mm-1469 === Subject: Re: Epistemology 202: Advanced Topics > >> I would say the basic concept of set theory is the concept of a set. > Can you explain that concept to me, keeping in mind that I am a > programmer and not a mathematician? The only set concept that I am > aware of is that of a container, meaning to me a closed boundary > isolating the things inside from the things outside, and containing no > duplicates. That closely corresponds to my notion of a set. I use the example of a dice bag when introducing the concept, where dice are the elements of the set, and the bag with its dice is the set. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Epistemology 202: Advanced Topics >> >>> I would say the basic concept of set theory is the concept of a set. >> Can you explain that concept to me, keeping in mind that I am a >> programmer and not a mathematician? The only set concept that I am >> aware of is that of a container, meaning to me a closed boundary >> isolating the things inside from the things outside, and containing no >> duplicates. > That closely corresponds to my notion of a set. I use the example of a > dice bag when introducing the concept, where dice are the elements of > the set, and the bag with its dice is the set. How does one place a boundary around that which is boundless? -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Epistemology 202: Advanced Topics > How does one place a boundary around that which is boundless? The boundary is neither physical nor geometrical. It is the distinction between what is in the set and what is not in the set. The boundary is the predicate that defines set membership. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics >> How does one place a boundary around that which is boundless? > The boundary is neither physical nor geometrical. It is the > distinction between what is in the set and what is not in the set. The > boundary is the predicate that defines set membership. Oh yeah, I forgot. You guys have magical powers. Abracadabra...Definition! -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Epistemology 202: Advanced Topics > Can you explain that concept to me, keeping in mind that I am a > programmer and not a mathematician? The only set concept that I am > aware of is that of a container, meaning to me a closed boundary > isolating the things inside from the things outside, and containing no > duplicates. Cantor -indicated- the meaning of set (as opposed to defining the meaning). He meant any grouping of objects. Grouping objects into sets, he regarded as a mental operation. For example assume there are ten pennies on the desck in front of you. Can can mentally group the pennies into a set. So you have created (in a a manner of speaking) the set of pennies on the the desck in front of you. Regarding the pennies as a collection or grouping does nothing physical to them. It is purely a thought operation. You notion of container has the essentions of a set. You can tell what is in the set and what is not in the set. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics >> Can you explain that concept to me, keeping in mind that I am a >> programmer and not a mathematician? The only set concept that I am >> aware of is that of a container, meaning to me a closed boundary >> isolating the things inside from the things outside, and containing no >> duplicates. >Cantor -indicated- the meaning of set (as opposed to defining the >meaning). He meant any grouping of objects. So points are objects now? > Grouping objects into sets, >he regarded as a mental operation. For example assume there are ten >pennies on the desck in front of you. Can can mentally group the pennies >into a set. So you have created (in a a manner of speaking) the set of >pennies on the the desck in front of you. Regarding the pennies as a >collection or grouping does nothing physical to them. It is purely a >thought operation. >You notion of container has the essentions of a set. You can tell what >is in the set and what is not in the set. >Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics > So points are objects now? They are indeed. So are lines. So are planes. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics >> So points are objects now? >They are indeed. So are lines. So are planes. So in what way are points objects apart from the intersection of lines? === Subject: Re: ed Topics >> Can you explain that concept to me, keeping in mind that I am a >> programmer and not a mathematician? The only set concept that I am >> aware of is that of a container, meaning to me a closed boundary >> isolating the things inside from the things outside, and containing no >> duplicates. > Cantor -indicated- the meaning of set (as opposed to defining the > meaning). He meant any grouping of objects. Grouping objects into sets, > he regarded as a mental operation. For example assume there are ten > pennies on the desck in front of you. Can can mentally group the pennies > into a set. So you have created (in a a manner of speaking) the set of > pennies on the the desck in front of you. Regarding the pennies as a > collection or grouping does nothing physical to them. It is purely a > thought operation. How does one mentally group an infinite set of points? Especially, when infinity cannot be conceived by a human? > You notion of container has the essentions of a set. You can tell what > is in the set and what is not in the set. > Bob Kolker -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: ed Topics > Especially, when infinity cannot be conceived by a human? Looks like most mathematicians aren't human. (Or, as I said, you're extrapolating your own limitations to all other human beings.) -- Giuseppe Oblomov Bilotta [W]hat country can preserve its liberties, if its rulers are not warned from time to time that [the] people preserve the spirit of resistance? Let them take arms...The tree of liberty must be refreshed from time to time, with the blood of patriots and tyrants. -- Thomas Jefferson, letter to Col. William S. Smith, 1787 === Subject: Re: ed Topics >>Especially, when infinity cannot be conceived by a human? > Looks like most mathematicians aren't human. Oh, they're human enough, just arrogant and self-deluded. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: ed Topics > How does one mentally group an infinite set of points? Especially, when > infinity cannot be conceived by a human? I can conceive of the set of integers, therefor I am not human. Bob Kolker === Subject: Re: ed Topics >> How does one mentally group an infinite set of points? Especially, >> when infinity cannot be conceived by a human? > I can conceive of the set of integers, No you can't. therefor I am not human. Very possibly. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Epistemology 202: Advanced Topics Lester Zick said: > >Lester has another problem: he won't accept the logical inference that > >if 3D space is defined in terms of three mutually orthogonal directions, > >then N-D space is defined by N mutually orthoganal directions. > I accept the logical inference. It just can't be made to work. Lester - What is it that doesn't work in higher dimensions? Can you give an example of things you would like to do but can't? I mean, We can even discover that one of Hilbert's axioms is wrong in space of higher than three dimensions. There are all sorts of ways to explore higher dimensions, even if our brains' visual centers aren't made to see them. So, what do you want to do in more than three dimensions? > > What does > >it mean to say that two directions orthogonal to each other? In a flat > >(Euclidian) space, it means that Pythagoras's theorem holds. But NB that > >you can define spaces in which the concepts of direction and distance > >are more abstract, so that Pythagoras's theorem is meaningless in such > >spaces, even if they are flat. Wolf - What kind of space are these? Is there any analogue of Pythagorean theorem? there must be...... > >That's enough for tonight. -- Smiles, Tony === Subject: Re: Epistemology 202: Advanced Topics >Lester Zick said: > >> >Lester has another problem: he won't accept the logical inference that >> >if 3D space is defined in terms of three mutually orthogonal directions, >> >then N-D space is defined by N mutually orthoganal directions. >> I accept the logical inference. It just can't be made to work. >Lester - >What is it that doesn't work in higher dimensions? Can you give an example of >things you would like to do but can't? I mean, We can even discover that one of >Hilbert's axioms is wrong in space of higher than three dimensions. There are >all sorts of ways to explore higher dimensions, even if our brains' visual >centers aren't made to see them. So, what do you want to do in more than three >dimensions? Tony, it isn't anything in specific that doesn't work. It's the dimensions themselves that don't and can't work. The reasoning is a little involved but is thoroughly consistent with the three basic spatial dimensions. I'm just trying to decide if this is a worthwhile topic at this point. I mean I'm so far down in arguing these kinds of points of view at this stage that I don't really see any significant upside to discussing this kind of problem right here. >> > What does >> >it mean to say that two directions orthogonal to each other? In a flat >> >(Euclidian) space, it means that Pythagoras's theorem holds. But NB that >> >you can define spaces in which the concepts of direction and distance >> >are more abstract, so that Pythagoras's theorem is meaningless in such >> >spaces, even if they are flat. >Wolf - >What kind of space are these? Is there any analogue of Pythagorean theorem? >there must be...... >> >That's enough for tonight. >-- >Smiles, >Tony === Subject: Re: Epistemology 202: Advanced Topics Lester Zick said: > >Lester Zick said: > > > >> >Lester has another problem: he won't accept the logical inference that > >> >if 3D space is defined in terms of three mutually orthogonal directions, > >> >then N-D space is defined by N mutually orthoganal directions. > >> > >> I accept the logical inference. It just can't be made to work. > >Lester - > >What is it that doesn't work in higher dimensions? Can you give an example of > >things you would like to do but can't? I mean, We can even discover that one of > >Hilbert's axioms is wrong in space of higher than three dimensions. There are > >all sorts of ways to explore higher dimensions, even if our brains' visual > >centers aren't made to see them. So, what do you want to do in more than three > >dimensions? > Tony, it isn't anything in specific that doesn't work. It's the > dimensions themselves that don't and can't work. The reasoning is a > little involved but is thoroughly consistent with the three basic > spatial dimensions. I'm just trying to decide if this is a worthwhile > topic at this point. I mean I'm so far down in arguing these kinds of > points of view at this stage that I don't really see any significant > upside to discussing this kind of problem right here. Well, Lester, this has come up before, and I have to say I am about totally diametrically opposed to you on this one. Having played with higher dimensions for 25 years now, I feel confident in saying that they work very well in a number of ways. If you could explain what doesn't seem to you to work, maybe I can help it work, or maybe you can alert me to a problem I haven't considered. I know this isn't the central thread of your thinking, but I have a feeling it's actually quite related, so if you feel like it.... Do you know Hilbert's postulate 1.7 is wrong in dimensions over 3? I figured this out recently trying to generalize them and integrate them with axioms from other areas. I wasn't sure at first, though I had a formula that made sense. I just couldn't picture two planes intersecting at only a single point. Can you? Well, it turns out they can. Picture a 4D space (okay, not in your head) with w,x,y,and z axes, and two planes. The first is the plane where y and z are zero, extending on the plane that includes the w and x axes, so each point is (w,x,0,0). The second lies on the plane with the y and z axes and has zero for all w and x, so all points are of the form (0,0,y,z). We can see that the only point which will lie on both these planes is the origin, (0,0,0,0). I can't picture it, but I know it works, and it ends up putting what we consider to be basic geometrical rules in a larger context as special cases. I know you hate the idea of dimension as a dependent variable, but you might want to try getting used to it. It's fun!! :D Okay I'm outta here for the day. G'night! -- Smiles, Tony === Subject: Re: Epistemology 202: Advanced Topics >Lester Zick said: >> >Lester Zick said: >> > >> >> >Lester has another problem: he won't accept the logical inference that >> >> >if 3D space is defined in terms of three mutually orthogonal directions, >> >> >then N-D space is defined by N mutually orthoganal directions. >> >> >> >> I accept the logical inference. It just can't be made to work. >> >Lester - >> >What is it that doesn't work in higher dimensions? Can you give an example of >> >things you would like to do but can't? I mean, We can even discover that one of >> >Hilbert's axioms is wrong in space of higher than three dimensions. There are >> >all sorts of ways to explore higher dimensions, even if our brains' visual >> >centers aren't made to see them. So, what do you want to do in more than three >> >dimensions? >> Tony, it isn't anything in specific that doesn't work. It's the >> dimensions themselves that don't and can't work. The reasoning is a >> little involved but is thoroughly consistent with the three basic >> spatial dimensions. I'm just trying to decide if this is a worthwhile >> topic at this point. I mean I'm so far down in arguing these kinds of >> points of view at this stage that I don't really see any significant >> upside to discussing this kind of problem right here. >Well, Lester, this has come up before, and I have to say I am about totally >diametrically opposed to you on this one. Having played with higher dimensions >for 25 years now, I feel confident in saying that they work very well in a >number of ways. If you could explain what doesn't seem to you to work, maybe I >can help it work, or maybe you can alert me to a problem I haven't considered. >I know this isn't the central thread of your thinking, but I have a feeling >it's actually quite related, so if you feel like it.... Actually it turns out to be quite central to my thinking, Tony. Would it surprize you if I said it's the same problem that proves FLT? In other words if more than three dimensions were possible FLT could not be true (for what it's worth this short form proof of FLT goes back to 1989 and antedates the thousand page proof by several years). Let me analogize the issue this way: if spatial dimensionality is put together with round pegs in round holes and suddenly changes to square pegs at dimension four, there is no way to progress beyond three dimensions (not that I expect you to take an analogical explanation). In other words I'm trying to highlight the fact that there is a strictly mechanical problem here: higher dimensionality just doesn't and can't fit together with the three lower dimensions. >Do you know Hilbert's postulate 1.7 is wrong in dimensions over 3? I figured >this out recently trying to generalize them and integrate them with axioms from >other areas. I wasn't sure at first, though I had a formula that made sense. I >just couldn't picture two planes intersecting at only a single point. Can you? >Well, it turns out they can. Picture a 4D space (okay, not in your head) with >w,x,y,and z axes, and two planes. The first is the plane where y and z are >zero, extending on the plane that includes the w and x axes, so each point is >(w,x,0,0). The second lies on the plane with the y and z axes and has zero for >all w and x, so all points are of the form (0,0,y,z). We can see that the only >point which will lie on both these planes is the origin, (0,0,0,0). I can't >picture it, but I know it works, and it ends up putting what we consider to be >basic geometrical rules in a larger context as special cases. I know you hate >the idea of dimension as a dependent variable, but you might want to try >getting used to it. It's fun!! :D >Okay I'm outta here for the day. G'night! >-- >Smiles, >Tony === Subject: Re: Epistemology 202: Advanced Topics >Lester Zick said: >> >Lester Zick said: >> > >> >> >Lester has another problem: he won't accept the logical inference that >> >> >if 3D space is defined in terms of three mutually orthogonal directions, >> >> >then N-D space is defined by N mutually orthoganal directions. >> >> >> >> I accept the logical inference. It just can't be made to work. >> >Lester - >> >What is it that doesn't work in higher dimensions? Can you give an example of >> >things you would like to do but can't? I mean, We can even discover that one of >> >Hilbert's axioms is wrong in space of higher than three dimensions. There are >> >all sorts of ways to explore higher dimensions, even if our brains' visual >> >centers aren't made to see them. So, what do you want to do in more than three >> >dimensions? >> Tony, it isn't anything in specific that doesn't work. It's the >> dimensions themselves that don't and can't work. The reasoning is a >> little involved but is thoroughly consistent with the three basic >> spatial dimensions. I'm just trying to decide if this is a worthwhile >> topic at this point. I mean I'm so far down in arguing these kinds of >> points of view at this stage that I don't really see any significant >> upside to discussing this kind of problem right here. >Well, Lester, this has come up before, and I have to say I am about totally >diametrically opposed to you on this one. Having played with higher dimensions >for 25 years now, I feel confident in saying that they work very well in a >number of ways. If you could explain what doesn't seem to you to work, maybe I >can help it work, or maybe you can alert me to a problem I haven't considered. >I know this isn't the central thread of your thinking, but I have a feeling >it's actually quite related, so if you feel like it.... Well, Tony, let me ponder the issue. If I do I'll need to get the expression right because I know it'll generate a lot of flack. >Do you know Hilbert's postulate 1.7 is wrong in dimensions over 3? I figured >this out recently trying to generalize them and integrate them with axioms from >other areas. I wasn't sure at first, though I had a formula that made sense. I >just couldn't picture two planes intersecting at only a single point. Can you? >Well, it turns out they can. Picture a 4D space (okay, not in your head) with >w,x,y,and z axes, and two planes. The first is the plane where y and z are >zero, extending on the plane that includes the w and x axes, so each point is >(w,x,0,0). The second lies on the plane with the y and z axes and has zero for >all w and x, so all points are of the form (0,0,y,z). We can see that the only >point which will lie on both these planes is the origin, (0,0,0,0). I can't >picture it, but I know it works, and it ends up putting what we consider to be >basic geometrical rules in a larger context as special cases. I know you hate >the idea of dimension as a dependent variable, but you might want to try >getting used to it. It's fun!! :D >Okay I'm outta here for the day. G'night! >-- >Smiles, >Tony === Subject: Re: Question about the minesweeper consistency problem (NP-complete problems) > > > > > > I probably should have used the word 'average' instead of > > > 'expected'; more precisely, there's an algorithm for an > > > NP-complete problem where, if you calculate the running > > > time for each instance of size n and take the average > > > (mean), the average running time is polynomial in n. > Actually, this is true for some NP-complete problems but not for > others. Lasse --- ca 1988 there were some folks at MIT trying to get a program to learn all available information. Now, with the web and search engines it should be so much easier for such a program to find stuff to learn. Of course, you could say search engines sorta do that.. but not quite.. any updates on what happened? - = - Vasos-Peter John Panagiotopoulos II, Columbia'81+, Bio$trategist BachMozart ReaganQuayle EvrytanoKastorian http://ourworld.compuserve.com/homepages/vjp2/vasos.htm ---{Nothing herein constitutes advice. Everything fully disclaimed.}--- [Homeland Security means private firearms not lazy obstructive guards] [Fooey on GIU,{MS,X}Windows 4 Bimbos] [Cigar smoke belongs in veg food group] %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > ca 1988 there were some folks at MIT trying to get a program to > learn all available information. Now, with the web and search > engines it should be so much easier for such a program to find stuff > to learn. Of course, you could say search engines sorta do that.. but > not quite.. any updates on what happened? Your mistake in logic is the belief that the information we learn is stuff you can put into books. Learning to stand up and walk is what we learn. You don't need the web to give a machine access to that problem. Just give it legs and arms. Build a machine that can learn to stand up and walk on it's own (without being programmed to walk by you) in order to get the things it needs in life, and you will be way on your way to understanding AI. If however, you are not interested in real AI, and just want to see what has been happening in the information sucking business, check out sites like www.cyc.com. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ >> ca 1988 there were some folks at MIT trying to get a program to >> learn all available information. Now, with the web and search >> engines it should be so much easier for such a program to find stuff >> to learn. Of course, you could say search engines sorta do that.. but >> not quite.. any updates on what happened? >Your mistake in logic is the belief that the information we learn is stuff >you can put into books. >Learning to stand up and walk is what we learn. You don't need the web >to give a machine access to that problem. Just give it legs and arms. >Build a machine that can learn to stand up and walk on it's own (without >being programmed to walk by you) in order to get the things it needs in >life, and you will be way on your way to understanding AI. >If however, you are not interested in real AI, and just want to see what >has been happening in the information sucking business, check out sites >like www.cyc.com. I always did like your style, Welch. You tell it like you see it. Louis Savain The Silver Bullet: Why Software Is Bad and What We Can Do to Fix it http://users.adelphia.net/~lilavois/Cosas/Reliability.htm <20050503141632.632$lh@newsreader.com> <0qlf71556icqlcvvasbhll6el52vb8o1a2@4ax.com> have faith! the Sony walking robots can mimic us quite well. how did two entirely different movement and structural and power and processing and manufactured systems result in 2 closely matched able bodies? the fact murphies law of doubling the speed of computers every 18 monhts is slowing down means the computers are nearly ready. a 10 GHZ serial processor VS 10 Billion neural processor. intelligent design + pinnacle of computer power = AI will be here soon Herc <20050503141632.632$lh@newsreader.com> <0qlf71556icqlcvvasbhll6el52vb8o1a2@4ax.com> I just realised the connection with the 2 varients of murpheys law there.. if it can happen it will. http://www.plyojump.com/qrio.html Herc > I just realised the connection with the 2 varients of murpheys law > there.. > if it can happen it will. > http://www.plyojump.com/qrio.html > Herc Herc, It's spelled 'Murphy's.' Later, Pepe le Pew -- PT Barnum was right ! %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > I always did like your style, Welch. You tell it like you see it. :) > Louis Savain > The Silver Bullet: Why Software Is Bad and What We Can Do to Fix it > http://users.adelphia.net/~lilavois/Cosas/Reliability.htm Hey, I just scanned over that stuff. Starting about 10 years ago after doing a lot of Smalltalk work, I got very excited about the idea of signal processing (message passing) as a programming paradigm. It always struck me as an approach that showed great promise in breaking free of our dominate single-thread view of software and moving into a truly parallel model of machine operation. I never got very far however on making the idea work. There always seemed to be fatal issues with deadlocks when messages queues filled up. However, I still have this gut feeling like it's the way to go and that I was just never looking at it correctly. So I'm excited to see you have been exploring similar idea. On the message-queue deadlock problem, I heard one idea from a friend that gave me hope but which I have not explored. It was the idea of a pull-based paradigm instead of push based paradigm. That is, instead of thinking of a computer as a sending a signal when you hit a key, which then, like a domino effect, explodes out into many different paths, to produce the correct response from the computer, think instead of the computer screen as sucking data out of the computer as it needs it. The point here is that with a push system, things can jam up - message queues can fill up and deadlock. But with a pull system, data only flows when it is asked for - when it's needed. Now, there's not much difference from a push system with flow control and a pull system, but yet I feel there's something to this approach which might allow one to design massively parallel processing systems that don't have deadlock problems. And your whole point about how a single processing system could inherently improve software reliability I think is an exciting idea. I don't know if I buy it, but I like the sound of the idea. With the size and complexity of software these days, it's getting damn hard not to have bugs in it. And, as you already know, my AI work has made great progress by switching to a temporal signal process paradigm (because of your suggestions a few years back). So the idea of applying some of that knowledge back to the software reliability problem sounds exciting to me. So, I'll have to look closer at what you have been able to do with your cosas stuff and see if there's something there I can get excited about as well... -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ <20050503141632.632$lh@newsreader.com> <0qlf71556icqlcvvasbhll6el52vb8o1a2@4ax.com> I just wanted to say that I have become obsessed with this thing. I can't stop. > ca 1988 there were some folks at MIT trying to get a program to > learn all available information. Now, with the web and search > engines it should be so much easier for such a program to find stuff > to learn. Of course, you could say search engines sorta do that.. but > not quite.. any updates on what happened? Yes, they let it loose on Usenet and the web, so it got dumber and dumber by the minute. So they had to shut it down in order to avoid core meltdown. They stripped the search engine off the program and made a fortune with that :-) Dirk Vdm > Of course, you could say search engines sorta do that.. but > not quite.. any updates on what happened? http://commonsense.media.mit.edu/cgi-bin/search.cgi There used to be an AI/NatLang DOS proggie (NDFILT) that took a file and made it more terse. ANy newer versions out there? - = - Vasos-Peter John Panagiotopoulos II, Columbia'81+, Bio$trategist BachMozart ReaganQuayle EvrytanoKastorian http://ourworld.compuserve.com/homepages/vjp2/vasos.htm ---{Nothing herein constitutes advice. Everything fully disclaimed.}--- [Homeland Security means private firearms not lazy obstructive guards] [Fooey on GIU,{MS,X}Windows 4 Bimbos] [Cigar smoke belongs in veg food group] === Subject: Re: how Herc sees sci.logic discussing Godel's proof > Bill Gates is the worlds most recognised BILLionaire IS A COINCIDENCE! No that's a crime. > Lady Di died IS A COINCIDENCE A lot of people died. In fact, everybody dies sooner or later. > HawKING is the smartest IS A COINCIDENCE There is much to be said about Hawking being the smartest. He's surely a very intelligent person, and given his physical situation his achievements are exceptional. But that has little to do with kings. I'd rather try to stress on the HAWK part, peery eyes and all. > Tiger WOODs the best golfer IS A COINCIDENCE Yes. > Ronald Raegun initiated space ray guns IS A COINCIDENCE No, that's a typo. Reagan is the correct spelling, and anyway he initiated shit. He financed research about space warfare with your tax money, yes. > Michelangelo painted angels IS A COINCIDENCE And a lot of other stuff. Plus, a lot of people painted angels. > TutanKham had the best tomb IS A COINCIDENCE Where's the connection? > Nic Cage steals cars and goes to prison IS A COINCIDENCE No, that's a movie. > Peano formalised the unit number (phallus) IS A COINCIDENCE No, it's a non-sequitur. > GODel formalised the unprovable IS A COINCIDENCE No, it's a typo. G.9adel or Goedel (depending on preferred spelling). > Cantor can't order IS A COINCIDENCE Cantor can order all right. -- Giuseppe Oblomov Bilotta Axiom I of the Giuseppe Bilotta theory of IT: Anything is better than MS === Subject: Re: CARDINALITY IS * E * N * T * I * R * E * L * Y DEBUNKED <4273894d$0$580$ed2e19e4@ptn-nntp-reader04.plus.net> > S = {1, 2} > P(S) = ( {}, {1},. {2}, {1,2} } > P_1 = {} > P_2 = {1} > P_3 = {2} > P_4 = {1, 2} > the elements that don't map to themselves = {1, 2} > P_4 = {1, 2} > there is an n outside of S that will list X. 4! What is your point here? There exists a function mapping the set N of natural numbers into its powerset such that every FINITE subset of N belongs to the image of this function? This is certainly true --- the set of FINITE subsets of N is countable, while the set of ARBITRARY subsets is not. > therefore its not 1-1. What is not 1-1? The mapping {1,2,3,4} -> P({1,2}) that you defined certainly seems to be bijective. > S = {1, 2} > F(S) = {1, 2, 0} What is F(S)??? > if S is finite, then its not 1-1, then its BIGGER! some proof, S + 0 > S. > if S is infinite, then it is 1-2, then its not BIGGER. Now here you lose me completely. What are you trying to say? Perhaps you are saying that the power set of any finite set is finite. Since the finite sets 'converge' to the natural numbers, the cardinality of the power set of N should be the same as that of N? There is no reason for this 'limit argument' to be valid --- and in fact, it is not. Perhaps you are thinking of FINITE subsets (see above). > Now comes your Russels set in the powerset mapping, its not 1-1 in > finite and infinite versions ?????????? Lasse === Subject: Re: CARDINALITY IS * E * N * T * I * R * E * L * Y DEBUNKED > define a list > define x(n) = NOT(list(n,n)).......... BEWARE CHILDREN IT MAY TAKE ON > ANOTHER FORM TO DISGUISE ITSELF > ITS THE MOST FUCKING STUPID CONSTRUCTION YOU COULD POSSIBLY MAKE. Shouting obscenities has never convinced anyone, Herc. > NOONE KNOWS WHAT THE FUCK THE FORUMALA REFERS TO BUT ITS NOT MEMBER > 1, AND ITS NOT MEMBER 2, AND ITS NOT MEMBER 3.... interpretation, but the actual construction is extremely explicit. I suppose you mean: given a sequence of sequences of natural numbers, the 'diagonal sequence' does not appear in this sequence. Well, if you produce explicitly any sequence of sequences of natural numbers, I can tell you explicitly what each entry of the diagonal sequence is. There is, in fact, no self-reference involved at all: from a given object (a 'list', as you seem to call it), we construct another object which depends on the first: namely the diagonal sequence. It's no more self-referential than taking the product of two natural numbers is. > THAT ALL IT IS.... ITS ( MEMBER = NOT MEMBER ) with a list > attached. ??? > RUSSELS SET IS AN IMPOSSIBLE SET.. IT ****HAS**** A DEFINTION. No, it does not. The axioms of set theory do NOT, in general, allow you to build a set { x : P(x) } for a given predicate P. What IS allowed is to form, given any set X, the set { x in X: P(x) }. The former would allow self-reference; the latter does not, and thus resolves Russel's 'paradox'. You seem to be under the impression that actual paradoxes exist in today's mathematics. This is not the case --- within the framework of modern logic, there are no (known) paradoxes, and it is believed there are none. Instead, the word 'paradox' is now often used to descrbe phenomena which, at first glance, seem to be paradoxical: either because two statements which at first seem contradictory do not, in fact, contradict each other, or because there is some step in the construction which is not justified (such as the formation of a 'set' whose existence is not guaranteed by the axioms of set theory). Lasse === Subject: Re: CARDINALITY IS * E * N * T * I * R * E * L * Y DEBUNKED Let me try to summarize the argument so far: Simplicio is trying to grasp the subtleties of the concept of infinity. He learns enough to make a few statements that seem to make sense to him, and working with these incomplete / imprecise / inaccurate statements he produces a conclusion that appears inconsistent with a basic concept in mathematics. Because the nature of the human mind is such that the errors of others are patently obvious, but one's own errors are invisible [axiomatic to any proofreader], Simplicio has concluded that he is right and a basic concept successfully used for thousands of years of mathematics is flawed. Let me inject a passage on this topic of cardinality written in 1638 (Discourses and Mathematical Demonstrations Relating to Two New Sciences - Galileo): +++ Simplicio: Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension. Salviati: This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another. To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty. I take it for granted that you know which of the numbers are squares and which are not. Simplicio: I am quite aware that a squared number is one which results from the multiplication of another number by itself; this 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by themselves. Salviati: Very well; and you also know that just as the products are called squares so the factors are called sides or roots; while on the other hand those numbers which do not consist of two equal factors are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not? Simplicio: Most certainly. Salviati: If I should ask further how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. Simplicio: Precisely so. Salviati: But if I inquire how many roots there are, it cannot be denied that there are as many as the numbers because every number is the root of some square. This being granted, we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said that there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers, Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part of all the numbers; up to 10000, we find only 1/100 part to be squares; and up to a million only 1/1000 part; on the other hand in an infinite number, if one could conceive of such a thing, he would be forced to admit that there are as many squares as there are numbers taken all together. Sagredo: What then must one conclude under these circumstances? Salviati: So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally the attributes equal, greater, and less, are not applicable to infinite, but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number. +++ HERC777 seems to be more interested in uttering profanities than in seeking wisdom, and for my nickel should be left to rot in a puddle of his own swill, unless he can gather the wit to learn the way out. Tom Davidson Richmond, VA === Subject: Re: CARDINALITY IS * E * N * T * I * R * E * L * Y DEBUNKED Well if it isn't text boi himself on his quest to immitate life with quotation. Now watch as Tom dissapears for fear of recognising this disproof of Cardinality in front of his peers. AN NON DIAGONALISABLE LIST say for arguments sake a UTM terminates on every input (row, col). the nth real is UTM(n, digit) mod 10. this makes a list of reals UTM(1, digit) == 0.23232.. UTM(2, digit) == 0.232323.. UTM(3, digit) == 0.3141592.. in the 1st digit position there are oo 1s, ie there are oo reals starting with 1. there are oo reals starting with 2 and so on. there are also oo reals with the 2nd digit is a 1, 2, and so forth. there are also oo reals starting with 0.11, 0.12, 0.14 etc. and so forth, and they are all listable. We want to make an infinite set of reals were the ordering is not important. STEP 1 start at digit 1 STEP 2 use a radioactive decay device to generate a random number from 0 to 9. STEP 3 run through the original list of reals from real 1 onwards until you find a real with the random digit matching that digit. STEP 4 remove that real from the list and add it to the second list. STEP 5 advance one digit position STEP 6 Go to STEP 2 Say the random generator gives the digit sequence <324643> then the diagonal on the second list will be that sequence. 