Talk:Controversy over Cantor's theory/Archive 1

The never-ending debate about Cantor's Theory in Usenet is very real, as can be verified using Google Groups. This article is an attempt to give an overview of the more sensible views on this  topic from a neutral point of view, and also to give some historical background. I (the original author) have to admit that I've never seen any published research on the Usenet debate, and hence, possibly this article is "original research", which does not belong in the encyclopedia. On the other hand, it's likely that many people, after seeing the debate in sci.math and sci.logic, will want to look in the Wikipedia for a level-headed analysis of the debate, and so I think the article is somewhat important.

Also, if anyone out there wants to edit this article, please be sure that you understand both sides of the debate so that you can give a neutral point of view. -Dave

Hi Dave. Dave Petry, I think, am I right? I'm Mike Oliver (as it says on my user page; not trying to hide anything).

So certainly there is room in Wikipedia for a discussion of the views of people "opposed" in whatever sense to modern set theory. If it's to be this article, it will need some serious revision, though; right now it does seem to be largely your personal observations. The fact that I don't agree with your characterization of the arguments is beside the point at the moment, unless I want to actually enter into the discussion, which I kind of do, but there are other things I ought to be doing. I'll post a note at Wikipedia talk:WikiProject Mathematics and we'll see what shakes out. --Trovatore 03:59, 3 October 2005 (UTC)

Yes, it's Dave Petry. The article is my best attempt to explain what the debate is all about, as objectively and neutrally as possible. I do not think the article should be dismissed as "original research". I assume you're the one who had that tag put in the article?
 * Actually, no. You can find out such things by clicking on the "history" tab. (While viewing the article, not the talk page. If you click on history while viewing the talk page, you'll se the history of the talk page.) --Trovatore 01:25, 6 October 2005 (UTC)


 * PS I would encourage you to register an account, and then to sign your remarks on talk pages by appending four tildes: ~ --Trovatore 01:32, 6 October 2005 (UTC)

Some naive questions
I have a couple of naive, use-net quality questions about the overlap of set theory with, e.g. parts of analysis, such as measure theory. Without measure theory, one has oddities like the Banach-Tarski paradox. With measure theory, one still has a horrid time trying to construct reasonable measures for things like fractals (which, besides just kleinian and fuchsian groups, includes things like lattices in Lie groups with ergodic behaviour, and so is of some interest to physicists). In my (somewhat limited) readings of ergodicty theory & etc. one can certainly sense that Banach-Tarski is "on the other side of the wall", but, for some reason (not clear to me) discussions of Cantorian set theory just don't seem to occur in this context. Why not? Am I simply reading in too narrow a region of topics? I mean, the "symbolic dynamics of a system with two letters" is something studied by chaos theorists and physicists (and even lowly fractalists), but no one ever seems to say "aha, now apply axiom of choice" in the way that its explicitly done in Banach-Tarski. Why is that? linas 01:33, 5 October 2005 (UTC)

I guess the above comments show how naive I am. I re-read the article and am now left confused. I would have thought that a close encounter with computers, fractals, and (for example) the incredibly filligree'd wave functions seen in computer simulations of quantum chaos, would make one into a Cantorain, not an anti-Cantorian. Yet the article implies the opposite. I mean, there are guys in lab coats with radar cavities shaped like Bunimovich stadiums, and they are measuring Cantor dusts and Cantor functions (devil's staircase)s and fractional quantum Hall effects out the wazoo, and this seems like a cantorian paradise to me. Many of these effects (e.g. buckling of a beam, tearing of a sheet, percolation) are in the domain of structural engineering, and are deep in the heart of fractal land, and so would be a breeding ground for cantorian beliefs. The article implies quite the opposite. So I guess I doubly don't get it. Maybe I don't know what "cantorian" means, or have an insane notion of what ZFC  and the continuum hypothesis might have to do with fractals. At a minimum, this article should delve into this, since right now I'm confused. linas 04:03, 5 October 2005 (UTC)


 * To answer Linas' questions: all of the mathematics you are talking about fits the view that mathematics has computation at its foundation. The Cantorian view implies there is something beyond that. That's what the anti-Cantorians object to. -- Dave


 * What do mean by computation Dave? Do you perhaps mean Turing computation? Barnaby dawson 20:25, 19 October 2005 (UTC)


 * Dave's answer was absurd. All of computation can be described in terms of grammars and Turing machines and semi-Thue systems and lambda calculi and L-systems and what not, whose outputs can be viewed as tilings of sorts, which can be given topologies and measures, and analyzed through homologies and what not. This is what folks who work in dynamical systems try to do. Computation is the topic of study, not the foundation. There's even that guy, Wolfram (?), who wrote a very fat pop-lit book on the topic. linas 00:02, 20 October 2005 (UTC)


 * No I think you've missed a trick linas. Firstly there is infinite computation (which you could be forgiven for not knowing about) which is much more general than Turing machines.  Secondly the constructible universe can be given entirely in terms of computation and the constructible universe is a possible foundation for mathematics (hence computation is too).  But the constructible universe is a model of ZFC so is certainly not 'anti-cantorian'.  Barnaby dawson 07:48, 20 October 2005 (UTC)


 * Never heard of infinite computation, have heard of parallel processing with clocks set at irrational frequencies. I have read Conway's book on numbers and games. If you want to say "all of constructive mathematics is the outcome of a set of production rules applied to a set of axioms", that's fine. However, let me speak naively: naively, all of constructive mathematics is just the collection of sentences generated by a formal grammar (whose primitives happen to be ZFC). Right? One can define a topology on a collection of sentences (using, e.g. cylinder sets) and then, because one has a topology, start asking about limits and functions and fixed points, etc. In various situations, this might lead one to contemplate Banach spaces and Hilbert spaces. In the contemplation thereof, one potentially has questions of commuting sets of observables on these spaces, i.e. complementarity (physics). There is a branch of fringe math (most of what I've seen gave the impression of being pseudo-math, but I don't know) that asks about these relations back into logic, and try to construct quantum logic. I'm not sure where this process stopped being constructive. Oh well. linas 15:15, 23 October 2005 (UTC)


 * See Infinite time turing machines by Joel Hamkins. Infinite time computers (generalizations of Hamkins design) and their relation to the constructible hierarchy are my PhD topics.


 * AFAIK quantum logic stops being constructive at the point where you want to operate with closed subspaces of Hilbert space. ZFC isn't constructive mathematics at all. Charles Matthews 15:23, 23 October 2005 (UTC)


 * Dohh, right, got carried away. linas 14:28, 25 October 2005 (UTC)


 * It occurs to me that you may be mixing up the notion of constructive reasoning with the notion of the constructible universe L. These are not the same notions.  As far as I can tell nowhere in Linas's above comment does he do anything inconsistent with the axiom V=L. Barnaby dawson 08:28, 26 October 2005 (UTC)


 * As far as I can tell, it's transfinite ordinals that offend Dave. --noösfractal 09:00, 6 October 2005 (UTC)

The recursive constructible universe
Barnaby dawson wrote: the constructible universe can be given entirely in terms of computation and the constructible universe is a possible foundation for mathematics (hence computation is too).
 * I don't know this result: do you have a pointer? Is it a realizability semantics? --- Charles Stewart 13:56, 26 October 2005 (UTC)


 * This result is, as far as I know, not yet published. However, myself and Peter Koeppe (at Bonn) have independently proved this result.  I may be able to locate a copy of an unpublished paper by Koeppe on the subject for you.  Alternatively I could send you a paper I intend to publish when I've got it into a sensible form.  From the point of view of this article, however, it is sufficient to note that there are multiple generalisations of Turing computation and that it is not clear that ZF or even ZFC goes beyond these more general computers.


 * Rereading what I wrote it sounds like I expected this result to be widely known. I didn't mean to give that impression.  Barnaby dawson 15:07, 26 October 2005 (UTC)


