Talk:Controversy over Cantor's theory/Archive 2

“Carl Friedrich Gauss” on Cantor

The very concept of a "countably infinite set" as employed by Cantor is suspect. What can be counted can not possibly be infinite, and what is infinite can not be counted. Karl Freidrich Gauss pointed this out when he wrote "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics". The flaw in the 'diagonal argument' has precisely this defect. Cantor asks us to think of his "diagonal" as having been 'drawn' through all members of the reals before he begins his contruction.

- see also: Ardeshir Mehta's article titled "A Simple Argument Against Cantor's Diagonal Proceedure": http://homepage.mac.com/ardeshir/ArgumentAgainstCantor.html —Preceding unsigned comment added by 66.67.96.142 (talk) 17:55, 28 May 2008 (UTC)

That reference is without merit. It's not affiliated with any university, academic organisation, or noted mathematician. The single community college link it offers is broken. Oh, and it makes a number of first year mistakes, such as:

• Claiming that the diagonally constructed decimal appears on the list after the n'th decimal. Where then, on the list? The k'th number on the list, for k even bigger? Wouldn't the k'th decimal places of these numbers differ?
• Claiming that a single point has positive non-zero probability within the interval [0,1]. Singletons sets have zero measure. Also, there's no such thing as an ifintesimally small probibility, only zero probability.
• Claiming that natural numbers are expressible as infinitely ling strings of digits. They're arbitrarily long but *finite*; the string 222..... doesn't represent any natural number.

The one working link it contains goes to some other irrelevant unaffiliated personal home page, which carries its own glaring mistakes:

• Claiming that a list contains all infinite binary strings when it's quite clearly missing 01010101.... (yet admitting that non-repeating decimals are necessary).
• Assuming that 11111111... appears somewhere on the list, when by construction the k'th number only zeros after the log_2(k)'th digit.

Gauss' incorectness can be forgiven since he died in 1855, some 20 years before Cantor published his results. The reference has no merit today. Oh, and by the way, the link *does* admit the sensibility of a countably infinite set - it uses the terminology in the opening sentence. Endomorphic (talk) 16:02, 21 August 2008 (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?

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 Wikipedia: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)

Why be a smart aleck? I certainly read and understood and accepted that line. I wrote an article which criticises the established view, and then "editors" who accept the established view distorted what I wrote to promote their own views. There is something wrong with that. David Petry 21:48, 21 November 2005 (UTC)

It is quite standard here, in the sense that the emphasis of articles may change. I think you might accept that some of the people showing interest in this page have actual expertise. That they may have something to add that does not 'distort'. I can tell you why your position is relatively weak: you are a single-issue editor. You are not a contributor over a range of topics. This is not something impossible to accommodate in Wikipedia. But it does mean that the learning curve has to be very steep. There is a strong 'house style', if you like. Clearly those who are familiar with the editing style and conventions find it much easier than those who say 'something very wrong here', when all that is happening is easily explicable and, frankly, fairly normal. I am not a 'smart aleck', in that I have admin responsibilities and am bound to explain patiently the facts on the ground, to anyone who appears to need that explanation. (The older form of that message used the term 'mercilessly', which was truer to life here.) I don't actually want the article dedicated to the kind of philosophical logic that dominates the User:Dbuckner version. I do want to see some article that deals even-handedly with the topic. I'm not convinced that your perpective has much original in it, in fact, but I am committed in principle to having this article, like everything else I get involved in, brought up to an acceptable standard of writing and balance. Charles Matthews 22:54, 21 November 2005 (UTC)

Revision by Dbuckner

I have cleaned up some parts, added a quote from Cantor, and removed a number of the Wittgenstein quotations, which clearly were out of balance with the rest.

>>> Matthews I don't actually want the article dedicated to the kind of philosophical logic that dominates the User:Dbuckner version.

