate mathematics to physics. > Dedekind admitted that he did not have any evidence for his guess > that the entity of all rational numbers can be split into larger and > smaller ones wrt his cut. Reference? Stertigkeit und irrationale Zahlen, Vieweg, Braunschweig (1872). Alas I don't read fascist languages :-( > For instance, Buridan's donkey has primarily to do with poorly understood > logical fundamentals rather than with physics. Buridan's ass has nothing to do with mathematics. Perhaps you are too ignorant. Not as ignorant in mathematics as you Herr Blumshcein. But this is typical: instead of demonstrating the relevance of the fable of Buridan's ass to mathematics you resort to insult. > Not because all sci.math readers had worked out what an arrogant, > conceited, self-satisfied, ineducable ignoramus Eckard Blumschein > is? If you are unable to reply factually, I am sorry.... I expect your display of contrition is a lie like the rest of your writings. Of course my quoted sentence is entirely factual. > Hilbert space is a topic in mathematics, and a very important > and fertile contribution. But why is it of interest whether or > not one particular mathematician may (or may not) have decided > not to believe in it? He was not just one mathematician. Really? He was more than one mathematician? Again, as with all your anti-mathematical > arguments, you do not isolate any contradiction in the mathematics Pointing to a putative very basic mistake is not anti-mathematical. There's the rub: you called your posting very basic mistakes yet now you admit that that very basic mistake is singular and only putative. And even so you have failed to point to anything that might fit that description. Victor Meldrew I don't believe it! === Subject: Re: A consideration concerning the diagonal argument of G. Cantor > Exactly so. Did I ever suggest that physics (especially astronomy) is > free from dogmas, oh well, let's say: questionable extrapolations ? > You have clearly suggested that it is the sole source of all truth. I've only suggested that mathematics could borrow some truth from that > source called physics. The trouble is that the objects that mathematics deals with (numbers, sets, etc.) are not physically observable phenomena. That's where any similarity between mathematics and physics breaks down. === Subject: last few digits of a^(a^(a^(a....z times) a and z integers < 1000 =K9O-DgoAAACfJNeR4QSYgwIBRN_weVgi Gecko/20080311 (Debian-1.8.1.13+nobinonly-0ubuntu1) Galeon/2.0.4 (Ubuntu 2.0.4-1ubuntu1),gzip(gfe),gzip(gfe) > hi, > If a is a positive integer, i Need few suggestions on how to find the > last 10 digits of tetration of a number (a < 1000). > a^a^a^a.......(x times), how do we find the last 10 digts of this huge > number.the amount of iterations in a tetration can be 500 .please > suggest how to proceed. what i knew is for numbers only number 1 , for > 2 , if we can know what > a^a^a.....x-1 times is %4 then we can tell the last digit like 2 , 4, > 8,6 but the numbers are quite huge. Any number theory stuff . > I got some hints saying euler phi function could help , but don't know how . Can any of you throw some light. === Subject: Re: last few digits of a^(a^(a^(a....z times) a and z integers < 1000 =HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) hi, > If a is a positive integer, i Need few suggestions on how to find the > last 10 digits of tetration of a number (a < 1000). > a^a^a^a.......(x times), how do we find the last 10 digts of this huge > number.the amount of iterations in a tetration can be 500 .please > suggest how to proceed. what i knew is for numbers only number 1 , for > 2 , if we can know what > a^a^a.....x-1 times is %4 then we can tell the last digit like 2 , 4, > 8,6 but the numbers are quite huge. Any number theory stuff . I got some hints saying euler phi function could help , but don't know > how . Can any of you throw some light. Does this help? 1. http://en.wikipedia.org/wiki/Euler%27s theorem 2. http://www.physicsforums.com/archive/index.php/t-116088.html === Subject: Solution Manual Required =stS0SAoAAACOhfSHYxRjKXEoBVIgsIwr Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Do you have the Solution Manual for the following book: Linear Algebra An Introductory Approach by Charles W. Curtis. Naveed === Subject: Re: Palin does not believe in Darwin Evolution, does McCain or Biden or Obama?? #17 Plutocracy (Science Council) will in the future replace our Democracy =d-ESTAkAAAAG0l03yI1WJgsTVXx4ebeJ Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > It is said that Palin tried to remove the science section of the > library in Alaska. I sincerely hope so! God the crap that passes for science in public education these days! It's no wonder America is down 25 th in the world in science education! Ripping out all that leftist propaganda that passes for science would be a wonderful start! > We know that George Bush does not believe in Darwin Evolution. We know > George Bush is antiscience. Hey I think Darwin's version of evolution is over-blown bull too. I guess that makes me anti-science as well. . All that education for nothing! > When a person is anti-science they have a difficult time of judgement > and priority-keeping and > assessement of reality of the world. Hey, you are hissing like a snake! Well it's a good thing that the British invented the concentration camp and the USSR showed us all the proper way to use them. All you need to do is just gather up all those suspected of anti-Soviet...er...I mean anti-science crimes and throw them in there never to be heard from again. And society keeps it priorities and everyone keeps on enjoying the reality of the world (unless you happen to be one of the ones in the camps, natch). Moron. === Subject: Linear Algebra problem =llxexwoAAADqntGDNkT4ytZG1pkEWVd9 SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; Zune 2.0; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) I've been stuck on this for awhile. Let x = (1,1,1,...1) and y = (1,2,3,...n). Both are in R^n. Let theta_n be the angle between x and y in R^n. Find limit as n-> inf of theta_n. Even though we are looking at infinite sets, it seems to me that I should approach this as I have other practice problems in this section. That means that I need to start by getting the dot product of these two vectors. Algebraically, this is simple since all elements in x are 1's, so I only need to sum the elements in y and I can do that with the formula sum(1,2,3...n) = n(n+1)/2. To find cos(theta), I need to get the magnitude of the two vectors, multiply them times cos(theta) and set that equal to n(n+1)/2, then solve for cos(theta) by dividing by the magnitudes and using the inverse cos function to obtain theta. So far, much of this looks like other practice problems but this is the first one that used infinite sets. Unless someone can tell me that my procedure so far has a flaw in it, it seems as though this should work for this problem just as it has for other problems. Calc II, I have the formula n(n+1)(n+2)/6 as the sum of the squares of the sequence 1^2+2^2+3^2...+n^2. So it looks to me like I need to take the square root of this to get the y magnitude and then multiply it by the sq root of the sum of the terms in the x vector which would simply be sqrt(n). Those two magnitudes times cos(theta) would then be set equal to n(n+a)/2. Then I would divide through and take the arccos of the result. But I keep messing something up. The answer is supposted to be pi/6. I cannot see how that result is achieved. Did I make some obvious error in this process? Some not so obvious error? Or does this look right and I just screwed up the algebra? Alan === Subject: Re: Linear Algebra problem =llxexwoAAADqntGDNkT4ytZG1pkEWVd9 SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; Zune 2.0; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) Calc II, I have the formula n(n+1)(n+2)/6 as the sum of the squares of >the sequence 1^2+2^2+3^2...+n^2. That's where the bug is! æLook that problem up again. -- > Angus Rodgers > Contains mild peril So you're saying that n(n+1)(2n+1)/6 is not the same as n(n+1)(n+2)/ 6 ???? Who'd a thunk it! :) formulas right. Alan === Subject: A distribution that solves u''+au = Delta_y =K8JCNAoAAAAZz7bBcYzz_X9G7WYwtN90 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Show that there exists a distribution u supported in [y,infinity) such that u'' + au = Delta_y where a is a C_infinity function and Delta_y is dirac delta at y. === Subject: Re: A distribution that solves u''+au = Delta_y =aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Show that there exists a distribution u supported in [y,infinity) such > that > u'' + au = Delta y > æ where a is a C infinity function and Delta y is dirac delta at y. If we can find a function satisfying the equation w'' + aw = 0 such that w(y) = 0 and w'(y) = c =/= 0 then let u(x) = -w(x)/c for x >= y and u(x) = 0 for x < y which works, I think. I believe that from the theory of ordinary differential equations we can find w satisfying the above conditions on a neighbourhood of y, but I don't think in general w can be defined over the whole of |R; in the case that it can't I don't know the answer to your question, sorry. To answer your later post in this thread, note that if F is a smooth solution to w'' + aw = 0 then F will not in general have compact support so is not a test function, thus attempting to take F as an argument of u'' + au is invalid. If we allow ourselves to take arbitrary functions as arguments of distributions then we can come up with similar paradoxes, e.g. if u(x) is the Heaviside function so that u' = delta 0 and F is a nonzero constant function with F(x) = c for all x then c = = - = 0 But of course this is incorrect since the proof that = makes use of the fact that F(x) -> 0 for x -> infinity. === Subject: Re: A distribution that solves u''+au = Delta_y > Show that there exists a distribution u supported in [y,infinity) such > that > u'' + au = Delta_y > where a is a C_infinity function and Delta_y is dirac delta at y. > Is this homework? === Subject: Re: A distribution that solves u''+au = Delta_y =K8JCNAoAAAAZz7bBcYzz_X9G7WYwtN90 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) No, its not. This is from an old exam which I'm preparing for. I am notsure how to show exist My point here is if there exists such u, then suppose F is the solution of w''+aw=0 and w(y)=b Then < u'' + au,F > = < u , F''+aF > = 0 and its not Delta y(F) On Sep 5, 5:34æpm, The World Wide Wade Show that there exists a distribution u supported in [y,infinity) such > that > u'' + au = Delta y > æ where a is a C infinity function and Delta y is dirac delta at y. > Is this homework? === Subject: Re: What if: the Church had NOT condemned Galileo What would have happened if the Church had not prosecuted or censured > Galileo? Would Newton have had the same incentive to develop his > comprehensive Copernican explanation of the Universe? Would society > have been destabilized by lack of faith in the Church and conventional > social order? Galileo would not have been under house arrest... might have had more collaborations with others. His influence was not very diminished by the Inquisition... in my opinion. === Subject: Re: What if: the Church had NOT condemned Galileo > Oh, certainly, I agree the Church's objections were psychological -- > or, shall we say, socio-psychological. They feared social > disruption from crticism of the current conception of things without a > comprehensive alternative being presented. As for Newton being just a > link in the chain, Newton's Principia is a pretty big link! And, > probably the final one. A truly comprenhensive system of things to be > put alongside Thomas Aquinas' work. I doubt Galileo was capable of > work of this type, I suspect he was more of an engineer than a > scientist. And, he got out of his depth, to his cost. Galileo's work with the inclined plane alone qualifies him as a great scientist. -- Michael Press === Subject: Re: What if: the Church had NOT condemned Galileo > If anyone wants to know a bit more about the development of the work of > Copernicus, Kepler and Galileo it's well worth finding the Koestler book.- Hide quoted text - - Show quoted text - insulting the officials. i don't know if he's right or not, ?but he > deserves the benefit of the doubt. Which he is the one who deserves the benefit of the doubt here? The > honored historical figure, or the writer expressing an opinion about him? > (Surely not actual historians, who don't rate a mention.) One way of > come back with, let's say, three clear instances of his insulting the > officials. The first, I know, will be the big famous one, which is actually false or > at least an unfounded charge. I'm curious to see what the other two will > be. -- > Dan Drake > d...@dandrake.comhttp://www.dandrake.com/ > porlockjr.blogspot.com Galileo was trying to advance his own views at the expense of those > with more power than he had. ?This does tend to get one into > difficulties, in general, unless your evidence is overwhelming -- > which Galileo's was not, at the time. ?He was able to challenge the > existing system, not establish a new one, as Newton did. Newton was just one more link in the chain of reason. The heliocentric > system already made mush more sense at the time of Kepler and Galileo. > All Newton was to do was to describe it all mathematically from a > suggestion made by Robert Hooke. > Of the 2 systems available the heliocentric one was far and away the > most elegant and simple. The church's objections were never scientific > but psychological. Psychological in two ways: first, was that it was > thought that divinely inspired ideology should and could not be > wrong; any gainsaying of church dogma was heretical, and secondly > the heliocentric hypothesis moved the earth away from its special > position at the centre of a relatively small universe to a subordinate > position which (with the evidential lack of stellar parallax evidence) > expanded the distance to the nearest star to unimaginable distances.- Hide quoted text - - Show quoted text - Oh, certainly, I agree the Church's objections were psychological -- > or, shall we say, socio-psychological. They feared social > disruption from crticism of the current conception of things without a > comprehensive alternative being presented. As for Newton being just a > link in the chain, Newton's Principia is a pretty big link! And, > probably the final one. A truly comprenhensive system of things to be > put alongside Thomas Aquinas' work. I doubt Galileo was capable of > work of this type, I suspect he was more of an engineer than a > scientist. And, he got out of his depth, to his cost. > I LOVE the way this is cross-posted to sci.physics so you can tell all the physicist just what a lousy scientist was the man whom so many of them consider the primary founder of their science. But I mean, after all, what do THEY know about phsyics? -- Dan Drake dd@dandrake.com http://www.dandrake.com/ porlockjr.blogspot.com === Subject: Re: What if: the Church had NOT condemned Galileo Galileo, I believe, simply underestimated the Pope. ?He thought his >opponents were fools, and they weren't. >No, he misunderstood his foes. He thought that they were rational and >they were not. ?They could not stand the idea that their views might >be wrong because it meant the bible was wrong. ?Rational people >accept change. The Roman Church has never been irrational. The reasons were just > political, as they are for all revisionisms, and as they are and have > always been in history. Irrational in the scientific sense means unwilling to accept reason. Irrational means not rational, that's all. They were rational in the political sense as they wanted to keep their > power and priviledge. You are simplifying too much. We are not talking about this or that > priest and not even about this or that Pope. We are talking about > events at the transition from the middle ages to the modern epoch. The church folk lived very well, for example > the Medici pope who spent all the church's money on wine, women and > song and had to start selling indulgances to finance his parties. This is not even a legend. I'm genuinely curious here: which part of that mini-capsule account is so seriously wrong (or, if you prefer, not even wrong)? That the Papacy was grossly corrupt, a treasure for powerful families who fought and killed each other for it? That Popes in and and before the time of Luther -- most famously named Borgia, but that's discriminating against the other families -- gave high ecclesiastical positions to their childre(Sons. The daughters were profitable to marry off.) That some of the Popes were as profligate with the money as with their morals, and badly needed money all the time? (For which we got some really fine art, but they still