 * Jaap van Oosten has talked about applying realizability semantics to ZFC, but it didn't sound as clean as what you seem to be saying, which sounds very nice. I'd like very much to see the paper when it is ready. --- Charles Stewart 15:14, 26 October 2005 (UTC)
 * Postscript: Van Oosten talks about Krivine's work on realizability for ZFC, which I think is a predecessor of this paper. --- Charles Stewart 15:28, 26 October 2005 (UTC)
 * Are you allowing these "computations" to take arbitrary ordinals as inputs, or something like that? If so the result is hardly surprising; there's a "canonical" way to globally wellorder L, and thereby to code any element of L by an ordinal, and the map is simple enough that something akin to "computation" might well be defined using it. But I doubt it's going to satisfy Dave. --Trovatore 17:01, 26 October 2005 (UTC)
 * You might run into difficulty with Fränkel's replacement axiom with that: how do you know that sets of pairs define functions under which the universe is closed? --- Charles Stewart 17:25, 26 October 2005 (UTC)
 * All I was doing was trying to imagine in what sense L could be considered to be described by "computation", given that, in general, ordinals can't be thus described (for the usual sense of "computation"), and of course all ordinals are in L. I wanted Barnaby to explain a little more what he meant by his comments. --Trovatore 17:36, 26 October 2005 (UTC)
 * The point is that the criticism of ZFC (that it goes beyond the purely computational) depends on the computer. Hence any mention of computation ought to specify the type of computer.  The result concerning the presentation of L from a computational viewpoint is IMHO irrelevant to the article.  Barnaby dawson 22:04, 26 October 2005 (UTC)
 * Well, you were the one who brought it up. Did you change your mind about its relevance? Anyway, you have me and Charles both curious, so please don't drop it now--you could use one of our talk pages if you think it's not contributing to the encyclopedic nature of this article. --Trovatore 22:09, 26 October 2005 (UTC)
 * OK, I was thinking you thought perhaps you could use the fact that the V=L construction uses the omega*2-th set as its model to find a constructive ordinal characterising ZFC: my gut feeling says that technique would work for ZC but probably not for ZFC. If you use a non-recursive ordinal, then it is hard to see how the model could be recursive. --- Charles Stewart 18:03, 26 October 2005 (UTC)
 * It's well-known that there is no computable (in Soare's terminology, "recursive" in the older nomenclature) model of ZFC. I would be quite surprised if there were such a model of Z. Even analysis strikes me as kind of unlikely.
 * If you're hoping to get a computable model by looking at L&omega;+&omega;, note that you're going to have to use some non-obvious way of coding the elements of the model by naturals. In particular if the association between real-wold naturals, and codes for naturals in the model, is itself computable, it can't work. That's because 0', the set of all Gödel numbers of Turing machines that halt, is an element of L&omega;+1; if you could computably decide what elements of the model are elements of that set in the sense of the model, and also what ordinary naturals they correspond to, then you could solve the halting problem. --Trovatore 19:06, 26 October 2005 (UTC)

Trovatore wrote: That's because 0', the set of all Gödel numbers of Turing machines that halt, is an element of L&omega;+1; if you could computably decide what elements of the model are elements of that set in the sense of the model, and also what ordinary naturals they correspond to, then you could solve the halting problem.
 * Z and ZC give you a lot of room to construct just such non-obvious models. See, eg. this result (postscript), and in the case of Z, I think the models might not be all that strange. --- Charles Stewart 20:06, 26 October 2005 (UTC)
 * I couldn't figure out how to view the article. The closest thing I could see that might have worked was "save to a binder", but it wouldn't let me do that even after I did the free registration.
 * Blast the ACM! Try this JSL preprint of a related article instead. --- Charles Stewart 20:42, 26 October 2005 (UTC)
 * Postcript: since you a seem au fait with recursion theory, might you give a hand filling out the subject around Turing degrees: I've been meaning to write up a bit about mass problems for a while, and a cocontributor might be just what I need to get started. --- Charles Stewart 20:11, 26 October 2005 (UTC)
 * I don't know an awful lot about it, to tell you the truth (and nothing about mass problems). I might be able to throw in something about the Martin measure, in the AD context. --Trovatore 20:23, 26 October 2005 (UTC)
 * BTW I went into Google Groups to see if I could find a reference, among the discussions I remembered, for the claim that there's no computable model of ZFC, and I'm no longer so sure about that. It may be one of those factoids that we (or at least I) tend to accumulate without ever sufficiently checking them. --Trovatore 20:23, 26 October 2005 (UTC)

Scholarly work on this topic?
A very, very short and basic test suggests, that even the concept of "anti-cantorian" doesn't exist in scholarly discourse, see: Of course, some USENET phenomena may become notable, that they may get their own article without any real life background.
 * http://scholar.google.com/scholar?q=anti-cantorian

Pjacobi 14:54, 6 October 2005 (UTC)

A very, very short and basic test suggests, that even the concept of "anti-cantorian" doesn't exist in scholarly discourse, see:

http://scholar.google.com/scholar?q=anti-cantorian Of course, some USENET phenomena may become notable, that they may get their own article without any real life background.

Pjacobi 14:54, 6 October 2005 (UTC)

- That's silly. There are plenty of things in scholarly books that aren't in Google. For example John Mayberry (student of Von Neumann, according to Wikepedia) writes in a recent-ish book (Mayberry, J.P., The Foundations of Mathematics in the Theory of Sets, Encyclopedia of Mathematics and its Applications, Vol. 82, Cambridge University Press, Cambridge, 2000) about what he calls the "anti-Cantorian" viewpoint. See chapter 8 ("The Serpent in Cantor's paradise).

E.g. "Historically, the anti-Cantorians have drawn the distinction between finite and infinite in operationalist terms" (p.262).

"The first prominent anti-Cantorian was the German algebraist and number theories Leopold Kronecker ... "God made the natural numbers, everything else is the work of man". To a modern mathematician a consequence of that slogan is that the basic principles of natural number arithmetic - the principle of proof by mathematical induction and the principle of definition by recursion - stand on their own as fundamental principles, and do not require justification, or formulation, in the theory of sets.  This point of view has been embraced by the whole ****anti-Cantorian**** movement in modern mathematics: Poincare, Weyl, Brouwer and the Dutch school of intuitionists, and, more recently, Bishop, have all followed Kronecker in this regard" (p. 270). Mayberry is critical of this school, but the question was whether there is or was such a school at all.


 * Thanks for providing that reference. David Petry 00:05, 14 November 2005 (UTC)

I like the idea of a page like this, but it has to be NPOV. There are some claims in this article that are hard to verify (where are the references for the claims about computer scientists, for example).


 * Constructive mathematics is quite popular among theoretical computer scientists (or at least the constructivists like to say that is so). It's easy to find lots of references mentioning this, but I'm not sure what reference I need to include in the Wikipedia article David Petry 00:05, 14 November 2005 (UTC)

There are the usenet groups, but these are mostly very cranky. Any references should be to papers in peer-reviewed journals.


 * The quotes in the article are mostly from expository writings and speeches from prominent mathematicians, and not from peer reviewed journals. I don't see anything wrong with that in an article on this topic. Would you say it would be wrong to quote from Mayberry's book (not peer reviewed), for example? David Petry 00:05, 14 November 2005 (UTC)


 * A further comment: after stating that references need to be from peer-reviewed journals, you have included a quote from the interlingua mailing list (on the internet) in your rewrite of the article (Sowa's quote). Nevertheless, I like the quote, and wish I could find an equivalent quote from a more respectable source. It definitely is the kind of thing the anti-Cantorians like to say David Petry 23:52, 14 November 2005 (UTC)

As I've mentioned before, Wittgenstein was a bitter opponent of Cantor's theory. His name ought to go on any such page.


 * I've always had the impression that Wittgenstein didn't really understand the issues, which is why I didn't mention him. I might be mistaken, and of course, you are free to edit this article. Do you know any concise quotes from him on this topic? David Petry 00:05, 14 November 2005 (UTC)


 * As your rewrite seems to show, including too much Wittgenstein just mucks up the issue. David Petry 23:52, 14 November 2005 (UTC)

dean

Article shortening
I have cut the length down, without, I hope, removing any essential arguments. Charles Matthews 21:23, 6 October 2005 (UTC)

Issues, proposed name change
Some issues with the article as it stands:
 * Modern set theory isn't grounded in Cantorian intuition, for the most part. Modern set theory is generally understood and justified in reference to the cumulative hierarchy, a semi-constructive idea of the generation of a transfinite realm of sets that vindicates ZFC plus many large cardinal axioms, which was, I believe, proposed in the 1920s, around the time Fraenkel's axiom was proposed, and gained widespread acceptance with Goedel's constructible universe, which shows that one of the stages of unfolding of this hierarchy is a model of ZFC.
 * User:198.104.0.100: It has it's roots in Cantor's intuitions.
 * I don't think vague talk about roots is good enough. --- Charles Stewart 17:12, 25 October 2005 (UTC)


 * It's laughable that a quote of Hermann Weyl from 1946 is the only citation in "Recent attacks", only adding to the impression of a straw man argument.
 * User:198.104.0.100: The original article was chopped up a bit. I have tried to clarify things
 * The introduction of a connections section is some sort of impropement, but I don't agree with the points bveing made. I'll tackle this when I have more time. --- Charles Stewart 17:12, 25 October 2005 (UTC)


 * Kline isn't saying what the article thinks he is saying, if I read him aright. See "New article on anti-Cantor newsgroup participants" on Wikipedia talk:WikiProject Mathematics for more.
 * There is nothing indicating what beliefs the group of anti-Cantorians have in common. Is Thurston an anti-Cantorian?  He doesn't seem to have any problem with formalisability in ZFC as the gold standard of mathematical demonstration, only with how ZFC is to be understood.
 * User:198.104.0.100: They think Cantor created a fantasy world, as I said right at the start. I'm not really arguing that Thurston is an anti-Cantorian, but only that he has expressed doubts abou the reality of set theory.