Then by all means add to the article to reduce the imbalance. I can only write about what I know best. But I still say, if A, B, C imply D, and there is controversy about D, then there must be controversy about one or more of A, B, C. If you question the reality of the transfinite numbers, you have to question the assumptions that prove they are real. Dbuckner 20:42, 23 November 2005 (UTC)

So I think Dave is not so interested in arguing about whether transfinite numbers are real (though he clearly has an opinion on the subject). His main claim seems to be that they're not part of mathematics, or anyway not part of practical mathematics. Not so much "they don't exist", but "just shut up about them, already". (Hey, I didn't claim I was a good David's advocate....) --Trovatore 22:10, 23 November 2005 (UTC)

Preface

On the contrary the preface whose addition I reverted is blatantly POV. If material from it is to be included consensus should be reached on the talk page for what is being added. Barnaby dawson 09:46, 7 December 2005 (UTC)

It is not *my* POV, but rather a POV shared by many "anti-Cantorians", and that is what the article is about. The idea that Cantor's ideas about infinity should be accepted as mathematics is also a "POV", which happens to be the current mainstream POV. David Petry 20:31, 7 December 2005 (UTC)

But the idea of the NPOV policy is that one should not be able to make significant inferences from how an article is written, to the writer's own opinions. Starting with the article's title, which should be the only given, I'd say that POV writing from a 'Cantorian' would be to dismiss the idea that there is controversy. It is certainly not POV to say that Cantor's set theory, as axiomatised, is widely accepted as a basis for mathematics. Charles Matthews 23:13, 7 December 2005 (UTC)

That's debatable. It seems to me that the NPOV policy is that one's opinions shouldn't distort the content. In what I have written, everything that is POV, I clearly state as being POV. By quoting Wilfred Hodges right at the start, Buckner clearly is setting a tone for the article, and that tone is a strong opinion. It's not neutral. You are applying the NPOV policy in a very one-sided manner - in favor of the side you happen to agree with David Petry 00:12, 8 December 2005 (UTC)

Well, no, I thought I was explaining to you the implications of the policy. You can debate the widespread acceptance of set theory all you like; but if you deny the acceptance in a WP article, that probably violates NPOV. I don't 'happen to agree with' the acceptance; I know as a factual matter what proportion of books in a good mathematical library accept set theory, and what proportion do not. Charles Matthews 22:54, 12 December 2005 (UTC)

I don't really get your point. I fully acknowledge that set theory is widely accepted as a basis for mathematics, as I pointed out in the original article I wrote. But the point of the article that is that there are some serious mathematicians and applied mathematicians who believe that set theory contains an element of fantasy which doesn't really belong in mathematics. Just because the vast majority of mathematicians do accept set theory as a basis for mathematics, does not mean that there is no place in the Wikipedia for an article about the minority of mathematicians who don't accept it. David Petry 00:03, 14 December 2005 (UTC)

I dispute that there is such a group of "anti-Cantorians" whose views are being expressed accurately by David Petry's preface. See my remarks on the usage of this terminology below. I recommend just deleting this preface, as the rest of the article doesn't appear so bad. --C S (Talk) 13:14, 9 February 2006 (UTC)

Ok, so is there anybody besides David Petry that supports this preface being in the article? --C S (Talk) 18:56, 11 February 2006 (UTC)

Anti-Cantorians

Your objection is that the rewrite misunderstands the ant-Cantorian argument. Indeed, you explicitly claim "Those who don't understand the anti-Cantorian argument generally insist that the anti-Cantorians must surely have some problem understanding the diagonal argument".

But I am baffled by what the anti-Cantorian point of view actually is. Here are six anti-Cantorian theses that you mention in you preface to the rewritten essay, with the authors identified in brackets.

• 1. Cantor introduced into mathematics, an element of fantasy that should be expunged (certain unnamed pure and applied mathematicians)
  2. Cantor applied the mathematics of finite sets to infinite sets (Weyl)
  3. Cantor built a theory which can be proven true when our domain of discourse is finite sets, and then claimed without actual proof that we can retain logical consistency if we extend our domain of discourse to include infinite sets. (the author)
  4. Though Cantor's Theory appears to be logically consistent there is, nevertheless, something wrong with it (unnamed anti-Cantorians)
  5. Cantor's Theory does not agree with the purpose of mathematics, which is to model real-world phenomena (unnamed anti-Cantorians)
  6. Cantor's theory introduced the notion of a completed infinity, which doesn't belong in mathematics (Gauss)

Is this a good summary of the AC position?