needed the money) That indulgences were sold for revenue? I'm curious where your revisionism comes in, and how you account for the Counter-Reformation trying to stop some of these mythological abuses. Although that did lead to the Protestant reformation which diluted > the power of the church. And this too: the reasons for the protestant reformation, and then the > anglican, are again mostly political! The power and influence of the > church had been overwhelming on all european countries for centuries. > The reason why those reformations could happen is simply because the > various kings and emperors had had enough: now they wanted their own > empires. Single-factor theories of huge historical movements are always more or less wrong. That includes the ones that attribute everything to high-monded philosophical ideas as well as those which make it all politics all the time. -- Dan Drake dd@dandrake.com http://www.dandrake.com/ porlockjr.blogspot.com === Subject: Re: What if: the Church had NOT condemned Galileo =F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Galileo, I believe, simply underestimated the Pope. ¯He thought his >opponents were fools, and they weren't. >No, he misunderstood his foes. He thought that they were rational and >they were not. ¯They could not stand the idea that their views might >be wrong because it meant the bible was wrong. ¯Rational people >accept change. The Roman Church has never been irrational. The reasons were just > political, as they are for all revisionisms, and as they are and have > always been in history. Irrational in the scientific sense means unwilling to accept reason. Irrational means not rational, that's all. They were rational in the political sense as they wanted to keep their > power and priviledge. You are simplifying too much. We are not talking about this or that > priest and not even about this or that Pope. We are talking about > events at the transition from the middle ages to the modern epoch. The church folk lived very well, for example > the Medici pope who spent all the church's money on wine, women and > song and had to start selling indulgances to finance his parties. This is not even a legend. I'm genuinely curious here: which part of that mini-capsule account is so > seriously wrong (or, if you prefer, not even wrong)? The fact that that is only the surface, something good for people to think about matters as if it were a soap opera, but historic analysis has nothing to do with it. It is actually even more complicated than that, as this shift of perspective from systemic dynamics to the behaviour of the single is just another aspect of the flattening of all perspectives to one-dimensional analysis. I'll be even (slightly) more specific: this was started on a systemic and global level in the 50's, when the Institute for World Mental Health, taking all cyberneticians advise and putting in practice the exact opposite of it, established that all anti-social/non-functional behaviour is an individual matter, rather than a systemic problem intrinsic to an unbalanced social system. So, if people are anti-social, is not because this society is un-human, it's because they need treatment or jail. Or, on a bigger scale, terrorism is not due to global dynamics of power and exploitation, no, it's because this or that guy is a religious fanatic. Etc. etc., you get the picture. (And let's don't tell Bin Laden is a bad guy. The point is indeed that they are the two sides of the very same coin.) > That the Papacy was grossly corrupt, a treasure for powerful families who > fought and killed each other for it? That Popes in and and before the time > of Luther -- most famously named Borgia, but that's discriminating against > the other families -- gave high ecclesiastical positions to their > childre(Sons. The daughters were profitable to marry off.) That some of > the Popes were as profligate with the money as with their morals, and > badly needed money all the time? (For which we got some really fine art, > but they still needed the money) That indulgences were sold for revenue? > I'm curious where your revisionism comes in, and how you account for the > Counter-Reformation trying to stop some of these mythological abuses. Again, that's just the surface and the soap opera. > Although that did lead to the Protestant reformation which diluted > the power of the church. And this too: the reasons for the protestant reformation, and then the > anglican, are again mostly political! The power and influence of the > church had been overwhelming on all european countries for centuries. > The reason why those reformations could happen is simply because the > various kings and emperors had had enough: now they wanted their own > empires. Single-factor theories of huge historical movements are always more or > less wrong. > That includes the ones that attribute everything to high-monded > philosophical ideas as well as those which make it all politics all the > time. It's a matter of what is relevant to a (true) historical analysis. The one-dimensionality is a modern thing, and it is a large-scale control strategy, so that people have no instruments to think *critically* (which is a scientific term here). -LV === Subject: Re: What if: the Church had NOT condemned Galileo > Nine years before, the Roman Catholics burned Giordano Bruno, locking > his mouth in an iron gag with a spike through his tongue to prevent > heretical observations like the stars are sums and planets orbit around > them. The apologists claim that, as an act of mercy, the Inquisition slit > Bruno's throat before they burned him. ?There is no factual declaration > that happened: ?just speculation by devout Catholics trying to > white-wash the non defenseable. The bureaucratic murderer whose victims included Giordano Bruno was > Robert Bellarmine, and that killer is a effing saint by Papal decree. By the way, Vatican toadies ought to mention the nature of Gaileo's > arrogance. ?Way back then, in imitation of Plato, scientific topics were > presented as dialogue between characters in a playlet. ?Gaileo was bold > enough to put in the words of a fool some arguments that the Pope > himself had privately presented to the scholar. That is enough to kill a > man over. And the Church demonstrated clemency just by ruining the guy's > life. The true testimony of what anyone thinks and does is when you give them > absolute power over a situation and observe what they say and do then, Giordano Bruno did everything short of begging the Inquisition to burn > him... Oh, but backing up a step: This, naturally, makes the Church's actions reasonable enough, from your point of view. I fear that among those who do *not* think that torturing people to death is an appropriate response to incorrect views -- even for stubbornness! -- even for a martyr complex!! -- it will not carry much weight. -- Dan Drake dd@dandrake.com http://www.dandrake.com/ porlockjr.blogspot.com === Subject: Re: What if: the Church had NOT condemned Galileo > Nine years before, the Roman Catholics burned Giordano Bruno, locking > his mouth in an iron gag with a spike through his tongue to prevent > heretical observations like the stars are sums and planets orbit around > them. The apologists claim that, as an act of mercy, the Inquisition slit > Bruno's throat before they burned him. ?There is no factual declaration > that happened: ?just speculation by devout Catholics trying to > white-wash the non defenseable. The bureaucratic murderer whose victims included Giordano Bruno was > Robert Bellarmine, and that killer is a effing saint by Papal decree. By the way, Vatican toadies ought to mention the nature of Gaileo's > arrogance. ?Way back then, in imitation of Plato, scientific topics were > presented as dialogue between characters in a playlet. ?Gaileo was bold > enough to put in the words of a fool some arguments that the Pope > himself had privately presented to the scholar. That is enough to kill a > man over. And the Church demonstrated clemency just by ruining the guy's > life. The true testimony of what anyone thinks and does is when you give them > absolute power over a situation and observe what they say and do then, Giordano Bruno did everything short of begging the Inquisition to burn > him. His introduction to De l'Infinito Universo et Mondi, probably > his most important work, in which he postulates an infinite universe, > actually paraphrases Isaiah's discussion of the Man of Sorrows, the > christ-like saviour who sacrifices himself for the good of humanity. > Bruno saw himself in these terms. He was out of the Inquisition's > power, and chose, voluntarily to return to Italy. I not to get much involved in the Bruno business, which often functions as a red herring; but sometimes there are matters of fact involved (one fact being that there is no copy extant of the Inquisition's judgment against him, so we really don't *know* what they condemned him for, the way we do with Galileo). But here's a fact: when he chose to return to Italy, it was by express invitation to Venice, where and when the Inquisition was *not* after him. He even applied for and got a professorship there (Padua) before somebody decided he was too hot to handle. You know, like Bertrand Russell getting a post in New York till the [ethnic and religious characterization omitted] Yahoos who dominated New York politics got him kicked for being an Atheist and Communist [do not correct me here as the facts; correct _them_]. (I'm not as big a fan of Russell as it may seem, but he does come up now and then.) Then the Venetian Inquisition -- put onto it by the very guy who'd invited him -- got after Bruno, and then handed him over to the Roman I. If his travels were raised to prove his martyr complex, they fail in that; rather, the betrayal does sort of remind you of somebody else who was executed for being too troublesome. He was imprisoned > for seven years during which he refused to renounce his views. He > knew that martyrdom would attract attention to his ideas. Galileo, I believe, simply underestimated the Pope. He thought his > opponents were fools, and they weren't. > Answered too many times to merit another. -- Dan Drake dd@dandrake.com http://www.dandrake.com/ porlockjr.blogspot.com === Subject: Re: What if: the Church had NOT condemned Galileo > If anyone wants to know a bit more about the development of the work of > Copernicus, Kepler and Galileo it's well worth finding the Koestler book.- Hide quoted text - - Show quoted text - insulting the officials. i don't know if he's right or not, but he > deserves the benefit of the doubt. Which he is the one who deserves the benefit of the doubt here? The > honored historical figure, or the writer expressing an opinion about him? > (Surely not actual historians, who don't rate a mention.) One way of > come back with, let's say, three clear instances of his insulting the > officials. The first, I know, will be the big famous one, which is actually false or > at least an unfounded charge. I'm curious to see what the other two will > be. Galileo was trying to advance his own views at the expense of those > with more power than he had. This does tend to get one into > difficulties, in general, unless your evidence is overwhelming -- > which Galileo's was not, at the time. He was able to challenge the > existing system, not establish a new one, as Newton did. Newton was just one more link in the chain of reason. The heliocentric > system already made mush more sense at the time of Kepler and Galileo. > All Newton was to do was to describe it all mathematically from a > suggestion made by Robert Hooke. > Of the 2 systems available the heliocentric one was far and away the > most elegant and simple. The church's objections were never scientific > but psychological. Psychological in two ways: first, was that it was > thought that divinely inspired ideology should and could not be > wrong; any gainsaying of church dogma was heretical, and secondly > the heliocentric hypothesis moved the earth away from its special > position at the centre of a relatively small universe to a subordinate > position which (with the evidential lack of stellar parallax evidence) > expanded the distance to the nearest star to unimaginable distances. Or it could be the RCC entered into a contract where they paid > Copernicus to develop a method to calculate the date of Easter. > Copernicus delivered a correct, easily calculated method. Implicit in, > but not necessary to, his method is a heliocentric model. Since > the project was funded by the RCC, they contended that they had > the rights; and prosecuted those who used it without permission. > The Intellectual Property theory. Oh joy, warring sect appears over the horizon! Write a book, and you could get some people believing it. -- Dan Drake dd@dandrake.com http://www.dandrake.com/ porlockjr.blogspot.com === Subject: Re: What if: the Church had NOT condemned Galileo If anyone wants to know a bit more about the development of the work of > Copernicus, Kepler and Galileo it's well worth finding the Koestler book.- Hide quoted text - - Show quoted text - insulting the officials. i don't know if he's right or not, but he > deserves the benefit of the doubt. Which he is the one who deserves the benefit of the doubt here? The > honored historical figure, or the writer expressing an opinion about him? > (Surely not actual historians, who don't rate a mention.) One way of > come back with, let's say, three clear instances of his insulting the > officials. The first, I know, will be the big famous one, which is actually false or > at least an unfounded charge. I'm curious to see what the other two will > be. Galileo was trying to advance his own views at the expense of those > with more power than he had. This does tend to get one into > difficulties, in general, unless your evidence is overwhelming -- > which Galileo's was not, at the time. He was able to challenge the > existing system, not establish a new one, as Newton did. Newton was just one more link in the chain of reason. The heliocentric > system already made mush more sense at the time of Kepler and Galileo. > All Newton was to do was to describe it all mathematically from a > suggestion made by Robert Hooke. > Of the 2 systems available the heliocentric one was far and away the > most elegant and simple. The church's objections were never scientific > but psychological. Psychological in two ways: first, was that it was > thought that divinely inspired ideology should and could not be > wrong; any gainsaying of church dogma was heretical, and secondly > the heliocentric hypothesis moved the earth away from its special > position at the centre of a relatively small universe to a subordinate > position which (with the evidential lack of stellar parallax evidence) > expanded the distance to the nearest star to unimaginable distances. Or it could be the RCC entered into a contract where they paid > Copernicus to develop a method to calculate the date of Easter. > Copernicus delivered a correct, easily calculated method. Implicit in, > but not necessary to, his method is a heliocentric model. Since > the project was funded by the RCC, they contended that they had > the rights; and prosecuted those who used it without permission. > The Intellectual Property theory. Oh joy, warring sect appears over the > horizon! Write a book, and you could get some people believing it. Intellectual property is a contentious matter today. Why not then? Or maybe Galileo was not taken to task for his work with the telescope, but rather his work with the microscope. -- Michael Press === Subject: Re: What if: the Church had NOT condemned Galileo =F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Intellectual property is a contentious matter today. Why not then? Because the notion of intellectual property makes sense only within the contemporary socio-economic system, based on social automization and private property. In ancient societies, all belonged to God, or to the King, or to some combination/variation of these notions. And the transition from the ancient model (which we still can see in some traditional societies of today) based on unifying spiritual principles (an organic point of view), to the modern model based on the contrary on autonomization and everybody on his own, corresponds in fact to the transition from the medioeval society to the mercantil society. -LV === Subject: Re: What if: the Church had NOT condemned Galileo If anyone wants to know a bit more about the development of the work of >Copernicus, Kepler and Galileo it's well worth finding the Koestler book.