 * There are people who doubt the consistency of ZFC, the modern constructivists. Martin-Loef's type theory is the only developed rival to ZFC as a foundation for (constructive) mathematics.  These people could, unlike Kline and Thurston, be seen as inheritors of the early doubts about set theory.
 * User:198.104.0.100: The consistency of set theory is not at issue here.
 * Before the mathematical status of the consistency of set theory was clarified, it was a central issue. This is one of several respects in which the modern ZFC-bashers differ from the early ones.  You are asserting a continuity where I see discontinuity. --- Charles Stewart 17:12, 25 October 2005 (UTC)

I think the material here divides into three sets of concerns: concern about the consistency of ZFC; concern about whether the axioms of ZFC, even if consistent, should be taken at "face value", and concern about the role of set theory in mathematical education and exposition. "Cantorianism" doesn't cover the issues well; better might be an article formulated as a question, such as "Is set theory the right foundation for mathematics?". I'm guessing tha with the right name, the problem of how to deal with the content will become easier.--- Charles Stewart 16:07, 13 October 2005 (UTC)
 * User:198.104.0.100: There would be nothing wrong with having both the current article and an article like "is set theory the right foundation for mathematics"
 * I repeat my objection below to Charles Matthew's suggestion (which is an improvement over the status quo): such contra articles are POV magnets. --- Charles Stewart 17:12, 25 October 2005 (UTC)


 * I have argued for criticisms of set theory. On such a page, surely, segregating (a) things people have against naive set theory (b) things people have against any axiomatic set theory (c) beefs about ZFC (d) finitist stuff (e) impredicativity, etc., can all be dealt with; as can pedagogy. And that's just a matter of sorting out a few headings. I'm pretty much against taking UseNet seriously as matter on which to comment (kook-magnet, need I say more?) Charles Matthews 16:52, 13 October 2005 (UTC)
 * All the quotes I have given are from reputable mathematicians. I'm only pointing out that the debate from the past is being continued on the internet.


 * I don't like "criticisms of" articles, because they have a kind of inbuilt tendency to becomes homes of slanted commentary. Better, I think, to have an article whose title names the issue of controversy.  The article Is logic empirical? is a good example of what such articles can be (it's a bit different, in that it is in part concerned with two articles by that name, but it is mainly concerned with the question there. --- Charles Stewart 17:24, 13 October 2005 (UTC)

Comments on purpose of article
Recall that Brouwer and Hilbert - two very reputable mathematicians - had a heated debate on this topic. Now imagine what would happen if they both wrote articles on their points of view, and then each of them edited the other's article. That's similar to what is happening here. The purpose of this article is to present the views of those who (like Brouwer, Poincare, Kronecker, Weyl, Bishop etc) think that Cantor created a fantasy world. Granted, that view is now definitely not mainstream. But some of you guys are editing an article about things you don't really understand, and don't like. In doing so, you are injecting you own point of view (which undeniably is the mainstream point of view). But there is nothing wrong with having an article which describes an alternate view taken by respected academics. When you guys insert rebuttals to points being made, right in the middle of a paragraph, the result is paragraph with a garbled message. I'd like to ask you to move your rebuttals to a separate paragraph, with a disclaimer such as "in contrast, the standard view is that ..." David Petry 21:26, 26 October 2005 (UTC)


 * A further comment: in the original article, I proposed that the reason for the controversy may be that applied mathematicians tend to view mathematics as a science, and the pure mathematicians tend to view mathematics as an art. Certainly mathematicians have told me that in person, and I've read comments similar to that in the informal literature. So I thought the idea should be presented. But Charles Matthews edited the whole paragraph out. I'd like to hear some discussion on this. David Petry 00:21, 14 November 2005 (UTC)
 * I am a pure mathematician, and view mathematics as a science; I'm far from alone in this among realist-leaning set theorists. You should look up some of Maddy's work where she discusses this view. Basically I agree with you about the applicability of empirical methods; I just disagree when you claim that set theory fails the test. I think that you're confounding empiricism with materialism, assuming beforehand that there can never be empirical evidence of non-physical objects. --Trovatore 00:47, 14 November 2005 (UTC)

Removal of OR tag
I intend to remove the "Original Research" tag soon. I'd also like to point out that if you put that tag on an article, you are expected to make comments in the discussion page about why you have placed that tag. The guy who put it there didn't leave any comments, and hence it's questionable whether it really should be there at all. I'll wait for comments before I remove the tag. David Petry 00:43, 14 November 2005 (UTC)

Some naive questions
I have a couple of naive, use-net quality questions about the overlap of set theory with, e.g. parts of analysis, such as measure theory. Without measure theory, one has oddities like the Banach-Tarski paradox. With measure theory, one still has a horrid time trying to construct reasonable measures for things like fractals (which, besides just kleinian and fuchsian groups, includes things like lattices in Lie groups with ergodic behaviour, and so is of some interest to physicists). In my (somewhat limited) readings of ergodicty theory & etc. one can certainly sense that Banach-Tarski is "on the other side of the wall", but, for some reason (not clear to me) discussions of Cantorian set theory just don't seem to occur in this context. Why not? Am I simply reading in too narrow a region of topics? I mean, the "symbolic dynamics of a system with two letters" is something studied by chaos theorists and physicists (and even lowly fractalists), but no one ever seems to say "aha, now apply axiom of choice" in the way that its explicitly done in Banach-Tarski. Why is that? linas 01:33, 5 October 2005 (UTC)

I guess the above comments show how naive I am. I re-read the article and am now left confused. I would have thought that a close encounter with computers, fractals, and (for example) the incredibly filligree'd wave functions seen in computer simulations of quantum chaos, would make one into a Cantorain, not an anti-Cantorian. Yet the article implies the opposite. I mean, there are guys in lab coats with radar cavities shaped like Bunimovich stadiums, and they are measuring Cantor dusts and Cantor functions (devil's staircase)s and fractional quantum Hall effects out the wazoo, and this seems like a cantorian paradise to me. Many of these effects (e.g. buckling of a beam, tearing of a sheet, percolation) are in the domain of structural engineering, and are deep in the heart of fractal land, and so would be a breeding ground for cantorian beliefs. The article implies quite the opposite. So I guess I doubly don't get it. Maybe I don't know what "cantorian" means, or have an insane notion of what ZFC  and the continuum hypothesis might have to do with fractals. At a minimum, this article should delve into this, since right now I'm confused. linas 04:03, 5 October 2005 (UTC)


 * To answer Linas' questions: all of the mathematics you are talking about fits the view that mathematics has computation at its foundation. The Cantorian view implies there is something beyond that. That's what the anti-Cantorians object to. -- Dave


 * What do mean by computation Dave? Do you perhaps mean Turing computation? Barnaby dawson 20:25, 19 October 2005 (UTC)


 * Dave's answer was absurd. All of computation can be described in terms of grammars and Turing machines and semi-Thue systems and lambda calculi and L-systems and what not, whose outputs can be viewed as tilings of sorts, which can be given topologies and measures, and analyzed through homologies and what not. This is what folks who work in dynamical systems try to do. Computation is the topic of study, not the foundation. There's even that guy, Wolfram (?), who wrote a very fat pop-lit book on the topic. linas 00:02, 20 October 2005 (UTC)


 * No I think you've missed a trick linas. Firstly there is infinite computation (which you could be forgiven for not knowing about) which is much more general than Turing machines.  Secondly the constructible universe can be given entirely in terms of computation and the constructible universe is a possible foundation for mathematics (hence computation is too).  But the constructible universe is a model of ZFC so is certainly not 'anti-cantorian'.  Barnaby dawson 07:48, 20 October 2005 (UTC)


 * Never heard of infinite computation, have heard of parallel processing with clocks set at irrational frequencies. I have read Conway's book on numbers and games. If you want to say "all of constructive mathematics is the outcome of a set of production rules applied to a set of axioms", that's fine. However, let me speak naively: naively, all of constructive mathematics is just the collection of sentences generated by a formal grammar (whose primitives happen to be ZFC). Right? One can define a topology on a collection of sentences (using, e.g. cylinder sets) and then, because one has a topology, start asking about limits and functions and fixed points, etc. In various situations, this might lead one to contemplate Banach spaces and Hilbert spaces. In the contemplation thereof, one potentially has questions of commuting sets of observables on these spaces, i.e. complementarity (physics). There is a branch of fringe math (most of what I've seen gave the impression of being pseudo-math, but I don't know) that asks about these relations back into logic, and try to construct quantum logic. I'm not sure where this process stopped being constructive. Oh well. linas 15:15, 23 October 2005 (UTC)


 * See Infinite time turing machines by Joel Hamkins. Infinite time computers (generalizations of Hamkins design) and their relation to the constructible hierarchy are my PhD topics.


 * AFAIK quantum logic stops being constructive at the point where you want to operate with closed subspaces of Hilbert space. ZFC isn't constructive mathematics at all. Charles Matthews 15:23, 23 October 2005 (UTC)


 * Dohh, right, got carried away. linas 14:28, 25 October 2005 (UTC)


 * It occurs to me that you may be mixing up the notion of constructive reasoning with the notion of the constructible universe L. These are not the same notions.  As far as I can tell nowhere in Linas's above comment does he do anything inconsistent with the axiom V=L. Barnaby dawson 08:28, 26 October 2005 (UTC)


 * As far as I can tell, it's transfinite ordinals that offend Dave. --noösfractal 09:00, 6 October 2005 (UTC)

The recursive constructible universe
Barnaby dawson wrote: the constructible universe can be given entirely in terms of computation and the constructible universe is a possible foundation for mathematics (hence computation is too).
 * I don't know this result: do you have a pointer? Is it a realizability semantics? --- Charles Stewart 13:56, 26 October 2005 (UTC)


 * This result is, as far as I know, not yet published. However, myself and Peter Koeppe (at Bonn) have independently proved this result.  I may be able to locate a copy of an unpublished paper by Koeppe on the subject for you.  Alternatively I could send you a paper I intend to publish when I've got it into a sensible form.  From the point of view of this article, however, it is sufficient to note that there are multiple generalisations of Turing computation and that it is not clear that ZF or even ZFC goes beyond these more general computers.