Dbuckner 21:42, 12 December 2005 (UTC)

Of course, I'm baffled by your bafflement. Here's the way I like to explain the "anti-Cantorian" position, although I can't really put it in the article since it seems to be "original" with me, at least to some extent. We can think of the computer as a microscope which helps us peer deeply into the world of computation, and then mathematics is the theory which makes falsifiable predictions about the phenomena seen in that world. When we do that, we are inexorably led to view mathematics the way the constructivists view mathematics. And when we do that, we notice that Cantor's Theory does not make falsifiable predictions about the phenomena seen in the world of computation, and hence it is not part of the "science" of mathematics (i.e. "science" requires falsifiability), and it is precisely the "science" of mathematics which has the potential to help us model phenomena in the real world. And, when looked at from that point of view, the comments made about Cantor's Theory (and the completed infinite) by guys like Gauss, Kronecker, Poincare, Brouwer, Bishop, Bridges, and Jaynes, make perfect sense, and the six "theses" you mention are all seen to be merely slightly different ways of saying essentially the same thing.

And let me add that when you write "Logician Wilfrid Hodges has commented on the energy devoted to refuting this 'harmless little argument'. What had it done to anyone to make them angry with it?", you are mocking and trivializing (and misunderstanding) the anti-Cantorian position. What you have written is definitely not written from a neutral point of view. David Petry 23:52, 13 December 2005 (UTC)

where I'm baffled is that sometimes (like now) you appear to accept Cantor's Theory that there are transfinite numbers, and you accept his argument for it. Your point is simply that there are areas of mathematics to which his argument does not apply. But then how is there any controversy over Cantor's Theory that there are transfinite numbers? And do the 'anti-Cantorians' who write in newsgroups actually agree with the view as you have characterised it? The question is simple. Do anti-Cantorians agree with the claim that there are transfinite numbers or not? DEan

It will probably seem to you that I am avoiding the question, but I really am not. As I pointed out in the article I wrote, for the anti-Cantorians, a statement about infinitary concepts makes sense if only if there is a way to translate the statement into an assertion having implications for concrete, finitary concepts. In other words, it wouldn't be honest for me to answer your question with a yes or no, since we have very different ideas about what the question even means. And as far as your question about the anti-Cantorians who write in newsgroups goes, the answer is: some do, but why is that question important? David Petry 21:30, 14 December 2005 (UTC)

PS you may be interested in this

http://www.math.rutgers.edu/~zeilberg/Opinion68.html

Yes, indeed, that was interesting. Thanks. If we ever get this mess of an article sorted out, we should include some kind of quote from Zeilberger -maybe, "It is a Paradise of Fools, and besides feels more like Hell". David Petry 21:30, 14 December 2005 (UTC)

Remove claim not supported by given quotes of Thurston and Kline

I removed:

Others believe that the assumptions of set theory lead to conclusions that are unreal or absurd.

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. (William P. Thurston)

[The pure mathematicians] have followed a gleam that has led them out of this world... The fact that mathematics is valuable because it contributes to the understanding and mastery of of nature has been lost sight of... the work of the idealist who ignores reality will not survive." (Kline, 1982)

The given claim of "...lead to conclusions that are unreal or absurd" is not claimed in the given quotes. Neither Thurston or Kline appear to be sayiing this. In fact, Thurston only makes the claim that there is "an air of unreality" about foundations, not that it leads to certain kinds of conclusions. Additionally, Kline only asserts that "the idealist who ignores reality will not survive"; he says nothing about whether an idealist of that sort's conclusions are of a certain kind.

One should also note that the Thurston quote is from an article in SciAm that should not be considered an authoritative source; they took his words out of context and misinterpreted some of what he said. He does not support the article's spin and title of "The death of proof?", as can clearly be seen from his article, "Proof and progress in mathematics", Bulletin of the AMS. He does believe in mathematical proof, just not in a formal notion of proof that does not involve human understanding. Even if one believes the SciAm article, there is nothing in the article about him being against set theory, Cantor, or supporting David Petry's claims. --C S (Talk) 12:36, 9 February 2006 (UTC)

I'm pretty sure that Thurston has never claimed that he was misquoted by Horgan David Petry 23:29, 10 February 2006 (UTC)

I didn't say "misquoted"; I said "misinterpreted" and words taken "out of context". And it'll be interesting if you say Thurston never claimed that, as you don't know him and I've talked to him about this! --C S (Talk) 17:58, 11 February 2006 (UTC)

Kline's "translation" of Poincare's alleged quote

The removed endnote is:

The quote "Later generations will regard set theory as a disease from

which one has recovered" is from Kline[1982], and is apparently his translation of a quote from Poincaré's speech "The future of mathematics" given in 1908. There has been considerable dispute about what Poincaré actually intended to imply. Another translation reads "I think, [...] that it is important never to introduce any conception which may not be completely defined by a finite number of words. Whatever may be the remedy adopted, we can promise ourselves the joy of the physician called in to follow a beautiful pathological case." So Poincaré's proposed cure for the disease, "never to introduce any conception which may not be completely defined by a finite number of words" would undermine Cantor's seminal idea underlying set theory; he was calling set theory a disease.