- Hide quoted text - >- Show quoted text - insulting the officials. i don't know if he's right or not, but he >deserves the benefit of the doubt. >Which he is the one who deserves the benefit of the doubt here? The >honored historical figure, or the writer expressing an opinion about him? >(Surely not actual historians, who don't rate a mention.) One way of >come back with, let's say, three clear instances of his insulting the >officials. >The first, I know, will be the big famous one, which is actually false or >at least an unfounded charge. I'm curious to see what the other two will >be. Galileo was trying to advance his own views at the expense of those >with more power than he had. This does tend to get one into >difficulties, in general, unless your evidence is overwhelming -- >which Galileo's was not, at the time. He was able to challenge the >existing system, not establish a new one, as Newton did. >Newton was just one more link in the chain of reason. The heliocentric >system already made mush more sense at the time of Kepler and Galileo. >All Newton was to do was to describe it all mathematically from a >suggestion made by Robert Hooke. >Of the 2 systems available the heliocentric one was far and away the >most elegant and simple. The church's objections were never scientific >but psychological. Psychological in two ways: first, was that it was >thought that divinely inspired ideology should and could not be >wrong; any gainsaying of church dogma was heretical, and secondly >the heliocentric hypothesis moved the earth away from its special >position at the centre of a relatively small universe to a subordinate >position which (with the evidential lack of stellar parallax evidence) >expanded the distance to the nearest star to unimaginable distances. Or it could be the RCC entered into a contract where they paid >Copernicus to develop a method to calculate the date of Easter. >Copernicus delivered a correct, easily calculated method. Implicit in, >but not necessary to, his method is a heliocentric model. Since >the project was funded by the RCC, they contended that they had >the rights; and prosecuted those who used it without permission. >The Intellectual Property theory. Oh joy, warring sect appears over the >horizon! Write a book, and you could get some people believing it. > Intellectual property is a contentious matter today. Why not then? Or maybe Galileo was not taken to task for his work with the telescope, > but rather his work with the microscope. > You do know that there are history books where you can look things up don't you? === Subject: Re: What if: the Church had NOT condemned Galileo Galileo, I believe, simply underestimated the Pope. He thought his >opponents were fools, and they weren't. >No, he misunderstood his foes. He thought that they were rational and >they were not. They could not stand the idea that their views might >be wrong because it meant the bible was wrong. Rational people >accept change. >The Roman Church has never been irrational. The reasons were just >political, as they are for all revisionisms, and as they are and have >always been in history. >Irrational in the scientific sense means unwilling to accept reason. >Irrational means not rational, that's all. >They were rational in the political sense as they wanted to keep their >power and priviledge. >You are simplifying too much. We are not talking about this or that >priest and not even about this or that Pope. We are talking about >events at the transition from the middle ages to the modern epoch. >The church folk lived very well, for example >the Medici pope who spent all the church's money on wine, women and >song and had to start selling indulgances to finance his parties. >This is not even a legend. > Sorry, I see this is actually on Wikipedia. But it is far from a > historic perspective: this is rather just the gossip and the folklore. > Those guys were not stupid, they have never been. -LV There is a differnence between being stupid and power hungry. They were very cunning in their politics. Including the one who had the father of Pope Clement VII killed in the Duomo in Milan. It is interesting that even though a Pope killed on Medici, two of his near family became Popes and nearly ruined the church. >Although that did lead to the Protestant reformation which diluted >the power of the church. >And this too: the reasons for the protestant reformation, and then the >anglican, are again mostly political! The power and influence of the >church had been overwhelming on all european countries for centuries. >The reason why those reformations could happen is simply because the >various kings and emperors had had enough: now they wanted their own >empires. >-LV === Subject: Re: What if: the Church had NOT condemned Galileo >Galileo, I believe, simply underestimated the Pope. æHe thought his >opponents were fools, and they weren't. No, he misunderstood his foes. He thought that they were rational and >they were not. æThey could not stand the idea that their views might >be wrong because it meant the bible was wrong. æRational people >accept change. >The Roman Church has never been irrational. The reasons were just >political, as they are for all revisionisms, and as they are and have >always been in history. Irrational in the scientific sense means unwilling to accept reason. >Irrational means not rational, that's all. They were rational in the political sense as they wanted to keep their >power and priviledge. >You are simplifying too much. We are not talking about this or that >priest and not even about this or that Pope. We are talking about >events at the transition from the middle ages to the modern epoch. The church folk lived very well, for example >the Medici pope who spent all the church's money on wine, women and >song and had to start selling indulgances to finance his parties. >This is not even a legend. Sorry, I see this is actually on Wikipedia. But it is far from a > historic perspective: this is rather just the gossip and the folklore. > Those guys were not stupid, they have never been. -LV There is a differnence between being stupid and power hungry. æThey > were very cunning in their politics. Including the one who had > the father of Pope Clement VII killed in the Duomo in Milan. It is > interesting that even though a Pope killed on Medici, two of his > near family became Popes and nearly ruined the church. What is interesting? That people are power hungry since the beginning of time? You still don't get my point: there is nothing weird or misterious going on here, you just don't know the *historical* reasons why that happened, and are left with speculations. For instance, that killing was in fact to get a Medici Pope: there just is nothing weird, as that was the very goal of it. It's rather true that 99% of the history books around aren't better than that brasilian soap opera. -LV >Although that did lead to the Protestant reformation which diluted >the power of the church. >And this too: the reasons for the protestant reformation, and then the >anglican, are again mostly political! The power and influence of the >church had been overwhelming on all european countries for centuries. >The reason why those reformations could happen is simply because the >various kings and emperors had had enough: now they wanted their own >empires. >-LV === Subject: Re: What if: the Church had NOT condemned Galileo > >Galileo, I believe, simply underestimated the Pope. He thought his >opponents were fools, and they weren't. >No, he misunderstood his foes. He thought that they were rational and >they were not. They could not stand the idea that their views might >be wrong because it meant the bible was wrong. Rational people >accept change. >The Roman Church has never been irrational. The reasons were just >political, as they are for all revisionisms, and as they are and have >always been in history. >Irrational in the scientific sense means unwilling to accept reason. >Irrational means not rational, that's all. >They were rational in the political sense as they wanted to keep their >power and priviledge. >You are simplifying too much. We are not talking about this or that >priest and not even about this or that Pope. We are talking about >events at the transition from the middle ages to the modern epoch. >The church folk lived very well, for example >the Medici pope who spent all the church's money on wine, women and >song and had to start selling indulgances to finance his parties. >This is not even a legend. >Sorry, I see this is actually on Wikipedia. But it is far from a >historic perspective: this is rather just the gossip and the folklore. >Those guys were not stupid, they have never been. >-LV >There is a differnence between being stupid and power hungry. They >were very cunning in their politics. Including the one who had >the father of Pope Clement VII killed in the Duomo in Milan. It is >interesting that even though a Pope killed on Medici, two of his >near family became Popes and nearly ruined the church. > What is interesting? That people are power hungry since the beginning > of time? You still don't get my point: there is nothing weird or > misterious going on here, you just don't know the *historical* reasons > why that happened, and are left with speculations. For instance, that > killing was in fact to get a Medici Pope: there just is nothing weird, > as that was the very goal of it. The killing was not to get a Medici pope, it was the pope's desire to get rid of that Medici. It's rather true that 99% of the > history books around aren't better than that brasilian soap opera. > I have no idea what a brasilian soap opera is but a lot of soap operas come from real life. Look at the US elections coming up as an example. > -LV >Although that did lead to the Protestant reformation which diluted >the power of the church. >And this too: the reasons for the protestant reformation, and then the >anglican, are again mostly political! The power and influence of the >church had been overwhelming on all european countries for centuries. >The reason why those reformations could happen is simply because the >various kings and emperors had had enough: now they wanted their own >empires. >-LV === Subject: Re: What if: the Church had NOT condemned Galileo > Galileo, I believe, simply underestimated the Pope. He thought his >opponents were fools, and they weren't. >No, he misunderstood his foes. He thought that they were rational and >they were not. They could not stand the idea that their views might >be wrong because it meant the bible was wrong. Rational people >accept change. >The Roman Church has never been irrational. The reasons were just >political, as they are for all revisionisms, and as they are and have >always been in history. >Irrational in the scientific sense means unwilling to accept reason. > Irrational means not rational, that's all. And being rational means accepting reason. >They were rational in the political sense as they wanted to keep their >power and priviledge. > You are simplifying too much. We are not talking about this or that > priest and not even about this or that Pope. We are talking about > events at the transition from the middle ages to the modern epoch. > They ALL wanted to keep their lifestyle going at whatever the period. The church folk lived very well, for example >the Medici pope who spent all the church's money on wine, women and >song and had to start selling indulgances to finance his parties. > This is not even a legend. > No, it is the truth. Read some history. Although that did lead to the Protestant reformation which diluted >the power of the church. > And this too: the reasons for the protestant reformation, and then the > anglican, are again mostly political! The power and influence of the > church had been overwhelming on all european countries for centuries. > The reason why those reformations could happen is simply because the > various kings and emperors had had enough: now they wanted their own > empires. > Read some real history. Martin Luther did not like the selling of indulgances and, for his opposition, he was excommunicated. The split started. > -LV === Subject: Re: What if: the Church had NOT condemned Galileo =F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Galileo, I believe, simply underestimated the Pope. æHe thought his >opponents were fools, and they weren't. >No, he misunderstood his foes. He thought that they were rational and >they were not. æThey could not stand the idea that their views might >be wrong because it meant the bible was wrong. æRational people >accept change. The Roman Church has never been irrational. The reasons were just >political, as they are for all revisionisms, and as they are and have >always been in history. >Irrational in the scientific sense means unwilling to accept reason. Irrational means not rational, that's all. And being rational means accepting reason. >They were rational in the political sense as they wanted to keep their >power and priviledge. You are simplifying too much. We are not talking about this or that > priest and not even about this or that Pope. We are talking about > events at the transition from the middle ages to the modern epoch. They ALL wanted to keep their lifestyle going at whatever the period. Lifestyle is for the simple man. We are talking about an empire that has yet today no equivalent. >The church folk lived very well, for example >the Medici pope who spent all the church's money on wine, women and >song and had to start selling indulgances to finance his parties. This is not even a legend. No, it is the truth. æRead some history. I have grown up there. My family comes from there. My ancestors were there. Some of them were priest eaters as we call them; while others, on the contrary, were priests or had studied in seminary. Also, Italy is a delirium country, no objection, but at least our education (Bologna is the most ancient university in the world) is still not yet completely based on performative skills. >Although that did lead to the Protestant reformation which diluted >the power of the church. And this too: the reasons for the protestant reformation, and then the > anglican, are again mostly political! The power and influence of the > church had been overwhelming on all european countries for centuries. > The reason why those reformations could happen is simply because the > various kings and emperors had had enough: now they wanted their own > empires. Read some real history. Martin Luther did not like the selling of > indulgances and, for his opposition, he was excommunicated. The > split started. Whyever Luther did what he did, he could do it because he had the necessary support. -LV === Subject: Re: Samir and Petra : The Sum and Product Problem =5cbwMAoAAAAZoYdmWfebkP1kkdh09sZi CLR 1.1.4322; .NET CLR 3.0.04506.30; InfoPath.1; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) To everyone out there, æ æ æI've been facing the following problem for weeks and haven't been > able to solver it : I have two numbers x,y such that x+y < 100 and 1 < x < y. I tell the product toPetra, and the sum toSamir. After that, the following conversation takes place : Petra: I don't know what x and y are. >Samir: I knew that already. >Petra: Aha, now I know what x and y are! >Samir: So do I. What are x and y? I have been able to figure out that both x and y cannot be prime at > the same time, because thenPetrawould be able to decompose xy into > it's prime factors and figure out what x and y are. I have also been > able to figure out that the sum of x and y cannot be even, as any even > number greater than two can be expressed as the sum of the two primes > (If the sum were even,Samirwould not be sure whether or notPetra > knows what x and y are.) How do I go from here? > You are starting well. æIn addition, the product cannot be the cube of > a prime, nor can the sum be 2 more than a prime for the same reasons. > As big numbers have more decompositions than small numbers, I would > look at small candidates for the sum, then see if the conversation > makes sense. æNote that the product must have more than one > decomposition, but that only one of those can meet the (sum not two > primes nor 2+prime) criterion, and also thatSamirdoesn't know the > numbers untilPetrasays she does- Hide quoted text - - Show quoted text - more than a prime, can the sum then be 2 more than an odd composite number? What about 4 or 6 or any even number more than an odd composite number? What aobut 4 or any even number more than a prime?? === Subject: Re: Mathematics: how to start again > I had studied mathematics for 16 years. Now I am away from it for 6 > years. I know I like it, so I want to start it again. No goals, no > deadlines, and I can allocate 4 hours a week. Now, I have some > questions in my mind like, from where I can start, how to proceed, > etc., so I thought I could ask here. You could restart here: Courant & Robbins, WHAT IS MATHEMATICS? -- My first suggestion was going to be exactly that. However, if he has studied maths for 16 years, I would hope that he already knows what is mathematics. Its still a good suggestion. === Subject: Re: GPS Math I saw it on the History Channel... three giagantic spheres looming over the planet Earth, and the narrator said, GPS is based on the intersection of three spheres from three satellite transmissions, at two points. All I did was calculate those two points. ALRIGHT? Include yourself among the others in instead on relying on others. I made a life out of Bouguer anomalies and corrections. WHO NEEDS THEM? -- Jon G. jon8338@peoplepc.com http://mypeoplepc.com/members/jon8338/math/index.html > Here is the math that calculates position from 3 satellites. There are 2 > points of intersection of 3 spheres of known radii, which are determined > by > time-stamped transmissions from the satellites. http://www.geocities.com/jongiff2000/GPS_math.xls Jon Giffen > jon8...