 * Rereading what I wrote it sounds like I expected this result to be widely known. I didn't mean to give that impression.  Barnaby dawson 15:07, 26 October 2005 (UTC)


 * Jaap van Oosten has talked about applying realizability semantics to ZFC, but it didn't sound as clean as what you seem to be saying, which sounds very nice. I'd like very much to see the paper when it is ready. --- Charles Stewart 15:14, 26 October 2005 (UTC)
 * Postscript: Van Oosten talks about Krivine's work on realizability for ZFC, which I think is a predecessor of this paper. --- Charles Stewart 15:28, 26 October 2005 (UTC)
 * Are you allowing these "computations" to take arbitrary ordinals as inputs, or something like that? If so the result is hardly surprising; there's a "canonical" way to globally wellorder L, and thereby to code any element of L by an ordinal, and the map is simple enough that something akin to "computation" might well be defined using it. But I doubt it's going to satisfy Dave. --Trovatore 17:01, 26 October 2005 (UTC)
 * You might run into difficulty with Fränkel's replacement axiom with that: how do you know that sets of pairs define functions under which the universe is closed? --- Charles Stewart 17:25, 26 October 2005 (UTC)
 * All I was doing was trying to imagine in what sense L could be considered to be described by "computation", given that, in general, ordinals can't be thus described (for the usual sense of "computation"), and of course all ordinals are in L. I wanted Barnaby to explain a little more what he meant by his comments. --Trovatore 17:36, 26 October 2005 (UTC)
 * The point is that the criticism of ZFC (that it goes beyond the purely computational) depends on the computer. Hence any mention of computation ought to specify the type of computer.  The result concerning the presentation of L from a computational viewpoint is IMHO irrelevant to the article.  Barnaby dawson 22:04, 26 October 2005 (UTC)
 * Well, you were the one who brought it up. Did you change your mind about its relevance? Anyway, you have me and Charles both curious, so please don't drop it now--you could use one of our talk pages if you think it's not contributing to the encyclopedic nature of this article. --Trovatore 22:09, 26 October 2005 (UTC)
 * OK, I was thinking you thought perhaps you could use the fact that the V=L construction uses the omega*2-th set as its model to find a constructive ordinal characterising ZFC: my gut feeling says that technique would work for ZC but probably not for ZFC. If you use a non-recursive ordinal, then it is hard to see how the model could be recursive. --- Charles Stewart 18:03, 26 October 2005 (UTC)
 * It's well-known that there is no computable (in Soare's terminology, "recursive" in the older nomenclature) model of ZFC. I would be quite surprised if there were such a model of Z. Even analysis strikes me as kind of unlikely.
 * If you're hoping to get a computable model by looking at L&omega;+&omega;, note that you're going to have to use some non-obvious way of coding the elements of the model by naturals. In particular if the association between real-wold naturals, and codes for naturals in the model, is itself computable, it can't work. That's because 0', the set of all Gödel numbers of Turing machines that halt, is an element of L&omega;+1; if you could computably decide what elements of the model are elements of that set in the sense of the model, and also what ordinary naturals they correspond to, then you could solve the halting problem. --Trovatore 19:06, 26 October 2005 (UTC)

Trovatore wrote: That's because 0', the set of all Gödel numbers of Turing machines that halt, is an element of L&omega;+1; if you could computably decide what elements of the model are elements of that set in the sense of the model, and also what ordinary naturals they correspond to, then you could solve the halting problem.
 * Z and ZC give you a lot of room to construct just such non-obvious models. See, eg. this result (postscript), and in the case of Z, I think the models might not be all that strange. --- Charles Stewart 20:06, 26 October 2005 (UTC)
 * I couldn't figure out how to view the article. The closest thing I could see that might have worked was "save to a binder", but it wouldn't let me do that even after I did the free registration.
 * Blast the ACM! Try this JSL preprint of a related article instead. --- Charles Stewart 20:42, 26 October 2005 (UTC)
 * Postcript: since you a seem au fait with recursion theory, might you give a hand filling out the subject around Turing degrees: I've been meaning to write up a bit about mass problems for a while, and a cocontributor might be just what I need to get started. --- Charles Stewart 20:11, 26 October 2005 (UTC)
 * I don't know an awful lot about it, to tell you the truth (and nothing about mass problems). I might be able to throw in something about the Martin measure, in the AD context. --Trovatore 20:23, 26 October 2005 (UTC)
 * BTW I went into Google Groups to see if I could find a reference, among the discussions I remembered, for the claim that there's no computable model of ZFC, and I'm no longer so sure about that. It may be one of those factoids that we (or at least I) tend to accumulate without ever sufficiently checking them. --Trovatore 20:23, 26 October 2005 (UTC)

Scholarly work on this topic?
A very, very short and basic test suggests, that even the concept of "anti-cantorian" doesn't exist in scholarly discourse, see: Of course, some USENET phenomena may become notable, that they may get their own article without any real life background.
 * http://scholar.google.com/scholar?q=anti-cantorian

Pjacobi 14:54, 6 October 2005 (UTC)

A very, very short and basic test suggests, that even the concept of "anti-cantorian" doesn't exist in scholarly discourse, see:

http://scholar.google.com/scholar?q=anti-cantorian Of course, some USENET phenomena may become notable, that they may get their own article without any real life background.

Pjacobi 14:54, 6 October 2005 (UTC)

- That's silly. There are plenty of things in scholarly books that aren't in Google. For example John Mayberry (student of Von Neumann, according to Wikepedia) writes in a recent-ish book (Mayberry, J.P., The Foundations of Mathematics in the Theory of Sets, Encyclopedia of Mathematics and its Applications, Vol. 82, Cambridge University Press, Cambridge, 2000) about what he calls the "anti-Cantorian" viewpoint. See chapter 8 ("The Serpent in Cantor's paradise).

E.g. "Historically, the anti-Cantorians have drawn the distinction between finite and infinite in operationalist terms" (p.262).

"The first prominent anti-Cantorian was the German algebraist and number theories Leopold Kronecker ... "God made the natural numbers, everything else is the work of man". To a modern mathematician a consequence of that slogan is that the basic principles of natural number arithmetic - the principle of proof by mathematical induction and the principle of definition by recursion - stand on their own as fundamental principles, and do not require justification, or formulation, in the theory of sets.  This point of view has been embraced by the whole ****anti-Cantorian**** movement in modern mathematics: Poincare, Weyl, Brouwer and the Dutch school of intuitionists, and, more recently, Bishop, have all followed Kronecker in this regard" (p. 270). Mayberry is critical of this school, but the question was whether there is or was such a school at all.


 * Thanks for providing that reference. David Petry 00:05, 14 November 2005 (UTC)

I like the idea of a page like this, but it has to be NPOV. There are some claims in this article that are hard to verify (where are the references for the claims about computer scientists, for example).


 * Constructive mathematics is quite popular among theoretical computer scientists (or at least the constructivists like to say that is so). It's easy to find lots of references mentioning this, but I'm not sure what reference I need to include in the Wikipedia article David Petry 00:05, 14 November 2005 (UTC)

There are the usenet groups, but these are mostly very cranky. Any references should be to papers in peer-reviewed journals.


 * The quotes in the article are mostly from expository writings and speeches from prominent mathematicians, and not from peer reviewed journals. I don't see anything wrong with that in an article on this topic. Would you say it would be wrong to quote from Mayberry's book (not peer reviewed), for example? David Petry 00:05, 14 November 2005 (UTC)


 * A further comment: after stating that references need to be from peer-reviewed journals, you have included a quote from the interlingua mailing list (on the internet) in your rewrite of the article (Sowa's quote). Nevertheless, I like the quote, and wish I could find an equivalent quote from a more respectable source. It definitely is the kind of thing the anti-Cantorians like to say David Petry 23:52, 14 November 2005 (UTC)

As I've mentioned before, Wittgenstein was a bitter opponent of Cantor's theory. His name ought to go on any such page.