This is mistaken as there is nothing like Kline's quote from his cited document by Poincare. In fact, Kline's mistake (he apparently relied on another source quoting the document) is well-known to math historians (and detailed in a Math Intelligencer article). The second quote given above as "another translation" was brought to David Petry's attention in this sci.math post. Here it's pointed out that the closest anybody has been able to find to Kline's "quote" is this comment by Poincare that indicates he thinks of set theory as "a beautiful pathological case" that it should be a "joy" to remedy. In the same discussion, Keith Ramsay also makes it clear that Poincare's philosophy and opinions on Cantor's set theory are quite different than Davis Petry's potrayal.

The entire quote, by the way is:

"... it has come to pass that we have encountered certain paradoxes, certain apparent contradictions that would have delighted Zeno the Eleatic and the school of Megara. And then each must seek the remedy. For my part, I think, and I am not the only one, that the important thing is never to introduce entities not completely definable in a finite number of words. Whatever be the cure adopted, we may promise ourselves the joy of the doctor called in to follow a beautiful pathologic case."

Changes the context a bit, eh? --C S (Talk) 12:54, 9 February 2006 (UTC)

Usage of "anti-Cantorians"

This term, "anti-Cantorians", should be considered as part of original research by David Petry. This usage is unsourced, and it is not indicated how or where it was used by critics of Cantor; it really is the tip of the iceberge as it appears to me that the page is a real mess, consisting of a mix of OR by Petry and various edits by people trying to clean it up to something resembling a reasonable article. --C S (Talk) 13:05, 9 February 2006 (UTC)

We've already discussed this in the talk pages. Buckner has given us at least one reference where the term "anti-Cantorian" appears, and besides, the meaning of the term is quite obvious, given that the term "Cantorian" is well used.

I agree that the page is a big mess. At this point, the only part of the article that I care about is the preface. The rest is messed up beyond repair, as I see it. Also, I don't think that you should have removed the endnote, nor the quotes from Thurston and Kline, but I'm not going to worry about it.

I'd really like you (Chan-Ho) to read the following link http://www.math.rutgers.edu/~zeilberg/Opinion68.html which should prove to you that there really are "anti-Cantorians" out there who are also well-respected mathematicians. David Petry 21:56, 9 February 2006 (UTC)

Huh. I've liked some (not all) of Zeilberg's other opinions, but he does use strong language. I admit I still don't understand what the anti-Cantorians object to, and that even after studying Conway's games. Zeilberg seems to say "I don't like countable infinity, and detest the others even more", but as a practical mathematician solving integrals and series, I don't understand what's not to like. linas 00:24, 10 February 2006 (UTC)

I agree with David Petry that the term is not a neologism (though since Zeilberger does not use the term, he's not chosen the right weapon to defend himself here). However, I do think there is an original research aspect here: David is trying to make out that there is a coherent, consistent, well-thought out school of thought that can be called anti-Canotorianism. I think that many mathematicians have an allergic reaction to foundational questions, and this manifests itself in a plurality of ways, with little intellectual or historical consistency. But I've said this all before: see Talk:Controversy over Cantor's theory/Archive1#Doubts about set theory. --- Charles Stewart^(talk) 15:13, 10 February 2006 (UTC)

Since apparently I have been unclear, let me clear up some confusion and explain what I mean by my comments asking "how and where" the term "anti-Cantorian" is used and about a group "whose views are being expressed accurately by David Petry's preface". I dispute that there is a cohesive group of anti-Cantorians who, as Charles Stewart has phrased, have a "well-thought out school of thought". In particular, it's not at all clear and probably not true, really, that David Petry's philosophy of anti-Cantorianism is all that similar to those of Zeilberger and others. What I see is David attempting to portray these others such as Thurston, Kline, etc., by finding excerpts and listing weblinks, as part of his group of "anti-Cantorians" although there is plenty of evidence that their philosophies are not compatible with his. That is a pretty strong indication to me of OR. --C S (Talk) 16:20, 10 February 2006 (UTC)