@peoplepc.com Pick up a book (see Spilker, Tsui, or Enge & Misra) and do some reading instead of relying on others or the Internet solely for your info. As you are hopefully learning, this is not a very reliable strategy. But if you find this all too taxing for your mind, then at last read http://en.wikipedia.org/wiki/Pseudorange (*read*, don't *skim*, got it) M === Subject: Re: GPS Math <57KdnZg2DpGqR1zVnZ2dnUVZ_vzinZ2d@earthlink.com> =3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs 1.7; .NET CLR 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) > I saw it on the History Channel... three giagantic spheres looming over the > planet Earth, and the narrator said, GPS is based on the intersection of > three spheres from three satellite transmissions, at two points. > Wow! The History Channel! What a solid scientific source. Dude, I am like so impressed. Forgive me for ever doubting you for even a time- stamped nano-second. BTW, did you ever figure out what the pseudorange was? M === Subject: Re: tensor analysis Apparently you didn't catch my drift. Separated by space, but space and time is UNITY; otherwise in a steady state among other Lorentz specs. The species is at ONE in a deepest subconscious way. Those few who have figured it out EXPLOIT IT, but are misinformed. The steeple of the cathedral can't swap with its foundation without repercussions. but tell them that. I think your statement is virtually true, if I could figure out what you were talking about. > Jon G. a .8ecrit : > What does this mean? > It means if you take a small but noninfinitesimal point, map an event > with > the point as the origin, turn the point inside-out, expand the point into > a > sphere of radius r, and map the event from the inner surface of the > sphere > normally inward.... then the experience is the same. > I suggest that space is everywhere expanding and contracting at a > frequency > that releases EMR to propagate along the path of least resistance. > I suggest that atoms manifest as inversions of higher matter that > collapsed > on itself and turned inside-out into probability fields. > I also suggest that light exists by inheritance from the singularity, and > is > an integral part of space and time, like a continuously generating gene > from > the past. It has enough momentum to continue radiating; the fabric of > space > and time remembers its place in the primordial universe. > I also suggest that the borogrovs ecorcomaliss themselves, and emparooy > you until your drabble. Don't laugh at me : I'm very serious ! > ;) === Subject: Re: tensor analysis <487a0eba$0$21143$7a628cd7@news.club-internet.fr> among other Lorentz specs. The species is at ONE in a deepest subconscious > way. Those few who have figured it out EXPLOIT IT, but are misinformed. > The steeple of the cathedral can't swap with its foundation without > repercussions. I miss Peter Olcott. === Subject: Re: Is Geometric Algebra a valid subject? =Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) On Sep 3, 6:32 am, Timothy Golden BandTechnology.com > The text by Harris never introduces any product that I can see. > So even the naming of this topic is a controversy. > How two groups can use the same title for their subject is beyond me. > As I read Harris, the constraints on the math up front don't look so > good. > But this is not a book for a one day review. > Regardless, Harris is not anything like Dorst and Mann. > So whatever criticism I have may not go back to Grassman. > Where is the geometric product introduced? > Where is the wedge product introduced? > Nowhere as far as I can tell within Harris. > The use of a graphical carat as a symbol is so far the only > confluence. my suggestion of harris was because it has some pretty good problems that can help get better geometric intuitions the very first exercise (quizzically called 6.2) works you through building the n-choose-4 quadratic relations that cut the plucker variety (the embedding) it works through computations with many of the geometric intuitions needed intersections of lines and planes with varieties general k-plane fibrations joins of varieties through parametrisation of line families basically the foundations of really understanding how grassmannians arise in the foliated constructions > What then should we call algebraic geometry? > Who is misusing this term? definitely the physicists it's the stupidest rebranding i've seen in a while just stick to calling them clifford algebras and use them as penrose and rindler and dirac and ... > My criticism of Dorst and Mann and their ilk can stand. > It is not challenged by Harris. > Nowhere yet have I seen any derivation of > (ei^ej)(ei^ej) = - 1 > in Harris. again the products differ from anything i was discussing > Unfortunately the subject Algebraic Geometry may be one of the first > falsifiable internet allocations. A robust barrage of supporting > information from wikipedia to wolfram to PhD papers to university > departments seem to be behind Algebraic Geometry but as I read it > all from my computer screen it leaves me wondering. This is sort of > like a religious breakaway. The integrity of this faction is > falsifiable and I predict that their push will wither. Should we burn > their books? Here is the trouble with rejection. Within a given > university the books might best be regarded in terms of acceptance or > rejection no different than one author might opt for one method over > another in the presentation of material. The library bookshelves will > eventually reflect these religious tendencies, particularly in lieu of > accumulation. Hence the short set of works that are considered the > best and complete will be a matter of contention amongst professors, > these professors then conglomerating in united fronts which form > factions. The shelves of libraries are not currently organized this > way so this is a prediction of the future. no algebraic geometry has already united huge swaths of math from number theory to differential geometry and many of their deep interactions (the theory of equations over number fields differential equations ...) from k-theory to hypergeometrics and mirror symmetry algebraic geometry has shown itself a powerful tool -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Is Geometric Algebra a valid subject? =Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) tg: > g: > (x ^ y) ^ (x ^ y) = -2 > v > (even if 3D) x^x = 0 > But this zero relation of v^v is apparently only true of a first rank > (1-blade) vector v^v. Since this is a second rank (2-blade) we cannot > admit this. Wolfram states that pretty clearly. i think i was responding to a typo which has now happened twice in this thread with two different posters possibly that is the error of this entire thread different products are being discussed and regularly wedges are getting introduced accidentally (at least where i choose to respond) the wedge is a product just like the clifford product (apparently being called a geometric product by some physicists who got so wowed by it they needed to promote it just wait for supergeometric algebras...) but the wedge product is also like a nail you use it to construct something namely a 2-form (a / b) and a 3-form (x / y / z) and so on these are very much like multivectors because you've just stuck a bunch of vectors together they have the additive structure inherited from the underlying vector space and they want to behave like an algebra in other words it would be nice for the addition to be be distributively compatible with the product (which is how algebraic structures with addition and a product behave) so x / (y + z) = (x / y) + (x / z) (x + y) / z = (x / z) + (y / z) it turns than consistently be used you can still visualise these as bivectors e.g. the product still acts like a nail to construct bivectors but now the notion of addition has been pulled into bivectors wedge products have a little more structure to distinguish them from other products (like the dot) that also commute with addition one of the big pieces of structure for a form is a / b = - b / a the mental picture here is again two vectors which are used to determine an area (of the parallelogram they form 2 sides of) and flipping vectors gives the same area just oriented the other way this way the negation is used here is what gives the wedge product it's huge role in connecting spatial measures length area volume and orientation now just using this antisymmetry law it is easy to derive x / x = - x / x 2 (x / x) = 0 (the zero 2-form) x / x = 0 (the zero 2-form) so this one sticking point is immediately true from just the algebra from the mental picture the two vectors still define a parallelogram it's just that it has become one of zero area you can't build an area with only one vector even if you use some integer multiple of the same vector x / (nx) (n e N) then x / (nx) = (x / x) + (x / x) + ... n times = 0 + 0 + ... = 0 similarly with just two vectors you can't get a volume x / y / (ax + by) = x / y / (ax) + x / y / (by) = - (x / (ax) / y) + x / 0 = 0 + 0 = 0 and so on for you to have a non-zero form you must be determining a positive measure in the n-parallelotope this is why it is different from the dot product which has the exact opposite symmetry (a . b = b . a) just like with wedge products where we aren't attempting to get scalars out one doesn't need to interpret the dot products as a scalar it's just like with the multivector/parallelotope/spatial measure interpretation of wedge scalars are just the natural mental picture to use working coordinate free means getting used to such abstraction we can define a partial ordering of closeness on grassmannians (using the standard topological definition with neighborhoods) and we see that the closer we are to the zero n-form the closer the n-multivector is to linear dependence with the dot product the closer to the zero dot the closer some of the n-vectors are to orthogonality does any of this help? > Anyway if it were true > then I think we'd get > (x^y)(x^y) = + 1 > instead of minus unity which is the trail that lead to this criticism. > We simply take the definition of the geometric product literally to > get > (x^y)(x^y) == (x^y) ^ (x^y) + (x^y) . (x^y) > where == is the definitional application. note that these are all using different products than the equation i responded to i still think that product confusion has been the biggest barrier to communication here -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Is Geometric Algebra a valid subject? =Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) > tg: g: > (x ^ y) ^ (x ^ y) = -2 > v > (even if 3D) x^x = 0 > But this zero relation of v^v is apparently only true of a first rank > (1-blade) vector v^v. Since this is a second rank (2-blade) we cannot > admit this. Wolfram states that pretty clearly. i think i was responding to a typo > which has now happened twice in this thread > with two different posters possibly that is the error of this entire thread different products are being discussed > and regularly wedges are getting introduced accidentally > (at least where i choose to respond) the wedge is a product > just like the clifford product > (apparently being called a geometric product > by some physicists who got so wowed by it > they needed to promote it just wait for supergeometric algebras...) but the wedge product is also like a nail > you use it to construct something namely a 2-form (a / b) > and a 3-form (x / y / z) > and so on these are very much like multivectors > because you've just stuck a bunch of vectors together they have the additive structure inherited from the underlying vector > space > and they want to behave like an algebra in other words > it would be nice for the addition > to be be distributively compatible with the product > (which is how algebraic structures with addition and a product behave) so x / (y + z) = (x / y) + (x / z) > (x + y) / z = (x / z) + (y / z) it turns than consistently be used it turns out that this can consistently be used somehow that this can -> than in editing > you can still visualise these as bivectors > e.g. the product still acts like a nail to construct bivectors > but now the notion of addition has been pulled into bivectors wedge products have a little more structure > to distinguish them from other products > (like the dot) > that also commute with addition one of the big pieces of structure for a form > is a / b = - b / a the mental picture here is again two vectors > which are used to determine an area > (of the parallelogram they form 2 sides of) > and flipping vectors gives the same area > just oriented the other way this way the negation is used here > is what gives the wedge product it's huge role > in connecting spatial measures > length > area > volume > and orientation now > just using this antisymmetry law > it is easy to derive x / x = - x / x > 2 (x / x) = 0 (the zero 2-form) > x / x = 0 (the zero 2-form) so this one sticking point is immediately true > from just the algebra from the mental picture > the two vectors still define a parallelogram > it's just that it has become one of zero area you can't build an area with only one vector even if you use some integer multiple of the same vector x / (nx) (n e N) > then x / (nx) = (x / x) + (x / x) + ... n times > = 0 + 0 + ... = 0 similarly > with just two vectors > you can't get a volume x / y / (ax + by) = > x / y / (ax) + x / y / (by) = > - (x / (ax) / y) + x / 0 > = 0 + 0 = 0 and so on for you to have a non-zero form > you must be determining a positive measure in the n-parallelotope this is why it is different from the dot product > which has the exact opposite symmetry > (a . b = b . a) just like with wedge products > where we aren't attempting to get scalars out > one doesn't need to interpret the dot products as a scalar > it's just > like with the multivector/parallelotope/spatial measure > interpretation of wedge > scalars are just the natural mental picture to use working coordinate free means getting used to such abstraction we can define a partial ordering of closeness on grassmannians > (using the standard topological definition with neighborhoods) > and we see that the closer we are to the zero n-form > the closer the n-multivector is to linear dependence with the dot product > the closer to the zero dot > the closer some of the n-vectors are to orthogonality does any of this help? Anyway if it were true > then I think we'd get > (x^y)(x^y) = + 1 > instead of minus unity which is the trail that lead to this criticism. > We simply take the definition of the geometric product literally to > get > (x^y)(x^y) == (x^y) ^ (x^y) + (x^y) . (x^y) > where == is the definitional application. note that these are all using different products > than the equation i responded to i still think that product confusion > has been the biggest barrier to communication here -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > === Subject: Series for PI pi = 4*Sum [((-1)^n ) /(2n + 1)] n=0 to infinity by Fourier === Subject: Re: Series for PI I guess you mean Using Fourier series ... which is interesting. One could challenge readers to find other proofs. === Subject: Re: Odds of a point being its closest neigbor's closest neigbor? > E.g. for the line, n=1, c=2 and k=1, so your answer is indeed 2/3. > For the plane, n=2, c=pi and k = 2 pi/3 - sqrt(3)/2 if I'm not > mistaken, so the answer would be 6 pi/(8 pi + 3 sqrt(3)). a computer simulation, so I'm sure it's correct. -- Keith F. Lynch - http://keithlynch.net/ Please see http://keithlynch.net/email.html before emailing me. === Subject: Re: Uncomputable Natural Numbers =Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) > PlanetMath defines recursive set as:http://planetmath.org/encyclopedia/RecursiveSet.html A subset, S, of the natural numbers, N, is said to be > recursive if its characteristic function is computable. > In other words, there is an algorithm (via Turing machine > for example) that determines whether an element > is in S or