 * I've always had the impression that Wittgenstein didn't really understand the issues, which is why I didn't mention him. I might be mistaken, and of course, you are free to edit this article. Do you know any concise quotes from him on this topic? David Petry 00:05, 14 November 2005 (UTC)


 * As your rewrite seems to show, including too much Wittgenstein just mucks up the issue. David Petry 23:52, 14 November 2005 (UTC)

dean

Article shortening
I have cut the length down, without, I hope, removing any essential arguments. Charles Matthews 21:23, 6 October 2005 (UTC)

Issues, proposed name change
Some issues with the article as it stands:
 * Modern set theory isn't grounded in Cantorian intuition, for the most part. Modern set theory is generally understood and justified in reference to the cumulative hierarchy, a semi-constructive idea of the generation of a transfinite realm of sets that vindicates ZFC plus many large cardinal axioms, which was, I believe, proposed in the 1920s, around the time Fraenkel's axiom was proposed, and gained widespread acceptance with Goedel's constructible universe, which shows that one of the stages of unfolding of this hierarchy is a model of ZFC.
 * User:198.104.0.100: It has it's roots in Cantor's intuitions.
 * I don't think vague talk about roots is good enough. --- Charles Stewart 17:12, 25 October 2005 (UTC)


 * It's laughable that a quote of Hermann Weyl from 1946 is the only citation in "Recent attacks", only adding to the impression of a straw man argument.
 * User:198.104.0.100: The original article was chopped up a bit. I have tried to clarify things
 * The introduction of a connections section is some sort of impropement, but I don't agree with the points bveing made. I'll tackle this when I have more time. --- Charles Stewart 17:12, 25 October 2005 (UTC)


 * Kline isn't saying what the article thinks he is saying, if I read him aright. See "New article on anti-Cantor newsgroup participants" on Wikipedia talk:WikiProject Mathematics for more.
 * There is nothing indicating what beliefs the group of anti-Cantorians have in common. Is Thurston an anti-Cantorian?  He doesn't seem to have any problem with formalisability in ZFC as the gold standard of mathematical demonstration, only with how ZFC is to be understood.
 * User:198.104.0.100: They think Cantor created a fantasy world, as I said right at the start. I'm not really arguing that Thurston is an anti-Cantorian, but only that he has expressed doubts abou the reality of set theory.


 * There are people who doubt the consistency of ZFC, the modern constructivists. Martin-Loef's type theory is the only developed rival to ZFC as a foundation for (constructive) mathematics.  These people could, unlike Kline and Thurston, be seen as inheritors of the early doubts about set theory.
 * User:198.104.0.100: The consistency of set theory is not at issue here.
 * Before the mathematical status of the consistency of set theory was clarified, it was a central issue. This is one of several respects in which the modern ZFC-bashers differ from the early ones.  You are asserting a continuity where I see discontinuity. --- Charles Stewart 17:12, 25 October 2005 (UTC)

I think the material here divides into three sets of concerns: concern about the consistency of ZFC; concern about whether the axioms of ZFC, even if consistent, should be taken at "face value", and concern about the role of set theory in mathematical education and exposition. "Cantorianism" doesn't cover the issues well; better might be an article formulated as a question, such as "Is set theory the right foundation for mathematics?". I'm guessing tha with the right name, the problem of how to deal with the content will become easier.--- Charles Stewart 16:07, 13 October 2005 (UTC)
 * User:198.104.0.100: There would be nothing wrong with having both the current article and an article like "is set theory the right foundation for mathematics"
 * I repeat my objection below to Charles Matthew's suggestion (which is an improvement over the status quo): such contra articles are POV magnets. --- Charles Stewart 17:12, 25 October 2005 (UTC)


 * I have argued for criticisms of set theory. On such a page, surely, segregating (a) things people have against naive set theory (b) things people have against any axiomatic set theory (c) beefs about ZFC (d) finitist stuff (e) impredicativity, etc., can all be dealt with; as can pedagogy. And that's just a matter of sorting out a few headings. I'm pretty much against taking UseNet seriously as matter on which to comment (kook-magnet, need I say more?) Charles Matthews 16:52, 13 October 2005 (UTC)
 * All the quotes I have given are from reputable mathematicians. I'm only pointing out that the debate from the past is being continued on the internet.


 * I don't like "criticisms of" articles, because they have a kind of inbuilt tendency to becomes homes of slanted commentary. Better, I think, to have an article whose title names the issue of controversy.  The article Is logic empirical? is a good example of what such articles can be (it's a bit different, in that it is in part concerned with two articles by that name, but it is mainly concerned with the question there. --- Charles Stewart 17:24, 13 October 2005 (UTC)

Comments on purpose of article
Recall that Brouwer and Hilbert - two very reputable mathematicians - had a heated debate on this topic. Now imagine what would happen if they both wrote articles on their points of view, and then each of them edited the other's article. That's similar to what is happening here. The purpose of this article is to present the views of those who (like Brouwer, Poincare, Kronecker, Weyl, Bishop etc) think that Cantor created a fantasy world. Granted, that view is now definitely not mainstream. But some of you guys are editing an article about things you don't really understand, and don't like. In doing so, you are injecting you own point of view (which undeniably is the mainstream point of view). But there is nothing wrong with having an article which describes an alternate view taken by respected academics. When you guys insert rebuttals to points being made, right in the middle of a paragraph, the result is paragraph with a garbled message. I'd like to ask you to move your rebuttals to a separate paragraph, with a disclaimer such as "in contrast, the standard view is that ..." David Petry 21:26, 26 October 2005 (UTC)


 * A further comment: in the original article, I proposed that the reason for the controversy may be that applied mathematicians tend to view mathematics as a science, and the pure mathematicians tend to view mathematics as an art. Certainly mathematicians have told me that in person, and I've read comments similar to that in the informal literature. So I thought the idea should be presented. But Charles Matthews edited the whole paragraph out. I'd like to hear some discussion on this. David Petry 00:21, 14 November 2005 (UTC)
 * I am a pure mathematician, and view mathematics as a science; I'm far from alone in this among realist-leaning set theorists. You should look up some of Maddy's work where she discusses this view. Basically I agree with you about the applicability of empirical methods; I just disagree when you claim that set theory fails the test. I think that you're confounding empiricism with materialism, assuming beforehand that there can never be empirical evidence of non-physical objects. --Trovatore 00:47, 14 November 2005 (UTC)

Removal of OR tag
I intend to remove the "Original Research" tag soon. I'd also like to point out that if you put that tag on an article, you are expected to make comments in the discussion page about why you have placed that tag. The guy who put it there didn't leave any comments, and hence it's questionable whether it really should be there at all. I'll wait for comments before I remove the tag. David Petry 00:43, 14 November 2005 (UTC)

Mathematical vs philosophical objections?
Oh dear. I don't think it would be a good idea to separate out the "philosophical" as against the "mathematical" objections.

The purpose of the rewritten article was


 * to explain in simple terms what Cantor's argument actually is. You can't really have an article about an objection unless you say what the objection is to
 * to divide the argument in a way that makes the main steps clear
 * Thus I divide the argument into the 4 steps: Cantor's Theorem, Hume's Principle, Axiom of Infinity, Power Set

The reason it doesn't make sense to split the article into "philosophical" versus "mathematical" objections is that anyone who accepts the four steps above accepts the whole argument, end of story. Anyone who has an objection, objects to one or more of the steps, or they object to the logic of the argument. Thus I split the rest of the article into 5, namely, those objections which question the logic nature of the argument (such as those who deny proof by contradiction), and those who object to other 4. End of story.

So I don't understand the split between "philosophical" and "mathematical".

You are very welcome to add "mathematical" objections to the article, but you need to supply references. Either peer-reviewed article, or a book (which by virtue of being a book as usually gone through a detailed review process, at least if an academic imprint such as Clarendon or Routledge &c).

You say e.g. that "Contemporary anti-Cantorians today are typically lay mathematicians (that is, not pure mathematicians), who professionally apply mathematics. Younger researchers in the field of artificial - intelligence seem to be especially attracted to the anti-Cantorian point - of view."

I deleted this because, while it is an interesting claim, it has not been verified. Who are these lay mathematicians? Who are the researchers in the field of artificial intelligence? You need to cite one. Dbuckner 20:19, 15 November 2005 (UTC)


 * I'm convinced that although you obviously know a great deal about Wittgenstein's opinions about Cantor's Theory, you really have little understanding of the issues from the point of view of the constructivists and applied mathematicians who adopt the anti-Cantorian view. Thus, I've taken the liberty of moving your version of this article to Philosophical objections to Cantor's Theory.

I want to emphasize that Dean Buckner is writing about a different topic from the original topic of this article. I have therefore, move his article to a new topic name. Please respect that, or at least explain why you don't.  David Petry 20:29, 16 November 2005 (UTC)

New Article
Dean Buckner's rewrite of this article actually changed the topic from the challenges to Cantor's Theory by the constructivists and applied mathematicians, to challenges from a philosophical point of view, especially from Wittgenstein. So, I have moved his article to a new place under the title "Philosophical objections to Cantor's Theory" David Petry 20:39, 16 November 2005 (UTC)