0.3.... 0..2... 0...4... 0....6... 0.....4... 0......3.. According to quantum physics, all reals on the original list will fit into the second list. The diagonal of the new set of computable numbers can be ANY RANDOM SEQUENCE. The anti-digaonal could similarly be ANY REAL NUMBER. Run the process a second time and the the missing reals will be entirely different. Hence, there are no missing reals to start with! Herc === Subject: Re: CARDINALITY IS * E * N * T * I * R * E * L * Y DEBUNKED In sci.logic, HERC777 on 3 May 2005 17:08:50 -0700 > Well if it isn't text boi himself on his quest to immitate life with > quotation. Now watch as Tom dissapears for fear of recognising this > disproof of Cardinality in front of his peers. > AN NON DIAGONALISABLE LIST > say for arguments sake a UTM terminates on every input (row, col). > the nth real is UTM(n, digit) mod 10. > this makes a list of reals > UTM(1, digit) == 0.23232.. > UTM(2, digit) == 0.232323.. > UTM(3, digit) == 0.3141592.. Here's an alternate construct which is far less ambiguous. T_10 = {b/10^k, b, k in N, b < 10^k}. One can walk through it in a fairly obvious fashion: T_10(0, digit) = 0.000000.... T_10(1, digit) = 0.100000.... ... T_10(9, digit) = 0.900000.... T_10(10, digit) = 0.01000000... T_10(11, digit) = 0.11000000... T_10(12, digit) = 0.21000000... ... T_10(98, digit) = 0.89000000... T_10(99, digit) = 0.99000000... T_10(100, digit) = 0.00100000... T_10(101, digit) = 0.10100000... T_10(102, digit) = 0.20100000... ... > in the 1st digit position there are oo 1s, ie there are oo reals > starting with 1. there are oo reals starting with 2 and so on. > there are also oo reals with the 2nd digit is a 1, 2, and so forth. > there are also oo reals starting with 0.11, 0.12, 0.14 etc. > and so forth, and they are all listable. So far, you're fine. > We want to make an infinite set of reals were the ordering is not > important. > STEP 1 > start at digit 1 > STEP 2 > use a radioactive decay device to generate a random number from 0 to 9. > STEP 3 > run through the original list of reals from real 1 onwards until you > find a real with the random digit matching that digit. > STEP 4 > remove that real from the list and add it to the second list. > STEP 5 > advance one digit position > STEP 6 > Go to STEP 2 > Say the random generator gives the digit sequence <324643> > then the diagonal on the second list will be that sequence. > 0.3.... > 0..2... > 0...4... > 0....6... > 0.....4... > 0......3.. > According to quantum physics, all reals on the original list will fit > into the second list. Not sure why QM is even an issue here; it's irrelevant. Spin a wheel if that's your fancy, or roll a 20-sided die (such die are common in D&D games). Or implement a simple electronic circuit consisting of a counter, an oscillator, a switch, and whatever is needed to drive a Nixie tube (there, I'm dating myself :-) ) or a 7-segment display or a dot matrix encoder, or even just 4 LEDs which blink in binary (or BCD; they're the same for a single digit). Then push the button, Frank. [*] :-) > The diagonal of the new set of computable numbers can be ANY RANDOM > SEQUENCE. The anti-digaonal could similarly be ANY REAL NUMBER. > Run the process a second time and the the missing reals will be > entirely different. Hence, there are no missing reals to start with! As usual, there's a few issues here. 1. #(P(S)) > #(S). [+] While not directly relevant to the #R problem, it nonetheless shows that there's an infinite number of transfinites. Your argument above only works on dense sets. N is not dense. 2. I'll see your UTM and raise you a T_10, which has exactly the same properties as your UTM, at least as enumerated in your proof here. To wit: a: T_10 has an infinite number of entries whose first digit is .1 . b: T_10 has an infinite number of entries whose first two digits are .12 . c: T_10 has an infinite number of entries whose first three digits are .123 . etc. (The proof is left to the interested reader, but should be trivial.) In short, T_10 contains an infinite number of entries whose first few, first few thousand, first googol, care to throw at it. This includes all finite prefixes of a certain number that is your b.92éte noire, as far as I can tell: 1/3. Is 1/3 in T_10? For any epsilon > 0, I can find an infinite number of elements in T_10 within that epsilon of 1/3, or any other real in [0,1). T_10 is quite dense over [0,1). In a very real sense T_10 covers all of [0,1] (since dots cannot have 0 radius in paper representations). Nevertheless, 1/3 is not in T_10. Why? Well, for starters, a simple inductive proof shows that 10^n mod 3 = 1 for any integer n > 0. For n = 0 it's obvious (1 = 1). If 10^n mod 3 = 1, then 10^(n+1) mod 3 = 10 * (10^n mod 3) = 10 * 1 = 1. Hence 10^n mod 3 is always 1. A corollary of this of course is that there is no possibility of 10^n / 3 being an integer, which means no integer b can be such that b / 10^n = 1/3, since that would imply 3b / 10^n = 1 or 10^n/3 = b. Therefore 1/3 is not in T_10 and T_10 cannot possibly cover all of [0,1), in the mathematical sense. Your proof doesn't work and has never worked properly. 3. There are a fair number of technical problems just generating UTM(k,d). For a halting machine it's not too difficult; just take the output string. However, for a non-halting machine there are at least two conditions. [a] The machine keeps on going and going and going and ... In this case, one can easily prove that the machine will generate a rational number (since it only has a finite number of states). method would take the first-ever digit written in that cell, though that has a number of issues. Given the rest of the proof these technical problems are comparatively minor. [4] The set of all computable reals (with blank input tape) is by necessity denumerable; there are only so many machines. I would posit that the anti-diagonal number is not among them, but it's never been a requirement that all reals in R be computable. If the antidiag D were machine-computable, then it would have to be with a machine M_D. Therefore UTM(M_D, digit) = D. The problem is that D(j) is constructed so that it never equals UTM(j, j). Therefore D(M_D) != UTM(M_D, M_D) and D is not machine-computable. [5] There are far too many technical problems in a satellite mindbeam-injection system. For starters, they'd have to *find* you while looking down -- an issue if you were to, say, be making love to a woman who is on top of you; where will the beam go? [-] Or perhaps you're a daddy with a small child riding on one's shoulders. (Daddy, why does my brain hurt?) Even without any salaciousness there are problems; consider a multilevel hotel, apartment building, parking garage, or mall. Presumably most hardtop cars are extremely good electromagnetic shields (ragtops of course are not, but somehow I doubt you own a convertible). :-) And then there's the 3-D problem. Say you had a tub of ball bearings, one of which is colored blue, and a pencil. How does one orient the pencil so that only the blue bearing is the one that gets touched, assuming that one can actually thrust the pencil through the bb's without disturbing them? One would need at least three pencils, and even then, there would be issues with partial stimulation of some of the bb's. Then there's the issue of actually *finding* the relevent ball bearing, since the satellite can only see part of one's head -- assuming the satellite can see you at all. And then there's the twinkle problem. Stars twinkle because the atmosphere distorts the beam of light. If you've ever seen a mirage of a lake on a hot roadway on a summer day, you'll appreciate in part the problem. I can see atmospheric distortions easily enough if I open my car door after it's been sitting, and the sun's positioned just right. One should also be able to see heated air distortions on a very cold day, as well. All this makes for a very tricky beam-positioning problem; that pencil of light is *wobbling*, and fairly badly. The best the government might do is to bounce a beam off a window nearby, and interpret thereby the noises inside of the building. (Such was actually been put into use, AIUI, somewhere on the Finno-Russian border.) Think quiet thoughts. > Herc [*] Mystery Science Theater 3000. [+] A more formal statement is that all mappings from S -> P(S) are not onto; there is an element X (the diagonalization construct!) such that f(s) != X for all s in S. [-] I guess it works for James Bond, though. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: ellipse: shortest distance to given point Hello I have an ellipse with x-axis-radius a and y-axis-radius b. Thus the ellipse fulfills the equation x^2/a^2 + y^2/b^2 = 1 for every (x,y) on that ellipse. For a point (x,y) not on this ellipse, is it right that the solution for x^2/(a + c)^2 + y^2/(b + c)^2 = 1 on variable c yields the minimal distance c from (x,y) to some point on the original ellipse (if not, how to interpret c then)? The idea is that i paint a second ellipse around the original ellipse that has a constant distance to the original ellipse and contains point (x,y). (I am only interested in the first quarter of the ellipse by the way.) Mathematica gives a very long solution on c (several pages) so it seams faster to compute it by newton approximation or secant approximation (for the solution datatype double). Any one know what approximation method yields at fastest the result? === Subject: Re: ellipse: shortest distance to given point The shortest distance from a point on an ellipse to a point not on the ellipse is a vector 90 deg from the ellipse. This is the normal unit vector of the ellipse. So to get this take the grad of that function to get this vector. ie grad(f) = df/dx (i) + df/dy (j) So the unit normal for an ellipse of the form x^2/a^2 + y^2/b^2 = c is: un = ( (2/a^2)x, (2/b^2)y ) / ||( (2/a^2)x, (2/b^2)y )|| Now if we have two point, p1 and p2 with p1 being of the ellipse its just nat (normal at point) = un (dot product) p1 then just find the vector from nat to p2 then find the length So, shortest distance = || p2 - nat || Hope that helps DP > Hello > I have an ellipse with x-axis-radius a and y-axis-radius b. > Thus the ellipse fulfills the equation > x^2/a^2 + y^2/b^2 = 1 > for every (x,y) on that ellipse. > For a point (x,y) not on this ellipse, is it right that the solution for > x^2/(a + c)^2 + y^2/(b + c)^2 = 1 > on variable c yields the minimal distance c from (x,y) to some point on > the original ellipse > (if not, how to interpret c then)? The idea is that i paint a second > ellipse around the original ellipse that > has a constant distance to the original ellipse and contains point (x,y). > (I am only interested in the first quarter of the ellipse by the way.) > Mathematica gives a very long solution on c (several pages) so it seams > faster to > compute it by newton approximation or secant approximation (for the > solution datatype double). > Any one know what approximation method yields at fastest the result? === Subject: Re: ellipse: shortest distance to given point Correction, dont (un (dot product) p1) just (un multiply p1), go on from there. DP > The shortest distance from a point on an ellipse to a point not on the > ellipse is a vector 90 deg from the ellipse. > This is the normal unit vector of the ellipse. So to get this take the > grad of that function to get this vector. > ie grad(f) = df/dx (i) + df/dy (j) > So the unit normal for an ellipse of the form x^2/a^2 + y^2/b^2 = c is: > un = ( (2/a^2)x, (2/b^2)y ) / ||( (2/a^2)x, (2/b^2)y )|| > Now if we have two point, p1 and p2 with p1 being of the ellipse its just > nat (normal at point) = un (dot product) p1 > then just find the vector from nat to p2 then find the length > So, > shortest distance = || p2 - nat || > Hope that helps > DP >> Hello >> I have an ellipse with x-axis-radius a and y-axis-radius b. >> Thus the ellipse fulfills the equation >> x^2/a^2 + y^2/b^2 = 1 >> for every (x,y) on that ellipse. >> For a point (x,y) not on this ellipse, is it right that the solution for >> x^2/(a + c)^2 + y^2/(b + c)^2 = 1 >> on variable c yields the minimal distance c from (x,y) to some point on >> the original ellipse >> (if not, how to interpret c then)? The idea is that i paint a second >> ellipse around the original ellipse that >> has a constant distance to the original ellipse and contains point (x,y). >> (I am only interested in the first quarter of the ellipse by the way.) >> Mathematica gives a very long solution on c (several pages) so it seams >> faster to >> compute it by newton approximation or secant approximation (for the >> solution datatype double). >> Any one know what approximation method yields at fastest the result? === Subject: Re: ellipse: shortest distance to given point ETAuAhUAo6JXEQSkZkpE7UWPuSbWcLuHFN4CFQCEc1nuR959uIoRYsu91qVMkA7/OQ== To begin with, your second curve is not an ellipse. It looks like one, but no cigar. What actualy happens can be seen most clearly by considering the limiting case where the eccentricity approaches 1 with fixed focal points and the ellipse thus becomes a line segment. The curve that's a constan distance from a line segment consists of two parallel line segments connected by semicircular caps. This is not an ellipse or any limiting case of the ellipse. Mathematica is right to give you a complicated expression for c. To find the point on the ellipse that's closest to a given point (x, y), one must solve what proves to be a 4th-degree equation for one of the coordinates of the point on the ellipse. Applying the standard technique symbolically then gives that unwieldy expression. It gets even worse that this, however, for the casus irreucibilis can rear its ugly head. In the context of 4th-degree equations this refers to the resolvent cubic having three real roots and thereby generating complex cube roots in that complicated expression from Mathematica. Geometrically, the casus irreducibilis occurs when there are four rather than two lines you can draw normal to an ellipse to a given point -- essentially when the ellipse has high eccentricity and your (presumably exterior) point is relatively close to the minor-axis diameter. (Kids, do try this at home. It's kind of neat to see this change in in the character of the nomals.) Given the fact that your constant-distance curve is not an ellipse and analytical determination of the constant distance is both unwieldy and uncertain, there are two ways to proceed. One, which you might find asthetically pleasing, is to draw a CONFOCAL ellipse about your original one. Given your ellipse (x/a)^2+(y/b)^2 = 1, you draw an ellipse of the form (x/a*)^2 + (y/b*)^2 = 1 with (a*)^2 - (b*)^2 = a^2 - b^2 (satisfying the confocality requirement) and then a* or b* set to match your external point. If a > b, so tht the foci ar on the x-axis then a* is simply half the sum of the sitances from your foci to your external point, with the foci at ((+/-)sqrt(a^2-b^2), 0). If a < b, then you use the same formulation as above only this time b* is half the sum of the distances to the foci, and the foci are at (0, (+/-)sqrt(b^2-a^2)). The seoond approach is to describe the true, nonelliptical constant-distance curve parametrically. First you must find c numerically, as the analytic solution (as we have seen) is totally impractical to work with. One way to do this: draw a line of slope m from the external point to the neaest point on the ellipse, intersecting at (u, v). This is found from the solution of a QUADRATIC equation, so it's not unweldy; you do need to pick the closer of the two roots. Then the slope of the tangent of the ellipse at that point is p = (-b^2u/a^2v). Compute 1+mp. Do this for several values of m and interpolate to approximate the one where 1+mp is zero, the condition for the line being perpendicular to the tangent. With c determined, we then can set up the parametric equations: x = a cos (theta) + c dcos (phi) y = b sin (theta) + c sin (phi) with theta and phi in the same quadrant and phi defined so that tan (phi) = (a/b) tan (theta). --OL === Subject: Re: ellipse: shortest distance to given point > To begin with, your second curve is not an ellipse. It looks like one, > but no cigar. What actualy happens can be seen most clearly by > considering the limiting case where the eccentricity approaches 1 with > fixed focal points and the ellipse thus becomes a line segment. The > curve that's a constan distance from a line segment consists of two > parallel line segments connected by semicircular caps. This is not an > ellipse or any limiting case of the ellipse. > Mathematica is right to give you a complicated expression for c. To > find the point on the ellipse that's closest to a given point (x, y), > one must solve what proves to be a 4th-degree equation for one of the > coordinates of the point on the ellipse. Applying the standard > technique symbolically then gives that unwieldy expression. It gets > even worse that this, however, for the casus irreucibilis can rear its > ugly head. In the context of 4th-degree equations this refers to the > resolvent cubic having three real roots and thereby generating complex > cube roots in that complicated expression from Mathematica. > Geometrically, the casus irreducibilis occurs when there are four rather > than two lines you can draw normal to an ellipse to a given point -- > essentially when the ellipse has high eccentricity and your (presumably > exterior) point is relatively close to the minor-axis diameter. (Kids, > do try this at home. It's kind of neat to see this change in in the > character of the nomals.) > Given the fact that your constant-distance curve is not an ellipse and > analytical determination of the constant distance is both unwieldy and > uncertain, there are two ways to proceed. One, which you might find > asthetically pleasing, is to draw a CONFOCAL ellipse about your original > one. Given your ellipse (x/a)^2+(y/b)^2 = 1, you draw an ellipse of > the form (x/a*)^2 + (y/b*)^2 = 1 with (a*)^2 - (b*)^2 = a^2 - b^2 > (satisfying the confocality requirement) and then a* or b* set to match > your external point. If a > b, so tht the foci ar on the x-axis then a* > is simply half the sum of the sitances from your foci to your external > point, with the foci at ((+/-)sqrt(a^2-b^2), 0). If a < b, then you > use the same formulation as above only this time b* is half the sum of > the distances to the foci, and the foci are at (0, (+/-)sqrt(b^2-a^2)). > The seoond approach is to describe the true, nonelliptical > constant-distance curve parametrically. First you must find c > numerically, as the analytic solution (as we have seen) is totally > impractical to work with. One way to do this: draw a line of slope m > from the external point to the neaest point on the ellipse, intersecting > at (u, v). This is found from the solution of a QUADRATIC equation, so > it's not unweldy; you do need to pick the closer of the two roots. Then > the slope of the tangent of the ellipse at that point is p = > (-b^2u/a^2v). Compute 1+mp. Do this for several values of m and > interpolate to approximate the one where 1+mp is zero, the condition for > the line being perpendicular to the tangent. With c determined, we then > can set up the parametric equations: > x = a cos (theta) + c dcos (phi) > y = b sin (theta) + c sin (phi) > with theta and phi in the same quadrant and phi defined so that tan > (phi) = (a/b) tan (theta). For some pictures see http://mathworld.wolfram.com/ParallelCurves.html -- Clive Tooth http://www.clivetooth.dk === Subject: Re: ellipse: shortest distance to given point > To begin with, your second curve is not an ellipse. It looks like one, Just to not confuse the troops, I think you mean the constant-distance-to the- first -ellipse curve he wants is not an ellipse. The second equation he was proposing: x^2/(a + c)^2 + y^2/(b + c)^2 = 1 is, as I'm sure you were not disputing, an ellipse. KeithK > but no cigar. What actualy happens can be seen most clearly by > considering the limiting case where the eccentricity approaches 1 with > fixed focal points and the ellipse thus becomes a line segment. The > curve that's a constan distance from a line segment consists of two > parallel line segments connected by semicircular caps. This is not an > ellipse or any limiting case of the ellipse. > Mathematica is right to give you a complicated expression for c. To > find the point on the ellipse that's closest to a given point (x, y), > one must solve what proves to be a 4th-degree equation for one of the > coordinates of the point on the ellipse. Applying the standard > technique symbolically then gives that unwieldy expression. It gets > even worse that this, however, for the casus irreucibilis can rear its > ugly head. In the context of 4th-degree equations this refers to the > resolvent cubic having three real roots and thereby generating complex > cube roots in that complicated expression from Mathematica. > Geometrically, the casus irreducibilis occurs when there are four rather > than two lines you can draw normal to an ellipse to a given point -- > essentially when the ellipse has high eccentricity and your (presumably > exterior) point is relatively close to the minor-axis diameter. (Kids, > do try this at home. It's kind of neat to see this change in in the > character of the nomals.) > Given the fact that your constant-distance curve is not an ellipse and > analytical determination of the constant distance is both unwieldy and > uncertain, there are two ways to proceed. One, which you might find > asthetically pleasing, is to draw a CONFOCAL ellipse about your original > one. Given your ellipse (x/a)^2+(y/b)^2 = 1, you draw an ellipse of > the form (x/a*)^2 + (y/b*)^2 = 1 with (a*)^2 - (b*)^2 = a^2 - b^2 > (satisfying the confocality requirement) and then a* or b* set to match > your external point. If a > b, so tht the foci ar on the x-axis then a* > is simply half the sum of the sitances from your foci to your external > point, with the foci at ((+/-)sqrt(a^2-b^2), 0). If a < b, then you > use the same formulation as above only this time b* is half the sum of > the distances to the foci, and the foci are at (0, (+/-)sqrt(b^2-a^2)). > The seoond approach is to describe the true, nonelliptical > constant-distance curve parametrically. First you must find c > numerically, as the analytic solution (as we have seen) is totally > impractical to work with. One way to do this: draw a line of slope m > from the external point to the neaest point on the ellipse, intersecting > at (u, v). This is found from the solution of a QUADRATIC equation, so > it's not unweldy; you do need to pick the closer of the two roots. Then > the slope of the tangent of the ellipse at that point is p = > (-b^2u/a^2v). Compute 1+mp. Do this for several values of m and > interpolate to approximate the one where 1+mp is zero, the condition for > the line being perpendicular to the tangent. With c determined, we then > can set up the parametric equations: > x = a cos (theta) + c dcos (phi) > y = b sin (theta) + c sin (phi) > with theta and phi in the same quadrant and phi defined so that tan > (phi) = (a/b) tan (theta). > --OL === Subject: Re: ellipse: shortest distance to given point > (if not, how to interpret c then)? The idea is that i paint a second ellipse > around the original ellipse that > has a constant distance to the original ellipse and contains point (x,y). But two ellipses *do not* have constant distance! Here is the standard method I would take: 1. Draw a circle around your point (x0,y0): (x-x0)**2+(y-y0)**2=c**2 2. It must be tangent to the ellipse x**2/a**2+y**2/b**2=1 - if it would intersect in two points a smaller circle would exist. 3. Eliminate x, get a quadratic for y, compute its discriminant. It must be zero to have one solution. Solve for c, your distance. -- Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn === Subject: Re: ellipse: shortest distance to given point >> (if not, how to interpret c then)? The idea is that i paint a second ellipse >> around the original ellipse that >> has a constant distance to the original ellipse and contains point (x,y). > But two ellipses *do not* have constant distance! > Here is the standard method I would take: > 1. Draw a circle around your point (x0,y0): > (x-x0)**2+(y-y0)**2=c**2 > 2. It must be tangent to the ellipse > x**2/a**2+y**2/b**2=1 > - if it would intersect in two points a > smaller circle would exist. > 3. Eliminate x, get a quadratic for y, compute > its discriminant. It must be zero to have one > solution. Solve for c, your distance. s/quadratic/quartic. (See other f'up of consequences) -- Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn === Subject: ergodicity I'm trying to understand ergodicity, but there are lots of different, difficult to reconcile explanations... Is an ergodic source the same as a stationary Markov Model as a source? (I understand the MM side of things, but not ergodicity.) If someone can explain, I'd appreciate it!! :o) JJ === Subject: Re: ergodicity > I'm trying to understand ergodicity, but there are lots of different, > difficult to reconcile explanations... > Is an ergodic source the same as a stationary Markov Model as a source? > (I > understand the MM side of things, but not ergodicity.) A non-stationary process must be non-ergodic, but an ergodic process can be stationary. Consider a Spherically Invariant Random Process (SIRP), for example: X(n) = S * Z(n), n = 0, 1, 2, ... where Z is Gaussian with mean zero and standard deviation sigma and S is from some distribution f. S is constant for each realization, but varies randomly across realizations of the SIRP. If you look at statistics for a single realization, you'll see Gaussian distributed data. But if you look at a single time sample across realizations, in general, you will not see Gaussian data. Is that helpful? -- Mostly economics: r c v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question fit perhaps to be discussed by n e . slaves in the hearing of their masters, but highly @ r c m unbecoming to reasonable and free men in search of d o the truth. -- Rousseau === Subject: On Programs That Output Themselves We all know from Recursion Theory that there is a program that outputs itself. But what is there about programs (the minimum requirement) that implies this fact? Here is a formal proof that: If substitution is recursive, then there is a program that outputs itself. Definitions: 1. For any functions M and N, M # N means that computer program M outputs N. 2. Let I and J represent input to these functions. 3. For any function f, f(I) means that f is recursive. 4. Function sub(I,J) = Program I with J substituted for its one free variable. Formal Assertions: A1. Definition of sub: M # f(I) => sub(M,N) # f(N) A2. Every recursive function is output by some program: f(I) => (exists M) M # f(I) A3. Assume that substitution is recursive: sub(I,J). A4. Substitution for input: f(I,J) => f(I,I). Formal Theorem: sub(I,J) => (exists N) N # N If substitution is recursive, then there is a program that outputs itself. Formal Proof: 1. sub(I,J) A3. Substitution is recursive. 2. sub(I,I) A4. Substitution for input. 3. M # sub(I,I) A2. Every recursive function is output by some program. 4. sub(M,M) # sub(M,M) A1. If M outputs f(I) then sub(M,M) outputs f(M). 5. (exists N) N # N qed BTW: If anyone thinks it's been done before, please support that by giving here the formal proof that already exists. C-B === Subject: Re: On Programs That Output Themselves > We all know from Recursion Theory that there is a program that outputs > itself. But what is there about programs (the minimum requirement) > that implies this fact? This can be done in any universal programming language that can output its own alphabet, as follows: Given any universal Turing machine M, then for any universal programming language P, by definition, there is a program MTOP, written for M that program that produces the same output as M(x), under some bijective mapping from strings in M's alphabet to strings in P's alphabet, so that there are no restrictions on the output that a converted program can produce. Given MTOP, it is possible to write TWICETOP, such that M(TWICETOP:x) produces the same output as M(MTOP:x:x), which is a P program that produces the same output as M(x:x). Note that we have only written M programs here -- no special features of P are required, except its ability to emulate UTMs. Even so, it is now a simple matter to verify that M(TWICETOP:TWICETOP) will write a P program that outputs itself: M(TWICETOP:TWICETOP) produces the same output as M(MTOP:TWICETOP:TWICETOP), which is a P program that produces the same output as M(TWICETOP:TWICETOP), which is exactly the P program that we want it to match. -- Matt === Subject: Re: On Programs That Output Themselves > We all know from Recursion Theory that there is a program that outputs > itself. But what is there about programs (the minimum requirement) > that implies this fact? Let P be a program that outputs its own text, then P is a program that outputs its own text is the minimum requirement that P outputs its own text. Nothing weaker will do, nothing stronger is required. In particular, no quantifiers are necessary. === Subject: Re: On Programs That Output Themselves > > We all know from Recursion Theory that there is a program that outputs > > itself. But what is there about programs (the minimum requirement) > > that implies this fact? > Let P be a program that outputs its own text, then > P is a program that outputs its own text > is the minimum requirement that P outputs its own text. Nothing weaker > will do, nothing stronger is required. In particular, no quantifiers > are necessary. Cool. But how do we know there is such a P (first line)? This is so sad. I ask a simple little question about proving something and people act like they don't even understand simple concepts of logic. But the answers are funny, anyway. :) C-B === Subject: Re: On Programs That Output Themselves Originator: harris@tcs.inf.tu-dresden.de (Mitchell Harris) >> > >> > We all know from Recursion Theory that there is a program that outputs >> > itself. But what is there about programs (the minimum requirement) >> > that implies this fact? ... >This is so sad. I ask a simple little question about proving something >and people act like they don't even understand simple concepts of >logic. But the answers are funny, anyway. :) Maybe they misunderstood the title? -- Mitch === Subject: Re: On Programs That Output Themselves > > > > > > We all know from Recursion Theory that there is a program that > outputs > > > itself. But what is there about programs (the minimum requirement) > > > that implies this fact? > > Let P be a program that outputs its own text, then > > P is a program that outputs its own text > > is the minimum requirement that P outputs its own text. Nothing > weaker > > will do, nothing stronger is required. In particular, no quantifiers > > are necessary. > Cool. But how do we know there is such a P (first line)? I can write you one in C if you like. How about this: char*s=char*s=%c%s%c;main(){printf(s,34,s,34);};main(){printf(s,34,s,34) ;} Compilers will object but the executable does as required. > This is so sad. I ask a simple little question about proving something > and people act like they don't even understand simple concepts of > logic. But the answers are funny, anyway. :) > C-B === Subject: Re: On Programs That Output Themselves > I can write you one in C if you like. How about this: char*s=char*s=%c%s%c;main(){printf(s,34,s,34);};main(){printf(s,34,s,34) ;} > Compilers will object but the executable does as required. You can get printf() to work without including the library (stdio.h, usually)? === Subject: Re: On Programs That Output Themselves > > I can write you one in C if you like. How about this: char*s=char*s=%c%s%c;main(){printf(s,34,s,34);};main(){printf(s,34,s,34) ; } > > Compilers will object but the executable does as required. > You can get printf() to work without including the library (stdio.h, > usually)? Yes. I have run it (Borland C/C++ 4.5, I think) but, as I say, the compiler may object. === Subject: Re: On Programs That Output Themselves > I can write you one in C if you like. How about this: char*s=char*s=%c%s%c;main(){printf(s,34,s,34);};main(){printf(s,34,s,34) ;} > Compilers will object but the executable does as required. Cool! Really. I'm sure one can (see my proof!) Now can you prove that it's possible in all languages? Formally? C-B === Subject: Re: On Programs That Output Themselves > > I can write you one in C if you like. How about this: char*s=char*s=%c%s%c;main(){printf(s,34,s,34);};main(){printf(s,34,s,34) ; } > > Compilers will object but the executable does as required. > Cool! Really. I'm sure one can (see my proof!) Now can you prove > that it's possible in all languages? Formally? Of course not. Because there is no formal definitrion of a programming language. That's why Church's thesis is just a thesis and not a theorem, by the way. Now, you can most certainly proove that it is doable in any language at least as powerfull as Turing Machines are. By the way, notice that no actual language is as powerful as TMs (due to bounded memory), so the requierment about programmin language to be able to do this are much lighter than being a powerful as TMs. Now to go further, you must define, formally, what a programming language is. Which is, I think, impossible. Hypocoristiquement, Jym. === Subject: Re: On Programs That Output Themselves > > I can write you one in C if you like. How about this: char*s=char*s=%c%s%c;main(){printf(s,34,s,34);};main(){printf(s,34,s,34) ; } > > Compilers will object but the executable does as required. > Cool! Really. I'm sure one can (see my proof!) Now can you prove > that it's possible in all languages? Formally? It isn't possible in all languages. For example it isn't possible in languages that have no output facility. === Subject: Re: On Programs That Output Themselves > > > I can write you one in C if you like. How about this: > > > > > > > > char*s=char*s=%c%s%c;main(){printf(s,34,s,34);};main(){printf(s,34,s,34) ; } > > > > > > Compilers will object but the executable does as required. > > Cool! Really. I'm sure one can (see my proof!) Now can you prove > > that it's possible in all languages? Formally? > It isn't possible in all languages. For example it isn't possible in > languages that have no output facility. Which I'll personnaly won't call languages. The goal of programming languages is, I think, to computes functions. This require the ability to output something. Hypocoristiquement, Jym. === Subject: Re: On Programs That Output Themselves ... > > It isn't possible in all languages. For example it isn't possible in > > languages that have no output facility. > Which I'll personnaly won't call languages. The goal of programming > languages is, I think, to computes functions. This require the ability to > output something. Algol 60 is not a programming language? The report does *not* define any I/O facility. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: On Programs That Output Themselves > ... > > > It isn't possible in all languages. For example it isn't possible in > > > languages that have no output facility. > > > > Which I'll personnaly won't call languages. The goal of programming > > languages is, I think, to computes functions. This require the ability to > > output something. > Algol 60 is not a programming language? The report does *not* define > any I/O facility. 'Zackly! Given the dependence of i/o on operating system and hardware, it's quite reasonable to _not_ define i/o facilities in the language. === Subject: Re: On Programs That Output Themselves > > Cool! Really. I'm sure one can (see my proof!) Now can you prove > > that it's possible in all languages? Formally? > It isn't possible in all languages. For example it isn't possible in > languages that have no output facility. HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA C-B === Subject: Re: On Programs That Output Themselves >> > Cool! Really. I'm sure one can (see my proof!) Now can you prove >> > that it's possible in all languages? Formally? >> It isn't possible in all languages. For example it isn't possible in >> languages that have no output facility. >HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA >HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA >HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA >HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA Now that you've had your fun, I'll give you a language that can't do it. ANSI C without any libraries. Proof. ANSI C without any libraries has only one output method available. The return value of main(). Since this is also required to be an integer, and the source for a C program is most certainly not an integer, It can't be returned. Martin === Subject: Re: On Programs That Output Themselves > > > Cool! Really. I'm sure one can (see my proof!) Now can you prove > > > that it's possible in all languages? Formally? > > It isn't possible in all languages. For example it isn't possible in > > languages that have no output facility. > HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA > HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA > HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA > HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA I'm trying to point out to you that all languages is silly, you need to say all languages of such-and-such a kind. === Subject: Re: On Programs That Output Themselves There's the Scheme program ((lambda (x)(x x))(lambda (x)(x x)) that runs the copy of itself that it creates. Of course, this gives a nonterminating process. ________________________________ Eric J. Wingler (wingler@math.ysu.edu) Dept. of Mathematics and Statistics Youngstown State University One University Plaza Youngstown, OH 44555-0001 330-941-1817 === Subject: Re: On Programs That Output Themselves <4277ae07$1@news.ysu.edu> > There's the Scheme program > ((lambda (x)(x x))(lambda (x)(x x)) > that runs the copy of itself that it creates. Of course, this gives a > nonterminating process. Where's the proof that every programming language has a program that outputs itself? (You haven't even formally proven it in that one language, actually, but that is really beside the point.) C-B > ________________________________ > Eric J. Wingler (wingler@math.ysu.edu) > Dept. of Mathematics and Statistics You're a college professor and you don't even know the difference between |-P(...) and |-(all X)P(X) ? Yikes! > Youngstown State University > One University Plaza > Youngstown, OH 44555-0001 > 330-941-1817 === Subject: Re: On Programs That Output Themselves > Where's the proof that every programming language has a program that > outputs itself? No such proof is possible. === Subject: Re: On Programs That Output Themselves >> There's the Scheme program >> ((lambda (x)(x x))(lambda (x)(x x)) >> that runs the copy of itself that it creates. Of course, this gives >> nonterminating process. >Where's the proof that every programming language has a program that >outputs itself? I don't think that's true. I'm not sure if there are any actual languages in which you can't do this, but it's easy enough to come up with contrived cases where it can't be done. Alan -- Defendit numerus === Subject: Re: On Programs That Output Themselves í Alan Morgan .97.8d.98.87.8b.8c .97.99.95 .92.86.94.9d.92.87 > >Where's the proof that every programming language has a program that > >outputs itself? > I don't think that's true. I'm not sure if there are any actual languages > in which you can't do this, but it's easy enough to come up with contrived > cases where it can't be done. Any programming language strong enough to have statements such as open(f,...), read(f,...) and write(f,...) (where f is a disk file buffer) can have a program that outputs itself as follows: open(f,this program's file name on disk); while not eof(f) do begin readln(f); writeln(f); end; close(f); > Alan > -- > Defendit numerus -- I. N. Galidakis http://users.forthnet.gr/ath/jgal/ Eventually, _everything_ is understandable === Subject: Re: On Programs That Output Themselves > .93 Alan Morgan [NonBreakingSpace].8b.96.87Ë.8c .97.99.95 .93fi.92.9b.93.87 >>> Where's the proof that every programming language has a program that >>> outputs itself? >> I don't think that's true. I'm not sure if there are any actual >> languages in which you can't do this, but it's easy enough to come >> up with contrived cases where it can't be done. > Any programming language strong enough to have statements such as > open(f,...), read(f,...) and write(f,...) (where f is a disk > file buffer) can have a program that outputs itself as follows: > open(f,this program's file name on disk); > while not eof(f) do > begin > readln(f); > writeln(f); > end; > close(f); That assumes that it's possible to find the source file for the program at runtime. This may not be possible in a compiled language -- the source file may not exist on this computer, or may not exist anywhere. In any case, opening the source file and reading it is in a sense cheating as another poster commented. The interesting issues in writing a self-outputing program arise when you do it the right way. --Mark === Subject: Re: On Programs That Output Themselves > Any programming language strong enough to have statements such as > open(f,...), read(f,...) and write(f,...) (where f is a disk file > buffer) can have a program that outputs itself as follows: > open(f,this program's file name on disk); > while not eof(f) do > begin > readln(f); > writeln(f); > end; > close(f); The above outputs into the same file, so a minor change is needed: open(f,this program's file name on disk); while not eof(f) do begin readln(f,...); writeln(output,...); end; close(f); -- I. N. Galidakis http://users.forthnet.gr/ath/jgal/ Eventually, _everything_ is understandable === Subject: Re: On Programs That Output Themselves <4277ae07$1@news.ysu.edu> > >Where's the proof that every programming language has a program that > >outputs itself? > I don't think that's true. I'm not sure if there are any actual languages > in which you can't do this, but it's easy enough to come up with contrived > cases where it can't be done. Unsubstantiated bullshit. C-B > Alan > -- > Defendit numerus === Subject: Re: On Programs That Output Themselves >> >Where's the proof that every programming language has a program that >> >outputs itself? >> I don't think that's true. I'm not sure if there are any actual >languages >> in which you can't do this, but it's easy enough to come up with >contrived >> cases where it can't be done. >Unsubstantiated bullshit. I did say contrived. If a language is required, say, to be in all upper case and can only output lower case letters then it will fall into this catagory (if APL couldn't print its own bizarre character set then it would be a real world example of this). Alan -- Defendit numerus === Subject: Re: On Programs That Output Themselves char *p=char *p=%c%s%c;main(){printf(p,34,p,34);};main(){printf(p,34,p,34);} [It's all one line, no trailing newline at the end] I think this was a result of W.V.O. Quine. (Look it up in Goedel Escher Bach.) Anyway, see http://www.nyx.net/~gthompso/quine.htm for examples in many computer languages. 10 DATA B$='DATA '+CHR$(34) 20 DATA FOR J=10 TO 180 STEP 10 30 DATA READ A$ 40 DATA PRINT J;B$;A$ 50 DATA IF J<>90 THEN 170 60 DATA RESTORE 70 DATA B$=' ' 80 DATA NEXT J 90 DATA END 100 B$='DATA '+CHR$(34) 110 FOR J=10 TO 180 STEP 10 120 READ A$ 130 PRINT J;B$;A$ 140 IF J<>90 THEN 170 150 RESTORE 160 B$=' ' 170 NEXT J 180 END to quine make r [pr [to quine] type [make r] pr list :r pr :r pr [end]] pr [to quine] type [make r] pr list :r pr :r pr [end] end -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: On Programs That Output Themselves <030520051047006328%edgar@math.ohio-state.edu.invalid> > char *p=char > *p=%c%s%c;main(){printf(p,34,p,34);};main(){printf(p,34,p,34);} > [It's all one line, no trailing newline at the end] > I think this was a result of W.V.O. Quine. (Look it up in Goedel > Escher Bach.) Anyway, see http://www.nyx.net/~gthompso/quine.htm > for examples in many computer languages. But that doesn't mean that it is possible in ALL programming languages. E.g. in the CBL programming language, $A is equal to WRITE $A (just as $PI is equal to 3.14159). Then there is the solution WRITE $A You have only shown (and not even formally proven) that it can be done in that programming language. C-B > 10 DATA B$='DATA '+CHR$(34) > 20 DATA FOR J=10 TO 180 STEP 10 > 30 DATA READ A$ > 40 DATA PRINT J;B$;A$ > 50 DATA IF J<>90 THEN 170 > 60 DATA RESTORE > 70 DATA B$=' ' > 80 DATA NEXT J > 90 DATA END > 100 B$='DATA '+CHR$(34) > 110 FOR J=10 TO 180 STEP 10 > 120 READ A$ > 130 PRINT J;B$;A$ > 140 IF J<>90 THEN 170 > 150 RESTORE > 160 B$=' ' > 170 NEXT J > 180 END > to quine > make r [pr [to quine] type [make r] pr list :r pr :r pr [end]] > pr [to quine] type [make r] pr list :r pr :r pr [end] > end > -- > G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: On Programs That Output Themselves On 3 May 2005 10:40:56 -0700, Charlie-Boo said: > But that doesn't mean that it is possible in ALL programming > languages. Right, it may not be possible in a computationally impoverished programming language, just as it is impossible to express certain concepts (like finitude) in representationally underpowered logical languages. -cm === Subject: Re: On Programs That Output Themselves <030520051047006328%edgar@math.ohio-state.edu.invalid> > On 3 May 2005 10:40:56 -0700, Charlie-Boo said: > > But that doesn't mean that it is possible in ALL programming > > languages. > Right, it may not be possible in a computationally impoverished > programming language, just as it is impossible to express certain > concepts (like finitude) in representationally underpowered logical > languages. > -cm Uh-oh. Somebody doesn't know what they're talking about. You doubt Stephen Kleene? So what's a computationally impoverished programming language, anyway? C-B === Subject: Re: On Programs That Output Themselves On 3 May 2005 12:31:33 -0700, Charlie-Boo said: >> On 3 May 2005 10:40:56 -0700, Charlie-Boo said: >> > But that doesn't mean that it is possible in ALL programming >> > languages. >> Right, it may not be possible in a computationally impoverished >> programming language, just as it is impossible to express certain >> concepts (like finitude) in representationally underpowered logical >> languages. > Uh-oh. Somebody doesn't know what they're talking about. You doubt > Stephen Kleene? Typical of you (not to mention ironic) simply to declare that someone is wrong, and to cite something authoritative out of the blue as if *you* know what you're talking about, with no accompanying argumentation. With what point do you think I'm at odds with Kleene? That the feat in question might not be possible in a computationally impoverished language? Surely that is indisputable, if the language is crippled in the right (or wrong, depending on perspective) sort of way. So is it that you think I know not whereof I speak regarding the impossibility of defining finitude in first-order logic? Oh, wait, that's right -- you're the one who thinks it *can* be defined in first-order logic; and I guess you think Kleene backs you up on this point. Well, gosh, all I can suggest is that you hit the books a little harder. > So what's a computationally impoverished programming language, anyway? You really have to ask? Bone up on Turing completeness. Chris Menzel === Subject: Re: On Programs That Output Themselves >> On 3 May 2005 10:40:56 -0700, Charlie-Boo said: >>> But that doesn't mean that it is possible in ALL programming >>> languages. >> Right, it may not be possible in a computationally impoverished >> programming language, just as it is impossible to express certain >> concepts (like finitude) in representationally underpowered logical >> languages. >> -cm > Uh-oh. Somebody doesn't know what they're talking about. You doubt > Stephen Kleene? > So what's a computationally impoverished programming language, anyway? How about a language that can process (and output) only numbers, not strings. Surely it's not possible to write a program that outputs itself in that language. --Mark === Subject: Re: On Programs That Output Themselves > We all know from Recursion Theory that there is a program that outputs > itself. $me = $0; open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; while () { print $_; } close INPUT; Dirk Vdm === Subject: Re: On Programs That Output Themselves >>We all know from Recursion Theory that there is a program that outputs >>itself. > $me = $0; > open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; > while () { print $_; } > close INPUT; > Dirk Vdm This is shorter: $_='$_=X;s/X/x27$_x27/;print';s/X/x27$_x27/;print === Subject: Re: On Programs That Output Themselves > >>We all know from Recursion Theory that there is a program that outputs > >>itself. > > $me = $0; > > open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; > > while () { print $_; } > > close INPUT; > > Dirk Vdm > This is shorter: > $_='$_=X;s/X/x27$_x27/;print';s/X/x27$_x27/;print Nice, but criminally dirty ;-) Dirk Vdm === Subject: Re: On Programs That Output Themselves >> We all know from Recursion Theory that there is a program that outputs >> itself. >$me = $0; >open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; >while () { print $_; } >close INPUT; Such a program is easy in Maple (and doesn't depend on disk files): f:= proc() eval(f) end proc; Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: On Programs That Output Themselves > >> We all know from Recursion Theory that there is a program that outputs > >> itself. > >$me = $0; > >open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; > >while () { print $_; } > >close INPUT; > Such a program is easy in Maple (and doesn't depend on disk files): > f:= proc() eval(f) end proc; I assume that you call it by entering the command f( ) and then it produces the output proc() eval(f) end proc; ? Now that is really nifty :-) Dirk Vdm === Subject: Re: On Programs That Output Themselves > > We all know from Recursion Theory that there is a program that outputs > > itself. > $me = $0; > open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; > while () { print $_; } > close INPUT; You realize of course that that's just cheating, right? C-B > Dirk Vdm === Subject: Re: On Programs That Output Themselves > > > We all know from Recursion Theory that there is a program that > > > outputs > > > itself. > > $me = $0; > > open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; > > while () { print $_; } > > close INPUT; > You realize of course that that's just cheating, right? Sure - but it works, doesn't it? ;-) Dirk Vdm === Subject: Re: On Programs That Output Themselves <1MOde.80277$R05.5131372@phobos.telenet-ops.be> > > > $me = $0; > > > open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; > > > while () { print $_; } > > > close INPUT; > > You realize of course that that's just cheating, right? > Sure - but it works, doesn't it? Here's my self-reference reply: I myself think it's all so silly! But come on people, this is serious. Can't anybody come up with a real proof (besides me)? C-B > ;-) > Dirk Vdm === Subject: Re: On Programs That Output Themselves > But come on people, this is serious. Can't anybody come up with a real > proof (besides me)? For any particular programming language (or more precisely, for any particular compiler), a real proof will consist of the program text that can be compiled, linked and run to output the program text. No proof can exist for all programming languages. === Subject: Re: On Programs That Output Themselves > But come on people, this is serious. Can't anybody come up with a real > proof (besides me)? Yes, Quine. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: On Programs That Output Themselves <1MOde.80277$R05.5131372@phobos.telenet-ops.be> <030520051457500707%edgar@math.ohio-state.edu.invalid> > > But come on people, this is serious. Can't anybody come up with a real > > proof (besides me)? > Yes, Quine. > -- > G. A. Edgar http://www.math.ohio-state.edu/~edgar/ LOL Don't you wish. I wonder what the real emotional attachment is . . . Colleges and college professors? Pretty books and journals? Famous people? Pride in their work? fame-worshipping. It's not just an emotional bonding, it's a religion! No wonder some people will do anything to say that their mentor is infallible! I think part of the problem is that my work as a Computer Programmer/Researcher must be for real. I must precisely formalize processes constantly - and they must work. People in professor-mode basically never have to produce anything that is actually verified as being correct - or even meaningful. The only requirement is that their professor friends (the Old Boy's Club) have to accept it for publication. Shows you what you get from inbreeding! BTW If one is interested in the prevailing thought on this problem, there is a program containing 4 parts that can be written in any language, that does the trick. But it is a bit messy and apparently nobody has even attempted to formalize it i.e. formally prove that it exists and outputs itself. In fact, my little proof is simpler than the program itself, whose formal proof (if anyone ever comes up with one) would be orders of magnitude more complex than my proof. C-B === Subject: Re: On Programs That Output Themselves > > > > $me = $0; > > > > open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; > > > > while () { print $_; } > > > > close INPUT; > > > > > > You realize of course that that's just cheating, right? > > Sure - but it works, doesn't it? > Here's my self-reference reply: I myself think it's all so silly! > But come on people, this is serious. Can't anybody come up with a real > proof (besides me)? Actually, I hadn't looked at what came beyond the first line of your original message. But okay, seriously now, let me give it an honest try: Set fso = CreateObject(Scripting.FileSystemObject) myname = fso.GetFile(WScript.ScriptFullName) Set infile = fso.OpenTextFile( myname, 1, false) allout = Do While infile.AtEndOfStream <> true allout = allout & infile.readline() & vblf Loop infile.close() MsgBox(allout) WScript.Quit Dirk Vdm [ Sorry, couldn't resist this one ;-P ] === Subject: Re: On Programs That Output Themselves <1MOde.80277$R05.5131372@phobos.telenet-ops.be> > But okay, seriously now, let me give it an honest try: > Set fso = CreateObject(Scripting.FileSystemObject) > myname = fso.GetFile(WScript.ScriptFullName) > Set infile = fso.OpenTextFile( myname, 1, false) > allout = > Do While infile.AtEndOfStream <> true > allout = allout & infile.readline() & vblf > Loop > infile.close() > MsgBox(allout) > WScript.Quit > Dirk Vdm > [ Sorry, couldn't resist this one ;-P ] Do you understand what the question is? What does the above prove? C-B === Subject: Re: On Programs That Output Themselves > > But okay, seriously now, let me give it an honest try: > > Set fso = CreateObject(Scripting.FileSystemObject) > > myname = fso.GetFile(WScript.ScriptFullName) > > Set infile = fso.OpenTextFile( myname, 1, false) > > allout = > > Do While infile.AtEndOfStream <> true > > allout = allout & infile.readline() & vblf > > Loop > > infile.close() > > MsgBox(allout) > > WScript.Quit > > Dirk Vdm > > [ Sorry, couldn't resist this one ;-P ] > Do you understand what the question is? What does the above prove? The above proves that two of the scripting languages I regularly use, can do the job :-) I promise, if I ever design my own language, that I will implement a Streak method, so I can write The Shortest Self-Outputting Program Of Them All ;-) Dirk Vdm === Subject: Re: On Programs That Output Themselves > The above proves that two of the scripting languages > I regularly use, can do the job :-) > I promise, if I ever design my own language, that I > will implement a Streak method, so I can write The > Shortest Self-Outputting Program Of Them All ;-) I think someone beat you to it: http://en.wikipedia.org/wiki/HQ9_Plus === Subject: Re: On Programs That Output Themselves <1MOde.80277$R05.5131372@phobos.telenet-ops.be> > > The above proves that two of the scripting languages > > I regularly use, can do the job :-) > > I promise, if I ever design my own language, that I > > will implement a Streak method, so I can write The > > Shortest Self-Outputting Program Of Them All ;-) > I think someone beat you to it: > http://en.wikipedia.org/wiki/HQ9_Plus No. There's shorter. Check out the entry smr in the 1994 winners of the IOCCC: http://www1.us.ioccc.org/years.html#1994 === Subject: Re: On Programs That Output Themselves > > > The above proves that two of the scripting languages > > > I regularly use, can do the job :-) > > > I promise, if I ever design my own language, that I > > > will implement a Streak method, so I can write The > > > Shortest Self-Outputting Program Of Them All ;-) > > I think someone beat you to it: > > http://en.wikipedia.org/wiki/HQ9_Plus > No. There's shorter. Check out the entry smr in the 1994 winners of > the IOCCC: > http://www1.us.ioccc.org/years.html#1994 Ah yes of course, this one definitely can't be beaten. And it works in many other languages as well :-) Dirk Vdm === Subject: Re: On Programs That Output Themselves <1MOde.80277$R05.5131372@phobos.telenet-ops.be> > > > But okay, seriously now, let me give it an honest try: > > > > > > Set fso = CreateObject(Scripting.FileSystemObject) > > > myname = fso.GetFile(WScript.ScriptFullName) > > > Set infile = fso.OpenTextFile( myname, 1, false) > > > allout = > > > Do While infile.AtEndOfStream <> true > > > allout = allout & infile.readline() & vblf > > > Loop > > > infile.close() > > > MsgBox(allout) > > > WScript.Quit > > > > > > Dirk Vdm > > > [ Sorry, couldn't resist this one ;-P ] > > Do you understand what the question is? What does the above prove? > The above proves that two of the scripting languages > I regularly use, can do the job :-) Oh, ok. Cool!!! > I promise, if I ever design my own language, that I > will implement a Streak method, so I can write The > Shortest Self-Outputting Program Of Them All ;-) How's that work? C-B Hey programmer, here's a little trick: Input the error message that is printed when a syntax error occurs! > Dirk Vdm === Subject: Re: On Programs That Output Themselves > > > > But okay, seriously now, let me give it an honest try: > > > > > > > > Set fso = CreateObject(Scripting.FileSystemObject) > > > > myname = fso.GetFile(WScript.ScriptFullName) > > > > Set infile = fso.OpenTextFile( myname, 1, false) > > > > allout = > > > > Do While infile.AtEndOfStream <> true > > > > allout = allout & infile.readline() & vblf > > > > Loop > > > > infile.close() > > > > MsgBox(allout) > > > > WScript.Quit > > > > > > > > Dirk Vdm > > > > [ Sorry, couldn't resist this one ;-P ] > > > > > > Do you understand what the question is? What does the above prove? > > The above proves that two of the scripting languages > > I regularly use, can do the job :-) > Oh, ok. Cool!!! > > I promise, if I ever design my own language, that I > > will implement a Streak method, so I can write The > > Shortest Self-Outputting Program Of Them All ;-) > How's that work? Source: Streak Output: Streak > C-B > Hey programmer, here's a little trick: Input the error message that is > printed when a syntax error occurs! Ouch. Dirty Boo. Dirk Vdm === Subject: Re: On Programs That Output Themselves <1MOde.80277$R05.5131372@phobos.telenet-ops.be> > > > > We all know from Recursion Theory that there is a program that > > > > outputs > > > > itself. > > > > > > $me = $0; > > > open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; > > > while () { print $_; } > > > close INPUT; > > You realize of course that that's just cheating, right? > Sure - but it works, doesn't it? Only in that one language, silly. C-B Ok, 2 languages. oo-2 to go. > ;-) > Dirk Vdm === Subject: Re: On Programs That Output Themselves yay. that was much shorter to read too :o) > > We all know from Recursion Theory that there is a program that outputs > > itself. > $me = $0; > open(INPUT,<$me) or die !!! Can't open myself ($me)!n $!; > while () { print $_; } > close INPUT; > Dirk Vdm === Subject: =?iso-8859-1?B?UmU6IEltcG9ydGFudCBwbGVhc2UgcmVhZCB0aGlzIFRoYW5rIHlvdSBgsLq3L i 4ut7qwYLC6ty4uLre6sGCwurcuhi4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLre6sG C wurcuLi63urBgsLq3LoYuLre6sGCwurcuLi63urBgsLq3Li63urBgIGCwurcuLi63urBgsLq3Li4 u t7qwYLC6ty6GLi63urBgsLq3Li4ut7qwYLC6ty4ut7qwYCBgsLq3Li4ut7qwYLC6ty4uLre6sGCw u rcuhi4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLre6sGCwurcuLi63urBgsLq3LoYuL r e6sGCwurcuLi63urBgsLq3Li63urBgIGCwurcuLi63urBgsLq3Li4ut7qwYLC6ty6GLi63urBgsL q 3Li4ut7qwYLC6ty4ut7qwYCBgsLq3Li4ut7qwYLC6ty4uLre6sGCwurcuhi4ut7qwYLC6ty4uLre 6 sGCwurcuLre6sGAgYLC6ty4uLre6sGCwurcuLi63urBgsLq3LoYuLre6sGCwurcuLi63urBgsLq3 L i63urBgIEdPT0dMRbdORVdTR1JPVVC3UE9TVLcxNTU=?= fuck you and the invisible man in the sky! HEY you might offend someone. Don <---------offending people since he started posting here > fuck you and the invisible man in the sky! === Subject: =?iso-8859-1?B?UmU6IEltcG9ydGFudCBwbGVhc2UgcmVhZCB0aGlzIFRoYW5rIHlvdSBgsLq3L i 4ut7qwYLC6ty4uLre6sGCwurcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLre6sG C wurcuLi63urBgsLq3Lj8uLre6sGCwurcuLi63urBgsLq3Li63urBgIGCwurcuLi63urBgsLq3Li4 u t7qwYLC6ty4/Li63urBgsLq3Li4ut7qwYLC6ty4ut7qwYCBgsLq3Li4ut7qwYLC6ty4uLre6sGCw u rcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLg==?= I don't care if I offend people - shit, I'm stuck all day being nice at work, on the train, to my wife, my family - everyone. If I feel like it, I will offend. Not a drag since last Friday... === Subject: Re: =?iso-8859-1?B?UmU6IEltcG9ydGFudCBwbGVhc2UgcmVhZCB0aGlzIFRoYW5rIHlvdSBgsLq3L i 4ut7qwYLC6ty4uLre6sGCwurcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLre6sG C wurcuLi63urBgsLq3Lj8uLre6sGCwurcuLi63urBgsLq3Li63urBgIGCwurcuLi63urBgsLq3Li4 u t7qwYLC6ty4/Li63urBgsLq3Li4ut7qwYLC6ty4ut7qwYCBgsLq3Li4ut7qwYLC6ty4uLre6sGCw u rcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLg==?= >If I feel like it, I will offend. >Not a drag since last Friday... you're just an asshole, junior grade. === Subject: =?iso-8859-1?B?UmU6IEltcG9ydGFudCBwbGVhc2UgcmVhZCB0aGlzIFRoYW5rIHlvdSBgsLq3L i 4ut7qwYLC6ty4uLre6sGCwurcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLre6sG C wurcuLi63urBgsLq3Lj8uLre6sGCwurcuLi63urBgsLq3Li63urBgIGCwurcuLi63urBgsLq3Li4 u t7qwYLC6ty4/Li63urBgsLq3Li4ut7qwYLC6ty4ut7qwYCBgsLq3Li4ut7qwYLC6ty4uLre6sGCw u rcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLg==?= suck a few dicks today? Or just you mother? === Subject: Re: =?iso-8859-1?B?UmU6IEltcG9ydGFudCBwbGVhc2UgcmVhZCB0aGlzIFRoYW5rIHlvdSBgsLq3L i 4ut7qwYLC6ty4uLre6sGCwurcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLre6sG C wurcuLi63urBgsLq3Lj8uLre6sGCwurcuLi63urBgsLq3Li63urBgIGCwurcuLi63urBgsLq3Li4 u t7qwYLC6ty4/Li63urBgsLq3Li4ut7qwYLC6ty4ut7qwYCBgsLq3Li4ut7qwYLC6ty4uLre6sGCw u rcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLg==?= >suck a few dicks today? Or just you mother? wow, are you clever! heard that out on the playground? you and your stupid mythology, plus the never nding stream go convert some cockroaches, they may be dumb enough for you. === Subject: =?iso-8859-1?B?UmU6IEltcG9ydGFudCBwbGVhc2UgcmVhZCB0aGlzIFRoYW5rIHlvdSBgsLq3L i 4ut7qwYLC6ty4uLre6sGCwurcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLre6sG C wurcuLi63urBgsLq3Lj8uLre6sGCwurcuLi63urBgsLq3Li63urBgIGCwurcuLi63urBgsLq3Li4 u t7qwYLC6ty4/Li63urBgsLq3Li4ut7qwYLC6ty4ut7qwYCBgsLq3Li4ut7qwYLC6ty4uLre6sGCw u rcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLg==?= actually, I made it up by myself... what mythology do you refer to? === Subject: =?iso-8859-1?B?UmU6IEltcG9ydGFudCBwbGVhc2UgcmVhZCB0aGlzIFRoYW5rIHlvdSBgsLq3L i 4ut7qwYLC6ty4uLre6sGCwurcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLre6sG C wurcuLi63urBgsLq3Lj8uLre6sGCwurcuLi63urBgsLq3Li63urBgIGCwurcuLi63urBgsLq3Li4 u t7qwYLC6ty4/Li63urBgsLq3Li4ut7qwYLC6ty4ut7qwYCBgsLq3Li4ut7qwYLC6ty4uLre6sGCw u rcuPy4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLg==?= sorry about that - it should've said or just your mother's! === Subject: Re: Groups of order 32 <1175c8qiss4qec6@corp.supernews.com> I know that a p-group G of order p^r always has a normal subgroup of order p^(r-1), and a subgroup of order p^k for k = 0,1,...,r, but is is true that there is a normal subgroup of G for all k <= r ? Marcel Wild's paper The Groups of Order 16 Made Easy, he proves that every group of order 16 other than (Z_2)^4 has Z_8 or Z_4 x Z_2 == K_8 as normal subgroups. He then shows that the automorphisms are A(Z_8) = Z_2 x Z_2 = (1,3,5,7) = Z*_8 and A(Z_4 x Z_2) = A(K_8) = D_4 ; There are 6 groups with normal subgroup Z_8 (4 non-abelian), and 7 with normal subgroup K_8 (5 non-abelian). I wonder if something similar can be done with groups of order 32, or how to tackled this, from a basic level if possible. I wonder if the following apprach is useful: Groups of order 32 have either Z_16 or Z_2 x Z_8 or Z_4 x Z_4 or Z_4 x Z_2 x Z_2 as normal subgroups. What are the automorphisms A(Z_2 x Z_8), A(Z_4 x Z_4), and A(Z_4 x Z_2 x Z_2) ? Van === Subject: Re: Groups of order 32 [...] > I wonder if the following apprach is useful: > Groups of order 32 have either Z_16 or Z_2 x Z_8 > or Z_4 x Z_4 or Z_4 x Z_2 x Z_2 as normal subgroups. I'm afraid not. For example, the group with presentation has no abelian subgroup of index 2. [...] -- Jim Heckman === Subject: Re: Groups of order 32 <1175c8qiss4qec6@corp.supernews.com> <117fpt328klt6b@corp.supernews.com> Looking at the group you gave; it looks like the groups of order 32 would be quite a job. I still don't see any systematic way of going about it. Van === Subject: Re: Groups of order 32 > I know that a p-group G of order p^r always has a normal subgroup > of order p^(r-1), and a subgroup of order p^k for k = 0,1,...,r, > but is is true that there is a normal subgroup of G for all k <= r ? Are you asking whether there is a normal subgroup of order p^k for each k with ) <= k <= r in each p-group of order p^r? The answer is yes. Assume r > 0. Consider the centre Z of G. Note that |Z| > 1. If p^k <= |Z| then Z has a subgroup H of order p^r and H is a normal subgroup of G. Suppose then p^k > |Z|. Then inductively G/Z has a normal subgroup of order p^k/|Z|. This pulls back to a normal subgroup of G with order p^k. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ === Subject: Is there a solution? for an equation: ax + by + cxy = d where a, b, c, d are constants and x and y are unknowns, and all of them are positive integers, Is there any easy way to solve this equation? Is there even a way to tell that there *is* a solution for a given combination of a,b,c,d? === Subject: Re: Is there a solution? Let x(t) = t then y(t) = ((-at+d)/(b+ct)) Solve for t based on selected a, b, c, d integers ((-at+d)/(b+ct)) = 0 Hope that helps. DP > for an equation: > ax + by + cxy = d > where a, b, c, d are constants and x and y are unknowns, and all of them > are positive integers, > Is there any easy way to solve this equation? Is there even a way to tell > that there *is* a solution for a given combination of a,b,c,d? === Subject: Re: Is there a solution? >for an equation: >ax + by + cxy = d >where a, b, c, d are constants and x and y are unknowns, and all of them >are positive integers, >Is there any easy way to solve this equation? Is there even a way to >tell that there *is* a solution for a given combination of a,b,c,d? (cx + b)(cy + a) = ab + cd So there's a solution for each divisor of ab+cd which is congruent to b modulo c. Mike Guy === Subject: Re: Is there a solution? >for an equation: >ax + by + cxy = d >where a, b, c, d are constants and x and y are unknowns, and all of them >are positive integers, >Is there any easy way to solve this equation? Is there even a way to >tell that there *is* a solution for a given combination of a,b,c,d? Multiply by c and add ab : you're asking whether ab+cd can be written as a product of two numbers, one congruent to a mod c and the other congruent to b mod c. If factoring a number like N = ab+cd is something you consider an easy step, then do it, write the list of all divisors of N, and scan to see whether there are any congruent to a mod c. Check that the cofactor is congruent to b mod c. If on the other hand you don't consider factoring N to be an easy step (e.g. if a,b,c,d all have over fifty digits or more) then no, there is in general no easy way to solve the original equation or even determine whether it _has_ solutions (as you will notice by contemplating the case c=1, a=b=0). dave === Subject: Re: Is there a solution? > for an equation: > ax + by + cxy = d > where a, b, c, d are constants and x and y are unknowns, and all of them > are positive integers, > Is there any easy way to solve this equation? Is there even a way to > tell that there *is* a solution for a given combination of a,b,c,d? I know very little about Diophantine equations, except to tell you that this is one (of second order because of the xy term), there are techniques for solving them and for determining if they have solutions, and that according to Mathworld, the Mathematica computer-math system can solve the general equation ax^2 + bxy + cy^2 + dx + ey + f = 0 when solutions exist. http://mathworld.wolfram.com/PellEquation.html (Yours is a special case of that with a=c=0). - Randy === Subject: Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum <42752913.8793B970@iw.net> I shall be leaving this newsgroup, rather than endure abuse by Herc. Here, then, are my parting words, re: Aristotle, Euclid, and proofs by contradiction. Enjoy! Following is the text based upon a translation of Heiberg's Greek edition of Euclid on the parallel postulate, with commentaries interpolated on the side, and additional comments. (Euclid. Euclidis Elementa . J. L. Heiberg. Leipzig. Teubner. 1883-1888): In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. Let ABC be an isosceles triangle having the side AB equal to the side AC, and let the straight lines BD and CE be produced further in a straight line with AB and AC. I.Def.20 Post.2 I say that the angle ABC equals the angle ACB, and the angle CBD equals the angle BCE. Take an arbitrary point F on BD. Cut off AG from AE the greater equal to AF the less, and join the straight lines FC and GB. I.3. Post.1 Since AF equals AG, and AB equals AC, therefore the two sides FA and AC equal the two sides GA and AB, respectively, and they contain a common angle, the angle FAG. Therefore the base FC equals the base GB, the triangle AFC equals the triangle AGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ACF equals the angle ABG, and the angle AFC equals the angle AGB. I.4 Since the whole AF equals the whole AG, and in these AB equals AC, therefore the remainder BF equals the remainder CG. C.N.3 But FC was also proved equal to GB, therefore the two sides BF and FC equal the two sides CG and GB respectively, and the angle BFC equals the angle CGB, while the base BC is common to them. Therefore the triangle BFC also equals the triangle CGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Therefore the angle FBC equals the angle GCB, and the angle BCF equals the angle CBG. I.4 Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG equals the angle BCF, the remaining angle ABC equals the remaining angle ACB, and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB, and they are under the base. C.N.3 Therefore in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. Q.E.D. There are two conclusions for this proposition, first that the internal base angles ABC and ACB are equal, second that the external base angles to prove the second conclusion from the first by simply subtracting the equal angles ABC and ACB the straight angles ABF and ACG, respectively. But Euclid doesn't accept straight angles, and even if he did, he hasn't proved that all straight angles are equal. Proposition I.13 would be enough, since it implies the sum of angles ABC and FBC equals two right angles, and the sum of angles ACB and GCB also equals two right angles, and so the two sums are equal effectively saying all straight angles are equal. Unfortunately, such an argument would be circular. I.13 depends on I.11, I.11 on I.8, I.8 on I.7, and I.7 on I.5. Thus, I.13 cannot be used in the proof of I.5. It may appear that I.7 only depends on the first conclusion of I.5, but a case of I.7 that Euclid does not discuss relies on the second conclusion of I.5. This proposition has been called the Pons Asinorum, or Asses' Bridge. Whether this name is due to its difficulty (which it isn't) or the resemblance of its figure to a bridge is not clear. Very few of the propositions in the Elements are known by names. Pappus' proof Pappus (fl. ca. 320 C.E.) gave a much shorter proof of the first conclusion, but it is also conceptually more difficult. The two triangles BAC and CAB have two sides equal to two sides, namely side BA of the first triangle equals side CA of the second triangle, and side AC of the first triangle equal to side AB of the second, and the contained angles are equal, namely angle BAC of the first triangle equals angle CAB of the second, therefore, by I.4, the corresponding parts of the two triangles are equal, in particular, the angle B in the first triangle equals the angle C of the second. The difficulty lies in treating one triangle as two, or in making a correspondence between a triangle and itself, but not the correspondence of identity. There is nothing wrong with this proof formally, but it might be more difficult for a student just learning geometry. Use of Proposition 5 This proposition is used in Book I for the proofs of several propositions starting with I.7 It is also used frequently in Books II, III, IV, VI, and XIII. end of quote from Heiberg. Granted, neither the terminology proof by contradiction nor reductio ad absurdum are employed. Netherless, is not Euclid's a proof by contradiction, whether called so or not? Aristotle had reductio ad absurdum, then credit (discredit?) should be attributed to Saccheri. I did NOT say that Saccheri actually created the myth, if myth it is. It seems more likely only that he (re-) parallel postulate, alm0ost from the outset, beginning at least with Proclus, geometers questioned it and its status as a postulate on a par with the previous four; and they therefore sought to derive it from the other four rather than treating it AS a postulate. Thus, they may not have articulated explicitly the notions of consistency and independence of axiom systems, but recognized that there was, indeed, a problem to be dealt with, albeit not in the same terms that, say, Hilbert, would have. (But one cannot fault Euclid, Proclus, Pappus, or other Hellenic Greeks for not being Hilbert!) These efforts continued among the medieval Arabic and Renaissance geometers as well. See Boris A. Rosenfeld's A History of Non-Euclidean Geometry: Evolution of the Concept of Geometric Space (New York/Berlin/Heidelberg/Vienna/London/Paris/Tokyo: Springer-Verlag, 1988) for a history, from ancient times to modern. Aristotle for his part in An. Pr. took earlier proofs of the parallel postulate to be grounded in a petitio principii (if I am allowed to use the Latin terminology for Aristotle's Greek), and apparently produced a text offering an alternative, requiring the parallel postulate to stand as an independent postulate within Euclid's system, and suggested what we would call proof by contradiction to show, e.g., that squares have parallel sides. This work of Aristotle's, no longer extant, was used by later Hellenic and Arabic mathematicians, in their works, as they refer to it in their own treatments. There's an entertaining little book on proof by contradiction and its history by Jean-Louis Gardies, Le raisonnement par l.92absurde (Paris: Presses Universitaires de France, 1991). Irving H. Anellis irvanellis@lycos.com === Subject: Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum <42752913.8793B970@iw.net> > I shall be leaving this newsgroup, rather than endure abuse by Herc. > Here, then, are my parting words, re: Aristotle, Euclid, and proofs by > contradiction. Enjoy! I second Archimedes Plutonium in urging you to stay. I think there's a small amount of vituperative posting, but a large amount of people really trying to learn and communicate. If you could kind of ignore the negative stuff... > Following is the text based upon a translation of Heiberg's Greek > edition of Euclid on the parallel postulate, with commentaries > interpolated on the side, and additional comments. > (Euclid. _Euclidis Elementa_. J. L. Heiberg. Leipzig. Teubner. > 1883-1888): > In isosceles triangles the angles at the base equal one another, and, > if the equal straight lines are produced further, then the angles under > the base equal one another. > Let ABC be an isosceles triangle having the side AB equal to the side > AC, and let the straight lines BD and CE be produced further in a > straight line with AB and AC. I.Def.20 > Post.2 > I say that the angle ABC equals the angle ACB, and the angle CBD equals > the angle BCE. > Take an arbitrary point F on BD. Cut off AG from AE the greater equal > to AF the less, and join the straight lines FC and GB. I.3. > Post.1 > Since AF equals AG, and AB equals AC, therefore the two sides FA and > AC equal the two sides GA and AB, respectively, and they contain a > common angle, the angle FAG. > Therefore the base FC equals the base GB, the triangle AFC equals the > triangle AGB, and the remaining angles equal the remaining angles > respectively, namely those opposite the equal sides, that is, the angle > ACF equals the angle ABG, and the angle AFC equals the angle AGB. I.4 > Since the whole AF equals the whole AG, and in these AB equals AC, > therefore the remainder BF equals the remainder CG. C.N.3 > But FC was also proved equal to GB, therefore the two sides BF and FC > equal the two sides CG and GB respectively, and the angle BFC equals > the angle CGB, while the base BC is common to them. Therefore the > triangle BFC also equals the triangle CGB, and the remaining angles > equal the remaining angles respectively, namely those opposite the > equal sides. Therefore the angle FBC equals the angle GCB, and the > angle BCF equals the angle CBG. I.4 > Accordingly, since the whole angle ABG was proved equal to the angle > ACF, and in these the angle CBG equals the angle BCF, the remaining > angle ABC equals the remaining angle ACB, and they are at the base of > the triangle ABC. But the angle FBC was also proved equal to the angle > GCB, and they are under the base. C.N.3 > Therefore in isosceles triangles the angles at the base equal one > another, and, if the equal straight lines are produced further, then > the angles under the base equal one another. > Q.E.D. Sorry, but I don't see the above as a reductio at all, but as a proof by 'construction' I suppose is the right term. Not that there aren't reductios in Euclid, just that I don't see this as one of them. Do other readers agree, or disagree? Ken === Subject: Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum (snipped) > Following is the text based upon a translation of Heiberg's Greek > edition of Euclid on the parallel postulate, with commentaries > interpolated on the side, and additional comments. > (Euclid. _Euclidis Elementa_. J. L. Heiberg. Leipzig. Teubner. > 1883-1888): > In isosceles triangles the angles at the base equal one another, and, > if the equal straight lines are produced further, then the angles under > the base equal one another. > Let ABC be an isosceles triangle having the side AB equal to the side > AC, and let the straight lines BD and CE be produced further in a > straight line with AB and AC. I.Def.20 > Post.2 > I say that the angle ABC equals the angle ACB, and the angle CBD equals > the angle BCE. > Take an arbitrary point F on BD. Cut off AG from AE the greater equal > to AF the less, and join the straight lines FC and GB. I.3. > Post.1 > Since AF equals AG, and AB equals AC, therefore the two sides FA and > AC equal the two sides GA and AB, respectively, and they contain a > common angle, the angle FAG. > Therefore the base FC equals the base GB, the triangle AFC equals the > triangle AGB, and the remaining angles equal the remaining angles > respectively, namely those opposite the equal sides, that is, the angle > ACF equals the angle ABG, and the angle AFC equals the angle AGB. I.4 > Since the whole AF equals the whole AG, and in these AB equals AC, > therefore the remainder BF equals the remainder CG. C.N.3 > But FC was also proved equal to GB, therefore the two sides BF and FC > equal the two sides CG and GB respectively, and the angle BFC equals > the angle CGB, while the base BC is common to them. Therefore the > triangle BFC also equals the triangle CGB, and the remaining angles > equal the remaining angles respectively, namely those opposite the > equal sides. Therefore the angle FBC equals the angle GCB, and the > angle BCF equals the angle CBG. I.4 > Accordingly, since the whole angle ABG was proved equal to the angle > ACF, and in these the angle CBG equals the angle BCF, the remaining > angle ABC equals the remaining angle ACB, and they are at the base of > the triangle ABC. But the angle FBC was also proved equal to the angle > GCB, and they are under the base. C.N.3 > Therefore in isosceles triangles the angles at the base equal one > another, and, if the equal straight lines are produced further, then > the angles under the base equal one another. > Q.E.D. > There are two conclusions for this proposition, first that the internal > base angles ABC and ACB are equal, second that the external base angles > to prove the second conclusion from the first by simply subtracting the > equal angles ABC and ACB the straight angles ABF and ACG, respectively. > But Euclid doesn't accept straight angles, and even if he did, he > hasn't proved that all straight angles are equal. Proposition I.13 > would be enough, since it implies the sum of angles ABC and FBC equals > two right angles, and the sum of angles ACB and GCB also equals two > right angles, and so the two sums are equal effectively saying all > straight angles are equal. > Unfortunately, such an argument would be circular. I.13 depends on > I.11, I.11 on I.8, I.8 on I.7, and I.7 on I.5. Thus, I.13 cannot be > used in the proof of I.5. It may appear that I.7 only depends on the > first conclusion of I.5, but a case of I.7 that Euclid does not discuss > relies on the second conclusion of I.5. > This proposition has been called the Pons Asinorum, or Asses' Bridge. > Whether this name is due to its difficulty (which it isn't) or the > resemblance of its figure to a bridge is not clear. Very few of the > propositions in the Elements are known by names. It is important that we name things we feel are important. It shows we are consciously aware of the item or event taking place. Important for historical tabulation. Example: we see a dog wandering through our yard but we do not name it for we do not expect it in our future. But when we raise a pet dog we name it because we expect much future encounters. Example: Linnaeus is the father of taxonomy or Classification of biology and lived 1707 to 1778. Now Schleiden and Schwann discovered Cell theory of biology in 1839. Now suppose all history of biology were on computers electronically storaged and we had a huge future blackout of computers. And all that survived in the future was some works of Linnaeus, much like the survival of Aristotle's work. So, are these future historians of biology going to say that Linnaeus discovered Cell theory because his taxonomy is all about the arrangement of cells? Likewise: are we going to say Aristotle discovered Reductio Ad Absurdum because of his Logic Classification of Baroco to Bocardo to Barbara. Example: One can say the importance of naming is very important for history because Fermi discovered the neutrino in that he named it neutrino which is a sign of conscious awareness of an entity. Can we say that some other physicist before Fermi discovered the neutrino? Certainly Pauli talked about the neutrino but he did not name it. > Pappus' proof > Pappus (fl. ca. 320 C.E.) gave a much shorter proof of the first > conclusion, but it is also conceptually more difficult. The two > triangles BAC and CAB have two sides equal to two sides, namely side BA > of the first triangle equals side CA of the second triangle, and side > AC of the first triangle equal to side AB of the second, and the > contained angles are equal, namely angle BAC of the first triangle > equals angle CAB of the second, therefore, by I.4, the corresponding > parts of the two triangles are equal, in particular, the angle B in the > first triangle equals the angle C of the second. > The difficulty lies in treating one triangle as two, or in making a > correspondence between a triangle and itself, but not the > correspondence of identity. There is nothing wrong with this proof > formally, but it might be more difficult for a student just learning > geometry. > Use of Proposition 5 > This proposition is used in Book I for the proofs of several > propositions starting with I.7 It is also used frequently in Books II, > III, IV, VI, and XIII. > end of quote from Heiberg. > Granted, neither the terminology proof by contradiction nor reductio > ad absurdum are employed. Netherless, is not Euclid's a proof by > contradiction, whether called so or not? No, I do not see Reductio Ad Absurdum above. What I do see is a huge leeway for the translator or the interpreter of Ancient Greek works to inject something that was never there in the first place. I see the above as direct geometrical proving methods. Keep in mind that the language of proofs in the Ancient Greek world were far different from the language used in modern proofs and that there is this huge compulsion to add modern day language into Ancient writings. > Aristotle had reductio ad absurdum, then credit (discredit?) should be > attributed to Saccheri. I did NOT say that Saccheri actually created I believe Reductio Ad Absurdum was borne by Saccheri and that the Ancient Greeks may have had hints of this method but were not consciously aware of it, nor used it. I base that assertion on the fact that Euclid's Infinitude of Primes proof was a direct method of increasing set cardinality and not a reductio-ad-absurdum. So if Euclid did not use reductio ad absurdum for Infinitude of Primes then probably no ancient Greek math proof ever used reductio ad absurdum and that this is a corruption of the history of mathematics to think the Ancients had reductio ad absurdum. Secondly, the sparsity and rarity of discussion of Consistency in the Ancients, considering that Consistency and Reductio ad Absurdum are concepts paralleling one another, or contributing concepts of one another. So if the Ancients did not have a developed concept of Consistency then surely they could not have a developed concept of Reductio Ad Absurdum. And thirdly, is this naming of things we deem important for the future. The fact that the Ancients never named reductio ad absurdum indicates they were not aware of it. They did name method of exhaustion and used it as a tool. So if they had known of reductio ad absurdum and given it a name would be proof that they knew it and used it as a method. > the myth, if myth it is. It seems more likely only that he (re-) No, considering that Euclid's Infinitude of Primes proof was not reductio ad absurdum but rather instead a geometrical direct proof by increasing set cardinality indicates that modern historians are injecting into their translations falsehoods about Euclid. > parallel postulate, alm0ost from the outset, beginning at least with > Proclus, geometers questioned it and its status as a postulate on a par > with the previous four; and they therefore sought to derive it from the > other four rather than treating it AS a postulate. Thus, they may not > have articulated explicitly the notions of consistency and independence Please tell, if the Ancient Greeks had a word for Consistency? Perhaps consistency is a Greek derived word? It is known that Euclid had discovered an Axiomatic System for geometry. So if Euclid was aware of consistency, what words would he have named it in Ancient Greek times? The word axiom is obviously Ancient Greek, but is the word Consistency Ancient Greek? > of axiom systems, but recognized that there was, indeed, a problem to > be dealt with, albeit not in the same terms that, say, Hilbert, would > have. (But one cannot fault Euclid, Proclus, Pappus, or other Hellenic > Greeks for not being Hilbert!) These efforts continued among the The Ancients were geniuses, no doubt about that. But we should not credit them with things they really did not have. I doubt they had the concept of Consistency and I doubt they had the concept and method of Reductio Ad Absurdum. But that does not diminish their genius for we have to remember that they discovered the Atomic theory which took 2,000 years for us to rediscover and they discovered the heliocentric theory which took us almost another 2,000 years to rediscover. Science has to get it exactly correct and even the history of science. > medieval Arabic and Renaissance geometers as well. See Boris A. > Rosenfeld's _A History of Non-Euclidean Geometry: Evolution of the > Concept of Geometric Space_ (New > York/Berlin/Heidelberg/Vienna/London/Paris/Tokyo: Springer-Verlag, > 1988) for a history, from ancient times to modern. > Aristotle for his part in An. Pr. took earlier proofs of the parallel > postulate to be grounded in a petitio principii (if I am allowed to use > the Latin terminology for Aristotle's Greek), and apparently produced a > text offering an alternative, requiring the parallel postulate to stand > as an independent postulate within Euclid's system, and suggested what > we would call proof by contradiction to show, e.g., that squares > have parallel sides. This work of Aristotle's, no longer extant, was > used by later Hellenic and Arabic mathematicians, in their works, as > they refer to it in their own treatments. More advanced than I expected. But considering it is not extant may lead to the conclusion that Aristotle never really did it in the first place. One must remember that Aristotle was a centerpiece of the christian religion and that these religionists ascribed much to Aristotle even though Aristotle never had those ideas. > There's an entertaining little book on proof by contradiction and its > history by Jean-Louis Gardies, _Le raisonnement par l.89absurde_ (Paris: > Presses Universitaires de France, 1991). > Irving H. Anellis > irvanellis@lycos.com Absurdum. I would guess that most of the book is erroneous. Historians of science should use the techniques of (1) caveat of lost in translation (2) parallel contributing concepts (cannot have reductio ad absurdum if there is no well developed concept of consistency (3) importance of naming entities or events for sake of history to prove of conscious awareness of the entity or event. Irving, it is a shame you are leaving this discussion for you have contributed alot. Use killfiles if others bother you and keep in mind that people are like radioactive decay 90% bad and 10% good and ignore the bad and focus only on that 10% good. I was hoping you contribute more to the data of Consistency used by the Ancient Greeks. What their word for consistency would have been. Whether Euclid set aside a chapter for consistency? Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum í Archimedes Plutonium .97.8d.98.87.8b.8c .97.99.95 .92.86.94.9d.92.87 > Please tell, if the Ancient Greeks had a word for Consistency? Perhaps > consistency is a Greek derived word? It is known that Euclid had discovered > an Axiomatic System for geometry. So if Euclid was aware of consistency, what > words would he have named it in Ancient Greek times? > The word axiom is obviously Ancient Greek, but is the word Consistency > Ancient Greek? The Ancient Greek word for consistency is synepeia. A consistent system would be a synepes systima. Specifically, synepeia (consistency)'s root is syn (plus) + epomai (to follow). In a way, the Greeks *assumed* that what they were talking about, as long as it was axiomatic and *deductive*, it was by definition synepes (consistent), as it followed from axioms and theorems. Your trying to relegate this explicit noetical dichotomy (of whether the notion was existent or not) is non-sensical at best, because by definition they were engaging in it (synepeia), whenever they argued using *any* axiomatic method. The word is still used today: An asynepes person, is someone who says one thing and does another, creating contradictions. > Archimedes Plutonium > www.iw.net/~a_plutonium > whole entire Universe is just one big atom where dots > of the electron-dot-cloud are galaxies -- I. N. Galidakis http://users.forthnet.gr/ath/jgal/ Eventually, _everything_ is understandable === Subject: Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum > The Ancient Greek word for consistency is synepeia. A consistent > system would be a synepes systima. > Specifically, synepeia (consistency)'s root is syn (plus) + epomai (to > follow). In a way, the Greeks *assumed* that what they were talking about, > as long as it was axiomatic and *deductive*, it was by definition synepes > (consistent), as it followed from axioms and theorems. > Your trying to relegate this explicit noetical dichotomy (of whether the > notion was existent or not) is non-sensical at best, because by definition > they were engaging in it (synepeia), whenever they argued using *any* > axiomatic method. > The word is still used today: An asynepes person, is someone who says one > thing and does another, creating contradictions. Yes, the words of synchronize and symphony and synergy come from that Ancient Greek word. I was looking up the word consistent to see its root and it seems to come from Latin meaning to stand together. So I think the concept of consistency was absent in Ancient Greek mathematics and philosophy and was borne from Saccheri sometime between 1667-1733. I suppose no other mathematician before Saccheri discuss a concept which we now know of as consistency for mathematics. So if Ioannis is correct in saying that the concept of consistency was subdued under or a subset of deduction and axiomatics, then the Ancient Greeks were unaware that their axioms may be inconsistent. Which tells me that the concept of Reductio Ad Absurdum could not be present in Ancient Greek times. To fully understand Reductio Ad Absurdum is to understand that its usage throws every axiom and every theorem into jeopardy. Reductio Ad Absurdum is a tool itself for checking upon consistency of axioms and theorems. So the place and time in mathematics history where a person fully understands the concept of consistency and then the concept of Reductio Ad Absurdum is Saccheri in the early 18th century. Saccheri probably gave it its name of consistency and reductio ad absurdum. Whether Saccheri was the one to blame for thence ascribing the Ancient Greeks with use of reductio ad absurdum is unknown. But it seems to me that Saccheri made the original discovery of consistency and reductio ad absurdum. This makes sense on a work list for the Ancient Greeks in that they founded a axiomatics for geometry and were so focused on using those axioms to prove theorems that they had no time to think about meta-issues of consistency. And another concept that the Ancient Greeks really did use and were unaware of it is modern day Set theory. Set theory is one of those simpleton concepts, those concepts so obvious that most thinkers would have bypassed or overlooked and not worthy of formalization. Set theory is one of those enigma's of science history in that it is so obvious that we are not consciously aware of using it. So can we say the Ancient Greeks discovered Set Theory because they used it but were not consciously aware that they were using it? I think the answer is no. I think the historical criteria of discovery has to have those three factors which I mentioned earlier (1) parallel concepts that need each other such as subset, union, membership, and most importantly number-systems such as Reals that necessitated the formalization of Set theory (remember the Ancient Greeks did not have a decimal number system) (2) naming of those concepts which discussion of them at length (3) not lost in translation or interpretation. We do not believe the Ancient Greeks of Pythagoras or Aristotle or Euclid or Archimedes had Set theory even though they used it in their work. And since the Ancient Greeks did not have a concept of Consistency then they did not have the Reductio Ad Absurdum. For anyone to claim that the Ancient Greeks had reductio ad absurdum and used would be tantamount to saying the Ancient Greeks had modern-Set theory and that is clearly not the case. I believe modern set theory was discovered in a environment where it had to be uncovered. An environment of distinguishing between different number systems of Rationals to Irrationals and that had to wait until the decimal number system. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum <42752913.8793B970@iw.net> <4277BAA6.FEA62A19@iw.net> <1115144201.422187@athnrd02> <427870FF.458AC519@iw.net> > > The Ancient Greek word for consistency is synepeia. A consistent > > system would be a synepes systima. > > Specifically, synepeia (consistency)'s root is syn (plus) + epomai (to > > follow). In a way, the Greeks *assumed* that what they were talking about, > > as long as it was axiomatic and *deductive*, it was by definition synepes > > (consistent), as it followed from axioms and theorems. > > Your trying to relegate this explicit noetical dichotomy (of whether the > > notion was existent or not) is non-sensical at best, because by definition > > they were engaging in it (synepeia), whenever they argued using *any* > > axiomatic method. > > The word is still used today: An asynepes person, is someone who says one > > thing and does another, creating contradictions. > Yes, the words of synchronize and symphony and synergy come from that Ancient > Greek word. > I was looking up the word consistent to see its root and it seems to come > from Latin meaning to stand together. > So I think the concept of consistency was absent in Ancient Greek mathematics > and philosophy and was borne from Saccheri sometime between 1667-1733. I > suppose no other mathematician before Saccheri discuss a concept which we now > know of as consistency for mathematics. > So if Ioannis is correct in saying that the concept of consistency was subdued > under or a subset of deduction and axiomatics, then the Ancient Greeks were > unaware that their axioms may be inconsistent. Which tells me that the concept > of Reductio Ad Absurdum could not be present in Ancient Greek times. To fully > understand Reductio Ad Absurdum is to understand that its usage throws every > axiom and every theorem into jeopardy. Reductio Ad Absurdum is a tool itself > for checking upon consistency of axioms and theorems. > So the place and time in mathematics history where a person fully understands > the concept of consistency and then the concept of Reductio Ad Absurdum is > Saccheri in the early 18th century. Saccheri probably gave it its name of > consistency and reductio ad absurdum. Whether Saccheri was the one to blame > for thence ascribing the Ancient Greeks with use of reductio ad absurdum is > unknown. But it seems to me that Saccheri made the original discovery of > consistency and reductio ad absurdum. > This makes sense on a work list for the Ancient Greeks in that they founded a > axiomatics for geometry and were so focused on using those axioms to prove > theorems that they had no time to think about meta-issues of consistency. > And another concept that the Ancient Greeks really did use and were unaware of > it is modern day Set theory. Set theory is one of those simpleton concepts, > those concepts so obvious that most thinkers would have bypassed or overlooked > and not worthy of formalization. Set theory is one of those enigma's of science > history in that it is so obvious that we are not consciously aware of using it. > So can we say the Ancient Greeks discovered Set Theory because they used it but > were not consciously aware that they were using it? I think the answer is no. I > think the historical criteria of discovery has to have those three factors > which I mentioned earlier (1) parallel concepts that need each other such as > subset, union, membership, and most importantly number-systems such as Reals > that necessitated the formalization of Set theory (remember the Ancient Greeks > did not have a decimal number system) (2) naming of those concepts which > discussion of them at length (3) not lost in translation or interpretation. > We do not believe the Ancient Greeks of Pythagoras or Aristotle or Euclid or > Archimedes had Set theory even though they used it in their work. And since the > Ancient Greeks did not have a concept of Consistency then they did not have the > Reductio Ad Absurdum. > For anyone to claim that the Ancient Greeks had reductio ad absurdum and used > would be tantamount to saying the Ancient Greeks had modern-Set theory and that > is clearly not the case. > I believe modern set theory was discovered in a environment where it had to be > uncovered. An environment of distinguishing between different number systems of > Rationals to Irrationals and that had to wait until the decimal number system. > Archimedes Plutonium > www.iw.net/~a_plutonium > whole entire Universe is just one big atom where dots > of the electron-dot-cloud are galaxies The notion of reductio ad absurdum is a notion that is part of our communal system of reasoning. For example, if someone accuses me of stealing a pie that's in a window 100 feet high, I could answer, 'that's absurd, I'd have to be 100 feet tall', and be confident my hearer would understand the point I was trying to make, my argument. Because I might not personally have a complete coherent and consistent system of logic doesn't mean I'm forbidden to use reductio, nor does it mean that using reductio would never occur to me, absent such a system. Nor does it mean that if I should use it, someone is entitled to say I wasn't really using it because I didn't understand all its logical implications - so therefore, my use of it never occurred, and whoever said I had used it must have misunderstood what I was saying, or misheard me, or is lying. I think attempting to understand all the logical implications of the usage of reductio is admirable, but it doesn't change history. It merely puts it in perspective. Ken === Subject: Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum í Archimedes Plutonium .97.8d.98.87.8b.8c .97.99.95 .92.86.94.9d.92.87 > Yes, the words of synchronize and symphony and synergy come from that Ancient > Greek word. As far as I am aware, these words only have their first syllable similar to synepeia. No other connection (in meaning) exists. > I was looking up the word consistent to see its root and it seems to come > from Latin meaning to stand together. In Latin, I have no idea. > So I think the concept of consistency was absent in Ancient Greek mathematics > and philosophy and was borne from Saccheri sometime between 1667-1733. I > suppose no other mathematician before Saccheri discuss a concept which we now > know of as consistency for mathematics. > So if Ioannis is correct in saying that the concept of consistency was subdued > under or a subset of deduction and axiomatics, then the Ancient Greeks were > unaware that their axioms may be inconsistent. It depends on how they viewed consistency. They probably believed that only truths could come out of their axiomatic and deductive systems, thus their unequivocal devotion to such reasoning. > Which tells me that the concept > of Reductio Ad Absurdum could not be present in Ancient Greek times. The above doesn't follow. I have read personally in the Ancient Greek version of Euclid's Elements the words oper atopon (thus contradiction). (Unfortunatelly my books are packed and elsewhere, so I cannot check the original versions right now and give you a reference). The word's root is topos (place/locus, same root as topology) meaning that their reasoning for a contradiction leads to no topos, or nowhere. Although they were taking the consistency of their system for granted, they could easily have used contradiction as a device which allowed them to differentiate truth from falsity. In particular, if they arrived at it, using deductive reasoning, this meant that their *reasoning* was faulty, and not necessarily that their axiomatic system was. The atopon sentence p/~p, could still have been seen as simply a false statement which could have resulted from ludicrous/mistaken reasoning. To summarize: synepeia (consistency) and atopon (contradiction) could well have been totally unrelated. I will also check on Monday here with a professor who specializes in Euclidian Geometry and will respond back with the definite answer to this. > Archimedes Plutonium > www.iw.net/~a_plutonium > whole entire Universe is just one big atom where dots > of the electron-dot-cloud are galaxies -- I. N. Galidakis http://users.forthnet.gr/ath/jgal/ Eventually, _everything_ is understandable === Subject: Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum <42752913.8793B970@iw.net> you know, he actually said that Saccheri did not recognize that he had shown a contradiction in Euclid, as opposed to Euclid's own conscious (apparent) use in hte proof. > Tartaglia but they were mostly algebraists. And Saccheri fits the > likelihood as the true discoverer of the method of Reductio Ad Absurdum as > he analyzed Aristotle. And Saccheri would then realize its importance and > give it a special name in Italian of Reductio-Ad-Absurdum. thus: uh, that is that the NYT weather miscellany box records lots of low-temperature records. just last week, on two days, they had a map with simultaneuos high and low records for the continental USA, which may occur every day, for all that I know. here's my favorite Sunday headline from the LAtribcoTimes, in the back of the Sunday paper a couple of years ago: A Hundred New Glaciers Dyscivered in the Rockies, the Continental Divide, to be more exact. alas, one cannot normally see those areas, because of near-constant cloudcover. --Chair Man George XOR Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://larouchepub.com http://members.tripod.com/~american_almanac === Subject: Re: Golden Number linked to 666 > Golden Number linked to 666 > Formal Proof needed > While working on my websites on 666 Myth ( http://www.666myth.co.nr/ > French = http://www.666mythe.co.nr/ ), on Pope Benedict 16 - Mark > 666 ( http://www.chez.com/cosmos2000/Forums/NEWS_MemoryPopeJohnPaul_II.html > ) and on Isomorphous Triplets ( > http://www.chez.com/cosmos2000/Numbers/IsomorphousTriplets.html ), I > have found recently direct links between the Beast Number 666 and the > famous Golden Number Phi, well-known by Pythagoras, Leonardo da Vinci > ... and in Sacred Geometry, in architecture and many natural > phenomenons. This unexpected and incredible relation 666 versus Phi > may constitute a way for the Rehabilitation of 666 !!! ... > First we have this curious relations: > 666 = 7^3 pi (Phi - 1) = 7^3 pi phi or 666 = (6/5) 7^3 Phi > with pi = 3.141593 Phi = 1.618034 phi = Phi -1 = > 0.618034 and 7^3 = cubic of 7 > Besides, after computing some sinus and cosines [a good online > Trigonometry calculator available at http://www.1728.com/trigcalc.htm > ], expressed in Degrees and absolute values, we obtain: > Phi /2 = sin 666.bc = cos 324.bc = cos 216.bc = cos 144.bc = cos 36.bc = > 0.80901699... > Phi = 2sin 666.bc = 2cos 324.bc = 2cos 216.bc = 2cos 144.bc = 2cos 36.bc = > 1,61803399... > Phi = sin 666.bc + cos 216.bc = sin 666.bc + cos (6x6x6).bc > Phi = cos 144.bc + cos 36.bc = cos [(6+6) x (6+6)].bc + cos (6 x 6).bc > Phi = sin 666.bc + cos 144.bc = sin 666.bc + cos [(6+6) x (6+6)].bc > But, I am wondering if there could exist formal mathematical proofs > for this 666 and Golden Number links. Like Tom already pointed out, the thing with 7^3 is rather silly, but we do have the exact formula Phi = 2 cos( pi/5 ) = -2 sin ( 666.bc ) See http://goldennumber.net/ I haven't seen a proof, but I'm sure someone on sci.math can help - if he wants ;-) I have just used Pari to check ( sqrt(5) + 1 )/2 and - 2 sin(666.bc) to 100000 digits. They match :-) Dirk Vdm [Copy and follow-up set to sci.math] === Subject: Re: Golden Number linked to 666 > Like Tom already pointed out, the thing with 7^3 is rather > silly, but we do have the exact formula > Phi = 2 cos( pi/5 ) = -2 sin ( 666.bc ) > See http://goldennumber.net/ > I haven't seen a proof, but I'm sure someone on sci.math > can help - if he wants ;-) > I have just used Pari to check > ( sqrt(5) + 1 )/2 > and > - 2 sin(666.bc) > to 100000 digits. > They match :-) > Dirk Vdm > [Copy and follow-up set to sci.math] Sorry for that - I hadn't noticed that he multi-posted. To the OP (unless you are a troll) Please do not multi-post. http://www.blakjak.demon.co.uk/mul_crss.htm Dirk Vdm === Subject: Re: Golden Number linked to 666 > Golden Number linked to 666 > Formal Proof needed Well I can do all integers: Theorum - all positive integers (plus 0) are 'linked' to 666 Proof (very informal - odviously) Create the set, S, of all such integers positive integers (plus 0) that are not 'linked' to 666. S is well-ordered with the standard order relation. If S contains one or more elements, then it's least element is the first number that is not 'linked' to 666 - BUT that's a link! # So S is the empty set. Do the same with the negative integers. Alun Harford === Subject: Re: Golden Number linked to 666 >> Golden Number linked to 666 >> Formal Proof needed > Well I can do all integers: > Theorum - all positive integers (plus 0) are 'linked' to 666 > Proof (very informal - odviously) > Create the set, S, of all such integers positive integers (plus 0) that > are > not 'linked' to 666. > S is well-ordered with the standard order relation. > If S contains one or more elements, then it's least element is the first > number that is not 'linked' to 666 - BUT that's a link! # > So S is the empty set. > Do the same with the negative integers. > Alun Harford well, doesn't the equation 666 = n + m or 666 = n - m have solutions given any n for any m? so ofcourse all integers are related to 666 ;) what is really cool is that 0 is related to 666 too since 666 = 666 + 0!!!!!!!!!!!!!! ;) === Subject: Re: Golden Number linked to 666 > But, I am wondering if there could exist formal mathematical proofs > for this 666 and Golden Number links. Take a look to http://mathworld.wolfram.com/BeastNumber.html for relation with 666 and a list of related papers. === Subject: Re: Golden Number linked to 666 > > But, I am wondering if there could exist formal mathematical proofs > > for this 666 and Golden Number links. > Take a look to http://mathworld.wolfram.com/BeastNumber.html for relation > with 666 > and a list of related papers. and read the sci.math thread The beast, again, starting at Hugo === Subject: Re: Question about integer partitions I see the error in what I said before. P(n) for n<0 is 0, but P(0) is 1, since an empty set can be 'partitioned' in only one way. Without this definition, any time k evenly divided n, I would undercount by 1. You didn't misunderstand, I just overlooked that point. Thus, for n=6, k=3, it now works, and (as another example), for n=6, k=2, you get: P(4) + P(2) + P(0) = 5 + 2 + 1 = 8, which is the number of times 2 appears in the list. === Subject: Re: Question about integer partitions This happens to work for n = 6, but is false in general. For example, by your argument for n = 7 and k = 3, you would have P(4) + P(1) = 5 + 1 = 6. But the correct answer is 10. > I see the error in what I said before. P(n) for n<0 is 0, but P(0) is > 1, since an empty set can be 'partitioned' in only one way. Without > this definition, any time k evenly divided n, I would undercount by 1. > You didn't misunderstand, I just overlooked that point. > Thus, for n=6, k=3, it now works, and (as another example), for n=6, > k=2, you get: > P(4) + P(2) + P(0) = 5 + 2 + 1 = 8, which is the number of times 2 > appears in the list. === Subject: Re: Question about integer partitions Hmm. Doesn't look like anyone got anywhere solid with this yet. Maybe some minor observations will knock an idea loose: Let K(n, i) be the number of n-partitions that contain an i-partition. So, as we saw before, K(6, 3) = 6: [neuron & ] > For example of the 11 partitions for n = 6: > 1 1 1 1 1 1 > 1 1 1 1 2 > 1 1 2 2 > 2 2 2 > 1 1 1 3 > 1 2 3 > 3 3 > 1 1 4 > 2 4 > 1 5 > 6 Via the argument last time, K(n, i) <= P(i)*P(n-i). There's an obvious symmetry: for all i, 0 < i < n, K(n, i) = K(n, n-i). That's simply because if we have an n-partition X that contains an i-partition, the rest of X must add to n-i, so the rest of X is an (n-i)-partition: the set of n-partitions containing an i-partition is equal to the set of n-partitions containing an (n-i)-partition. The endpoints are easy too: for n>1, K(n, 1) = K(n, n-1) = P(n-1) (= P(1)*P(n-1) -- equality holds in the inequality above for these cases). By the symmetry above, it suffices to show that this is true for K(n, 1). Any n-partition containing a 1 can be changed to an (n-1)-partition by dropping a 1, and contrarily any (n-1)-partition can be changed to an n-partition by adding an isolated 1 -- there's an obvious bijection between (n-1)-partitions and n-partitions with smallest part 1. Above, K(6, 1) = K(6, 5) = P(6-1) = 7. For K(6, 2), an inclusion/exclusion thingie works: comb(P(2), 1)*P(6-2) - comb(P(2), 2)*P(6-2*2)) = 2*5 - 1*2 = 10-2 = 8 That counts the number of ways to extend [1 1] and [2] to 6-partitions, less the number of ways to extend [1 1 2] (which are the double counting cases from the first term). K(6, 4) = 8 then too by symmetry. Saw last time that the same kind of argument works for K(3, 3): comb(P(3), 1)*P(6-3) - comb(P(3), 2)*P(6-2*3) = 3*3 - 3*1 = 6 Having exhasted n=6, now it's your turn . === Subject: Re: Question about integer partitions Tim, I appreciate your interest in this problem, and your efforts in clearly explaining your ideas. I've gone down this path of trying to use counting and inclusion-exclusion arguments, but I have not been smart enough to find a reasonable general solution by these methods. Each larger n seems to yield cases that aren't accounted for by hypothesized patterns based on the previous n-1 cases. I'm thinking that it might be quite difficult to solve this problem by these methods alone. Rather, it might be better to work directly with generating functions. Perhaps, if you're interested, you might try your hand at the complementary problem of finding the number of n-partitions that do **NOT** contain an i-partition. There are far fewer cases to consider for each (n,i) than in the original problem. > Hmm. Doesn't look like anyone got anywhere solid with this yet. Maybe some > minor observations will knock an idea loose: > Let K(n, i) be the number of n-partitions that contain an i-partition. So, > as we saw before, K(6, 3) = 6: > [neuron & ] > > For example of the 11 partitions for n = 6: > > 1 1 1 1 1 1 > > 1 1 1 1 2 > > 1 1 2 2 > > 2 2 2 > > 1 1 1 3 > > 1 2 3 > > 3 3 > > 1 1 4 > > 2 4 > > 1 5 > > 6 > Via the argument last time, K(n, i) <= P(i)*P(n-i). > There's an obvious symmetry: for all i, 0 < i < n, K(n, i) = K(n, n-i). > That's simply because if we have an n-partition X that contains an > i-partition, the rest of X must add to n-i, so the rest of X is an > (n-i)-partition: the set of n-partitions containing an i-partition is equal > to the set of n-partitions containing an (n-i)-partition. > The endpoints are easy too: for n>1, K(n, 1) = K(n, n-1) = P(n-1) (= > P(1)*P(n-1) -- equality holds in the inequality above for these cases). By > the symmetry above, it suffices to show that this is true for K(n, 1). Any > n-partition containing a 1 can be changed to an (n-1)-partition by dropping > a 1, and contrarily any (n-1)-partition can be changed to an n-partition by > adding an isolated 1 -- there's an obvious bijection between > (n-1)-partitions and n-partitions with smallest part 1. Above, K(6, 1) = > K(6, 5) = P(6-1) = 7. > For K(6, 2), an inclusion/exclusion thingie works: > comb(P(2), 1)*P(6-2) - comb(P(2), 2)*P(6-2*2)) = > 2*5 - 1*2 = 10-2 = 8 > That counts the number of ways to extend [1 1] and [2] to 6-partitions, less > the number of ways to extend [1 1 2] (which are the double counting cases > from the first term). K(6, 4) = 8 then too by symmetry. > Saw last time that the same kind of argument works for K(3, 3): > comb(P(3), 1)*P(6-3) - comb(P(3), 2)*P(6-2*3) = > 3*3 - 3*1 = 6 > Having exhasted n=6, now it's your turn . === Subject: A bounded polyhedral convex subset having exactly one extreme point I am learning Linear Programming and this is one of the exercizes. Could you give me an example that satisfies this proposition? === Subject: Re: A bounded polyhedral convex subset having exactly one extreme point this is that? that is this? > I am learning Linear Programming and this is one of the exercizes. > Could you give me an example that satisfies this proposition? --Martha? http://tarpley.net/bush12.htm et nonsequentia et al ad vomitorium http://larouchepub.com http://members.tripod.com/~american_almanac === Subject: Re: A bounded polyhedral convex subset having exactly one extreme point The simplest (maybe the only) example is a one point set. === Subject: Re: A bounded polyhedral convex subset having exactly one extreme point <29915140.1115152852111.JavaMail.jakarta@nitrogen.mathforum.org> > The simplest (maybe the only) example is a one point set. A proof would go like this: If you have such a set S which is a nonempty bounded polyhedron, then let x be in S. S is the convex hull of a finite set of points {y_1, y_2, ..., y_k}, by definition. If you choose this set to be minimal, then every y_i is an extreme point. We can write x as a convex combination of the y_i's: x = a_1 y_1 + ... + a_k y_k. By assumption, k = 1. Now you should be able to finish the proof. --- Christopher Heckman === Subject: Solution for x^x=a ? Anybody here knows how to solve the equation: x ^ x = a ? How many roots does it have? NoBody === Subject: Re: Solution for x^x=a ? >Anybody here knows how to solve the equation: > x ^ x = a ? >How many roots does it have? >NoBody Complete solutuion sent to the e-mail address you had the courtesy to provide. === Subject: Re: Was: Solution for x^x=a ?. f^(-1)(x) ? > Anybody here knows how to solve the equation: > x ^ x = a ? Does it have inverse? NoBody === Subject: Re: Was: Solution for x^x=a ?. f^(-1)(x) ? í NoBody .97.8d.98.87.8b.8c .97.99.95 .92.86.94.9d.92.87 > > Anybody here knows how to solve the equation: > > x ^ x = a ? > Does it have inverse? What do you get if you set f(x)=y=x^x and solve for x in terms of y? That's the inverse, wherever it's defined: f^(-1)(y)=log(y)/W(log(y)) > NoBody -- I. N. Galidakis http://users.forthnet.gr/ath/jgal/ Eventually, _everything_ is understandable === Subject: Re: Solution for x^x=a ? >Anybody here knows how to solve the equation: > x ^ x = a ? >How many roots does it have? Let b = exp(-exp(-1)) For a < b there are no real solutions. For a = b or a > 1 there is one real solution. For b < a <= 1 there are two real solutions. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Solution for x^x=a ? > >Anybody here knows how to solve the equation: > > x ^ x = a ? > >How many roots does it have? > Let b = exp(-exp(-1)) > For a < b there are no real solutions. ??? Let a = -1. Then a < b and yet we obviously have the real solution x = -1. > For a = b or a > 1 there is one real solution. > For b < a <= 1 there are two real solutions. I think that, wherever you said real, you may have intended nonnegative real instead. It depends on how one chooses to define ^, but I'd say that for some values of a, there are three real solutions. [Example: With a = (5/6)^(6/5), I'd say that there are three real solutions, the negative one being x = -6/5.] David === Subject: Re: Solution for x^x=a ? <20050503183333.876$pt@newsreader.com> >> >Anybody here knows how to solve the equation: >> > >> > x ^ x = a ? >> > >> >How many roots does it have? >> Let b = exp(-exp(-1)) >> For a < b there are no real solutions. >??? >Let a = -1. Then a < b and yet we obviously have the real solution x = -1. >> For a = b or a > 1 there is one real solution. >> For b < a <= 1 there are two real solutions. >I think that, wherever you said real, you may have intended nonnegative >real instead. Oops, I left out the negative integers. Yes there are real values of a {-1, 1/4, -1/27, 1/256, ...} which will give solutions for x which are negative integers. Negative non-integers only have solutions for complex values of a, I think. >It depends on how one chooses to define ^, but I'd say that for some values >of a, there are three real solutions. [Example: With a = (5/6)^(6/5), I'd >say that there are three real solutions, the negative one being x = -6/5.] All definitions of exponentiation that I have seen (until now) define fractional powers of negative numbers as either complex or undefined. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Solution for x^x=a ? look up lamberts W function, you'll find your answers following easily from that. === Subject: Re: what is charge fundamentally Re: mass/energy density Maxwell Eq Re: Maxwell Equations become NonLinear Re: Cosmic Coulombizing (gravity a fiction force) the Maxwell Equations <426D2948.7A55AEAD@iw.net> <426E9128.81A2435A@iw.net> In message <426E9128.81A2435A@iw.net>, Archimedes Plutonium ... >Charge is energy but charge is different than mass. You mean Charge is energy but charge is different *from* mass. Partially wrong. >Mass is a measure of weight of an object. Wrong. >Energy seems to be mass in motion. Wrong. ... >Archimedes Plutonium >www.iw.net/~a_plutonium >whole entire Universe is just one big atom where dots >of the electron-dot-cloud are galaxies No-one really knows what charge really is. This means that your pontificating about Plutonium is a waste of time. Archimedes is long dead. Choose a new name (or something). -- Jeremy Boden === Subject: Re: I'm Looking for Old TI Calculators > I'm trying to do some research on technology in math education. It would > make my life much easier if I actually had most of the technology I was > trying to study. And so, I am trying to acquire TI calculators (since TI > is probably the most popular brand). Specifically, I am looking for the > following, in working condition: > TI-80 > TI-81 > TI-83 (not plus) > TI-83 Plus > TI-85 > TI-92 (not plus) > TI-92 Plus > TI-CBL (not CBL 2) > - Tim > -- > Timothy M. Brauch > NSF Fellow > Department of Mathematics > University of Louisville Gee, here I was excited by topic until I read posting and realized you did not really mean Old TI calculators (TI-58, TI-58C, TI-59). Always looking for someone new to talk about old TI calculators .... Bradley Slavik === Subject: Re: I'm Looking for Old TI Calculators > > I'm trying to do some research on technology in math education. It would > > make my life much easier if I actually had most of the technology I was > > trying to study. And so, I am trying to acquire TI calculators (since TI > > is probably the most popular brand). Specifically, I am looking for the > > following, in working condition: > > TI-80 > > TI-81 > > TI-83 (not plus) > > TI-83 Plus > > TI-85 > > TI-92 (not plus) > > TI-92 Plus > > TI-CBL (not CBL 2) > > - Tim > > -- > > Timothy M. Brauch > > NSF Fellow > > Department of Mathematics > > University of Louisville > Gee, here I was excited by topic until I read posting and realized you > did not really mean Old TI calculators (TI-58, TI-58C, TI-59). Always > looking for someone new to talk about old TI calculators .... Same goes with the old HP calculators: http://www.hpmuseum.org And obvioulsy some =-fanatic has crated a TI museum as well: http://www.TImuseum.org [ Sorry - couldn't resist ;-) ] Better try this one: http://www.datamath.org/ Good luck, Dirk Vdm > Bradley Slavik === Subject: Re: I'm Looking for Old TI Calculators >> I'm trying to do some research on technology in math education. It would >> make my life much easier if I actually had most of the technology I was >> trying to study. And so, I am trying to acquire TI calculators (since TI >> is probably the most popular brand). Of course, the *first* real hand-held scientific calculator wasn't made by TI at all, but by Hewlett Packard. They had a monopoly with the HP-35 for a couple of years, so they charged over $400 for it. Have you looked on E-Bay for the calculators you are seeking? --Lynn === Subject: cheated by differential calculus course hi any one felt cheated by differential calculus course? I felt so cheated taking differential calculus course! All your do is switch sigma and intergration, all day long! (given continuity condition) You learn that in calculus 101! or maybe 201. any one felt the same way? I mean treating a function as a value, is that so revolutionary? === Subject: Re: cheated by differential calculus course > hi > any one felt cheated by differential calculus course? > I felt so cheated taking differential calculus > course! > All your do is switch sigma and intergration, all day > long! > (given continuity condition) > You learn that in calculus 101! or maybe 201. > any one felt the same way? > I mean treating a function as a value, is that so > revolutionary? Then I take it you are saying you didn't learn much. I can see a teacher emphasizing setting up an integral, in a given application, by writing the Riemann sum and converting to an integral, but that usually takes only a few minutes, not all day long! And, by the way, why are you distinguishing between differential calculus and Calculus 101 and Calculus 201? === Subject: translate a sentence into predicate logic? Can anyone help me with the following? For all containers, each container is a sum of all features, and for all features there is a corresponding delta. I'm not sure if I whould say something like: For all containers there exists a delta, such that ... and this is where I break down. Jim. (jharlan_not at iwon dot com) (remove the _not) === Subject: Re: translate a sentence into predicate logic? > Can anyone help me with the following? > For all containers, each container is a sum of all features, and for > all features there is a corresponding delta. > I'm not sure if I whould say something like: > For all containers there exists a delta, such that ... and this is > where I break down. Not knowing what it means, it is difficult to say, but something like this: (for all x)(if x is a container then x is the sum of all features) and (for all x)(if x is a feature then (exists y)(y is a delta and y corresponds to x)). If I knew what features are I might be able to analyse is the sum of all features further and one might end up with something like: (for all z)(if z is a feature then (...)) were ... is some alternative to the stuff above. You might also consider a many-sorted logic in which f, g, h, ... are individual variables ranging over containers; p, q, r, ... are individual variables ranging over features; x, y, z, ... are individual variables ranging over deltas; and get rid of if x is a container then, if x (or z) is a feature then, y is a delta and above. === Subject: Re: translate a sentence into predicate logic? days. My association with the Department is that of an alumnus. >Can anyone help me with the following? >For all containers, each container is a sum of all features, and for >all features there is a corresponding delta. >I'm not sure if I whould say something like: >For all containers there exists a delta, such that ... and this is >where I break down. There exist {F_i} features such that for all containers C, C is a sum of the F_i, and for all i there exist delta corresponding to F_i. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: translate a sentence into predicate logic? Arturo, I want to use logic symbols to express this sentence. Does the {f_i} notation translate to f-sub-i? That being said, would I write your sentence as: 3fiVc(c=sum(fi) A Vi3delta->fi), where fi = f-sub-i, 3 = existential quantifier, V = universal quantifier, A = logical and. I am trying to use Microsoft Equation Editor. After add'l thought, I am also trying to represent that each feature has x-number of elements, where an element is a face, edge or vertex, and x can be zero. === Subject: Re: translate a sentence into predicate logic? days. My association with the Department is that of an alumnus. clicking on the reply link at the bottom of the message, click on More options at the top and then click on the reply link that appears then. This will allow you to quote the message you are replying to. >I want to use logic symbols to express this sentence. Does the {f_i} >notation translate to f-sub-i? My problem is that I did not know what facets were. I would say that there exist some things, called facets, which I am calling F_i (yes, F-sub-i, where the varying i represents different facets), with i index by some index set i, such that every feature is the sum of all facets (i.e., sum, over all i in I, of F_i), and for each i, for the facet F_i there is a delta (possibly delta_i, since it depends on the facet). >That being said, would I write your sentence as: >3fiVc(c=sum(fi) A Vi3delta->fi), >where fi = f-sub-i, 3 = existential quantifier, V = universal >quantifier, A = logical and. I am trying to use Microsoft Equation >Editor. I don't see why you would write delta->fi. I don't know what it means for a delta to exist for the facet. If this corresponds to delta having some property P with respect to fi (so that P(delta,fi) means that delta 'works' for facet fi), then I would write everything as: 3I(Vi(i in I -> facet(fi)) A Vc(feature(c)-> c = sum(fi)) A Vi (i in I -> 3 delta ( P(fi,delta)))) So: there is a set I such that for every i in I we have a thing called fi which is a facet, with the property that for every c, if c is a feature then c is the sum of all fi, and for each i in I there is a delta which belongs to the corresponding fi. However, I do NOT think this is what you want to say. Because surely want to say that each feature is the sum of SOME facets, so you would need to say something like 3I(Vi(i in I -> facet(fi)) A Vc(feature(c)->3J((J subset I) A (J not= emptyset) A (c = sum{fj: j in J}))) A Vi (i in I -> 3 delta ( P(fi,delta)))) >After add'l thought, I am also trying to represent that each feature >has x-number of elements, where an element is a face, edge or vertex, >and x can be zero. Since I don't really know what you are talking about, and I don't really know what you are trying to say, and what your audience is, or just how formal you want to be, I cannot help you further. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Cardinality question <426fac60.65196758@netnews.att.net> <426ff58d$1_4@newsfeed.slurp.net> <427136e4.78743819@netnews.att.net> <427172ad$1_1@newsfeed.slurp.net> <42727e73.3546744@netnews.att.net> <4272b1b4.10436838@netnews.att.net> <42739790.13821342@netnews.att.net> <4273bdf9.17952661@netnews.att.net> <4273d56c.20149109@netnews.att.net> <42741d01.22262879@netnews.att.net> <1114921043.a8ab8960a232990e173a73395b45880b@teranews> <5G5de.328$gc6.2@okepread04> > As a toddler, I never started counting with zero. In grade > school, I was taught that zero was a place holder for a number, > therefore was not a number itself, but rather was an artifact of > positional notation of numerals. I taught a course about mathematics for elementary school teachers (California), and I understand why you got this misconception. That is because these future teachers thought so too and despite my trying to convince them otherwise, many still probably do, and some are likely now teaching kids in grade school. These are also people that had trouble adding fractions and subtracting numbers with different number of digits after the decimal. I had my primary education in Czech, and I don't at any moment remember thinking that zero was not a number. My later education was unfortunately in the US, and I learned many stupid (read: wrong) things. For example I remember my high school calculus teacher showing us a function that did not have a first derivative but was twice differentiable. Just because a teacher at school tells you the moon is made of green cheeze, doesn't mean it is (it is by the way, I've been there, but it's rather stale). Jiri === Subject: Re: Cardinality question >> As a toddler, I never started counting with zero. In grade >> school, I was taught that zero was a place holder for a number, >> therefore was not a number itself, but rather was an artifact of >> positional notation of numerals. > I taught a course about mathematics for elementary school teachers > (California), and I understand why you got this misconception. That is > because these future teachers thought so too and despite my trying to > convince them otherwise, many still probably do, and some are likely > now teaching kids in grade school. I find it interesting how Albert was so annoyed by the people on sci.math not being able to have an original thought, simply repeating what they were taught, and then coming out with a statement such as the above. -- Giuseppe Oblomov Bilotta They that can give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety. Benjamin Franklin === Subject: Re: Cardinality question > I learned to count as a toddler. I also learned that numbers > don't have 'behaviour'. They simply are. Did you also learn about Santa Claus, the tooth fairy, the stork and all that? -- Giuseppe Oblomov Bilotta They that can give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety. Benjamin Franklin === Subject: Re: Cardinality question >> >Since Hilbert never says what he means by between, there is no reason >> >assume that he includes the ability for points to overlap and still be >> >considered between each other. So at a minimum, his system is >> >incompletely defined. >> That is definitely (and purposefully) true in a colloquial sense. I'm >> not sure if it is in the mathematical sense, but I believe that is >> (incomplete) there as well. >> It's purposefully done that way so that its results can be used for >> *any* system that meets the axioms, regardless of whether or not that >> system is about lines or not. >But it can't, can it? Since it requires the system to define between in a >way that satisfies the axioms. If the system defines between in a certain >way, then the axioms would be violated. But the term between is only >introduced in the axioms themselves. You handle it the same way you handle the rest of the undefined terms. You find some way to define between in your system, so that the axioms hold for it. > So it is incomplete because the axioms >are ambiguous. To put it better, how can we tell if a system meets the >axioms if the axioms are ambiguous? In mathematics, you would prove the axioms. I.e take the undefined terms. Map them to some terms in your system. Prove the statements the axioms map to in your system. In physics, you do your mapping. Make some predictions based on this, and run some experiments. >> >Here's another question for you: how much can they overlap before they >are >> >no longer considered between? >> They are considered between as long as the overlap is less than 100%. >Which axiom specifies this? None of them do. There are no axioms that specify a particular model[*]. That is just *one* model of the axioms. [*] There is a sense in which this isn't quite right. I'm afraid I'm not quite good enough to explain it though. >> Because if we don't say that there is some >> >point where the overlap is too much to say that they are between each >> >other, then three points that all completely overlap each other -- ie >they >> >are all in the same location -- can fit into Hilbert's axioms. >Considering >> >between to not include overlap precludes this using Hilbert's axioms, >but >> >if it can overlap then we have a problem. >> > >> >And note that even if we ignore these problems we would still have the >issue >> >that a line segment has a minimum length (5 at least, right?). >> > >> Depends on how you measure the length of the line segment. You can >> get the same length measurements that you would expect by measuring >> from the center of the endpoints. >This adds its own difficulty, since you can have all of the centres of the >endpoints actually being all in the same point, right? Then how can that be >the length of the line segment, in that case? Without overlap, there's no >problem since it can't happen. With overlap, it's a problem. A B / / | | . . | | / / |-| d Here the distance is d. The overlap causes no difficulty. Martin === Subject: Re: Cardinality question >> You know, after 35+ years as a programmer, primarily in OOP, I cannot >> help but try and translate mathtalk into programming terms I am familiar >> with. All I hear you saying is that succ() is a method called by a >> recursive method, but without specifying the content of the succ() method. Aha! A means to communicate! The whole axiomatic method is really just a specification of some aspects of the behaviour of an external call within a program. We can make, for example, a program which computes m + n for two natural numbers m and n, which relies only on the two things mentioned in the Peano axioms: zero and succ() . Pardon my programming skills (which are nearly nonexistent) but I think you might get the picture here if I write this in pseudo-code (a la Maple): pred := proc(n) # compute the predecessor of n if n = zero then return FAIL else for k from zero do if n = succ(k) then return k end_if end_do end_if end_proc: plus := proc(m,n) if n = zero then return m else return succ( plus(m, pred(n)) ) end_if end_proc: Note that pred is built with the assumption that (1) zero is not the successor of anything (2) every other number is the successor of precisely one other thing. (The _existence_ of a predecessor for every element can be proven to be a consequence of the last of the Peano axioms.) We also use (2) in the definition of plus. Here it _absolutely does not matter_ what sorts of objects n and m are, as long as zero and the succ() operation have been previously defined. Will Twentyman had earlier proposed > As an alternate interpretation of exactly the same > axioms, I could use 1 as the starting element, and > dividing by 2 as the successor operation. My numbers > are then 1, 1/2, 1/4, 1/8, ... Nothing in the axioms > prevents me from making this rather confusing and > pointless choice. That would work just fine, that is, he could have defined zero := 1; and succ := proc(n) return n/2 end_proc: and the program would work as it should (should meaning that here we want plus(1/4, 1/8) to be 1/32 !) Or he could have defined the natural numbers to be sets via zero := {}; and succ := proc(n) return { n } end_proc: (so that the item commonly called two would be {{{}}} .) Philosophically, I suppose, one would like to know what numbers _are_ but since there seems to be no one who can tell us when we've got the right answer, that seems an impractical enterprise even by mathematician standards. It is sufficient to know how numbers _behave_, and the Peano axioms -- i.e. the specifications of the external routines zero and succ -- give us the answer we need. One can prove, for example, that the outputs of plus(m,n) and plus(n,m) will be identical for every m and n. Interestingly, we may also define times := proc(m,n) if n = zero then return zero else return plus(times(m, pred(n)),m) end_if end_proc: and then it is possible to prove all the associative, commutative, and distributive laws will hold. One can even define power := proc(m,n) if n = zero then return succ(zero) else return times(power(m, pred(n)),m) end_if end_proc: and then one gets the familiar laws of exponents. But one can alternatively define power := proc(m,n) if m = zero then return zero else_if n = zero then return succ(zero) else return times(power(m, pred(n)),m) end_if end_proc: There's really nothing wrong with this, I mean, it's a perfectly valid function and it even satisfies some identities like power(m,plus(n,k))=times(power(m,n),power(m,k)) (I mean, you can _prove_ that this will hold for all m,n,k .) It has the property that it effectively defines 0^0 = 0 , which some people take to be true and others take to be false. I don't know how you can unambiguously discern which is the real nature of exponentiation and so to determine whether 0^0 should equal 0 or 1. It is, when you come right down to it, an arbitrary definition. We all agree that the axioms a^0 = 1 (for a>0) and a^(n+1) = a . a^n are useful properties for an exponentiation function, and so those are built in to the two definitions of power( ) . Apart from the nature of 0^0, though, the axioms (i.e. the procedures defined here) unambiguously determine the nature of + , *, and ^ on the set of natural numbers -- whatever you consider the natural numebrs to BE, you will find that if you accept the Peano axioms and the axioms for + * ^ here, then your theorems about N will agree with those of anyone else who has a different model for N but agrees to the same axioms, always with the exception of results concerning 0^0 . It all just comes from these completely unknown calls to zero and succ(). (In practice, I think mathematicians nearly always opt for the first definition of power() for natural numbers.) dave === Subject: Re: Cardinality question >>>You know, after 35+ years as a programmer, primarily in OOP, I cannot >>>help but try and translate mathtalk into programming terms I am familiar >>>with. All I hear you saying is that succ() is a method called by a >>>recursive method, but without specifying the content of the succ() method. > Aha! A means to communicate! > The whole axiomatic method is really just a specification of some > aspects of the behaviour of an external call within a program. > We can make, for example, a program which computes m + n for two > natural numbers m and n, which relies only on the two things > mentioned in the Peano axioms: zero and succ() . Pardon my programming > skills (which are nearly nonexistent) but I think you might get > the picture here if I write this in pseudo-code (a la Maple): [...] Nicely done, Dave. I could follow it, even though I don't know Maple as such. Shows that Maple is a transparent language. NB that pseudo-code could be compiled with a suitable compiler. In fact, I think it's fairly obvious any high-level programming language is a pseudo-code until a compiler is written for it. Ie, pseudo-code means language for which we haven't written a compiler yet. IMO, this is relevant for AI (I stumbled into sci.math via other people's crosspostings to CAP): a lot of AI discussion seems to me to be really at the pseudo-code level, and formalising it at that level might result in clarifying many issues. Eg, the talk about behaviour could (IMO should) be seen as talk about procedures, tasks, and data-passing, among other things, which are nicely handled by high level languages. The modules that some people envision as handling varous perceptual task are, it seems to me, equivalent to objects, hence an object oriented pseudo-code should be helpful. There is a tendency to want to get right down to the hardware witohotu first undersdtanding the hardware's functions abstractly. Pseudo-code enables abstarction becasue it is abstraction. -- But I've probably just reinvented a wheel or two. :-) === Subject: Re: Cardinality question >>>> You know, after 35+ years as a programmer, primarily in OOP, I >>>> cannot help but try and translate mathtalk into programming terms I >>>> am familiar with. All I hear you saying is that succ() is a method >>>> called by a recursive method, but without specifying the content of >>>> the succ() method. >> Aha! A means to communicate! >> The whole axiomatic method is really just a specification of some >> aspects of the behaviour of an external call within a program. >> We can make, for example, a program which computes m + n for two >> natural numbers m and n, which relies only on the two things >> mentioned in the Peano axioms: zero and succ() . Pardon my programming >> skills (which are nearly nonexistent) but I think you might get >> the picture here if I write this in pseudo-code (a la Maple): > [...] > Nicely done, Dave. I could follow it, even though I don't know Maple as > such. Shows that Maple is a transparent language. > NB that pseudo-code could be compiled with a suitable compiler. In fact, > I think it's fairly obvious any high-level programming language is a > pseudo-code until a compiler is written for it. Ie, pseudo-code means > language for which we haven't written a compiler yet. Most pure OOP languages are not compiled because much binding cannot be done until run time. 'Compilation' is usually into a byte code rather than machine code. > IMO, this is relevant for AI (I stumbled into sci.math via other > people's crosspostings to CAP): a lot of AI discussion seems to me to be > really at the pseudo-code level, and formalising it at that level might > result in clarifying many issues. Eg, the talk about behaviour could > (IMO should) be seen as talk about procedures, tasks, and data-passing, > among other things, which are nicely handled by high level languages. > The modules that some people envision as handling varous perceptual > task are, it seems to me, equivalent to objects, hence an object > oriented pseudo-code should be helpful. There is a tendency to want to > get right down to the hardware witohotu first undersdtanding the > hardware's functions abstractly. Pseudo-code enables abstarction becasue > it is abstraction. -- But I've probably just reinvented a wheel or two. :-) -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question : Philosophically, I suppose, one would like to know what numbers _are_ : but since there seems to be no one who can tell us when we've got : the right answer, that seems an impractical enterprise even by : mathematician standards. It is sufficient to know how numbers _behave_, : and the Peano axioms -- i.e. the specifications of the external : routines zero and succ -- give us the answer we need. One can : prove, for example, that the outputs of plus(m,n) and plus(n,m) : will be identical for every m and n. Similarly, in a programming language we do not need to know how numbers are actually represented. For example, it should not affect how a program behaves if integers are represented in one's complement, two's complement or a packed decimal format. Stephen === Subject: Re: Cardinality question >: Philosophically, I suppose, one would like to know what numbers _are_ >: but since there seems to be no one who can tell us when we've got >: the right answer, that seems an impractical enterprise even by >: mathematician standards. It is sufficient to know how numbers _behave_, >: and the Peano axioms -- i.e. the specifications of the external >: routines zero and succ -- give us the answer we need. One can >: prove, for example, that the outputs of plus(m,n) and plus(n,m) >: will be identical for every m and n. >Similarly, in a programming language we do not need to >know how numbers are actually represented. For example, >it should not affect how a program behaves if integers >are represented in one's complement, two's complement >or a packed decimal format. This is the case if you are only doing the usual arithmetic operations on the integers, and the computer can handle exactly the integers it gets. If either is not the case, it can misbehave, and this does happen. >Stephen -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Cardinality question :>: Philosophically, I suppose, one would like to know what numbers _are_ :>: but since there seems to be no one who can tell us when we've got :>: the right answer, that seems an impractical enterprise even by :>: mathematician standards. It is sufficient to know how numbers _behave_, :>: and the Peano axioms -- i.e. the specifications of the external :>: routines zero and succ -- give us the answer we need. One can :>: prove, for example, that the outputs of plus(m,n) and plus(n,m) :>: will be identical for every m and n. :>Similarly, in a programming language we do not need to :>know how numbers are actually represented. For example, :>it should not affect how a program behaves if integers :>are represented in one's complement, two's complement :>or a packed decimal format. : This is the case if you are only doing the usual arithmetic : operations on the integers, and the computer can handle : exactly the integers it gets. If either is not the case, : it can misbehave, and this does happen. I purposely restricted my comments to integers and put the word should in there just for that reason. When writing code, your code should not be platform dependent, and should not make assumptions about how integers are represented. Of course this does not prevent people from doing that, and in many cases it does not matter because the way integers are represented is fairly uniform. Regarding floating point calculations the current state of affairs is not at all ideal. It would be nice if we did not have to know about any of the details of how real numbers are stored, or how the operations work, and still be able to use them and get the results we expect in all cases. However I do not know if a practial solution exists for this problem. Stephen === Subject: Re: Cardinality question > : Philosophically, I suppose, one would like to know what numbers _are_ > : but since there seems to be no one who can tell us when we've got > : the right answer, that seems an impractical enterprise even by > : mathematician standards. It is sufficient to know how numbers _behave_, > : and the Peano axioms -- i.e. the specifications of the external > : routines zero and succ -- give us the answer we need. One can > : prove, for example, that the outputs of plus(m,n) and plus(n,m) > : will be identical for every m and n. > Similarly, in a programming language we do not need to > know how numbers are actually represented. For example, > it should not affect how a program behaves if integers > are represented in one's complement, two's complement > or a packed decimal format. > Stephen Well, if I remember some of my machine-language course correctly, the representation of the numbers can affect bounds and precision of calculations, since representation is closely tied to the hardware. IOW, how a machine instantiates numbers (or other objects) in hardware will have an effect of just what can be done with them***. That's why Mueckenheim keeps claiming that only numbers which can be represented in hardware of some sort are real. *** One effect of this is that when large (and later, very large) hard drives appeared, some older BIOSs and/or OSs couldn't correctly calculate their sizes, since they didn't have enough space to represent the larger numbers. Some even reported negative capacities. === Subject: Re: Cardinality question >> : Philosophically, I suppose, one would like to know what numbers _are_ >> : but since there seems to be no one who can tell us when we've got >> : the right answer, that seems an impractical enterprise even by : >> mathematician standards. It is sufficient to know how numbers _behave_, >> : and the Peano axioms -- i.e. the specifications of the external >> : routines zero and succ -- give us the answer we need. One can >> : prove, for example, that the outputs of plus(m,n) and plus(n,m) >> : will be identical for every m and n. >> Similarly, in a programming language we do not need to >> know how numbers are actually represented. For example, >> it should not affect how a program behaves if integers >> are represented in one's complement, two's complement >> or a packed decimal format. >> Stephen > Well, if I remember some of my machine-language course correctly, the > representation of the numbers can affect bounds and precision of > calculations, since representation is closely tied to the hardware. IOW, > how a machine instantiates numbers (or other objects) in hardware will > have an effect of just what can be done with them***. That's why > Mueckenheim keeps claiming that only numbers which can be represented in > hardware of some sort are real. > *** One effect of this is that when large (and later, very large) hard > drives appeared, some older BIOSs and/or OSs couldn't correctly > calculate their sizes, since they didn't have enough space to represent > the larger numbers. Some even reported negative capacities. One interesting concept in the early Smalltalks is that all numbers were represented as integers and fractions (numerator and denominator as a pair of integers) and software, rather than hardware, representation of integers that had no size limit other than available memory. This had the advantage of being very precise. It had the disadvantage of being very slow. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question > :> :> Where did I ever claim that? I do not talk about truth. > :> :> I leave that for the philosophers. > :> > :> : I can find the post, if you really want, where you claimed that axioms > : were > :> : necessarily true ... > :> > :> They are true in the sense that they are assumed to be > :> true. They are not necessarily true in the sense that > :> they match something in the real world. > : Nothing in the real world is necessarily true in the strong logical sense. > : At any rate, YOU were the one who used that phrase, so I'll take your > : correction to heart. > I did not claim the axioms were necessarily true. Please > produce the post in which I said that, or retract your claim. Not going to reply to the rest of it now (I have no time) but I will stand corrected. You did not use that phrase. You DID say true by definition which I translated to necessarily true, since those two phrases are generally co-joined. But you never did say necessarily true. (Basically this is the case because something that is true by definition can never be false.) === Subject: Re: Cardinality question :> :> :> Where did I ever claim that? I do not talk about truth. :> :> :> I leave that for the philosophers. :> :> :> :> : I can find the post, if you really want, where you claimed that : axioms :> : were :> :> : necessarily true ... :> :> :> :> They are true in the sense that they are assumed to be :> :> true. They are not necessarily true in the sense that :> :> they match something in the real world. :> : Nothing in the real world is necessarily true in the strong logical : sense. :> : At any rate, YOU were the one who used that phrase, so I'll take your :> : correction to heart. :> I did not claim the axioms were necessarily true. Please :> produce the post in which I said that, or retract your claim. : Not going to reply to the rest of it now (I have no time) but I will stand : corrected. You did not use that phrase. You DID say true by definition : which I translated to necessarily true, since those two phrases are : generally co-joined. But you never did say necessarily true. : (Basically this is the case because something that is true by definition can : never be false.) Something that is true by definition is only true as long as you are using those definitions. The question was about the implications of Hilbert's axioms, and in that context, the axioms are true, by definition. Otherwise you are not talking about the implications of Hilbert's axioms, but something else. Stephen === Subject: Re: Cardinality question stephen@nomail.com said: > :> > :> :> :> Where did I ever claim that? I do not talk about truth. > :> :> :> I leave that for the philosophers. > :> :> > :> :> : I can find the post, if you really want, where you claimed that > : axioms > :> : were > :> :> : necessarily true ... > :> :> > :> :> They are true in the sense that they are assumed to be > :> :> true. They are not necessarily true in the sense that > :> :> they match something in the real world. > :> > :> : Nothing in the real world is necessarily true in the strong logical > : sense. > :> : At any rate, YOU were the one who used that phrase, so I'll take your > :> : correction to heart. > :> > :> I did not claim the axioms were necessarily true. Please > :> produce the post in which I said that, or retract your claim. > : Not going to reply to the rest of it now (I have no time) but I will stand > : corrected. You did not use that phrase. You DID say true by definition > : which I translated to necessarily true, since those two phrases are > : generally co-joined. But you never did say necessarily true. > : (Basically this is the case because something that is true by definition can > : never be false.) > Something that is true by definition is only true as long > as you are using those definitions. The question was about > the implications of Hilbert's axioms, and in that context, > the axioms are true, by definition. Otherwise you are not talking > about the implications of Hilbert's axioms, but something else. > Stephen Hilbert space is only in three dimensions, right? Otherwise postulate 1.7 is not true. Two planes can intersect at a single point in 4 or more dimensions. -- Smiles, Tony === Subject: Re: Cardinality question [...] > Hilbert space is only in three dimensions, right? Otherwise postulate 1.7 > is not true. Two planes can intersect at a single point in 4 or more > dimensions. for which the Hilbert axioms of geometry hold. In fact Hilbert space means something else (at least to mathematicians) - it is a term from functional analysis. Here is a not atypical definition: Let (H,<>) be an inner product space and let ||*|| be its associated norm. If (H,||*||) is a complete normed space, then we shall say that (H,<>) is a Hilbert space. Incidentally, it is said that David Hilbert once asked a colleage what a Hilbert space was and when he was told he remarked 'Oh, is that all.' If you have studied linear algebra - i.e. vector space - then you are probably familiar with the notion of the dot product of two vectors. An inner product is just a generalized version of the dot product. In other words the dot product is just a particular variety of inner product. The inner product of two vectors u and v is commonly written as . This is the source of the <> portion of (H,<>) above. So, (H,<>) just means some vector space H with a given inner product <>, and it is called an inner product space. Capice? Now, in physics - you've had some physics, right? - we are taught that a vector is a quantity that has both a magnitude and a direction, and when we represent vectors graphicaly we make the length of the vector proportional to its magnitude. The norm of a vector generalizes the notion of its length, size, or magnitude. The norm of a vector v is commonly written as ||v||. This is the source of the ||*|| portion of (H,||*||) above. So (H,||*||) just means some vector space H with a given norm ||*||, and it is called a normed space. You with me so far? Next up, we need to understand what is meant by the associated norm of an inner product space. The basic idea is simple - given an inner product space (H,<>) we can always define a function ||*|| in terms of the inner product <> in a canonical way such that ||*|| is in fact a norm. Hence every inner product space has a canonical associated norm that turns it into a normed space. (The converse, by the way, is not true.) Let's cut to the chase, let's just define the norm (aka length, magnitude) of a vector v in an inner product space (H,<>) as: ^(1/2). That is we compute the inner product of v with itself, take the square root of the result, and call that the norm (length, magnitude) of v. This function, ^(1/2), is guaranteed to meet all the criteria of a norm so long as <> meets all the criteria of an inner product. So if you imagine your plain old vanilla 2-D Euclidean vector space, where the inner product is defined as the plain old vanilla dot product: (x,y) DOT (u, v) = x*u + y*v then the canonical norm ^(1/2) recovers the Euclidean length of the vector v, where v = (x,y), = x*x + y*y, and ^(1/2) aka ||v|| is by definition sqrt(x*x + y*y). Of course all these definitions, generalizations, and machinery have been deveoloped for greater things than just recovering classical results, but it should be an aid to intuition (the growth of mathematical maturity) to see that the familiar notions of dot product and magnitude lead to familiar results as special cases of the more general concepts of inner product and norm. Does that make sense, Tony? We are almost there - almost have arrived at what the definition of Hilbert space states. We just need one more notion - the notion of a complete normed space. Every inner product space has an associated norm that makes it a normed space, but that is not quite enough to make it a Hilbert space - one more hurdle to clear: it has to be a complete normed space before it can wear the jacket of team Hilbert. A normed space (H,||*||) is complete if every Cauchy sequence of vectors in H converges to a vector in H, where a sequence of vectors {v_sub_k} is a Cauchy sequence if, given any epsilon > 0, there is an integer K such that ||v_sub_j - v_sub_i|| < epsilon whenever i and j are both >= K. Got that? It just means that a sequence is a Cauchy sequence if elements in the sequence become arbitrarily close the farther out in the sequence we go. And if every such sequence has a limit that is in the vector space H then H is complete, under the given norm ||*||. Simple, no? OK, maybe you might have to mull that over a bit before you get it. But that is what a mathematician means - all the forgoing text, rolled into a simple compact label: Hilbert space. By now, Tony, you have probably surmised that Hilbert space, in the forgoing sense, is certainly not limited to 3 dimensions. In fact most of the Hilbert spaces of interest in functional analysis are infinite dimensional spaces - function spaces. An example of an infinite dimensional Hilbert space is L^2, the set of square summable functions, all functions f: R -> R such that the integral of f^2 over the whole real line is finite, with the inner product defined as the IntegralOf(-inf, inf, f(x)g(x) dx). So the term Hilbert space packs a lot of punch, but basicaly it is just a vector space with some added structure. Why single out this particular conjunction of attributes? Cuz it's useful, of course. Heard of JPEG? That is at its core just applied Hilbert space. How about active noise engine noise in the cockpits of aircraft. The noise from the engine is well modeled as a square summable function, of time; dead silence is also a square summable function - so if engine noise is the function e, and such that e + c = s. So monitor the engine noise (function e) and create silence by blasting more noise c into the cockpit! It works. Applied Hilbert space. My own interests lie in artificial vision systems, and in that realm the decomposition of an image into the sum of waves is a central enabling technology - in unison now: applied Hilbert space. But that is not what you were asking. I was going to say something about your assertion about the intersection of planes in 4-d space, but I'll save it for later. Fair enough? === Subject: Re: Cardinality question > .... The noise from the engine is well modeled as a square summable > function... Did I say that? Nah, I could not have, could I? What was I thinking? Probably was thinking that engine noise has finite power and is well modeled still applied functional analysis, still an example of the motivation behind the particular aggregate of attributes called Hilbert space.., === Subject: Re: Cardinality question > Hilbert space is only in three dimensions, right? Otherwise postulate 1.7 is > not true. Two planes can intersect at a single point in 4 or more dimensions. Correct. That is quite so. He completed what Euclid started. Bob Kolker === Subject: Re: Cardinality question stephen@nomail.com said: > :> > :> :> :> Where did I ever claim that? I do not talk about truth. > :> :> :> I leave that for the philosophers. > :> :> > :> :> : I can find the post, if you really want, where you claimed that > : axioms > :> : were > :> :> : necessarily true ... > :> :> > :> :> They are true in the sense that they are assumed to be > :> :> true. They are not necessarily true in the sense that > :> :> they match something in the real world. > :> > :> : Nothing in the real world is necessarily true in the strong logical > : sense. > :> : At any rate, YOU were the one who used that phrase, so I'll take your > :> : correction to heart. > :> > :> I did not claim the axioms were necessarily true. Please > :> produce the post in which I said that, or retract your claim. > : Not going to reply to the rest of it now (I have no time) but I will stand > : corrected. You did not use that phrase. You DID say true by definition > : which I translated to necessarily true, since those two phrases are > : generally co-joined. But you never did say necessarily true. > : (Basically this is the case because something that is true by definition can > : never be false.) > Something that is true by definition is only true as long > as you are using those definitions. The question was about > the implications of Hilbert's axioms, and in that context, > the axioms are true, by definition. Otherwise you are not talking > about the implications of Hilbert's axioms, but something else. > Stephen I agree. Deifnitions are like axioms: necessary, but not conclusive. They need to be justified, and if definitions themselves create inconsistencies, then they are stated poorly, and need reworking. It's very like the revising of scientific theories as the result of new experimental evidence that contradicts them. The empirical test of mathematics is in its consistency with other fields of mathematics, and with science. So far, I like Hilbert's axioms, though I haven't analyzed and generalized them all yet. Postulate 1.7 is correct only in 3 dimensions. I have yet to get through most of them in any careful way. -- Smiles, Tony === Subject: Re: Cardinality question > I agree. Deifnitions are like axioms: necessary, but not conclusive. They need > to be justified, and if definitions themselves create inconsistencies, then > they are stated poorly, and need reworking. It's very like the revising of > scientific theories as the result of new experimental evidence that contradicts > them. The empirical test of mathematics is in its consistency with other > fields of mathematics, and with science. A mathematical system might have an aesthetic appeal with no particular connection to other systems or physics. Think of impressionism or Pollock's drippings. There is rational basis for taste. People like what they like. > So far, I like Hilbert's axioms, though I haven't analyzed and generalized them > all yet. Postulate 1.7 is correct only in 3 dimensions. I have yet to get > through most of them in any careful way. And well you should. Hilbert perfected Euclids axiomatization of three dimenensional space. Hilbert made explicit several things Euclid took for granted without making them plain in the axioms. Bob Kolker === Subject: Re: Cardinality question >> I agree. Deifnitions are like axioms: necessary, but not conclusive. >> They need to be justified, and if definitions themselves create >> inconsistencies, then they are stated poorly, and need reworking. It's >> very like the revising of scientific theories as the result of new >> experimental evidence that contradicts them. The empirical test of >> mathematics is in its consistency with other fields of mathematics, >> and with science. > A mathematical system might have an aesthetic appeal with no particular > connection to other systems or physics. Think of impressionism or > Pollock's drippings. There is rational basis for taste. People like what > they like. >> So far, I like Hilbert's axioms, though I haven't analyzed and >> generalized them all yet. Postulate 1.7 is correct only in 3 >> dimensions. I have yet to get through most of them in any careful way. > And well you should. Hilbert perfected Euclids axiomatization of three > dimenensional space. Hilbert made explicit several things Euclid took > for granted without making them plain in the axioms. Hilbert simply opened the door to absurdities not possible with Euclid's axioms. However, he made mathematics much more /interesting/ to other mathematicians. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question >> It's much easier to win an argument when you ignore the replies >> of others and answer your own questions, isn't it? >I wouldn't use win or lose in describing my interchanges >with you. Wow, is that how they do it over there in the other newsgroups? Silly me, an old sci.math academic, thinking that the point of internet exchanges was to actually learn something, not to win. fsbr (<- Oops. That's what happens to dave when you're too tired to notice where your fingers are.) (PS -- speakers of Russian might enjoy noting that mail=nauk under the same equivalence relation!) === Subject: Re: Cardinality question >>>It's much easier to win an argument when you ignore the replies >>>of others and answer your own questions, isn't it? >>I wouldn't use win or lose in describing my interchanges >>with you. > Wow, is that how they do it over there in the other newsgroups? Which other newsgroups? > Silly me, an old sci.math academic, thinking that the point of > internet exchanges was to actually learn something, not to win. pulling teeth. Most of the posters who reply to me from sci.math are unnaturally contentious and insulting, so, yes, an element of win/lose almost inevitably enters in. But it would be a gross mistake to assume that the blame is all on 'other newsgroups'. I assume that you are not reading all posts, so for nice clean example of what I am talking about, read the replies of Guenther Von Krackpot. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question >> Wow, is that how they do it over there in the other newsgroups? >Which other newsgroups? I'm pretty sure TB allows you to view all the headers of your post before it leaves your machine for the wilds of Usenet. One of the headers of your most recent message is > Most of the posters who reply to me from sci.math > are unnaturally contentious and insulting, Yes but we love them just the way they are. It's the CAP and SPM posters I sometimes can't figure out. (There's no reason mathematicians' people skills need to be different from anyone else's, but as for the _professional content_ of their messages, it's usually quite clear whose statements are correct and whose aren't. That's the nice thing about mathematical proof. Non-mathematicians dislike the dry, abstract tenor of the conversation, but there are these rigid definitions and clear axioms that make everything either true or false. Well, there are subtleties behind that claim, which would require a whole course in Mathematical Logic to clarify, but my statement should be Good Enough For Gummint Work.) dave === Subject: Re: Cardinality question >>>Wow, is that how they do it over there in the other newsgroups? >>Which other newsgroups? > I'm pretty sure TB allows you to view all the headers of your post > before it leaves your machine for the wilds of Usenet. One of the > headers of your most recent message is >>Most of the posters who reply to me from sci.math >>are unnaturally contentious and insulting, > Yes but we love them just the way they are. Forewarned is forearmed. > It's the CAP and SPM > posters I sometimes can't figure out. > (There's no reason mathematicians' people skills need to be different > from anyone else's, but as for the _professional content_ of their > messages, it's usually quite clear whose statements are correct and > whose aren't. For other mathematicians. > That's the nice thing about mathematical proof. > Non-mathematicians dislike the dry, abstract tenor of the conversation, > but there are these rigid definitions and clear axioms that make > everything either true or false. That hasn't been my experience. > Well, there are subtleties behind > that claim, which would require a whole course in Mathematical Logic > to clarify, but my statement should be Good Enough For Gummint Work.) -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question > Wolf Kirchmeir said: [...] >>>Imagine adding up a sequence of numbers. Now keep adding them. And imagine >>>that you are adding all members of an infinite set. I have no concept of >>>limit, but I now know that I can have an infinite sum. But I don't have an >>>answer. Limit gives me one. >>Allan, Allan, you've forgotten your high school math here. Tsk, tsk. You >>can do much better than this, I think. A typo of the mind, I guess. >>An infinite sum is not a limit. The sum of an infinite series may be a >>limit - if the series converges. If it doesn't, then the sum has no >>limit. Some infinite series ahve indefinite sums - eg, sums that depend >>on whether you have summed an odd or even number of terms. Etc. > Wolf, I think his point is clear, even if his terminology isn't. If you sum an > infinity of 1's you get an infinite sum. True, but he seems to conflate that with the sum of a converging series. He appears to be using limit to mean some arbitrary ending of the summing. === Subject: Re: Cardinality question Wolf Kirchmeir said: > > Wolf Kirchmeir said: > > > [...] > >>>Imagine adding up a sequence of numbers. Now keep adding them. And imagine > >>>that you are adding all members of an infinite set. I have no concept of > >>>limit, but I now know that I can have an infinite sum. But I don't have an > >>>answer. Limit gives me one. > >> > >>Allan, Allan, you've forgotten your high school math here. Tsk, tsk. You > >>can do much better than this, I think. A typo of the mind, I guess. > >> > >>An infinite sum is not a limit. The sum of an infinite series may be a > >>limit - if the series converges. If it doesn't, then the sum has no > >>limit. Some infinite series ahve indefinite sums - eg, sums that depend > >>on whether you have summed an odd or even number of terms. Etc. > >> > > > > Wolf, I think his point is clear, even if his terminology isn't. If you sum an > > infinity of 1's you get an infinite sum. > True, but he seems to conflate that with the sum of a converging series. > He appears to be using limit to mean some arbitrary ending of the > summing. Yes, well, I guess what he means is more of an integral, when speaking of sums of infinite numbers, than a limit. Is the point that an infinite set of natural numbers must contain infinite numbers, since the last members are counts of infinite 1's? If so, then I am compelled to agree. -- Smiles, Tony === Subject: Re: Cardinality question > Wolf Kirchmeir said: >>>Wolf, I think [Allan's] point is clear, even if his terminology isn't. If you sum an >>>infinity of 1's you get an infinite sum. >>True, but [Allan] seems to conflate that with the sum of a converging series. >>He appears to be using limit to mean some arbitrary ending of the >>summing. > Yes, well, I guess what [Allan] means is more of an integral, when speaking of sums > of infinite numbers, than a limit. Is the point that an infinite set of natural > numbers must contain infinite numbers, since the last members are counts of > infinite 1's? If so, then I am compelled to agree. An integral is a limit - see its standard definition in any calculus text. There are no last members of an infinite set - if there is a last member, then the set is finite. I don't understand the reference to counts of infinite 1's. As written, it's nonsense: 1s aren't infinite. (BTW, don't use an apostrophe to denote a plural - that's me as English teacher speaking :-)). === Subject: Re: Cardinality question Wolf Kirchmeir said: > > Wolf Kirchmeir said: > > > >>>Wolf, I think [Allan's] point is clear, even if his terminology isn't. If you sum an > >>>infinity of 1's you get an infinite sum. > >> > >> > >>True, but [Allan] seems to conflate that with the sum of a converging series. > >>He appears to be using limit to mean some arbitrary ending of the > >>summing. > >> > > > > Yes, well, I guess what [Allan] means is more of an integral, when speaking of sums > > of infinite numbers, than a limit. Is the point that an infinite set of natural > > numbers must contain infinite numbers, since the last members are counts of > > infinite 1's? If so, then I am compelled to agree. > An integral is a limit - see its standard definition in any calculus text. Okay. The limit is corect as far as I can see. > There are no last members of an infinite set - if there is a last > member, then the set is finite. That's not necessarily true. You may be able to define a first and last, but not all the middle, but that's for another day, after we have established some sanity in the set arena. In any case, Allan's point is that an infinite set of naturals includes elements which are the result of infinite numbers of incrementations, since we always increment both the set size and the largest value in the set, equally, as we add each natural to the set during the enumeration. The result of an infinite number of incrementations of 1 is infinite. > I don't understand the reference to counts of infinite 1's. As > written, it's nonsense: 1s aren't infinite. counting by 1's an infinite number of times. > (BTW, don't use an apostrophe to denote a plural - that's me as English > teacher speaking :-)). Yeah, well if it were an English paper, I would say ones. 1s looks too much like Is. So I'll continue using apostrophes in this case. And don't ever put a preposition at the end of a sentence. That is something up with which I absolutely will not put! English is a mutt language and deserves an occasional smack. -- Smiles, Tony === Subject: Re: Cardinality question > That's not necessarily true. You may be able to define a first and last, but > not all the middle, but that's for another day, after we have established some > sanity in the set arena. Define sanity. I propose that a theory of sets is sane if and only if it is consistent and is sufficient to ground the theory of fuctions of a real or complet variable. > In any case, Allan's point is that an infinite set of naturals includes > elements which are the result of infinite numbers of incrementations, since we > always increment both the set size and the largest value in the set, equally, > as we add each natural to the set during the enumeration. The result of an > infinite number of incrementations of 1 is infinite. But each natural number is gotten by a finite sequence of successions from 1 or 0 (depending on where you start). So while each natural is finite, there is no largest natural so the -set- of natural numbers is infinite. Bob Kolker === Subject: Re: Cardinality question >> Wolf Kirchmeir said: >>>>Wolf, I think [Allan's] point is clear, even if his terminology isn't. If you sum an >>>>infinity of 1's you get an infinite sum. >>>True, but [Allan] seems to conflate that with the sum of a converging series. >>>He appears to be using limit to mean some arbitrary ending of the >>>summing. >> Yes, well, I guess what [Allan] means is more of an integral, when speaking of sums >> of infinite numbers, than a limit. Is the point that an infinite set of natural >> numbers must contain infinite numbers, since the last members are counts of >> infinite 1's? If so, then I am compelled to agree. >An integral is a limit - see its standard definition in any calculus text. >There are no last members of an infinite set - if there is a last >member, then the set is finite. >I don't understand the reference to counts of infinite 1's. As >written, it's nonsense: 1s aren't infinite. >(BTW, don't use an apostrophe to denote a plural - that's me as English >teacher speaking :-)). I don't agree with this last comment, Wolf, in regard to unlettered text. 1's is preferable to 1s in my estimation although I can't, as usual, cite authority on the subject. === Subject: Re: Cardinality question > (BTW, don't use an apostrophe to denote a plural - that's me as English > teacher speaking :-)). Are you really an English teacher, Wolf? HS or college level? -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question <427172ad$1_1@newsfeed.slurp.net> <42727e73.3546744@netnews.att.net> <4272b1b4.10436838@netnews.att.net> <77Lde.5462$3U.448647@news20.bellglobal.com> > > > (BTW, don't use an apostrophe to denote a plural - that's me as English > > teacher speaking :-)). > Are you really an English teacher, Wolf? HS or college level? Does it matter? This is an elementary rule, probably taught by 8th grade. I'm NOT an English teacher, but I have my grammatical peeves also, such as confusion between it's and its. - Randy === Subject: Re: Cardinality question >> >>>(BTW, don't use an apostrophe to denote a plural - that's me as > English >>>teacher speaking :-)). >>Are you really an English teacher, Wolf? HS or college level? > Does it matter? This is an elementary rule, probably > taught by 8th grade. > I'm NOT an English teacher, but I have my grammatical > peeves also, such as confusion between it's and its. > - Randy For the record: I taught English language (history of, etc) and literature at university level in Alberta and Ontario. (US college could include what we mean by senior high school in Canada, depending on the State and Province.) I taught English language and literature at the high school level in Ontario (which at one time included what is 1st year college in some States.) I taught computer studies at the high school level. I am qualified to teach math at the high school level, but never did. I included a unit on symbolic logic in all senior high school English courses I taught, and always gave formal grammar lessons, even when they weren't, er, encouraged. I have two years of engineering, a year of history, a year of philosophy, an honours BA and MA in English. OI stratedon a Ph D program, but the need to make a living and help my wife support our family put an end to that. I have credits in three computer binary code and filled in rectangles on cards that then went through a card reader which produced a punched tape. I taught myself BASIC, Comal, and Fortran (and have forgotten 90%.) I speak, read, and write German almost as well as English, and can read French and Spanish reasonably well. I read about 75-100 books a year, about 1/3rd fiction, the rest current on are neurology, physics (pure and applied), genetics, linguistics, AI, computers (I build my own), railroads, economics/politics/history, psychology and gardening. My attitude has always been that if you can understand Shakespeare, you can understand anything, and I have found that to be mostly true -- that is, while I have limits, I know what they are, and don't fret when I run into them. When I meet someone and get a chance to find out what they do for a living, I ask questions - people are always interesting when they talk about their work, and I always learn something I didn't know before. I have been married to the same woman for 45 years, have three children and six grandchildren, and more friends and acquaintances than I can list, including, I am gratified to say, many of my former students. I try to be charitable and kind, but when I am attacked repeatedly, I counterattack at least as viciously as I was attacked. I am in good health, live in one of the best countries in the world in one of its loveliest and most interesting landscapes, and am able to grow roses that survive our winters. I never learned to type, and don't always use a spell checker, as many of you have no doubt noticed. Life is good. === Subject: Re: Cardinality question > For the record: of what I inferred. I too have been married to the same woman, (not the same as your wife) for 42 years and have 6 grandchildren. We have many of the same interests. What a shame usenet threads always deteriorate to insults. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question >> >>>(BTW, don't use an apostrophe to denote a plural - that's me as > English >>>teacher speaking :-)). >>Are you really an English teacher, Wolf? HS or college level? > Does it matter? Yes, to me. I think the teaching of English is a fine occupation which has my utmost respect. > This is an elementary rule, probably > taught by 8th grade. > I'm NOT an English teacher, but I have my grammatical > peeves also, such as confusion between it's and its. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question <42727e73.3546744@netnews.att.net> <4272b1b4.10436838@netnews.att.net> <77Lde.5462$3U.448647@news20.bellglobal.com> > > I'm NOT an English teacher, but I have my grammatical > > peeves also, such as confusion between it's and its. Your welcome. Grammar is it's own reward. Its nice to find someone who appreciates it's niceties. - Randy === Subject: Re: Cardinality question > Your welcome. Grammar is it's own reward. You mean you're. === Subject: Re: Cardinality question <4272b1b4.10436838@netnews.att.net> <77Lde.5462$3U.448647@news20.bellglobal.com> > > Your welcome. Grammar is it's own reward. > You mean you're. I tried to figure out a way to work in looser for loser but couldn't. - Randy === Subject: Re: Cardinality question Yes, very sloppy reading on my part. === Subject: Re: Cardinality question <426aa1c6.21541974@netnews.att.net> <426d7608$1_3@newsfeed.slurp.net> <426e94fb.48673528@netnews.att.net> <426eb55c_3@newsfeed.slurp.net> <426fac60.65196758@netnews.att.net> <426ff58d$1_4@newsfeed.slurp.net> <427136e4.78743819@netnews.att.net> <427172ad$1_1@newsfeed.slurp.net> <42727e73.3546744@netnews.att.net> <4272b1b4.10436838@netnews.att.net> <1Bdde.563$3U.121396@news20.bellglobal.com> > > > According to the mathematical view. When you talk about the > > value of an infinite sum, that value is a number (if it exists). > > The value of that number is the limit of the sequence of > > partial sums. That is the definition that those of us who > > have had formal training in limits have seen. It is the only > > definition that I am aware of. This is not a personal opinion > > I derived. I do not know another meaning for an infinite sum. > > If you ask what value it has, the only way I know to assign it > > a value is the value of the limit. > This is the saddest thing I have ever read. A mind is a terrible > thing to waste. Poor idiot wagner. If this is the saddest thing you have ever read then you are in dire need of two things. An education, and more importantly, a life. Poor idiot wagner, your incompetence is dwarfed by the sordidness of your being. Poor idiot wagner. === Subject: Re: Cardinality question Lester Zick said: > >Wolf Kirchmeir said: > >> [...] > >> >>You are apparently in the camp that believes that the infinite > >> >>sums can be discovered separately from their definition > >> >>as limits of sequences of partial sums. > >> > > >> > > >> > I am? That's nice to know [grin]. > >> > > >> > > >> >>That there is a separate > >> >>true meaning and that the sequence of partial sums is just > >> >>a calculation method. > >> >> > >> >>What is that other meaning? > >> > > >> > > >> > I can show how they can be discovered without limits: > >> > > >> > Imagine adding up a sequence of numbers. Now keep adding them. And imagine > >> > that you are adding all members of an infinite set. I have no concept of > >> > limit, but I now know that I can have an infinite sum. But I don't have an > >> > answer. Limit gives me one. > >> > >> Allan, Allan, you've forgotten your high school math here. Tsk, tsk. You > >> can do much better than this, I think. A typo of the mind, I guess. > >> > >> An infinite sum is not a limit. The sum of an infinite series may be a > >> limit - if the series converges. If it doesn't, then the sum has no > >> limit. Some infinite series ahve indefinite sums - eg, sums that depend > >> on whether you have summed an odd or even number of terms. Etc. > >> > >Wolf, I think his point is clear, even if his terminology isn't. If you sum an > >infinity of 1's you get an infinite sum. > Hey Tony - It's great to know you're still out there. At some point I figured I got labeled as a crank and wasn't getting any responses from anyone on anything. I had to bow out for a while. Got into a long offline math discussion regarding Cantor and cardinality, and developed an alternative system, and found that mathematicians need axioms to work with, I guess, so now I'm axiomatizing it. You'll love the part where it defines spaces as sets of points. ;) Nice to see you're still in there slugging. Rock on! -- Smiles, Tony === Subject: Re: Cardinality question >Lester Zick said: >> >Wolf Kirchmeir said: >> >> [...] >> >> >>You are apparently in the camp that believes that the infinite >> >> >>sums can be discovered separately from their definition >> >> >>as limits of sequences of partial sums. >> >> > >> >> > >> >> > I am? That's nice to know [grin]. >> >> > >> >> > >> >> >>That there is a separate >> >> >>true meaning and that the sequence of partial sums is just >> >> >>a calculation method. >> >> >> >> >> >>What is that other meaning? >> >> > >> >> > >> >> > I can show how they can be discovered without limits: >> >> > >> >> > Imagine adding up a sequence of numbers. Now keep adding them. And imagine >> >> > that you are adding all members of an infinite set. I have no concept of >> >> > limit, but I now know that I can have an infinite sum. But I don't have an >> >> > answer. Limit gives me one. >> >> >> >> Allan, Allan, you've forgotten your high school math here. Tsk, tsk. You >> >> can do much better than this, I think. A typo of the mind, I guess. >> >> >> >> An infinite sum is not a limit. The sum of an infinite series may be a >> >> limit - if the series converges. If it doesn't, then the sum has no >> >> limit. Some infinite series ahve indefinite sums - eg, sums that depend >> >> on whether you have summed an odd or even number of terms. Etc. >> >> >> >Wolf, I think his point is clear, even if his terminology isn't. If you sum an >> >infinity of 1's you get an infinite sum. >> Hey Tony - It's great to know you're still out there. >At some point I figured I got labeled as a crank and wasn't getting any >responses from anyone on anything. I had to bow out for a while. Yeah, tell me about it, Tony. > Got into a >long offline math discussion regarding Cantor and cardinality, and developed an >alternative system, and found that mathematicians need axioms to work with, I >guess, so now I'm axiomatizing it. You'll love the part where it defines spaces >as sets of points. ;) Maybe you missed my definition of universal truth as the set of all points which are universally true. Albert got a kick out of it. >Nice to see you're still in there slugging. Rock on! Yeah, I expect my general approach is pretty much fixed in stone. === Subject: Re: Cardinality question >> >> No these are axioms by which you define suc( ). >> > >> >No they aren't. succ() is an arbitrary name. It isn't >> >defined here. This doesn't say what the successor operation >> >is, it just says we have an operation called successor >> >(it could be called anything) and we will define the >> >things called numbers in terms of this operation. >> Whatever. You're just deliberately misconstruing my remarks. >Not deliberately. What did I misconstrue? The idea that I was talking about suc( ) as an aribtrary name. I'm not. When I use a technical name like e or pi or suc( ) I'm referring to the thing itself and not its name. >> >> You have no idea >> >> whether they define numbers >> > >> >They don't. They define a completely abstract framework >> >in which numbers is only a convenient word to help our >> >intuition. This is what abstraction looks like. There's >> >no necessary connection between numbers and numbers. >> Yeah, a number is when we deal numerically with something we intuit >> be numbers. Real good. >You're deliberately misconstruing my remarks. No I'm exemplifying your remarks by illustrating a definition for numbers comparable to yours. You can't define numbers by defining suc( ). >> >> How do you know what you're defining isn't elephants and that >other >> >> definitions of numbers are correct instead or yours? >> > >> >Correct doesn't come into axioms, only self-consistent. >> >You have to do a lot of work, actually, to connect these >> >to numbers. But when you connect this abstract model to >> >the everyday numbers, you have number theory. >> And when you connect it to elephants you have even more. >No, you have a description of objects whose properties >bear little or no resemblance to what we know of >elephants. So they don't make a very good model >of elephants, and using the word elephants for them >serves no purpose. But using intuitive definitions for numbers is ok? >However, these numbers do have a direct connection >to our rudimentary intuitions about counting. Why do they have a direct connection to counting when there are other things besides counting which numbers refer to and describe? You're just regressing an intuitive notion of numbers to intuitive notions of counting. > They >take the entirely empirical, limited, vague thing >we grow up with and put it on a rigorous footing, >and the initial set of properties corresponds >exactly to what we all call numbers. Yeah but counting doesn't describe numbers exhaustively. Nor does suc(). Suc( ) only describes counting but doesn't even do that right because we don't start counting with zero. > So they ARE a >useful model of numbers, in a way that they aren't >a useful model of elephants. At most they're a useful model of counting. >But they're more than that. Once you established >the rigorous foundation, you can derive unexpected >things from that foundation. Technically speaking, >what you get are theorems about things defined by >the Peano axioms (which are rigorously-defined >objects) and not about intuitive numbers, which are >not rigorously-defined and have no associated theorems. >But the former is a lot more interesting and useful. That's generally true of any abstract system. It's not peculiar to numbers. >> >No, there's nothing in the above that says Peano's >> >numbers are numbers, that zero is what we call 0, >> >that successor is the same as adding 1. That's all >> >additional stuff you have to add on to make the connection. >> Yeah well it would really help to know what they're supposed to be >> connected to. >They're supposed to be connected to counting numbers, >but whether you start at 0 or 1 is completely arbitrary. No it isn't completely aribtrary if you regress Peano axioms to natural numbers and counting especially children counting. >Some authors prefer 0, some 1. There's no great confusion >created by taking either convention. All it takes is >a statement up front about your convention. It isn't a matter of convention if your regression for Peano axioms is to natural numbers and counting especially childre counting. >If you take a convention which is wildly different >from ordinary usage, such as calling them elephants >and then using symbols written in Tamil, you won't >be wrong exactly, but you'll have created confusion >in your readers for no apparent purpose. More to the point you won't have anything to do with numbers. >Rather like your unconventional usage of language, come >to think of it. My usage of language is unconventional only because there is a basic confusion of terms, and when my usage of language goes unconventional it does so only for very good reasons which are clearly explained. >> >That's a choice. Peano chose to start with 0. Nothing in >> >the above prevents you starting with 1. >> Oh I expect a lot of mathematikers would really get bent out of shape >> if you started with 1. >No, they wouldn't. So why all the fuss about starting with zero? >> >Nowhere. There's a different, but similar axiomization >> >in terms of sets, but nothing above says sets. There's >> >no single definition of set theory either, and there's >> >plenty of discussion in this newsgroup (most of which >> >I can't follow) about the different models of sets one >> >can construct by adopting different axioms. >> So why use such terms to regress your intuition regarding natural >> numbers? >Regress your intuition is a Zickian phrase that carries >no meaning for me, so I have no answer to this collection >of symbols that appears to be in the form of a question. And yet you want me to regress my terminology to yours. Go figure. === Subject: Re: Cardinality question > It isn't a matter of convention if your regression for Peano axioms is > to natural numbers and counting especially childre counting. Reression to Peano Axioms is a meaningless nonsensical phrase. If you want to talk about deduction from the Peano axioms then do so. As to the successor function suc(), suc(x) is intended to mean the smallest integer which is greater than the integer x. So suc(0) = 1 makes perfectly good sense. Zero is the cardinality of the empty set so it is a perfectly good counting number. Bob Kolker === Subject: Re: Cardinality question >>>> >>>>>No these are axioms by which you define suc( ). >>>> >>>>No they aren't. succ() is an arbitrary name. It isn't >>>>defined here. This doesn't say what the successor operation >>>>is, it just says we have an operation called successor >>>>(it could be called anything) and we will define the >>>>things called numbers in terms of this operation. >>>Whatever. You're just deliberately misconstruing my remarks. >>Not deliberately. What did I misconstrue? > The idea that I was talking about suc( ) as an aribtrary name. I'm > not. When I use a technical name like e or pi or suc( ) I'm referring > to the thing itself and not its name. >>>>>You have no idea >>>>>whether they define numbers >>>> >>>>They don't. They define a completely abstract framework >>>>in which numbers is only a convenient word to help our >>>>intuition. This is what abstraction looks like. There's >>>>no necessary connection between numbers and numbers. >>>Yeah, a number is when we deal numerically with something we intuit >>to >>>be numbers. Real good. >>You're deliberately misconstruing my remarks. > No I'm exemplifying your remarks by illustrating a definition for > numbers comparable to yours. You can't define numbers by defining > suc( ). >>>>>How do you know what you're defining isn't elephants and that >>other >>>>>definitions of numbers are correct instead or yours? >>>> >>>>Correct doesn't come into axioms, only self-consistent. >>>>You have to do a lot of work, actually, to connect these >>>>to numbers. But when you connect this abstract model to >>>>the everyday numbers, you have number theory. >>>And when you connect it to elephants you have even more. >>No, you have a description of objects whose properties >>bear little or no resemblance to what we know of >>elephants. So they don't make a very good model >>of elephants, and using the word elephants for them >>serves no purpose. > But using intuitive definitions for numbers is ok? >>However, these numbers do have a direct connection >>to our rudimentary intuitions about counting. > Why do they have a direct connection to counting when there are other > things besides counting which numbers refer to and describe? You're > just regressing an intuitive notion of numbers to intuitive notions of > counting. >> They >>take the entirely empirical, limited, vague thing >>we grow up with and put it on a rigorous footing, >>and the initial set of properties corresponds >>exactly to what we all call numbers. > Yeah but counting doesn't describe numbers exhaustively. Nor does > suc(). Suc( ) only describes counting but doesn't even do that right > because we don't start counting with zero. >> So they ARE a >>useful model of numbers, in a way that they aren't >>a useful model of elephants. > At most they're a useful model of counting. >>But they're more than that. Once you established >>the rigorous foundation, you can derive unexpected >>things from that foundation. Technically speaking, >>what you get are theorems about things defined by >>the Peano axioms (which are rigorously-defined >>objects) and not about intuitive numbers, which are >>not rigorously-defined and have no associated theorems. >>But the former is a lot more interesting and useful. > That's generally true of any abstract system. It's not peculiar to > numbers. >>>>No, there's nothing in the above that says Peano's >>>>numbers are numbers, that zero is what we call 0, >>>>that successor is the same as adding 1. That's all >>>>additional stuff you have to add on to make the connection. >>>Yeah well it would really help to know what they're supposed to be >>>connected to. >>They're supposed to be connected to counting numbers, >>but whether you start at 0 or 1 is completely arbitrary. > No it isn't completely aribtrary if you regress Peano axioms to > natural numbers and counting especially children counting. >>Some authors prefer 0, some 1. There's no great confusion >>created by taking either convention. All it takes is >>a statement up front about your convention. > It isn't a matter of convention if your regression for Peano axioms is > to natural numbers and counting especially childre counting. >>If you take a convention which is wildly different >>from ordinary usage, such as calling them elephants >>and then using symbols written in Tamil, you won't >>be wrong exactly, but you'll have created confusion >>in your readers for no apparent purpose. > More to the point you won't have anything to do with numbers. >>Rather like your unconventional usage of language, come >>to think of it. > My usage of language is unconventional only because there is a basic > confusion of terms, and when my usage of language goes unconventional > it does so only for very good reasons which are clearly explained. >>>>That's a choice. Peano chose to start with 0. Nothing in >>>>the above prevents you starting with 1. >>>Oh I expect a lot of mathematikers would really get bent out of shape >>>if you started with 1. >>No, they wouldn't. > So why all the fuss about starting with zero? >>>>Nowhere. There's a different, but similar axiomization >>>>in terms of sets, but nothing above says sets. There's >>>>no single definition of set theory either, and there's >>>>plenty of discussion in this newsgroup (most of which >>>>I can't follow) about the different models of sets one >>>>can construct by adopting different axioms. >>>So why use such terms to regress your intuition regarding natural >>>numbers? >>Regress your intuition is a Zickian phrase that carries >>no meaning for me, so I have no answer to this collection >>of symbols that appears to be in the form of a question. > And yet you want me to regress my terminology to yours. Go figure. === Subject: Re: Cardinality question <427136e4.78743819@netnews.att.net> <427172ad$1_1@newsfeed.slurp.net> <42727e73.3546744@netnews.att.net> <4272b1b4.10436838@netnews.att.net> <42739790.13821342@netnews.att.net> <4273bdf9.17952661@netnews.att.net> <4273d56c.20149109@netnews.att.net> <427519bc.30044504@netnews.att.net> <42764a8b.39897749@netnews.att.net> <42767cfa.45062163@netnews.att.net> in > >> >> No these are axioms by which you define suc( ). > >> > > >> >No they aren't. succ() is an arbitrary name. It isn't > >> >defined here. This doesn't say what the successor operation > >> >is, it just says we have an operation called successor > >> >(it could be called anything) and we will define the > >> >things called numbers in terms of this operation. > >> > >> Whatever. You're just deliberately misconstruing my remarks. > >Not deliberately. What did I misconstrue? > The idea that I was talking about suc( ) as an aribtrary name. ***I*** made the statement that succ() is an arbitrary name. What I objected to was your statement that these axioms define succ(). They don't. The reason for using the name successor is to make the obvious connection, but that it involves adding 1 or corresponds to the successor in the way we count, IS NOT DEFINED HERE. These axioms don't define successor. They just say there's something called successor, and the only thing we're going to say about it is that everything in the set of interest has one. As an alternate interpretation of exactly the same axioms, I could use 1 as the starting element, and dividing by 2 as the successor operation. My numbers are then 1, 1/2, 1/4, 1/8, ... Nothing in the axioms prevents me from making this rather confusing and pointless choice. > I'm > not. When I use a technical name like e or pi or suc( ) I'm referring > to the thing itself and not its name. Never said you were. It beats me how you could read the paragraph above and think I was attributing succ() is an arbitrary name to you. > >> Yeah, a number is when we deal numerically with something we intuit > >to > >> be numbers. Real good. > >You're deliberately misconstruing my remarks. > No I'm exemplifying your remarks by illustrating a definition for > numbers comparable to yours. You can't define numbers by defining > suc( ). I'm not defining succ(). > >No, you have a description of objects whose properties > >bear little or no resemblance to what we know of > >elephants. So they don't make a very good model > >of elephants, and using the word elephants for them > >serves no purpose. > But using intuitive definitions for numbers is ok? No, if you want something which is mathematically OK, you need axioms. Aren't we talking about the axioms I originally quoted? How is that an intuitive definition of numbers. Intuitive definitions aren't definitions in the mathematical sense. But intuitions are helpful in understanding mathematics, in making it something you truly understand rather than a mechanical set of procedures. > >However, these numbers do have a direct connection > >to our rudimentary intuitions about counting. > Why do they have a direct connection to counting when there are other > things besides counting which numbers refer to and describe? Because counting also starts with 1 (or 0), has a successor operation (adding 1) and the property that every counting number has a successor. But that's a weird question. Just because it can be a model of other things than counting, it can't be a model of counting? > You're > just regressing an intuitive notion of numbers to intuitive notions of > counting. I'm afraid I tune out every time you write a sentence with regress in it. I've never yet seen one that made any sense out of the hundreds of times you've written it. > > They > >take the entirely empirical, limited, vague thing > >we grow up with and put it on a rigorous footing, > >and the initial set of properties corresponds > >exactly to what we all call numbers. > Yeah but counting doesn't describe numbers exhaustively. Nor does > suc(). Suc( ) only describes counting but doesn't even do that right > because we don't start counting with zero. Depends on what you're doing. For almost any programming task I've ever done, counting starts at 0, which is why C's conventions are a lot more convenient for most algorithms than FORTRAN's. > > So they ARE a > >useful model of numbers, in a way that they aren't > >a useful model of elephants. > At most they're a useful model of counting. Numbers have more uses than counting. > >But they're more than that. Once you established > >the rigorous foundation, you can derive unexpected > >things from that foundation. Technically speaking, > >what you get are theorems about things defined by > >the Peano axioms (which are rigorously-defined > >objects) and not about intuitive numbers, which are > >not rigorously-defined and have no associated theorems. > >But the former is a lot more interesting and useful. > That's generally true of any abstract system. It's not peculiar to > numbers. Is that supposed to be a counter to the Peano axioms give you number theory? I'm missing why the axioms of X give you the abstract theory of X implies that the Peano axioms don't give you Peano arithmetic. Or what are you saying? What was the point of that last remark? > >They're supposed to be connected to counting numbers, > >but whether you start at 0 or 1 is completely arbitrary. > No it isn't completely aribtrary if you regressx#@$%(signal lost) Yes it is completely arbitrary. It prevents nothing in number theory. And number theory is the point. > >Some authors prefer 0, some 1. There's no great confusion > >created by taking either convention. All it takes is > >a statement up front about your convention. > It isn't a matter of convention if your regressionx$%@(signal lost) - Randy === Subject: Re: Cardinality question > ***I*** made the statement that succ() is an arbitrary > name. What I objected to was your statement that these > axioms define succ(). They don't. The reason for using > the name successor is to make the obvious connection, > but that it involves adding 1 or corresponds to the > successor in the way we count, IS NOT DEFINED HERE. > These axioms don't define successor. They just say > there's something called successor, and the only > thing we're going to say about it is that everything > in the set of interest has one. > As an alternate interpretation of exactly the same > axioms, I could use 1 as the starting element, and > dividing by 2 as the successor operation. My numbers > are then 1, 1/2, 1/4, 1/8, ... Nothing in the axioms > prevents me from making this rather confusing and > pointless choice. You know, after 35+ years as a programmer, primarily in OOP, I cannot help but try and translate mathtalk into programming terms I am familiar with. All I hear you saying is that succ() is a method called by a recursive method, but without specifying the content of the succ() method. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question Albert Wagner said: > > > ***I*** made the statement that succ() is an arbitrary > > name. What I objected to was your statement that these > > axioms define succ(). They don't. The reason for using > > the name successor is to make the obvious connection, > > but that it involves adding 1 or corresponds to the > > successor in the way we count, IS NOT DEFINED HERE. > > > > These axioms don't define successor. They just say > > there's something called successor, and the only > > thing we're going to say about it is that everything > > in the set of interest has one. > > > > As an alternate interpretation of exactly the same > > axioms, I could use 1 as the starting element, and > > dividing by 2 as the successor operation. My numbers > > are then 1, 1/2, 1/4, 1/8, ... Nothing in the axioms > > prevents me from making this rather confusing and > > pointless choice. > You know, after 35+ years as a programmer, primarily in OOP, I > cannot help but try and translate mathtalk into programming terms > I am familiar with. All I hear you saying is that succ() is a > method called by a recursive method, but without specifying the > content of the succ() method. > Psssst! Albert! Increment! Think increment! -- Smiles, Tony === Subject: Re: Cardinality question > You know, after 35+ years as a programmer, primarily in OOP, I > cannot help but try and translate mathtalk into programming terms > I am familiar with. All I hear you saying is that succ() is a > method called by a recursive method, but without specifying the > content of the succ() method. In OOP talk succ() could be seen as a virtual method (with some constraints: for example, it should never give 0 as a result). When you devirtualize it, you get a *model* of the natural numbers. -- Giuseppe Oblomov Bilotta Da grande lotter.98 per la pace A me me la compra il mio babbo (Altan) (When I grow up, I will fight for peace I'll have my daddy buy it for me) === Subject: Re: Cardinality question >>You know, after 35+ years as a programmer, primarily in OOP, I >>cannot help but try and translate mathtalk into programming terms >>I am familiar with. All I hear you saying is that succ() is a >>method called by a recursive method, but without specifying the >>content of the succ() method. > In OOP talk succ() could be seen as a virtual method (with some > constraints: for example, it should never give 0 as a result). When > you devirtualize it, you get a *model* of the natural numbers. I don't consider C++ an OOP language but rather an OOP-like language. Pure OOP languages do not have 'virtual' methods. Hence, I don't understand what you are saying. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question <10c50lomse4ya$.1mkncvhofdc3v.dlg@40tude.net> >>>You know, after 35+ years as a programmer, primarily in OOP, I >>>cannot help but try and translate mathtalk into programming terms >>>I am familiar with. All I hear you saying is that succ() is a >>>method called by a recursive method, but without specifying the >>>content of the succ() method. >> In OOP talk succ() could be seen as a virtual method (with some >> constraints: for example, it should never give 0 as a result). When >> you devirtualize it, you get a *model* of the natural numbers. > I don't consider C++ an OOP language but rather an OOP-like > language. Pure OOP languages do not have 'virtual' methods. > Hence, I don't understand what you are saying. The succ() method is simply not defined. Some of its properties are defined (like, 0 can never be returned, two distinct numbers cannot give the same value for succ() etc), but nothing more. The reason why this is so is that: if you have a set (any set you like) on which you can define a function (any function you like) such that the set and the function satisfy the Peano axioms, then that set behaves exactly like the natural numbers. It's thus possible to define on it an addition, a product, and you can map each element to a natural number in such a way that the defined operations map to the actual natural operations. -- Giuseppe Oblomov Bilotta [W]hat country can preserve its liberties, if its rulers are not warned from time to time that [the] people preserve the spirit of resistance? Let them take arms...The tree of liberty must be refreshed from time to time, with the blood of patriots and tyrants. -- Thomas Jefferson, letter to Col. William S. Smith, 1787 === Subject: Re: Cardinality question > >> ***I*** made the statement that succ() is an arbitrary >> name. What I objected to was your statement that these >> axioms define succ(). They don't. The reason for using >> the name successor is to make the obvious connection, >> but that it involves adding 1 or corresponds to the >> successor in the way we count, IS NOT DEFINED HERE. >> These axioms don't define successor. They just say >> there's something called successor, and the only >> thing we're going to say about it is that everything >> in the set of interest has one. >> As an alternate interpretation of exactly the same >> axioms, I could use 1 as the starting element, and >> dividing by 2 as the successor operation. My numbers >> are then 1, 1/2, 1/4, 1/8, ... Nothing in the axioms >> prevents me from making this rather confusing and >> pointless choice. > You know, after 35+ years as a programmer, primarily in OOP, I cannot > help but try and translate mathtalk into programming terms I am familiar > with. All I hear you saying is that succ() is a method called by a > recursive method, but without specifying the content of the succ() method. That's about it. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Cardinality question >> >>> ***I*** made the statement that succ() is an arbitrary >>> name. What I objected to was your statement that these >>> axioms define succ(). They don't. The reason for using >>> the name successor is to make the obvious connection, >>> but that it involves adding 1 or corresponds to the >>> successor in the way we count, IS NOT DEFINED HERE. >>> These axioms don't define successor. They just say >>> there's something called successor, and the only >>> thing we're going to say about it is that everything >>> in the set of interest has one. >>> As an alternate interpretation of exactly the same >>> axioms, I could use 1 as the starting element, and >>> dividing by 2 as the successor operation. My numbers >>> are then 1, 1/2, 1/4, 1/8, ... Nothing in the axioms >>> prevents me from making this rather confusing and >>> pointless choice. >> You know, after 35+ years as a programmer, primarily in OOP, I cannot >> help but try and translate mathtalk into programming terms I am >> familiar with. All I hear you saying is that succ() is a method >> called by a recursive method, but without specifying the content of >> the succ() method. > That's about it. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question > Intuitive definitions aren't definitions in the > mathematical sense. But intuitions are helpful in > understanding mathematics, in making it something you > truly understand rather than a mechanical set of > procedures. Perhaps this is where I am having trouble with math. No one bothers to explain the intuition that makes something truely understandable rather than a mechanical set of procedures. I have no use for rote memorization of mechanical sets of procedures. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question Albert Wagner said: > > > Intuitive definitions aren't definitions in the > > mathematical sense. But intuitions are helpful in > > understanding mathematics, in making it something you > > truly understand rather than a mechanical set of > > procedures. > Perhaps this is where I am having trouble with math. No one > bothers to explain the intuition that makes something truely > understandable rather than a mechanical set of procedures. > I have no use for rote memorization of mechanical sets of procedures. > I agree, Albert. Axioms are wonderful things for deduction, but simply declaring them doesn't make them true. They should all really be justified somehow, at least intuitively, before being accepted. I think I decided not to become a mathematician when it wandered into the axiomatic mire, especially when I ran into Cantor. I see the vlaue of them now, more than then, and yet I seek understanding of things through more visual and constructive means. I think intuition is very important. -- Smiles, Tony === Subject: Re: Cardinality question > I agree, Albert. Axioms are wonderful things for deduction, but simply > declaring them doesn't make them true. They should all really be justified > somehow, at least intuitively, before being accepted. I think I decided not to > become a mathematician when it wandered into the axiomatic mire, especially > when I ran into Cantor. I see the vlaue of them now, more than then, and yet I > seek understanding of things through more visual and constructive means. I > think intuition is very important. intuition is very important in the search for consistent axioms but serves only as sanded putty over a botched job when it is used to leap over logical absurdities. A good craftsman should require no sanded putty to cover up gaps. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question > intuition is very important in the search for consistent axioms > but serves only as sanded putty over a botched job when it is > used to leap over logical absurdities. A good craftsman should > require no sanded putty to cover up gaps. Put otherwise when the intuition of others matches mine, it's fine. otherwise it's absurd. Eh. -- Giuseppe Oblomov Bilotta Da grande lotter.98 per la pace A me me la compra il mio babbo (Altan) (When I grow up, I will fight for peace I'll have my daddy buy it for me) === Subject: Re: Cardinality question >>intuition is very important in the search for consistent axioms >>but serves only as sanded putty over a botched job when it is >>used to leap over logical absurdities. A good craftsman should >>require no sanded putty to cover up gaps. > Put otherwise when the intuition of others matches mine, it's fine. > otherwise it's absurd. Eh. That is not a rephrasing of what I said. Your strawman. Destroy it at your leisure. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question > I agree, Albert. Axioms are wonderful things for deduction, but simply > declaring them doesn't make them true. They should all really be justified > somehow, at least intuitively, before being accepted. I think I decided not to > become a mathematician when it wandered into the axiomatic mire, especially > when I ran into Cantor. I see the vlaue of them now, more than then, and yet I > seek understanding of things through more visual and constructive means. I > think intuition is very important. Intuition is a treacherous guide at times. When Wierstrass came up with a function that was everywhere continuous and nowhere differentiable the mathematical community had a small fit, but they got over it. Intuitively when we draw curves the only places where tangetns are not well definted is where the curve comes to a point like the blade of a saw. //. There is not tangent defined at the peaks and valleys. The Weirstrass functions is sawtooth at every point. It is virtually impossible to visualized this, never the less the function is well defined by a convergent infinite series of functions. Intuition is a useful servant but a fearful master. Bob Kolker === Subject: Re: Cardinality question > They should all really be justified > somehow, at least intuitively, before being accepted The problem is, my intuition might be different from yours and accept (or deny) something that you would deny (or accept). -- Giuseppe Oblomov Bilotta I'm never quite so stupid as when I'm being smart --Linus van Pelt === Subject: Re: Cardinality question >>They should all really be justified >>somehow, at least intuitively, before being accepted > The problem is, my intuition might be different from yours and accept > (or deny) something that you would deny (or accept). Perhaps Lester is offering you a way out of that conundrum by basing logic on something other than your intuition as a starting point. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question <5t198fj0dk1g$.tye552ng8fj9$.dlg@40tude.net> >>>They should all really be justified >>>somehow, at least intuitively, before being accepted >> The problem is, my intuition might be different from yours and accept >> (or deny) something that you would deny (or accept). > Perhaps Lester is offering you a way out of that conundrum by > basing logic on something other than your intuition as a starting > point. Lester is just offering his own intuition as universal truth. -- Giuseppe Oblomov Bilotta Axiom I of the Giuseppe Bilotta theory of IT: Anything is better than MS === Subject: Re: Cardinality question >>>>They should all really be justified >>>>somehow, at least intuitively, before being accepted >>>The problem is, my intuition might be different from yours and accept >>>(or deny) something that you would deny (or accept). >>Perhaps Lester is offering you a way out of that conundrum by >>basing logic on something other than your intuition as a starting >>point. > Lester is just offering his own intuition as universal truth. It is an unfair projection on your part to assume that Lester arrives at universal truth the same way that mathematicians arrive at 'truth', by intuition. You should get out more. The real world is so much bigger and more complicated than the little abstract one in your head. However, I caution you: intuition is not accepted as a legitimate way to close logical gaps. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Cardinality question > >> Intuitive definitions aren't definitions in the >> mathematical sense. But intuitions are helpful in >> understanding mathematics, in making it something you >> truly understand rather than a mechanical set of >> procedures. > Perhaps this is where I am having trouble with math. No one > bothers to explain the intuition that makes something truely > understandable rather than a mechanical set of procedures. > I have no use for rote memorization of mechanical sets of procedures. Because intuitions are intuitions. You (generic) don't explain intuitions. You can try to describe them to someone else, you can even justify them *a posteriori*, but for this you need some common ground to start from. An example: your pet peeve of lines of points being impossible on the basis that points with no width, length, thickness. cannot sum up to lines with a length. For me, instead, it's pretty obvious, intuitively, that it's very well possible, by generalizing the idea that a very large number of very small things can add up to huge things, so infinitely many infinitely small things can give us finite. This is why set-based geometry is ok by me, but the intuition for which it works for me might not work for you. (And I'm not even talking about the cases when the intuitions are plain *wrong*, like is the case, for example, with many visual tricks) -- Giuseppe Oblomov Bilotta E la storia dell'umanit.88, babbo? Ma niente: prima si fanno delle cazzate, poi si studia che cazzate si sono fatte (Altan) (And what about the history of the human race, dad? Oh, nothing special: first they make some foolish things, then you study what foolish things have been made) === Subject: Re: Cardinality question >> >>>Intuitive definitions aren't definitions in the >>>mathematical sense. But intuitions are helpful in >>>understanding mathematics, in making it something you >>>truly understand rather than a mechanical set of >>>procedures. >>Perhaps this is where I am having trouble with math. No one >>bothers to explain the intuition that makes something truely >>understandable rather than a mechanical set of procedures. >>I have no use for rote memorization of mechanical sets of procedures. > Because intuitions are intuitions. You (generic) don't explain > intuitions. You can try to describe them to someone else, you can even > justify them *a posteriori*, but for this you need some common ground > to start from. > An example: your pet peeve of lines of points being impossible on > the basis that points with no width, length, thickness. cannot sum up > to lines with a length. For me, instead, it's pretty obvious, > intuitively, that it's very well possible, by generalizing the idea > that a very large number of very small things can add up to huge > things, so infinitely many infinitely small things can give us finite. > This is why set-based geometry is ok by me, but the intuition for > which it works for me might not work for you. Didn't you agree with Stephen that 5 diameter points are OK? 5 diameters are /not/ small. So how does you intuition work with large points? When you say, a very large number of very small things can add up to huge things, what is the dimension of a very small thing? At what point on the continuum of finite dimensioned large things down to dimensionless things does a very small thing exist? > (And I'm not even talking about the cases when the intuitions are > plain *wrong*, like is the case, for example, with many visual tricks) -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Roots of algebraic expressions zounds. first, after fully digesting your response, Ken, I realized that you were gently pointing me in the direction of a common, reliable, and very useful method that I know from Gullberg (Mathematics from the Birth of Numbers, p.318) for finding the general solution of cubic equations. That's ironic, since what prompted my posting in the first place was my interest in the (alternative) method described at mathworld.com, which involves the Viete substitution. I'd been having trouble getting satisfactory results with the Viete method because even for simple roots (my full original test equation was z^3 - 4z^2 + 6z - 8 = 0, with 2 as a root) I was having to find cubic roots of values that contain square roots. So why the heck am I messing around with the Viete method when there's a shorter, simpler, practical method available? Easy enough answer. I was curious. Now I know. By the way, how the devil does a math guru happen to know so much about archaic grammar? And what IS the plural form of thee? You must be racking up big-time zen berries. --S. >>Oh gods of yonder usenet group, I invoke and beseech thee, > .... beseech _you_. The old word thee was singular, so > shouldn't be used to address more than one god. >>...would someone explain, please, how to obtain roots of >>algebraic expressions? Or point me in the direction of >>information (a FAQ, web page, book, whatever) that can >>help me determine how to do this. >>Here, this may give you a better idea of what I'm trying >>to do. >>Given, for instance, this: >>(1) z = (1 + sqrt5) / 2 >> z = 1/2 + (sqrt5)/2 (same thing) >>I can find: >>(2) z^3 = 2 + sqrt5 >>So with: >>(3) x = 2 + sqrt5 >>I would like to be able to be able to obtain: >>(4) x^(1/3) = (1 +|- sqrt5) / 2 > Not +/- for a _cube_ root. Try cubing (1 - sqrt5)/2 and see > what you get. >>Seems simple enough. I'm familiar with the process of >>taking complex roots--or thought I was. But although >>it's *generally* quite simple to obtain the value for >>the argument, there remains the matter of obtaining >>the root for the modulus, no? .... > The obvious answer is to use a numerical method (or even a > calculator :-) to find as many decimal places as you need. But I > suspect you may be looking for a solution within the field Q[sqrt5] > containing all numbers of the form a + b.sqrt5 with rational a and > b. Is that it? > If so, a simple approach is to write your unknown root as > a + b.sqrt5 and then work forward. In your example, > (a + b.sqrt5)^3 = 2 + sqrt5, so > a^3 + 3(a^2)b.sqrt5 + 15a(b^2) + 5(b^3)sqrt5 = 2 + sqrt5. > Equating the rational and irrational terms, then factorizing, gives two > equations > a(a^2 + 15(b^2)) = 2 > b(3(a^2) + 5(b^2)) = 1. > Dividing the first of these equations by the second, writing c = > a/b, and tidying up, gives the cubic equation > c^3 - 6(c^2) + 15c - 10 = 0. > Now remember you want c to be rational, so you need only test numbers > of the form (factor of the last term)/(factor of the first > coefficient), which in this case means 1, -1, 2, -2, 5, -5, 10, -10. > In fact c = 1 works, so put a = b in either of the two equations > above, to get a = b = 8^(1/3) which luckily is rational. If it > weren't, then your cube root wouldn't be in Q[sqrt5]. > A lot more high-brow things can be said about all this, but is the > above elementary algorithm the sort of thing you wanted? > Ken Pledger. === Subject: Re: Roots of algebraic expressions [ snip ] > By the way, how the devil does a math guru happen to know so much > about archaic grammar? And what IS the plural form of thee? ye > You must be racking up big-time zen berries. > --S. [ snip ] >> Ken Pledger. -- -- Geo. Michael Henry Eat dessert first. Life is uncertain. === Subject: what is this equal to Let C_N^i denote choose i out of N then what's this equal to sum_{i>T} C_N^i I know sum_{i=1}^N C_N^i = 2^N but if I'd like to know the sum of when i>T. === Subject: Re: Epistemology 202: Advanced Topics >> Axiomatic definitions are all well and good, as long as you are sure your >> axioms are correct wherever you are applying them. >Axioms per se are posits or conventions. They are neither true nor >false. They become true only if -interpreted- to be assertions about the >world. In other words axioms are only true if mathematikers can point them out and if not they are neither true nor false because mathematikers have no standard of truth except what they want to be true. >Is it true that two points determine a unique line? Only in Euclidean geometry. > It dependes on what >you mean by point or line. No shit? Who helped you figure that out? > Is it true that given a line and a point not >on it only one parallel to the given line can be drawn? It depends on >the the space, what is meant by a line and what is meant by a point. We >have geometries where the parallel postulate holds and we have >geometries where it does not. Both kinds of geometries are consistent, >or are at least equally consistent within themselves. So to say the >parallel postulate is true (or false) without specifying the context in >which it is asserted is absurd. What's absurd is you as salad chef. === Subject: Re: Epistemology 202: Advanced Topics >> Probably why he made the egregious blunder with geometric contraction >> in SR. >The contraction (so called) is an artifact of the co-ordinates, not a >deformation of the body involved. That's funny since the effect of the contraction is supposedly measured in relation to the body involved according to your analysis of the Lorentz transform. It seems that mathematical effects are only real when mathematikers can point them out. Mathematical effects mathematikers can't point out are merely inexplicable artifacts of coordinate systems. [. . .] > The contraction, properly understood (that lets you out, >by the way) is a consequence of the Lorentz transformation. In other words the contraction properly understood doesn't exist and never has and you're way too lame to figure out why the contraction isn't a consequence of the Lorentz transformation at all but is instead nothing but an artifact of your rather tepid imagination. === Subject: Re: Epistemology 202: Advanced Topics >>>> >>>>> >>>>> >>>>>> >>>>>> >>>>>>> >>>>>>> >>>>>>>>Lester Zick said: >>>>>>>> >>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>>> >>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>>>>Of course because I'm interested in the properties of not in itself, >>>>>>>>>>>>>>>that is the consequences of it's tautological regression. It's not a >>>>>>>>>>>>>>>lot different from studying the properties of differences as such or >>>>>>>>>>>>>>>of differences between differences or the properties of addition >>>>>>>>>>>>>>>without designating operands. It's the general idea of contradiction >>>>>>>>>>>>>>>I'm interested in. >>>>>>>>>>>>>> >>>>>>>>>>>>>>Addition only has properties in conjunction with how it interacts with >>>>>>>>>>>>>>its operands. Addition of integers/reals is different from addition >>>>>>>>>>>>>>defined on permutations of the Rubik's cube. One is commutative, the >>>>>>>>>>>>>>other is not. >>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>>But both depend on the concept of succession. >>>>>>>>>>>> >>>>>>>>>>>>What is the successor of rotate top layer right counter-clockwise 90 >>>>>>>>>>>>degrees as viewed from above? >>>>>>>>>>> >>>>>>>>>>>Are you suggesting there is no successor or that the concept of >>>>>>>>>>>angular addition isn't based on it? >>>>>>>>>> >>>>>>>>>>I am suggesting that addition defined on permutations of the Rubik's >>>>>>>>>>cube is not based on the concept of succession. Since you claim it is, >>>>>>>>>>I challenge you to demonstrate how. >>>>>>>>> >>>>>>>>>Actually I'm not claiming that what you call addition defined on >>>>>>>>>permutations of Rubik's cube is based on the concept of succession. >>>>>>>>>I'm claiming addition is based on the concept of succession. I think >>>>>>>>>there's a reasonable question as to whether permutations of Rubik's >>>>>>>>>cube is addition or not. If you want to call it addition I think you >>>>>>>>>have to demonstrate in what way it qualifies as addition. And if that >>>>>>>>>qualification doesn't include some concept of succession I'm not sure >>>>>>>>>it would qualify. If there is some rigorous sequencing discipline >>>>>>>>>involved in the manipulation of Rubik's cube (which I assume there is >>>>>>>>>but have yet to discover in my own cube manipulation) I'm confident >>>>>>>>>that discipline can be analyzed mathematically. But whether it >>>>>>>>>qualifies as addition or is another matter. >>>>>>>>> >>>>>>>>> >>>>>>>> >>>>>>>>I have to agree with Lester on this point. I really don't see how the Rubik's >>>>>>>>cube qualifies as an analog to addition. What are you adding and where is the >>>>>>>>sum? >>>>>>> >>>>>>>You are adding sequences of rotations. The sum is the concatenation of >>>>>>>the sequences of rotations. >>>>>>> >>>>>>> >>>>>>> >>>>>>> >>>>>>>>It's really more of a pattern manipulation where order changes in >>>>>>>>dimensional slices by rotation. Addition does seem to rest upon succession as >>>>>>>>far as I can see, even though some areas of number and quantity don't seem to >>>>>>>>require succession. >>>>>>> >>>>>>>Addition can be abstracted out beyond succession. >>>>>> >>>>>> >>>>>>Which was my point to begin with, that addition can be analyzed >>>>>>independently of operands. So can not or contradiction. >>>>> >>>>>However it still needs operands. Notice that I still included operands >>>>>in the above. The same is true of vector addition. >>>> >>>> >>>>Sure just as not or contradiction needs operands. However that >>>>still doesn't prevent us from getting at the idea itself separately. >>>You missed my point. You cannot talk about properties of addition until >>>you know what the set of operands are and how addition operates on those >>>operands. There is no such thing as properties of addition. There is >>>such a thing as properties of addition of the natural numbers or >>>properties of addition over the integers or properties of addition >>>over permutations of a Rubik's Cube. >> And I'm still not sure why you think there is such a thing as the >> properties of ADDITION . . . for the cardinal integers or Rubik's >> cube or anything else. What is it that makes you think there is >> anything like addition for any operands at all? >When I have defined addition on permutations of a Rubik's cube, I can >then determine if it satisfies certain properties, such as the >commutative property (it doesn't) or the associative property (it does). And all I'm asking is how or why you define addition on permutations of Rubik's cube at all? > This is different from looking at the addition on the integers where >both properties hold. Of course. The problem still remains as to in what sense properties are added or additive? Given that the question becomes whether and which properties of one kind of addition hold in relation to other kinds of addition. There are other operations which associate and/or commute without being called addition. >> Your claim is simply that one cannot talk about things without regard >> to the things to which they apply. >When talking about operators, yes. In other cases, no. An operator is >a function. Part of its definition is the domain, range, and how it >maps objects from the domain to the range. Well this is your claim but I see no justification for the claim. > > This is absurd on the face of it >> since we talk about things all the time in the absence of operands and >> we could never talk about anything if we had to wait until all aspects >> of every operand were established, which is a never ending process. >Not everything is an operator, and therefor to talk about operands for >those things is silly. Car has no operands. So basically you're saying that those things you want to restrict consideration of to their operands you just call operators and other things that you don't want to or don't know how to restrict consideration of in terms of operands are not called operators. How do you know car is not an operator? Roads might disagree. How do you know red is not an operator? Red cars might disagree. Not is a logical operator. That doesn't mean it cannot be analyzed as a function or operator in its own right. >> All I'm trying to show is that there is some idea which I call not >> or contradiction whose tautological regression is self contradictory >> and hence univerally false and whose properties with respect to any >> kind of operands depends first and foremost on that general property. >You have to finish defining the operator. I didn't say I didn't have to finish defining operands for the not or contradiction function or operator. I just said I could define certain properties or characteristics of the funtion or operator without consideration of which operands it operates on or how.