not in S. Like most definitions of recursive, this one assumes > the input, for example, a tape to a Turing machine, > contains a natural number and the TM determines > if the input is a member of the set, S. Assuming the input is a natural number is equivalent > to assuming the set of all natural numbers, N, is recursive. The characteristic function for N is a TM > that simply halts and always returns True. > The TM doesn't even have to read the input tape > since we already assume the tape contains a natural number. What happens if we don't assume N is recursive? I will adapt a definition for Turing machine recognizable > I found in a posting on comp.theory: A language, L, is said to be Sequential Computer DECIDABLE > if there exists a Sequential Computer, M, which, given any > string w that is a member of L, halts and accepts w. > If w is not a member of L the TM halts and rejects w. I define a Sequential Computer, SC, as any machine that > from/to a tape in a single operation. I define the language of natural number representations > to be any length string of 1's followed by a blank > position. Any other string of characters is not a > natural number representation. I prove this language of natural number representations > is not sequential computer decidable. Assume we are given an input tape long enough to > contain any possible natural number representation. For simplicity, I will assume the input tape > can be infinitely long, that a SC can perform > an infinite number of operations, and that the > SC reads the input tape sequentially from the > start of the tape one position at a time. Assume we are given an input tape that > contains an infinite string of 1's. > Obviously, this string is not a member of L. Not even an infinitely fast sequential computer > can halt and reject this input tape. Proof: For each read operation, i, performed by the SC > let P_i be the set of unread positions. P_i can not be empty for any i. Assume P_z is the empty set. > This means the input tape has z positions. > Since z is a natural number and we assumed > the input tape was infinitely long, > the input tape can not have finite length, z. If P_i is never empty, the sequential computer > can never halt and reject the input tape. > There is always the possibility there is a > blank in an unread position. This proves the language of natural number > representations is not SC decidable. The original definition of Turing machine > recognizable allows the TM to loop forever > if w is not a member of L. This proof shows why we can never say > a SC loops forever. Even an infinitely > fast SC can only read a finite number of > positions from the inut tape. > We can never assume the SC will never halt > (assuming there is an input the SC will halt on). there are models of the natural numbers that contain nonstandard elements and there is no nonstandard model of arithmetic in which addition is recursive just pointing this out as there are a lot of things being stated as obvious in this thread that aren't so obvious -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Uncomputable Natural Numbers ... > Remember, it is a TM, not an SC. Regardless of how long the tape is, the > initial string of 1 symbols is finite. My TM only looks at that string > and one immediately following tape cell. OK I know it will reach one of its > halting states, having made its decision, without any external > intervention or time limit. I can afford to wait. How do you know the TM will reach a halting state? By mathematical induction on the length of the input string of 1 symbols. > Even if you wait forever, there will be an infinite > number of positions on the tape that haven't been read. But I'll never have to wait forever, because the initial 1 string must be of finite length. I don't care about any part of the tape other than any initial string of 1 symbols and the first symbol that is not a 1. That is all my TM needs to visit to know whether it has found the unary representation of a natural number or not. Incidentally, if the alphabet is limited to 1 and blank, the situation is even simpler. My TM can transition unconditionally from the Initial state to its Accept state, always halting after one step. Patricia === Subject: Re: Uncomputable Natural Numbers > ... > The language of unary notation natural numbers is TM-decidable by the > following TM: > There are three states, an Initial state, an Accept state, and a Reject > state. > ... > It reaches Accept if, and only if, the input begins with a sequence of > 1's followed by a blank - that is, the input is a natural number in > unary notation. In all other cases, it reaches the Reject state. This TM can not decide whether an arbitrary tape has a finite unary > representation of a natural number. Assume I give you a tape long enough to hold any unary > representation of a natural number. > ... > You give my tape to the ZM and wait one second. > The ZM still hasn't halted. Can you assume the ZM > will never halt? No. > Can you assume the input tape does not have a > unary representation? No. You halt the ZM and find the ZM has read an unimaginably > huge, but finite, number of positions. There are still an > infinite number of positions the ZM has not read and > any one of these positions could be blank. See below The only situation in which my TM halts is when it reaches Accept or > Reject, and at that point it has decided whether the input is the unary > representation of a natural number. There exist unary representations that your TM will never halt on. Remember, it is a TM, not an SC. Regardless of how long the tape is, the > initial string of 1 symbols is finite. My TM only looks at that string > and one immediately following tape cell. OK I know it will reach one of its > halting states, having made its decision, without any external > intervention or time limit. I can afford to wait. How do you know the TM will reach a halting state? If it is a proper TM, the tape will contain at least one blank in the direction that the TM is reading. If it is not a proper TM, then its behavior proves nothing about the behavior of proper TMs. > Even if you wait forever, there will be an infinite > number of positions on the tape that haven't been read. Name one! Incidentally, it would be really helpful if you would give a full > definition for your SC model. Assuming it is intended to be similar to a > Turing machine, you could start from a TM definition and list the > differences. It is clear that the rule that a TM does not have infinite > non-blank input on its tape is not part of the SC definition. If it makes you feel better, I can add the requirement that > there only be a finite number of non-blank characters on > the input tape. Is that > the only difference? A sequential computer is any computer that reads one > position at a time from the input tape. > This would include most of the definitions I have seen > for Turing machines. But would also include a lot of other machines whose behavior is irrelevant to the working of real TMs. === Subject: Re: Uncomputable Natural Numbers ... > Incidentally, it would be really helpful if you would give a full > definition for your SC model. Assuming it is intended to be similar to a > Turing machine, you could start from a TM definition and list the > differences. It is clear that the rule that a TM does not have infinite > non-blank input on its tape is not part of the SC definition. If it makes you feel better, I can add the requirement that > there only be a finite number of non-blank characters on > the input tape. Is that > the only difference? A sequential computer is any computer that reads one > position at a time from the input tape. > This would include most of the definitions I have seen > for Turing machines. However, all the definitions I've seen of Turing machines are much more specific than that. It's a bit like using the properties of ellipses to probe the question of whether a circle is a good shape for a wheel. If you intend to prove things about Turing machines, why not just do so, rather than muddling the situation with a vaguely defined model of your own? Patricia === Subject: Re: Uncomputable Natural Numbers > ... The language of unary notation natural numbers is TM-decidable by the > following TM: There are three states, an Initial state, an Accept state, and a Reject > state. Initial state: In all cases, rewrite the current symbol. If the current > cell contains 1, remain in state 0 and move the head right. If the > current cell contains a blank, transition to the Accept state. If the > current cell contains any other symbol, transition to the Reject state. The Accept and Reject states are simple sinks - once it reaches one of > them, the TM remains in that state. The initial tape, by definition, contains at most some finite number, n, > of 1 symbols. The number of steps spent in the Initial state is at most > n, and so the TM must reach Accept or Reject within a finite number of > time steps. It reaches Accept if, and only if, the input begins with a sequence of > 1's followed by a blank - that is, the input is a natural number in > unary notation. In all other cases, it reaches the Reject state. > This TM can not decide whether an arbitrary tape has a finite unary > representation of a natural number. Assume I give you a tape long enough to hold any unary > representation of a natural number. You give my tape to your TM. Your TM doesn't halt after 10^googol > reads. > Can you now say the TM will never halt? No. > Can you say my tape doesn't contain a unary represetation? No. There are still an infinite number of unread positions on the tape > and > any one of them could be a blank position. Assume you use a Zeno machine. The first read > takes 1/2 second, the next 1/4 second, etc. > We know this ZM will preform an infinite number of > reads in one second. You give my tape to the ZM and wait one second. > The ZM still hasn't halted. Can you assume the ZM > will never halt? No. Whyever not? What part of the tape remains unread by the ZM after 1 second? > Can you assume the input tape does not have a > unary representation? No. If that ZM is as advertised, I can. You halt the ZM and find the ZM has read an unimaginably > huge, but finite, number of positions. After one second, your ZM will have have run out of tape, and its built in out of tape tester will have turned it off to save its internal mechanism. === Subject: Re: Uncomputable Natural Numbers Obviously, TM and SC are significantly different. TM is a subset of SC. A proper subset if a subset at all. You have already > written about a very important language, the natural numbers in unary > notation, that is TM-decidable but not SC-decidable. My proof shows the language of natural number representations is not > TM decidable. It does not address TM's at all. Russell > - 2 many 2 count === Subject: Re: Uncomputable Natural Numbers I show there are always unread positions > that could be blank. You show no such thing. The TM that halts on blank never halts when > there are no blanks. > How do we know the TM never halts? The absence of the necessary condition for halting should be your first clue. The TM can read, at most, a finite number of positions from the tape. And cannot halt at any of them. If the TM reads every position on the tape then I can prove > the input tape has a finite length Not unless you assume that the TM can stop other than on a blank entry. As long as the TM keeps going without having read all positions, you cannot prove anything, and until it reaches a blank it must keep going. So your proof must wait upon the end of the universe to be completed.. If the tape is unbounded and can hold any finite representation, > there will always be an infinite number of unread positions > left on the tape. Any of these positions could be blank, > proving the input is a finite representation of a natural number. There is no (finite) position which will not be eventually read unless the TM stops before reaching that position. At least with standard TM's. Even if we assume the TM can read an infinite number > of positions, I prove there are still an infinite number > of unread positions on the tape. Given an unbouded tape, no TM can halt and say > the input tape does not contain the representation > of a natural number. Not even if the tape does contain the representation of a natural number? You really should get your TM's from a better supplier. === Subject: Re: Quadratic Diophantine Theorem =aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Quadratic Diophantine Theorem: In the ring of integers, given the quadratic expression c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy where the c's are constants, for solutions to exist it must be true > that ((c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3))v^2 + (2(c_2 - 2c_1)*(c_6 - > c_5) + 4c_5*(c_2 - c_1 - c_3))v + (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - > c_3) = n^2 mod p (1) for some n, where p is any prime coprime to z for a given solution, > when v = -(x+y)z^{-1} mod p. [...] The theorem is proven easily using what I call tautological spaces. I don't see that you need to use tautological spaces to prove this - it's just a matter of school level arithmetic to verify that the left hand side of (1) is given by z^{-2} ((c_2 - 2*c_1)x - (c_2 - 2*c_3)y - (c_6 - c_5)z)^2 which is clearly a square. But I don't see what the point of this is, and furthermore it is easy to generate a load of other such expressions; just start with a Diophantine equation, take the square of some arbitrary expression involving the variables and constants occurring in the expression and then make some substitutions. Can you present some use of your theorem, i.e. can you use it to prove something which isn't easy to prove by other means? === Subject: Re: Quadratic Diophantine Theorem > Quadratic Diophantine Theorem: > The theorem is proven easily using what I call tautological spaces. The famous spaces in which whatever pops into James Harris' mind is correct by proxy. Dirk Vdm === Subject: Re: Quadratic Diophantine Theorem reply_in_group-A24F6C.22040805092008@news.supernews.com Quadratic Diophantine Theorem: The theorem is proven easily using what I call tautological spaces. The famous spaces in which whatever pops into > James Harris' mind is correct by proxy. Do you happen to have a counterexample to his alleged theorem, or some > other way to show that it is not correct? I never even *look* at anything between his opening and closing line anymore. Not after this: http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/ArmyMath.html Dirk Vdm === Subject: Re: Quadratic Diophantine Theorem > I never even *look* at anything between his opening and closing > line anymore. > Not after this: > http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/ArmyMath.html His non sequiturs are half the reason I haven't kill-filed him. The mathematical content doesn't interest me too much--I've not studied higher-level algebras, so the only thing I've really paid close attention to is his TSP work. In any case, that posting is quite enlightening, given conversations with certain [insert favorite branch of the armed forces] officers. Also, JSH being relatively empty from your list of... postings does seem to reinforce what I've seen, i.e., he's relatively light compared to other people. You should probably put Androcles on a separate page, BTW. === Subject: Re: Quadratic Diophantine Theorem =aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Quadratic Diophantine Theorem: > The theorem is proven easily using what I call tautological spaces. The famous spaces in which whatever pops into > James Harris' mind is correct by proxy. Do you happen to have a counterexample to his alleged theorem, or some > other way to show that it is not correct? No, he doesn't; on this occasion James' theorem is true (see my earlier post where I point out how to prove it). === Subject: Re: Quadratic Diophantine Theorem =n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.27 Safari/525.13,gzip(gfe),gzip(gfe) On Sep 5, 3:00æpm, Dirk Van de moortel > Quadratic Diophantine Theorem: > The theorem is proven easily using what I call tautological spaces. The famous spaces in which whatever pops into > James Harris' mind is correct by proxy. Dirk Vdm Nope. The theorem is absolutely correct. The only question is, how useful is it? James Harris === Subject: Integral of mixed partial derivative with change of variables =JVPKNAoAAADT8GA9RydVIuuyOWRYhdVD .