 * Well, I noticed the changes. I was more amused than outraged, really. From something that really didn't engage with Cantor's argument, to something that (in a sense) only engages with the diagonal argument, and ignores the wider implications. It is all still 'controversy' though; so I see no justification for the move. Charles Matthews 20:47, 16 November 2005 (UTC)
 * The article is not intended to "engage with Cantor's argument" (whatever that really means). What the anti-Cantorians object to is the acceptance of a completed infinity, which is implicit in Cantor's ideas. David Petry 00:26, 17 November 2005 (UTC)
 * It is reasonable to want to consider in one place the argument that set theory has no importance to practical mathematics, and yes, this is a distinct argument from Wittgenstein's. However the article still looks very much like a personal essay and is very POV; Dave needs to accept changes aimed at rectifying that.
 * I believe the article accurately reflects the views of a large number of people who believe that Cantor's Theory doesn't really belong in mathematics. It definitely is not merely my own personal point of view, which would not be appropriate in the Wikipedia. David Petry 00:26, 17 November 2005 (UTC)
 * For example, Sowa's "rant" (as Dave himself calls it) is presented unchallenged, even though it seems to exclude all of measure theory and functional analysis from "analysis". It should also be mentioned that the theory of Hilbert spaces has found application in quantum mechanics, and that this theory is developed using complete metric spaces, which are uncountable by Cantor's argument. --Trovatore 21:04, 16 November 2005 (UTC)
 * The purpose of presenting Sowa's rant was merely to illustrate the kinds of anti-Cantorian articles that are posted in discussion groups and mailing lists. Furthermore, the constructivists (especially Bishop and his followers) have done a very good job of developing measure theory, functional analysis, and Hilbert spaces. They have been able to show that the applicable parts of those topics can be done constructively. You seem to be unaware of that. To point out that complete metric spaces are uncountable is no more significant than pointing out that the set of reals is uncountable. David Petry 00:26, 17 November 2005 (UTC)
 * If you're claiming they have been able to show that "the applicable parts of those topics can be done constructively" as easily and clearly as they can be done set-theoretically, I simply don't believe you. I know lots of people have tried. No one has yet succeeded. Set theory is in fact eminently practical; its expositional and intuitive clarity is as yet unmatched for explaining these things. --Trovatore 01:27, 17 November 2005 (UTC)
 * First of all, the constructive way of thinking is not "intuitive" for people who have spent years studying classical set theory, for almost obvious reasons. Part of the reason that set theory seems to have superior expositional and intuitive clarity is that literally thousands of people have worked hard to give it that clarity, whereas only perhaps dozens of people have worked on the constructive counterparts. But futhermore, that "intuitive" clarity is somewhat illusional: it takes years of study to develop the proper intuitions. In many ways, the constructive mathematics is far more intuitive for those who apply mathematics. David Petry 21:28, 17 November 2005 (UTC)
 * I think you're kidding yourself. --Trovatore 21:35, 17 November 2005 (UTC)
 * Re CM's point about the article only engaging the diagonal argument: it's really the thin end of the wedge. I don't suppose that many people would accept the extravagant ontology of set theory without the diagonal argument telling you that no ceiling is high enough. --- Charles Stewart 21:47, 16 November 2005 (UTC)


 * Since the article collects together early doubts about set theory, and later doubts about the value of mathematical foundations without any attempt to show a historical continuity between these positions, I don't see on what grounds David Petry is saying that Dean Buckner's content is not appropriate here.  I see a case for using this article to document various negative positions against set theory, but if the article is to be Dave Petry's personal take on what he calls Cantorianism, then I'll list the article on AfD. --- Charles Stewart 21:36, 16 November 2005 (UTC)
 * I'm claiming that most of the anti-Cantorians on the internet have developed their ideas independently of the historical arguments, but then upon examination, it is seen that they are essentialy the same arguments. To show a "historical continuity" would be to make up something that is not there. On the other hand, the philosophical arguments that Buckner introduced are not the same, and hence belong in a separate article (we could start a separate section for the philosophical arguments, if it is seen as important to have only one article). Also, once again, the anti-Cantorian view is a minority view, but it is not merely my own personal view, and also, I have *never* used the term "Cantorianism" David Petry 00:26, 17 November 2005 (UTC)


 * I'm going to put my foot down now, David. You didn't in fact 'move' the Buckner version, it turns out. You copied-and-pasted it to a new page, where it therefore appears as your work. This is against all good practice here, because that page now appears without history or accreditation. In effect (ironically) you have claimed credit for the User:Dbuckner work. Yes, it was taking a liberty to do that. So, to sort this out, that page needs to be redirected here, and I'm going to revert to an earlier version so that we can get on with working here. Can I make it absolutely plain that neither you nor anyone else has the slightest ownership of this title: you are without a leg to stand on, in claiming that this article ought in any way to conform to what you had first in mind. Charles Matthews 08:09, 17 November 2005 (UTC)
 * I shouldn't have talked of listing the article on AfD, it was unnecessarily combative, so sorry to Dave. Charles M is quite right to reverse the farming out: it is a license requirement of the GDFL that content is attributed, and the current understanding of this is that we should maintain a connection between article content and edit history.  If you want to do this kind of thing in future, you have to ask an admin to do it, who has access to special tools.--- Charles Stewart 16:06, 17 November 2005 (UTC)
 * Well, I am new to Wikipedia, and I agree I didn't do things the right way. But something needs to be done. We can't keep playing a game of reverting to the article we each think is the better one. Also, your apology is certainly accepted, but your actions do suggest that you are reacting to this situation emotionally, which needs explanation. David Petry 21:28, 17 November 2005 (UTC)
 * I'm not sure why I got hot under the collar, but I guess I was irritated by the disrespect shown your fellow editors, especially Dean, by this move. You surely knew it would be unwelcome, but you did it anyway.  This isn't, or at least it shouldn't be, a battle.  --- Charles Stewart 23:26, 17 November 2005 (UTC)

Similarity and continuity
From Dave's point above:
 * I'm claiming that most of the anti-Cantorians on the internet have developed their ideas independently of the historical arguments, but then upon examination, it is seen that they are essentialy the same arguments. - To have conceptual similarity without historical continutity is fine for your argument, although that this is what you were claiming is not at all obvious from the article. There's a raft of problems you have to overcome in making your claims of which some are:
 * There's no consensus that Cantor's conception of set theory is like the modern one grounded in the cumulative hierarchy. Certainly our formal understanding of set theory is dramatically different; Charles Stewart 21:41, 17 November 2005 (UTC)
 * The irrelevance of set theory to applications hasn't been changed by the new formal understandings David Petry 21:04, 17 November 2005 (UTC)
 * One of the things that has come out of our sharpened understanding is how much of normal mathematics depends on strong claims, as Trovatore has documented. Are you aware of Harvey Friedman's body of work showing that large cardinal axioms are needed to establish a broad class of claims of normal mathematics and computer science (eg. finite Ramsey theory, Boolean relation theory, greedy algorithms)? Charles Stewart 21:41, 17 November 2005 (UTC)
 * Set theory implies the existence of a world beyond what can be observed. It is very difficult to believe that any applicable mathematics is dependent upon believing in the existence of that world. I don't know enough about the topics you mention to know exactly what is going on, but I also have no reason to believe that Friedman et al are doing anything more than playing clever games. David Petry 23:50, 17 November 2005 (UTC)
 * While there is a modern group whose objections to set theory have some similarity to the early objections, namely the constructivists, most of the names that you cite in the article are not (Bishop is one, Kline and Thurston are not). That you perceive a similarity is not fit for inclusion in an encyclopedia article;
 * I gather that you and others view pure mathematics as the only real mathematics. The fact that many accomplished applied mathematicians have spoken out about the irrelevance of Cantor's Theory is noteworthy. I included quotes from Kline and Thurston because they have spoken out about the unreality of the set theoretic world (and that's the core issue), not because I'm suggesting that they are trying to find replacement foundations, or necessarily support the constructivists. David Petry 21:04, 17 November 2005 (UTC)
 * As it happens, I am sympathetic to the Martin-Loef crowd, but I think the work of Harvey Friedman demonstrates the need for strong foundations in mathematics, and I think there is a problem of how to reconcile the kind of account the former say mathematics should have, with the kind of mathematics the latter say we need. --- Charles Stewart 21:41, 17 November 2005 (UTC)
 * Why not treat these "strong foundations" as speculative? Certainly they're not saying that those who apply mathematics need to learn this new theory. David Petry 23:50, 17 November 2005 (UTC)
 * Your claim of an anti-Cantorian movement on the internet is going to be hard to render in an acceptable form.
 * I'm certainly not claiming there is a "movement" in the sense of collusion. But rather, lots of people independently come to the conclusion that Cantor's Theory is irrelevant to the mathematics that has applications, and the internet is the only place they can speak out about their views (for most of them) David Petry 21:04, 17 November 2005 (UTC)
 * Earlier in this page I listed a number of alternate ways that somneone might be dubious about the achievements of set theory. I think we should have an article along those lines, which would look quite different from the current article. --- Charles Stewart 21:41, 17 November 2005 (UTC)
 * If you can't establish your claim in a way that justifies an article devoted to it, then you are going to have to accept changes to the scope of the article.
 * But I think I have. David Petry 21:04, 17 November 2005 (UTC)
 * Convincing yourself is not enough. You have to establish a consensus. --- Charles Stewart 21:41, 17 November 2005 (UTC)
 * That's not in the spirit of WP, is it? This is not just an encyclopedia for "consensus" views. David Petry 23:50, 17 November 2005 (UTC)
 * It most certainly is in the spirit of wikipedia to create a text which all reasonable people agree with. Now of course it is valuable to have an article on minority views but it must still be written in an NPOV style.  Barnaby dawson 10:04, 18 November 2005 (UTC)
 * Sure, so now let's have a really big argument over what a "reasonable" person is :-) David Petry 00:15, 19 November 2005 (UTC)


 * On the other hand, the philosophical arguments that Buckner introduced are not the same, and hence belong in a separate article (we could start a separate section for the philosophical arguments, if it is seen as important to have only one article). - I agree with Dean when he says that there is no principled distinction between mathematical and philosophical argumentation here.
 * What exactly does "no principled distinction" mean? Neither the constructivists nor the classical mathematicians pay any attention to Wittgenstein's arguments, as far as I know David Petry 21:04, 17 November 2005 (UTC)
 * Much of Dean's arguments were a simple carving up of the problem space, as accessible to mathematicians as philosophers. Wittgenstein is important in the Martin-Loef school, which can be seen as a dissident fraction of the Bishop school.
 * Did you actually read Dean's rewrite? David Petry 23:50, 17 November 2005 (UTC)