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) I guess I learned the basic concepts here at one point, but never all put together in the kind of problem I have here. I want to figure out how to take the integral of a mixed partial derivative with respect to variables that are a change of variables, so I suppose they're functions of x and t. Imagine I have a mixed partial derivative d^2(u)/dadb. Suppose that I have used a change of variables to arrive at this, using a = x + ct, b = x - ct. Now suppose I want to find the integral of this mixed partial with respect to a. I assume that the mixed partial represents d/da (du/db) so the integral would treat du/db as a constant. So we could pull du/ db outside the integral, and then integration gives du/db multipled by u. Then u = f ( a (x, t), b (x, t)), which is f ((x + ct), (x - ct)) and du/db = du/db * db/dx + du/db * db/dt, which is du/db ^ 1 + du/db ^ (- c). I'm supposed to find that the integral of the mixed partial with respect to a gives a function g(b) and that the integral of the mixed partial with respect to b gives u = F(a) + G(b), where G is the antiderivative of g. How do I find my way out of this thicket? === Subject: Re: Two of the Milky Way's Spiral Arms Go Missing Quite a big and sudden change in our picture of the world ? Please take a look. http://www.jpl.nasa.gov/news/news.cfm?release=2008-094 By the way is it possible that our Milky Way is now > in the process of splitting its nucleus into two ? Best Regads, Hannu Poropudas This is all very interesting but why has this been posted in this news group? Sci.bio.paleontology has nothing to do with astronomy. === Subject: Number appeal =33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) 3^2 + 4^2 = 5^2 3^3 + 4^3 + 5^3 = 6^3 Which next relation (as an equal sum of like powers) appeals to you most as being in formulatable continuation of the above two? Narasimham === Subject: Re: Number appeal =Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) > 3^2 + 4^2 = 5^2 > 3^3 + 4^3 + 5^3 = 6^3 Which next relation (as an equal sum of like powers) appeals to you > most as being in formulatable continuation of the above two? i like the generalisation n-1 n-3 --- --- n-3 / 1 / n - 2 n-2-j n-3 n-3 / j = | ----- / | | B n | - 1 - 2 - n --- n - 2 --- j / j / j=3 j=0 definitely one of the classics... -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Number appeal > 3^2 + 4^2 = 5^2 > 3^3 + 4^3 + 5^3 = 6^3 Which next relation (as an equal sum of like powers) appeals to you > most as being in formulatable continuation of the above two? 3^4 + 4^4 + 5^4 + 6^4 = 7^4 but it doesn't hold. Then 3^5 + 4^5 + 5^5 + 6^5 = 7^5 but it doesn't hold either. === Subject: Re: Number appeal > 3^2 + 4^2 = 5^2 > 3^3 + 4^3 + 5^3 = 6^3 Which next relation (as an equal sum of like powers) appeals to you > most as being in formulatable continuation of the above two? sum(k=3,n+2) k^n = n^n ---- === Subject: Re: Number appeal =Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) > 3^2 + 4^2 = 5^2 > 3^3 + 4^3 + 5^3 = 6^3 Which next relation (as an equal sum of like powers) appeals to you > most as being in formulatable continuation of the above two? sum(k=3,n+2) k^n = n^n if wishes were horses... -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: ultrafinitism and ultraformalism Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) While browsing the Web, I read some papers by mathematician > Edward Nelson of Princeton. He's had a few papers published > in the Annals of Mathematics and has worked mostly > a book on Predicative arithmetic. If I understand correctly, > his view includes that > - thinking of N = {0, 1, 2 , ...} as a definite, unique > entity with addition and multiplication > is a point of faith. > - Peano Arithmetic could be inconsistent. > Then, separately, there is ultrafinitism. > Cf. The Foundations of Math. mailing list, November 2006http://www.cs.nyu.edu/pipermail/fom/2006-November/date.html#11062 > and in particular: >http://www.cs.nyu.edu/pipermail/fom/2006-November/011085.html > Exponentiation is non-predicative, in PFA: > x^(y+1) := x* (x^y) , and x^0 := 1. > I don't understand what's different with exponentiation > compared to multiplication in predicative arithmetic ... > David Bernier > Have a look at the book, which is available for free online. The point > is that every finitely axiomatisable fragment of Bounded Arithmetic is > interpretable in Robinson Arithmetic; this is not the case with > Exponential Function Arithmetic. You can't justify induction for > formulas which use the exponential function without using > impredicative second-order reasoning. He discusses this point in some > detail. Herbzet was saying it might have been thoughtful to post the link http://www.math.princeton.edu/~nelson/books/pa.pdf on-line book, and I could see that the logical > aspects weren't too easy. Second order arithmetic has variables for collections > of natural numbers as well as individual numbers. > First, I need to know how Robinson Arithmetic works. > There is a definition in Nelson's book and also on Wikipedia. It is a very weak theory, it cannot prove the commutativity of addition, but all recursive functions are representable in it, that is the main point of it. > This seems a good point to ask how to express FLT in Peano Arithmetic, > for example, the statement that > For all x, y, z, n in N, if n>2 and x^n + y^n = z^n , then x*y*z = 0. > I think it was G.9adel who first figured out how to do this. If you have a look at his famous 1931 paper on incompleteness, I think it's Proposition VII where he proves that every primitive recursive relation is arithmetical. > David Bernier P.S. We can't handle infinite sets like bags of marbles. A skeptic > could say something like: when doing mathematics, we work > with ideas, intuitions and memories, among other things. > The mind and brain aren't well understood. Just > because two mathematicians agree that clearly, there's > one standard model of arithmetic, and we both know > what it is; it's obvious... doesn't mean that > they have a clear model in mind. We can check that > optical illusions are misleading sensations with rulers > and other devices. In math., there's no ultimate > authority, intelligence or teacher. So one is > left with one's own beliefs and doubts. > Anyway, that's my opinion. === Subject: Re: ultrafinitism and ultraformalism =Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) > On Sep 2, 5:59 pm, G. A. Edgar Arithmetic is a proven mathematical theorem. It is quite complicated, > but is nevertheless a theorem. As I recall, the proof uses induction up to epsilon_0 ... There are proofs using the so-called transfinite induction. But the > point is, the basic axioms for numbers, sets of numbers, sets of sets > of numbers, etc, (and ordered n-tuples) suffice to give a proof. Even > the fragment known as second order arithmetic will suffice. For > example, one can prove in second order arithmetic a result known as > glorified finite Ramsey's theorem, and this implies con(PA) over > PA. and you don't find it strange that all such proofs (perhaps most obviously the second order arithmetic one) are impredicative? have you read poincare's admonition on predicativity? particularly in terms of consistency? (of course if arithmetic is inconsistent it could still prove it's consistency i'm sure you understand this but if you do i have no idea why you are appealing to proofs) -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: ultrafinitism and ultraformalism > On Sep 2, 5:59 pm, G. A. Edgar The arithmetic statement that encodes the consistency of Peano > Arithmetic is a proven mathematical theorem. It is quite complicated, > but is nevertheless a theorem. > As I recall, the proof uses induction up to epsilon_0 ... > There are proofs using the so-called transfinite induction. But the > point is, the basic axioms for numbers, sets of numbers, sets of sets > of numbers, etc, (and ordered n-tuples) suffice to give a proof. Even > the fragment known as second order arithmetic will suffice. For > example, one can prove in second order arithmetic a result known as > glorified finite Ramsey's theorem, and this implies con(PA) over > PA. and you don't find it strange > that all such proofs > (perhaps most obviously the second order arithmetic one) > are impredicative? have you read poincare's admonition on predicativity? particularly in terms of consistency? (of course > if arithmetic is inconsistent > it could still prove it's consistency i'm sure you understand this > but if you do > i have no idea why you are appealing to proofs) The point that using glorified finite Ramsey's theorem to establish Con(PA) is like trying to lift oneself by his/her bootstraps, is one that stands out for me. Concerning Poincar.8e, he didn't believe in the well-ordering theorem for the case of the real numbers. But around 1900 to 1910, there was first a flawed proof of WOT, and subsequently a correct one. In learning and knowledge, at any given time one may have misconceptions; the boundary between a known thing and a belief is not really clear. I did a search for common misconceptions, and I thought: He will get his just deserts was written He will get his just desserts, ( e.g. a piece of cake) According to some Web page, deserts relates to the verb to deserve: Cf.: http://listverse.com/miscellaneous/top-10-common-misconceptions/ required using only finistic methods of reasoning. This allows the Propositional Calculus, Predicate Calculus and a bit more. Usual problems in number theory don't require a long string of alternating existential and for all quantifiers. The Goldbach conjecture can be turned into a sentence with one alternation, I think : not( exists n, exists m, for all a, b, p, q: n>2 && n == 2*m && p+q = n (b=1 OR b=q) ) ) [ the not ( Prop(n) ) is supposed to say: n is a GC counterexample ] Twin Primes seems to require two alternations: (for all) .... (there exists) .... (for all) ... I'd be interested in learning about questions in elementary number theory where 3 or more quantifier alternations are required. David Bernier David Bernier === Subject: Re: ultrafinitism and ultraformalism =EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Ok. So, if I get the notation correctly, in *these* contexts (like > plain Z, etc.) Russell's paradox is not a paradox and that set is in > fact the empty set. In any ordinary set theory or class theory it's not a paradox. And, IF we use the Fregean method, then we get {x | ~xex} = 0. > OTOH, FOL proves the negation of Russell's > paradox, which I get to mean there does not exit a set R such > that.., No. R is okay; there is not a question of its existence since it is just 'R' is just a 2-place predicate symbol. And the word 'set' does not come in. Rather there is not an x such that Ay(Ryx <-> ~Ryy). Whether 'R' stands for membership or not, and whether 'x' ranges over sets or not, we have, in any case, the theorem ~ExAy(Ryx <-> ~Ryy). > and so FOL and --say-- plain Z set theory are incongruent one > another on this... No, not at all. Every theorem of plain first order logic (in a language with just the 2-place relation 'e') is also a theorem of Z set theory. We write 'yex' instead of 'eyx' only for visual convenience. Officially, the formula is 'eyx'. And 'e' is just a 2-place relation symbol, so it's just like 'R' in that respect. So first order logic has the theorem: ~ExAy(Ryx <-> ~Ryy) and if, 'e' is in the languae of any first order theory, we have the theorem: ~ExAy(yex <-> ~yey). 'Ryx' and 'yey' instead of 'Ryy'. There's no substantive diffference. For ANY 2-place relation symbol (whether 'R' or 'e' or whatever, and no matter whether we write 'Ryx' or 'yRx' or 'eyx' or 'yex', depending on our conventions) we have as a theorem of pure first order logic (and thus ALSO of set theory): ~ExAy(yex <-> ~yey) >I guess the fact is that I have no very clear idea > about the relationship between 1st and 2nd order logic on a side, and > the myriad of set-based axiomatizations. We don't need to concern ourselves about 2nd order logic here (though the result also holds for second order logic) nor with any axioms of set theory. No matter what axioms of set theory, we get from first order logic alone that ~ExAy(yex <-> ~yey) so we get that as a theorem of set theory too. Remember, EVERY theorem of pure first order logic (in the language of set theory) is a theorem of set theory. EVERY theory includes the set of theorems of pure first order logic (in the langauge of the theory). > Sure we often use the word 'set'. But do we HAVE to use the word > 'set'? I tend to think not. What we do need is some 2-place predicate > to refer to. And, getting back to the original question of Russell's > paradox, in a formal theory, it can be expressed without mentioning > the word 'set'. Of course it can, as indeed the notion of set remains undefined at the > formal level. Though, we can define it formally if we like. > Still, at the informal or metalogical level, I'd have my > doubts. But I am surely a beginner here, so never mind, this is not > crucial anyway. Also, don't forget that the meta-thoery can be formalized. And I would agree that never using the word 'set' in an informal meta-language while also communicating the mathematics of ZFC (or whatever theory) would be quite a strain, quite awkard, and sometimes cause some rather long and unattractive circumlocutions. But I don't see that it can't be done. > But the paradox is not unique to the membership relation. ANY 2-place > predicate symbol 'R' will do: Consider ixAy(Ryx <-> ~Ryy). The existence of such an x gives us Rxx <-> ~Rxx. So it is a theorem of pure first order logic: ~ExAy(Ryx <-> ~Ryy). And we don't even need the 'i' operator to prove that. Rather, we have this constructive proof: 1 ExAy(Ryx <-> ~Ryy) ... supposition > 2 Ay(Ryx <-> ~Ryy) ... existential instantiation > 3 Rxx <-> ~Rxx ... universal instantiation > 4 ~ExAy(Ryx <-> ~Ryy) ... from 1 and 3. whether this theorem, asserting that there is no x such that etc. > etc., is equivalent to the above theorem R = 0 in Z set theory, No, not 'R = 0'. Rather ixAy(Ryx <-> ~Ryy) = 0. Also, ixAy(Ryx <-> ~Ryy) = {x | ~xex} = 0. And this holds IF we use the Fregean method. As to equivalence, I didn't say this an equivalent theorem to ~ExAy(Ryx <-> ~yey). Though, they are equivalent (if we use the Fregean method) in the trivial sense that all theorems (relative to whatever axioms) are equivalent with one another. > or > instead means that x is not a set at all, like -- as I get it -- would > be the case in ZFC. What I said for Z holds for ZF, ZFC, etc, IF we use the Fregean method with any of those theories. Now, it's true that the usual way of describing the situation is to say ixAy(yex <-> ~yey) is not a set (or 'is not a class' in class theories) or {x | ~xex} is not a set. (or 'is not a class' in class theories) That's quite fine and adequate for informal purposes. But when we require a fully formal way of dealing with improper descriptions, then one such way is the Fregean method, by which we can avoid the informality of saying is not a set and instead get the exact (though not very useful) theorem: ixAy(yex <-> ~yey) = {x | ~xex} = 0. To properly understand this requires a good solid understanding of the first order predicate calclus and of the definitite description operator 'i' and the set abstraction operator '{ | }', along with a good explanation of the Fregean method for handling those. MoeBlee === Subject: Re: ultrafinitism and ultraformalism =F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Ok. So, if I get the notation correctly, in *these* contexts (like > plain Z, etc.) Russell's paradox is not a paradox and that set is in > fact the empty set. In any ordinary set theory or class theory it's not a paradox. And, IF > we use the Fregean method, then we get {x | ~xex} = 0. OTOH, FOL proves the negation of Russell's > paradox, which I get to mean there does not exit a set R such > that.., No. R is okay; there is not a question of its existence since it is > just 'R' is just a 2-place predicate symbol. And the word 'set' does > not come in. Rather there is not an x such that Ay(Ryx <-> ~Ryy). > Whether 'R' stands for membership or not, and whether 'x' ranges over > sets or not, we have, in any case, the theorem ~ExAy(Ryx <-> ~Ryy). and so FOL and --say-- plain Z set theory are incongruent one > another on this... No, not at all. Every theorem of plain first order logic (in a language with just the > 2-place relation 'e') is also a theorem of Z set theory. We write 'yex' instead of 'eyx' only for visual convenience. > Officially, the formula is 'eyx'. And 'e' is just a 2-place relation > symbol, so it's just like 'R' in that respect. So first order logic has the theorem: ~ExAy(Ryx <-> ~Ryy) and if, 'e' is in the languae of any first order theory, we have the > theorem: ~ExAy(yex <-> ~yey). 