 * Also, once again, the anti-Cantorian view is a minority view, but it is not merely my own personal view, and also, I have *never* used the term Cantorianism  - I've no probem with minority, dissenting or simply awkward views providing they are documented in an appropriate manner. The problem is making the content of articles complement each other, so that WP as a whole (or at least it's logic pages) is a coherent resource.  You've brought together a nice collection of quotes and ideas, and WP will have better coverage of dissenting views from set-theoretic foundationalism, but you will have to give up the idea that you are in control of the shape of this article.   I'd recommend spending a bit of time with some existing logic articles if you want to get a better idea of what this involves (I see you edited Georg Cantor) --- Charles Stewart 16:06, 17 November 2005 (UTC)
 * I'm afraid that your desire for "making the content of articles complement each other, so that WP as a whole (or at least it's logic pages) is a coherent resource" might lead you to justify censorship. The anti-Cantorians are claiming that the set theorist and logicians are creating a silly fantasy world. That will never "complement" other articles, but it is worthy of inclusion in WP.
 * If you define the anti-Cantorians to mean those people who think set theory is a silly fantasy world, very well, but it is a neologism. But it is precisely my point that there is no such coherent group of people as the anti-Cantorians picked out in the article, but rather mathematicians who object to set theory do so for a multiplicity of reasons, that not all of them agree with all the reasons, and that the reasons change through the history of the subject.
 * But that's not true. As far as I can see, there is exactly one central reason for objecting to Cantor's Theory: the world of the completed infinite is a fantasy world which cannot be relevant to understanding the world we live in. David Petry 23:50, 17 November 2005 (UTC)
 * Let me remind you of my point: the argument I was making is the argument that most applied mathematicians make when they question the relevance of Cantor's Theory. Buckner muddled up that argument and replaced it with philosophical arguments which those applied mathematicians would view as gibber. If Buckner had merely added a new section with "the philosphers also have reasons to object to Cantor's Theory ...", I wouldn't have had any objection. David Petry 21:04, 17 November 2005 (UTC)
 * Again, I think that there is no such argument that most applied mathematicians make, and that while you might say of a given argument that it is more mathematical or philosophical in nature, they all appear all mixed: mathematicians know some philosophical ones and some philosophers know some mathematical reasons. --- Charles Stewart 21:41, 17 November 2005 (UTC)


 * I'm going to let you guys "win" for now. I think there's something very wrong with what's going on, but that's life. David Petry 23:50, 17 November 2005 (UTC)

Reverting
I've have reverted to the article before Buckner's revisions. If you want his views to be in the article, please start a new section on philosphical objections to Cantor's Theory. Certainly his explanation of Cantor's proof has no business being in this article David Petry 21:28, 17 November 2005 (UTC)

Doubts about set theory
Earlier I wrote:
 * I think the material here divides into three sets of concerns: concern about the consistency of ZFC; concern about whether the axioms of ZFC, even if consistent, should be taken at "face value", and concern about the role of set theory in mathematical education and exposition. "Cantorianism" doesn't cover the issues well; better might be an article formulated as a question, such as "Is set theory the right foundation for mathematics?". I'm guessing tha with the right name, the problem of how to deal with the content will become easier.

which I repeat here, since I think it gets to the crux of the matter here. Most ways of combining these three concerns make sense, and one might have a number of reasons for any of these concerns. I say this to highlight the multiplicity of possible critical views. Dave's original article had what I understood to be exemplars of all three concerns: Poincare was concerned about consistency, Thurston about the Platonist interpretation of the system, and Kline about the ill effects of set theory.

This is the heart of my objection to the original article: it paints a unity where there is diversity. If Dave believes there is a unity, after all, fine. But the NPOV requirement means that this article cannot be an essay on what follows given Dave is right.

I'd like Dave to stay around and comment on changes. I don't think he is so far from being a good editor, and there is no shame in not grasping what WP policy demands from articles stright off. And I'm personally sorry if I have been discouraging in the tone of my comments: it is certainly not because I found the material to be dull. --- Charles Stewart 00:20, 18 November 2005 (UTC)


 * Just want to add my voice to Charles' here (although Dave's been contributing, so maybe it's not necessary). There is absolutely a place for these arguments in the Wikipedia. They just have to be presented in an encyclopedic and NPOV way, not as a personal essay, and with serious counterarguments given (an early version offered as a counterargument "For the Cantorians, the bottom line is that they have won", which is quite beside the point). --Trovatore 02:20, 19 November 2005 (UTC)

From Dean
David
 * > the argument I was making is the argument that most applied mathematicians
 * > make when they question the relevance of Cantor's Theory.

What argument? What is this argument that applied mathematicians are making? I don't see any arguments in the sense that philosophers use the word "argument" i.e. giving reasons for conclusions.


 * Maybe the "argument" is too simple for philosophers to see it as an argument. The "argument" is that mathematics is a tool that helps us understand the world we live in (i.e. the observable world, what we call "reality"). However, the "completed" infinite is not something we can observe (if you doubt that, the burden is on you to explain how we can possibly observe a completed infinite); we can only observe finite approximations to the infinite, and hence, the only access we have to the infinite is through the notion of limits (i.e. only a potential infinite truly "exists"). And hence, the notion of a completed infinite, which is implicit in Cantor's Theory, isn't really part of mathematics (it's philosophy or theology, as Kronecker said). That is the *whole* argument. And in your (Dean's) rewrite of the article, that simple message has been almost completely drowned out by philosophical "arguments" David Petry 00:02, 19 November 2005 (UTC)


 * Can you clarify? One also cannot "observe" the number 2, although one can observe 2 items and come to a pragmatic, common understanding as to the meaning of "two-ness".  I don't believe we can "observe" (with a microscope, telescope) a "countable infinity" of objects, although many mathematicians seem to have a pragmatic, common understanding of "contable-infinity-ness". Indeed, high-school students don't suffer too much over the limit 1/n, n-> infty; it seems accepted often enough, and no one particularly is miffed that the limit appears to be "zero".  Certainly, the "Buddha-nature" of "uncountable infinity" is harder to comprehend, but that's not an argument against it.  In math, "observation" is always noetic, and the observed objects are denizens of the noosphere.  linas


 * As I further explained in the article, it's reasonable to think of the computer as a microscope which helps us peer deeply into the world of computation, and then we can accept that world of computation as something real and concrete (not merely "noetic" in nature). And then mathematics is the science which studies the phenomena seen in that world of computation. That "paradigm" is sufficient for all of applicable mathematics. The idea here is that the "real" world is observable, and so if mathematics is to be the language we use to understand the real world, then mathematics itself should have the notion of "observability" incorporated in a fundamental way. (Note: that is in fact my own personal way of explaining things. I've never heard anyone else explain things in exactly that way. However, when I do explain things that way, other people who accept the "anti-Cantorian" view usually accept my explanation as a good one) David Petry 01:42, 19 November 2005 (UTC)
 * Dave keeps repeating these things, but the fact is that uncountable set theory passes all the criteria that he sets. It does make predictions about the behavior of computers. For example, it predicts that a properly programmed and properly functioning computer will not discover a proof of 0=1 from ZFC. The (realist) set theorist has a very natural explanation for this observation, namely that ZFC is true--true of real, though non-physical, objects, the sets of the von Neumann universe. For Dave, on the other hand, it must simply be a "brute fact"; he seems to have hopes of finding an explanation that doesn't go outside his ontology, but he hasn't done so, and nor has anyone else.
 * Well, we've discussed this at length in sci.logic, without coming to any agreement, so we won't have any better luck now. It's the assertion that "ZFC is consistent", which is a constructively meaningful statement, that makes the predictions, not ZFC itself. To say that we can "know" that ZFC is consistent by "believing" that ZFC is true, is to live in a dream world. David Petry 23:22, 19 November 2005 (UTC)
 * Your first prediction above I think is probably correct :-/ But for those who haven't followed sci.logic; "ZFC is consistent" is a more complicated/less motivated claim than that ZFC is simply true. Passing to the consistency is analogous to the "epicycle" explanation of planetary orbits. --Trovatore 23:27, 19 November 2005 (UTC)
 * Oh, I should have said, all criteria except the one that all the objects considered (I guess that was tacitly intended) should be observable. But even physics doesn't satisfy that criterion; it explains observable phenomena all the time via inferred objects that are not directly observable. Dave hasn't, to my knowledge, made a case why mathematics should be different. --Trovatore 02:28, 19 November 2005 (UTC)
 * As for his claim that the computabilist paradigm suffices for all applicable mathematics, that is just simply false. At best, again, he has hopes that it will do so at some time in the future. --Trovatore 02:07, 19 November 2005 (UTC)


 * The argument keeps getting circular. This just rehashes the conversation above: among other things, computers are not just manipulators of floating point numbers, they are also theorem provers and machines for evaluating formal grammars. All sorts of (self-consistent) axioms can be used to generate finite statements in a finite amount of time; including grammars with axioms of infinity. I can "observe" theorems about uncountable infinities in a finite amount of time with a computer. If one beleives some set of axioms (about infinity, for example) to be inconsistent, one can use a computer to examine the first million or billion theorems, looking for a contradiction. If a contradiction is not found in a finite amount of time, what should one conclude? I conclude that using the word "computer" in this argument only clouds the issue. linas 22:56, 19 November 2005 (UTC)

If someone writes "In some ways, the foundations of mathematics has an air of unreality" that is an opinion not an argument.


 * It's a revealing opinion David Petry 00:02, 19 November 2005 (UTC)


 * > Buckner muddled up that argument and replaced it with philosophical
 * > arguments which those applied mathematicians would view as gibber.

All I did was to state what most people to take Cantor's theory to be (i.e. that there are sets which have a greater number than the set of finite integers, and by implication there is no ceiling to the number that can apply to any set) It's common practice in an encyclopedia to tell a possibly uninformed reader what the subject matter of the article is about.


 * I defined "Cantor's Theory" in the very first sentence. "Cantor's THEOREM" plays a central role in the theory, but the two must not be equated. David Petry 00:02, 19 November 2005 (UTC)

Is it that I have not correctly characterised "Cantor's Theory"? My biographical dictionary lists his main achievement as follows:

"He worked out a highly original theory of the infinite, extending the concept of cardinal number and ordinal numbers to infinite sets [i.e. Hume's Principle]"

Then I divide all possible objections into

1. Those that are naïve or silly (such as arguments that beg the question, plus Cantor's own comments on the question-begging nature of such arguments.  Plus Hodges' comments.

David, Wilfrid Hodges is a philosopher but he is also a respected mathematical logician who has written standard texts on model theory.

2. Objections to "Hume's Principle". Most philosophical objections fall into this category - they do not deny Cantor's argument, but they deny it has anything to do with the concept "number".

3. Objections to "Cantor's Theorem".

3. Objections to the Axiom of Infinity. Some of the quotations mentioned by David, such as "Actual infinity does not exist." belong here, although set theory does not have the concept of "actual infinity" as such.

All "finitist" objections - made by philosophers and mathematicians alike - fall into this category.

4. Objections to power set.

I mention one argument of W's, someone can surely supply others?

--

To repeat, any objection to "Cantor's Theory" has to fall into one of these classes. Otherwise one accepts all the assumptions of his argument, and therefore accepts his argument. I don't see where the distinction between "philosophical" and "mathematical" here. And where is the mathematical (or logical) content in the following quotations, all given originally by David.

"Later generations will regard [Cantor's] set theory as a disease from which one has recovered"

"Set theory is based on polite lies, things we agree on even though we know they're not true. In some ways, the foundations of mathematics has an air of unreality."

" I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there"

I'm happy to remove the quotations by Wittgenstein if it's felt these give it too much of a philosophical flavour. The main point was to give the article some structure. If you are objecting to a theory, you have to say what's wrong with it.

Charles:
 * > better might be an article formulated as a question, such as "Is set
 * > theory the right foundation for mathematics?".

But that would be a different article. "Controversy over Cantor's Theory" is the title of this article.

Dbuckner 19:11, 18 November 2005 (UTC)

I like the current article in that I finally understand it. The earlier versions stated that there was a controversy, but were vague as to what the actual controversy was. Now its a lot more clear. linas 23:03, 18 November 2005 (UTC)

-

P>>>> And hence, the notion of a completed infinite, which is implicit in Cantor's Theory, isn't really part of mathematics (it's philosophy or theology, as Kronecker said). That is the *whole* argument. And in your (Dean's) rewrite of the article, that simple message has been almost completely drowned out by philosophical "arguments" >>>>

That is wrong. Cantor's theory of the "completed infinite" is what we would now call the axiom of infinity, as is clear from sec 11 of the Grundlagen. I have already created a space in the article for objections to that Axiom of Infinity.


 * You are not right, but I don't think there's anything I could say that would force you to understand the point I have been making. David Petry 23:22, 19 November 2005 (UTC)

One of these objections is Wittgenstein's. I  have already said that, if you feel there is over-emphasis on Wittgenstein, then remove it. We could always have a separate article on W's finitist views.

>>>> Note: that is in fact my own personal way of explaining things. I've never heard anyone else explain things in exactly that way. However, when I do explain things that way, other people who accept the "anti-Cantorian" view usually accept my explanation as a good one)

This has already been pointed out. Wikipedia is not a primary source of information.


 * I don't believe I am providing any new (primary) information, although the way I present the information is my own (but that's always the case when we write). Nothing I have said would be seen as "new" to most anti_Cantorians. David Petry 23:22, 19 November 2005 (UTC)

>>>>> I defined "Cantor's Theory" in the very first sentence. "Cantor's THEOREM" plays a central role in the theory, but the two must not be equated. David Petry 00:02, 19 November 2005 (UTC)

You said it was a form of naïve set theory. That is questionable, as someone has already pointed out. In any case, if you are asked to sum up Cantor's main theory in one bullet point, even if there are a few bullet to choose from, and you are forced to choose, you will say that


 * First of all, Charles Mathews edited what I actually wrote, and I didn't change it back, because I felt it was close enough. That's certainly adding an element of confusion to this discussion. David Petry 23:22, 19 November 2005 (UTC)


 * Cantor's theory is that there are transfinite numbers

Why do you make the point about "Cantor's THEOREM". In my rewrite of the article, I pointed out that very few people have objected to Cantor's THEOREM itself. Cantor's THEOREM merely says that a set cannot be placed in one one correspondence with all its subsets. Cantor's THEOREM says nothing about completed infinities or numbers or anything like that.


 * The question is, why do *you* even mention Cantor's Theorem? That merely distracts attention away from the real issues. David Petry 23:22, 19 November 2005 (UTC)

P >>>> And in your (Dean's) rewrite of the article, that simple message has been almost completely drowned out by philosophical "arguments"

Then remove them if you like, but keep the framework. It is essential to begin with a short explanation of what "Cantor's Theory" is, then an explanation of what the controversy is about, preferably with some sort of grouping of the objections.

>>>> the notion of a completed infinite, which is implicit in Cantor's Theory, isn't really part of mathematics (it's philosophy or theology, as Kronecker said).

So why are you complaining that I included "philosophical" arguments?

user: dbuckner


 * This has gotten to be such a mess that I am ready to wash my hands of the whole situation, and try to have my name removed from the history page and the discussion page, if that's possible. (Is it?). To sum up everything I wanted to say in the article, I'd say: "The anti-Cantorians view mathematics as a tool which helps us create models of real world phenomena, and (according to the anti-Cantorians) Cantor's Theory has no role to play in that view of mathematics". If you are going to take out or hide that simple assertion (which you have done, in my opinion), then I want nothing to do with the article. David Petry 23:22, 19 November 2005 (UTC)

Dave, on the one hand, I like that one sentence summary: its short and seems to be to the point. On the other hand, it begs the question "what is Cantor's theory?" (and how does it have no role)? If I look at the world of physics, I see lots of persistent and recurring questions about infinities (e.g. in quantum mechanics) and differentiability and continuity (e.g. in dyanamical systems and fractals) and both paths lead to topology which seems to be based on set theory and Cantorian ideas. Even after all of the above debate, I still don't understand what "Cantor's theory" is, and in what way its not applicable to physics. linas 04:58, 20 November 2005 (UTC)
 * If mathematics is merely formal word games, then there is nothing wrong with Cantor's Theory, and in fact, as you point out, it's hard to see how Cantor's Theory is anything different from any other part of mathematics. But if you ask yourself the question, "what is the 'reality' underlying mathematics?", then eventually you will start to understand what this argument is all about. I don't think I can usefully say any more than that. David Petry 22:54, 20 November 2005 (UTC)


 * (Responding to Dave, not Linas.) As far as I know, short of deletion of the article, there is no provision for removing entries from the history. There is a somewhat unusual recourse, which requires the intervention of a "bureaucrat" (a step above an administrator; analogous in some ways to a Catholic bishop), to have your username changed, including past references to it. See Changing username. Seems a bit extreme to me; no reasonable person will attribute the views of an article to you just because you contributed to it, but you asked so I responded. Alternatively, you can nominate the article at WP:AfD, and though I doubt your reason would be adjudged grounds for deletion, people might vote for the proposal to start over clean. (I wouldn't, but I might not vote against it either.)
 * But I do wish you wouldn't take this tack. It seems to me that you're taking things a bit personally. Obviously I disagree with you sharply on the merits, but it doesn't mean I don't want your views represented or that I don't want you to contribute. --Trovatore 05:36, 20 November 2005 (UTC)
 * If someone looks at the history page, they might be led to conclude that I have given my "seal of approval" to the article, which I definitely haven't. I've merely given up on the article as hopeless, so I'd prefer not to have my name appear on the history page. I don't see any point to trying to have the article deleted. As far as taking things "personally", I did put some thought and effort into the article I wrote, and I was even proud of the article, and now I feel that all that effort was a total waste of time, although I am not entirely surprise by what has happened. What more can I say? David Petry 22:54, 20 November 2005 (UTC)


 * I know people never read these things. But where it says, under the edit box, If you don't want your writing to be edited and redistributed by others, do not submit it, it means every word. Charles Matthews 22:58, 20 November 2005 (UTC)