'Ryx' and 'yey' instead of 'Ryy'. There's no substantive diffference. For ANY 2-place relation symbol (whether 'R' or 'e' or whatever, and > no matter whether we write 'Ryx' or 'yRx' or 'eyx' or 'yex', depending > on our conventions) we have as a theorem of pure first order logic > (and thus ALSO of set theory): ~ExAy(yex <-> ~yey) I guess the fact is that I have no very clear idea > about the relationship between 1st and 2nd order logic on a side, and > the myriad of set-based axiomatizations. We don't need to concern ourselves about 2nd order logic here (though > the result also holds for second order logic) nor with any axioms of > set theory. No matter what axioms of set theory, we get from first > order logic alone that ~ExAy(yex <-> ~yey) so we get that as a theorem of set theory too. Remember, EVERY theorem of pure first order logic (in the language of > set theory) is a theorem of set theory. EVERY theory includes the set > of theorems of pure first order logic (in the langauge of the theory). Sure we often use the word 'set'. But do we HAVE to use the word > 'set'? I tend to think not. What we do need is some 2-place predicate > to refer to. And, getting back to the original question of Russell's > paradox, in a formal theory, it can be expressed without mentioning > the word 'set'. Of course it can, as indeed the notion of set remains undefined at the > formal level. Though, we can define it formally if we like. Still, at the informal or metalogical level, I'd have my > doubts. But I am surely a beginner here, so never mind, this is not > crucial anyway. Also, don't forget that the meta-thoery can be formalized. And I would > agree that never using the word 'set' in an informal meta-language > while also communicating the mathematics of ZFC (or whatever theory) > would be quite a strain, quite awkard, and sometimes cause some rather > long and unattractive circumlocutions. But I don't see that it can't > be done. But the paradox is not unique to the membership relation. ANY 2-place > predicate symbol 'R' will do: Consider ixAy(Ryx <-> ~Ryy). The existence of such an x gives us Rxx <-> ~Rxx. So it is a theorem of pure first order logic: ~ExAy(Ryx <-> ~Ryy). And we don't even need the 'i' operator to prove that. Rather, we have this constructive proof: 1 ExAy(Ryx <-> ~Ryy) ... supposition > 2 Ay(Ryx <-> ~Ryy) ... existential instantiation > 3 Rxx <-> ~Rxx ... universal instantiation > 4 ~ExAy(Ryx <-> ~Ryy) ... from 1 and 3. whether this theorem, asserting that there is no x such that etc. > etc., is equivalent to the above theorem R = 0 in Z set theory, No, not 'R = 0'. Rather ixAy(Ryx <-> ~Ryy) = 0. Also, ixAy(Ryx <-> ~Ryy) = {x | ~xex} = 0. And this holds IF we use the Fregean method. As to equivalence, I didn't say this an equivalent theorem to ~ExAy(Ryx <-> ~yey). Though, they are equivalent (if we use the Fregean method) in the > trivial sense that all theorems (relative to whatever axioms) are > equivalent with one another. or > instead means that x is not a set at all, like -- as I get it -- would > be the case in ZFC. What I said for Z holds for ZF, ZFC, etc, IF we use the Fregean method > with any of those theories. Now, it's true that the usual way of describing the situation is to > say ixAy(yex <-> ~yey) is not a set (or 'is not a class' in class theories) or {x | ~xex} is not a set. (or 'is not a class' in class theories) That's quite fine and adequate for informal purposes. But when we > require a fully formal way of dealing with improper descriptions, then > one such way is the Fregean method, by which we can avoid the > informality of saying is not a set and instead get the exact (though > not very useful) theorem: ixAy(yex <-> ~yey) = {x | ~xex} = 0. To properly understand this requires a good solid understanding of the > first order predicate calclus and of the definitite description > operator 'i' and the set abstraction operator '{ | }', along with a > good explanation of the Fregean method for handling those. Right, your point is still not 100% clear to me, and that's of course because indeed I do not have that solid understanding yet. I will keep studying the matter. In the meantime, I thank you very much for taking the time to explain things to me. === Subject: Re: ultrafinitism and ultraformalism ZFC. I believe that phrases such as let X be a> set or let X be a class make no sense.> You can do ZFC all up and down without ever having to utter any phrase> that has the word 'set' or 'class'.Can you? Beyond sheer formalism? æ> Russell's paradox is a clue that one> should treat as meaningless the let X be a set talk,> Russell's paradox can be formulated without the use of the word 'set'.> The negation of Russell's paradox is a theorem of plain first order> logic.> MoeBlee Well, I never said that made sense either. === Subject: Re: ultrafinitism and ultraformalism > You can do ZFC all up and down without ever > having to utter any phrase> that has the word 'set' or 'class'. > Can > you? Beyond sheer formalism? Formally, it's trivial, since we don't need to define 'set' or 'class' in ZFC. And in working in ZFC itself, but with informal English, I don't know of any instance in which the words 'set' or 'class' are used in ZFC but couldn't be avoided (of course, in class theories, we do need to distinguish between sets specifically and classes generally, though, in a formal theory, 'is a set' need not be taken any more literally than being a nickname for a 1-place predicate symbol). And in an informal meta-language for an informal meta-theory ABOUT ZFC, I would think it very awkard and tedious always to avoid the words 'set' and 'class', but I don't see that it can't be done. > The negation of Russell's paradox is a theorem of plain first > order logic. > Well, I never said that made sense either. Then, if you don't think first order logic makes sense, it seems to me that any critiques about set theory is not even the point, since your critique is much more fundamental. Perhaps you might say whether your critique is even more fundamental than regarding first order logic as nonsensical. That is, do you find formal logic itself nonsensical? Formal languages nonsensical? If not, then what formal logic(s) or what formal language(s) do you find sensical? MoeBlee === Subject: Re: ultrafinitism and ultraformalism =FeQf_goAAABjohysG-nJ6IFLUfjTwKeK Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) You can do ZFC all up and down without ever > having to utter any phrase> that has the word 'set' or 'class'. > Can > you? Beyond sheer formalism? Formally, it's trivial, since we don't need to define 'set' or 'class' > in ZFC. And in working in ZFC itself, but with informal English, I > don't know of any instance in which the words 'set' or 'class' are > used in ZFC but couldn't be avoided (of course, in class theories, we > do need to distinguish between sets specifically and classes > generally, though, in a formal theory, 'is a set' need not be taken > any more literally than being a nickname for a 1-place predicate > symbol). > And in an informal meta-language for an informal meta-theory ABOUT > ZFC, I would think it very awkard and tedious always to avoid the > words 'set' and 'class', but I don't see that it can't be done. Well, OK, though I thought people thought it was about sets. > The negation of Russell's paradox is a theorem of plain first > order logic. > Well, I never said that made sense either. Then, if you don't think first order logic makes sense, it seems to me > that any critiques about set theory is not even the point, since your > critique is much more fundamental. Perhaps you might say whether your > critique is even more fundamental than regarding first order logic as > nonsensical. That is, do you find formal logic itself nonsensical? > Formal languages nonsensical? If not, then what formal logic(s) or > what formal language(s) do you find sensical? Yes, you are probably right. I am not sure exactly what you mean by first order logic, but I am fairly sure whatever you mean, it does not make sense to me. Formal languages? It does come down to language (making sense or not making sense). I think sentences like 3 exists in the set of numbers {1,2,3} or 561 exists in the set of numbers {Carmichael numbers} make sense. Sentences like 3 exists do not make sense. In general, quantification full stop, does not make sense. You quantify (there is, for every) over a set of numbers, or whatever. You also can't compare numbers to sets of numbers, for example, just like you can't say or denounce 2>{the set of primes}. This is the way I first thought Russell's paradox is no such thing. Thus I think someone trying to peddle Russell's paradox, is not wrong (disproven by a theorem) but rather not even talking properly. The way of talking I find that makes sense may seem restrictive, but it's not really. === Subject: Re: ultrafinitism and ultraformalism > I am not sure exactly what you mean by > first order logic, but I am fairly sure whatever you mean, it does not > make sense to me. See http://en.wikipedia.org/wiki/First_order_logic to get started === Subject: Re: ultrafinitism and ultraformalism =EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > You can do ZFC all up and down without ever > having to utter any phrase> that has the word 'set' or 'class'. > Can > you? Beyond sheer formalism? Formally, it's trivial, since we don't need to define 'set' or 'class' > in ZFC. And in working in ZFC itself, but with informal English, I > don't know of any instance in which the words 'set' or 'class' are > used in ZFC but couldn't be avoided (of course, in class theories, we > do need to distinguish between sets specifically and classes > generally, though, in a formal theory, 'is a set' need not be taken > any more literally than being a nickname for a 1-place predicate > symbol). > And in an informal meta-language for an informal meta-theory ABOUT > ZFC, I would think it very awkard and tedious always to avoid the > words 'set' and 'class', but I don't see that it can't be done. Well, OK, though I thought people thought it was about sets. Nothing I said contradicts that most people think that ZFC is about sets. That was not in question. The negation of Russell's paradox is a theorem of plain first > order logic. > Well, I never said that made sense either. Then, if you don't think first order logic makes sense, it seems to me > that any critiques about set theory is not even the point, since your > critique is much more fundamental. Perhaps you might say whether your > critique is even more fundamental than regarding first order logic as > nonsensical. That is, do you find formal logic itself nonsensical? > Formal languages nonsensical? If not, then what formal logic(s) or > what formal language(s) do you find sensical? Yes, you are probably right. I am not sure exactly what you mean by > first order logic, I mean exactly first order logic. To be excruciatingly exact would be to specify some specific formulation of first order logic. But 'first order logic' is pretty clear onto itself. > but I am fairly sure whatever you mean, it does not > make æsense to me. You don't know what it is, but you're fairly sure it doesn't make sense to you. Brilliant. > Formal languages? It does come down to language > (making sense or not making sense). I think sentences like 3 exists in > the set of numbers {1,2,3} or 561 exists in the set of numbers > {Carmichael numbers} make sense. Sentences like 3 exists do not make > sense. Very fine. And such phrases as 'x exists' are entirely dispensable in such mathematics as ZFC. > In general, quantification full stop, does not make sense. You > quantify (there is, for every) over a set of numbers, or whatever. Indeed, just a quantifier without a matrix following it is not even syntactical in the language for ZFC. Though, contrary to your preference, it is not required that every quantifier be relativized; however a quantifier does only range over a given set per an interpretation for the language. > You also can't compare numbers to sets of numbers, for example, just > like you can't say or denounce 2>{the set of primes}. Predicate symbols such as '>' can be defined in a way that eliminates such problems as how to regard such formulas as you mention above. > This is the > way I first thought Russell's paradox is no such thing. Thus I think > someone trying to peddle Russell's paradox, is not wrong (disproven by > a theorem) but rather not even talking properly. They shouldn't be trying to peddle it. That's a big mistake, because it's free anyway, and it's tough to get people to buy what they can get for free anyway. More suitable items for peddling are used housewares, fabric by the yard, or freshed baked knishes. More notably, your scare quotes around the word 'theorem' are silly. As to 'talking properly', you've given no basis, other than a statement of your UNinformed preferences, for a conclusion that talking about Russell's paradox is not proper. > The way of talking I find that makes sense may seem restrictive, but > it's not really. That's reassuring. MoeBlee === Subject: Re: ultrafinitism and ultraformalism > Not even the sun's core could enlighten your vow to blindness. However blind julio thinks me, I still see more than he. === Subject: Re: ultrafinitism and ultraformalism =F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Not even the sun's core could enlighten your vow to blindness. However blind julio thinks me, I still see more than he. Hehehe... Well, one is for sure: should even the whole universe at once disappear, you'll still be there alice and kicking! -LV === Subject: Re: ultrafinitism and ultraformalism <48c09dca$0$311$b45e6eb0@senator-bedfellow.mit.edu> =FeQf_goAAABjohysG-nJ6IFLUfjTwKeK 1.1.4322; .NET CLR 2.0.50727; InfoPath.2),gzip(gfe),gzip(gfe) > true. æWe see that it is obviously true, and say that it is true.Is it? Which kind of obvious you have in mind?> Precisely the kind of obvious that kleptomaniac666 had in mind, whatever> that happens to be.> If you have been following the thread, you'll see that my main goal is not> to argue directly for ZFC, but to argue that kleptomaniac666 is being facile> in dismissing Nelson. ækleptomaniac666 holds that certain mathematical> statements are obvious and that denying them makes no sense, while> simultaneously making denials of other mathematical statements that others> declare to be obvious. æA double standard if there ever was one.> --> Tim Chow æ æ æ tchow-at-alum-dot-mit-dot-edu> The range of our projectiles---even ... the artillery---however great, will> never exceed four of those miles of which as many thousand separate us from> the center of the earth. æ---Galileo, Dialogues Concerning Two New Sciences No. I said that reasons Nelson gave for denying them made no sense. I clarified this but you appear to have failed to notice. For that reason I will keep this post (almost) as short as possible. === Subject: Re: ultrafinitism and ultraformalism =FeQf_goAAABjohysG-nJ6IFLUfjTwKeK 1.1.4322; .NET CLR 2.0.50727; InfoPath.2),gzip(gfe),gzip(gfe) the Web, I read some papers by mathematician> Edward Nelson of Princeton. æHe's had a few papers published> in the Annals of Mathematics and has worked mostly> in analysis and mathematical understand correctly,> his view includes that> - thinking of N = {0, 1, 2 , æ...} as a definite, unique> æ entity with addition and multiplication> æ is a point of faith.> - Peano Arithmetic could be inconsistent.> Then, separately, there is ultrafinitism. Cf. æThe Foundations of Math. mailing list, November 2006http:// www.cs.nyu.edu/pipermail/fom/2006-November/date.html#11062> and in particular:>http://www.cs.nyu.edu/pipermail/fom/2006-November/ 011085.html> Exponentiation is non-predicative, in PFA:> x^(y +1) := æx* (x^y) , æ and x^0 := 1.> I don't understand what's different with exponentiation> compared to multiplication in predicative arithmetic ...> David Bernier> Have a look at the book, which is available for free online. The point> is that every finitely axiomatisable fragment of Bounded Arithmetic is> interpretable in Robinson Arithmetic; this is not the case with> Exponential Function Arithmetic. You can't justify induction for> formulas which use the exponential function without using> impredicative second-order reasoning. He discusses this point in some> detail.> Herbzet was saying it might have been thoughtful to post the link>http://www.math.princeton.edu/~nelson/books/pa.pdf> book, and I could see that the logical> aspects weren't too easy.> Second order arithmetic has variables for collections> of natural numbers as well as individual numbers.> First, I need to know how Robinson Arithmetic works.> This seems a good point to ask how to express FLT in Peano Arithmetic,> for example, the statement that> For all x, y, z, n in N, æif n>2 æand x^n + y^n = z^n , then x*y*z = 0.> David Bernier> P.S. We can't handle infinite sets like bags of marbles. A skeptic> could say something like: æwhen doing mathematics, we work> with ideas, intuitions and memories, among other things.> The mind and brain aren't well understood. æJust> because two mathematicians agree that clearly, there's> one standard model of arithmetic, and we both know> what it is; it's obvious... doesn't mean that> they have a clear model in mind. æWe can check that> optical illusions are misleading sensations with rulers> and other devices. æIn math., there's no ultimate> authority, intelligence or teacher. æSo one is> left with one's own beliefs and doubts.> Anyway, that's my opinion. Whereas we can handle {1,2,3} like a bag of marbles? P.S. Models, schmodels. === Subject: Re: ultrafinitism and ultraformalism > true. æWe see that it is obviously true, and say that it is true.Is it? Which kind of obvious you have in mind?> Precisely the kind of obvious that kleptomaniac666 had in mind, whatever> that happens to be.> If you have been following the thread, you'll see that my main goal is not> to argue directly for ZFC, but to argue that kleptomaniac666 is being facile> in dismissing Nelson. ækleptomaniac666 holds that certain mathematical> statements are obvious and that denying them makes no sense, while> simultaneously making denials of other mathematical statements that others> declare to be obvious. æA double standard if there ever was one.No. I said that the *reasons* Nelson had given for denying them made no sense. I clarified this before but you appear to have failed to notice. For that reason I will keep this post (almost) as short as possible. === Subject: Re: ultrafinitism and ultraformalism Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >No. I said that the *reasons* Nelson had given for denying them >made no sense. I clarified this before but you appear to have failed >to notice. For that reason I will keep this post (almost) as short as >possible. I did not fail to notice. Nelson is one step ahead of you; I'm trying to get you to that point. For example, one of your original complaints was that the notion of creating the number 5 is a spatiotemporal one, which doesn't mesh with what we mean by 5. But Nelson's point is that what *you* (along with most of the rest of us, including myself) mean by 5 doesn't make sense. And if we are faced with this challenge, what do we say? Your response is to say that these basic properties of numbers and so forth are obvious. We simply look at them and declare them to be true. Well, if you're just going to make such declarations and obstinately stick to your guns, there's nothing I can do to *force* you to do otherwise. But what I can try to do is to get you see that your declarations of obviousness are arbitrary. If someone else were to declare that the existence of inaccessible cardinals is just obvious, you would balk. Can't you see that someone else is equally justified in balking at *your* dogmas? You lump the consistency of PA along with Lagrange's theorem and so forth as if there were an uncontroversial, universal notion of what proven mathematical facts are, but the truth is there is no such uncontroversial notion. In fact, your insistence on obviousness is surely just an admission that you don't have any coherent defense of said properties of numbers, etc. It's just something you take for granted, as a basic *philosophical* stance. If Nelson takes a different philosophical stance about what numbers are, he's not making any less sense than you are. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: ultrafinitism and ultraformalism <48c1da51$0$298$b45e6eb0@senator-bedfellow.mit.edu> =lwNmEAoAAAB9yKqyOQ9ijwax7bRvEMO6 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) No. I said that the *reasons* Nelson had given for denying them >made no sense. I clarified this before but you appear to have failed >to notice. For that reason I will keep this post (almost) as short as >possible. I did not fail to notice. æNelson is one step ahead of you; I'm trying to get > you to that point. For example, one of your original complaints was that the notion of > creating the number 5 is a spatiotemporal one, which doesn't mesh with > what we mean by 5. æBut Nelson's point is that what *you* (along with most > of the rest of us, including myself) mean by 5 doesn't make sense. æ Out of interest, where does he say that? I thought Nelson's critique concerned numbers like 2^65536. I didn't realize Nelson said anything about the number 5, or that he thought of 5 differently than most of the rest of us. What I know of Nelson's approach strikes me as incoherent. He *complains* about our assumption that the natural number system is given, but then he goes right out and uses Q as a base theory, which assumes that all the natural numbers are there. === Subject: Re: ultrafinitism and ultraformalism <48c06f7e$0$298$b45e6eb0@senator-bedfellow.mit.edu> =FeQf_goAAABjohysG-nJ6IFLUfjTwKeK 1.1.4322; .NET CLR 2.0.50727; InfoPath.2),gzip(gfe),gzip(gfe) I'm not sure you understood exactly what I was saying. But now you >mention it, I don't accept choice principles generally, and don't >regard anything depending on such principles as a fully proven >mathematical theorem. But the reasons you give for rejecting choice principles, or let X be a set, > are nonsensical. æThe axiom of choice, or more generally ZFC, is obviously > true. æWe see that it is obviously true, and say that it is true. æDitto with > the existence of an inaccessible cardinal. æSince the consistency of ZFC > follows from this true statement, con(ZFC) is true. æSince con(ZFC) is true, > and Lagrange's theorem from group theory is true, which one you are denying > matters little. Beyond the second sentence, we are stuck on the first issue of nonsensicality (choice principles are coherent to state, I just don't accept them). One can only talk about obviously true w.r.t something that makes sense. Given that, let me consider your first sentence. Hmm. But I never gave any reasons why I thought let X be a set as is talked about in ZF did not make sense. If I recall correctly, I more or less just said it did not make sense. >Russell's paradox is a clue that one >should treat as meaningless the let X be a set talk, I believe. I >actually wouldn't be surprised if ZFC was inconsistent, although >equally I wouldn't be surprised if it was. It's just a meaningless >formal system as far as I'm concerned. Yet if Nelson or someone else were to say, The incompleteness theorems are a > cluse that one should treat as meaningless the `let N be the natural numbers' > talk, I believe---I actually wouldn't be surprised if PA were inconsistent, > although equally I wouldn't be surprised if it was, since it's just a > meaningless formal system as far as I'm concerned, you would consider that > nonsense? No, just wrong. I'm well aware that the declaration what you are saying is meaningless doesn't give much room for argument or discussion. === > I am having a few problems in calculating some limits: lim (2^n - 1) ^ (1/n) > lim(n->oo) ? lim(n -> -n) ?? log (2^n - 1)^(1/n) = (log (2^n - 1))/n in the limit to oo has the form oo/oo. Use l'Hospital's rule (log 2)2^n / (2^n - 1) -> log 2 Answer is 2. Generlize to lim(n->oo) (a^n - 1)^(1/n) = a, 0 < a Simlar thoughts . . (a^n - 1)^(1/n) = a(1 - a^-n)^(1/n) Show . . lim(n->oo) (1 - a^-n)^(1/n) = 1 -- > lim [ (2n)! / (n!)^2 ] ^ (1/n) Try using Sterling's approximation . . n! ~ (n/e)^n sqr 2pi.n If it works, try finding . . lim(n->oo) [(kn)! / (n!)^k]^(1/n) ---- === =K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) I am having a few problems in calculating some limits: lim (2^n - 1) ^ (1/n) It helps to write 2^n - 1 = 2^n *[1 - 2^(-n)]. This gives a limit of two factors; the first is easy, and the second is easy too, especially by first taking logarithms. lim [ (2n)! / (n!)^2 ] ^ (1/n) Use Stirling's formula x! ~ St(x) = sqrt(2*pi) * x^(x + 1/2) * exp(- x); this is an asymptotic result, in the sense that limit x!/St(x) = 1 as x --> infinity; in fact, the limit is approached like 1 + 1/(12*x). R.G. Vickson I feel that there might be a rule for this kind of limits and also for > this type of limits with factorials but I can't find it. Miguel === Subject: parameterization of like cube power sum number sets =33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) What is the parameterization for a, b, c as functions of u,v or u,v,w? E.g., which pair or triplet generates the quartet (3,4,5,6) in 3^3 + 4^3 + 5^3 = 6^3? May be well known in number theory. Narasimham === Subject: Re: Existance of a distribution =K8JCNAoAAAAZz7bBcYzz_X9G7WYwtN90 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) But then cut-off function cuts the good test functions also. So u and f does not agree on R^n{0}. I think without a cut-off function it works if m is sufficiently large to guarantee the integrability near infinity of int_R [g - (m^th taylor approx. of g at 0)]*f. But for the original question (that is arbitrary m and n) it doesnt seem working. === Subject: maximal number of points in a set such that no more than three lie in same direction? =00fBxwoAAADNo9O-CEyB0GYRQaWsAiCr Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Is the below a question with a known answer? If not, I would appreciate any suggestions as to what the appropriate mathematics to use might be. Let x = (x1,...,xn) and y=(y1,.83,yn) be points in the positive orthant of R^n. Perform a componentwise comparison, xi [CapitalEth] yi, yielding ñ+î when xi > yi, ñ-ñ when xi < yi, and ñ0î when xi = yi. Thus, if x=(1,0,0) and y=(1/2,1/2,0), the componentwise comparison is ñ+-0ñ. Call each possible componentwise comparison (ñ+++î, ñ++0,î.83,î---[Cap italOGrave]) a direction. What, then, is the largest set of such points in R^n so that no more than three of them lie in the same direction with respect to each other? That is, given any w,x,y, and z in such a set, it cannot be that the direction from w to x to y to z is the same. Here is an example in R^3 for which no more than three points lie in the same direction; this may not be the maximal set: {(1,0,0), (0,1,0), (0,0,1), (1/2,1/2,0), (1/2,0,1/2), (0,1/2,1/2), (1/2,1/4,1/4), (1/4,1/2,1/4), (1/4, 1/4, 1/2)}. Colin Rowat Economics, University of Birmingham === Subject: Brownian motion on multidimensional space Consider the Brownian motion, it is well known that it can be described using a path integral approach (wiener measure). What I need is something similar but where the Brownian motion is a field in the sense that the parameter space is not R but rather R^d d>1. I guess, that in terms of path integral one has to replace the squared 'velocity' in the path integral with the square of the gradient. Is this right? Is it possible to do a similar construction in a Minkowskian parameter space so as to respect lorentz invariance? I know that in quantum field theory there is a similar construction but the path integral is in complex space. Also I have studied both the path integral approach by physicist and the SDE point of view of mathematicians but the relation between the two seems obscure to me. For instance, what is the counterpart of the ito rule, or what is a brownian bridge in the path integral approach? The books I found are not clear in this respect. === Subject: orbits of conjugation in the general linear group My question is about the orbits of the conjugation action of the general linear group on itself. I'm only interested in the complex case, of finite dimenension. Say the invertible nxn-matrices with complex coefficients. How can one calculate the dimensions of the orbits? I know that one can represent an orbit by its Jordan normal form, which is unique up to ordering the blocks. But does this say anything about the dimension of the orbit? And what about the dimension of the stabiliser of an element? Can you calculate that as well? How are the two related? Is it true that the dimensions of the orbit of an element and that of its stabiliser add up to n^2? Finally, what does the orbit space look like? I can't find anything on the web, so any advise is welcome. Also singau === Subject: Re: orbits of conjugation in the general linear group My question is about the orbits of the conjugation action of the > general linear group on itself. I'm only interested in the complex > case, of finite dimenension. Say the invertible nxn-matrices with > complex coefficients. > How can one calculate the dimensions of the orbits? I assume that you mean the dimension as a manifold - provided that you already know that the orbits _are_ manifolds (which is true). > I know that one > can represent an orbit by its Jordan normal form, which is unique up > to ordering the blocks. But does this say anything about the dimension > of the orbit? > And what about the dimension of the stabiliser of an element? Can you > calculate that as well? How are the two related? Is it true that the > dimensions of the orbit of an element and that of its stabiliser add > up to n^2? Yes, this is a good point where I would start from as well. Say we are interested in the orbit of matrix B in G=GL(n,C). If G_B denotes the stabilizer of B then we have a bijection G/G_B -> G.B. G.B denotes the orbit of B. G_B is a manifold, more generally it is a closed Lie group of the Lie group G, and G/G_B is a manifold. Furthermore it is true that G.B is a manifold and the bijection is a diffeomorphism. In particular we have that dim(G/G_B) = dim (G.B). Since dim(G/G_B) = dim(G) - dim(G_B) your guess on the dimensions is right if you know that dim(G) = n^2. Now all depends on the calculation of dim(G_B). This dimension is equal to the dimension of the tangent space of G_B at every point of G_B, especially I (the identity matrix). Since the tangent space at I is (isomorphic to) the kernel of the commutator map [B,.] considered as endomorphism of M(nxn,C) we are done with the general stuff here. Using your idea on the Jordan normal form should give a good starting point to calculate the kernel of [B,.]. Note that the commutator nicely behaves under conjugation, so you can assume that B is already in Jordan normal form, w.l.o.g. > Finally, what does the orbit space look like? Does the representation as G/G_B suffice? ;) > I can't find anything on the web, so any advise is welcome. Also Topic: Matrix groups in textbooks on Lie groups. > singau HTH. -- Best wishes, J. === Subject: Sagan's Identity: Billions and billions of solutions =-ACVjwoAAAAVqSiDl929-Pe1jSK2zs-Q SV1),gzip(gfe),gzip(gfe) 1. Sagan's Identity For Carl Sagan (1934-1996), an astronomer, science popularizer, novelist, and humanist: 1 + 5^k + (3+2y)^k + (3-2y)^k + (-3+3y)^k + (-3-3y)^k = (-2+x)^k + (-2- x)^k + (5-y)^k + (5+y)^k for k = 1,3,5,7 if x^2-10y^2 = 9. This has, er, billions and billions (Sagan's favorite catchphrase) of solutions. Actually, there are an infinite number of rational solutions to the conditional equation, as well as integral ones if it is to be treated as a Pell equation. Identities like these are in the context of the Prouhet-Tarry-Escott Problem. It is easily shown that equations valid for k = 1,3,5,7 can be tweaked (with an increase in the number of terms) such that it will be valid for k = 1,2,3,...8. while. 2. X's Identity: Selling an equation in eBay? It seems naming conventions for scientific objects can be whimsical. In biology, some species are named after movie actors: an ant for Harrison Ford, a beetle for Schwarzenegger, etc. See: http://www.mentalfloss.com/blogs/archives/16839 MACHOs (massive compact halo objects), etc. In genetics, we have the sonic hedgehog gene (go figure), and others. I was thinking of doing an experiment. I have another identity for k =1,3,5,7 just like Sagan's Identity. (In fact, both are special cases of a more general one.) Can I sell the right to name it in eBay? The winning bidder can then name it after their favorite scientist (or movie star, politician, or after himself/herself, as the case may be). After all, it is said that, Politics is for the moment, but an equation is forever. Just a thought I'm considering. (I wonder how much it would go?).... :-) P.S. I hope I didn't post this twice in sci.math Yours, Tito === Subject: somewhat imperfect odd numbers If d(n) is the sum of the divisors of n which are less than n, I get for odd n: d(n)-n n ---------------------------- excess = 270 for 103367745 excess = -434 for 105771225 excess = -558 for 112183575 excess = 910 for 116103225 excess = 954 for 124405155 excess = 846 for 124607385 excess = 774 for 124742205 excess = 630 for 125011845 excess = 450 for 125348895 excess = 306 for 125618535 excess = 198 for 125820765 excess = 90 for 126022995 excess = 270 for 126498105 excess = -306 for 126764505 excess = -414 for 126966735 excess = -810 for 127021365 excess = -450 for 127034145 excess = -630 for 127371195 excess = -810 for 127609425 excess = -882 for 127843065 excess = -954 for 127977885 excess = 462 for 132335385 excess = -42 for 132701205 excess = -294 for 132884115 excess = 750 for 146418825 excess = 390 for 154364925 excess = 18 for 159030135 What is known about small values of |d(n)-n| as n grows without bound? David Bernier