We've discussed this stuff before. I am not claiming that there is a "coherent, consistent, well-thought out school of thought that can be called anti-Canotorianism", but rather I am merely pointing out that there are many mathematicians who (often independently) come to the conclusion that Cantorian set theory includes an element of fantasy (I think this clearly includes Thurston and Kline, by the way), and also that the ones who do think it through, tend to agree with the basic idea summarized by Hermann Weyl's quote. I also think that that is enough to justify a Wikipedia article, and I also think that Buckner's revision of the article tries to dismiss all objection the Cantorian set theory as a misunderstanding of Cantor's theorem, which is a mockery of serious people, and is extremely POV. David Petry 23:28, 10 February 2006 (UTC)

--

Consensus has two common meanings. One is a general agreement among the members of a given group or community. The other is as a theory and practice of getting such agreements (for infomation on the practice of achieving formal consensus, see Consensus decision-making).

Achieving consensus requires serious treatment of every group member's considered opinion. Once a decision is made it is important to trust in members' discretion in follow-up action. In the ideal case, those who wish to take up some action want to hear those who oppose it, because they count on the fact that the ensuing debate will improve the consensus. In theory, action without resolution of considered opposition will be rare and done with attention to minimize damage to relationships.

Where has this wikipedic definition of consensus over the preface been reached???? karl malbrain

Petry's claim on "no consensus"

David Petry, and one mysterious Karl Malbrain (with a total of three edits), has claimed there is no consensus to remove the preface. What I see are more than a few editors that don't want the preface, and are a little too polite to remove it themselves. So I guess it's come down to this: people are going to have to voice agreement or disagreement with my revert, otherwise David (and Karl) get to dictate what consensus is. Anyway, I'm leaving for Scotland, so that's it for now. --C S (Talk) 02:56, 11 March 2006 (UTC)

Indicate whether or not you support removing the preface:

• Support. --C S (Talk) 02:56, 11 March 2006 (UTC)
• Do not Support. As I understand it, Wikipedia is not a place where the majority can silence the minority view, so what's the point of this vote?David Petry 22:43, 13 March 2006 (UTC)

Comment: I'll take this as a tacit admission that you recognize you are the minority here. --C S (Talk) 07:04, 18 March 2006 (UTC)

• Equivocate. I sympathize to some extent with David's claim that the article without the preface fails to represent the view that contemporary set theory has lost all connection with the real world, and there is no doubt that this is a distinct current of thought held by some serious people. (They're wrong, but that's beside the point.) On the other hand David has clearly constructed his own gloss on the thought of various disparate people, which is original research unless someone else has isolated this supposed commonality among them. Moreover I think he's throwing various net.kooks into the mix (plenty of the latter really don't understand the diagonal argument; David keeps hoping they'll eventually turn out to be supporters of his philosophy, but it's a forlorn hope, as the truth is that they just don't get it). --Trovatore 03:16, 11 March 2006 (UTC)

Comment: I have no objection to valid subjects, e.g. the kind of good article this article could become. But I think despite your equivocation, we agree that the current state really is unacceptable. --C S (Talk) 07:04, 18 March 2006 (UTC)

• Response Yeah, I agree with you there. Ideally I'd like Dave to fix it, by sourcing the material in the literature and removing the personal-essay wording. I'm not certain where to find such source material but the place I'd look, as I've tried to suggest to him, is philosophy journals; whether he considers it philosophy or not, the people writing seriously about it probably do. If he flat refuses and insists on treating the "preface" as his blog, then we can consider what to do about it (I'm not sure the rest of the article has a good rationale for a separate existence from older articles). But I really would like to avoid such a confrontational outcome; I don't agree with Dave at all, but he's not a bad guy and I think he could contribute. --Trovatore 06:50, 19 March 2006 (UTC)

• • First of all, if you state without hesitation that "they're wrong", perhaps that indicates that you don't understand what they are saying, and hence, you are not really qualified to say to that I am throwing my own gloss on their thoughts. Second, could you explain your comment that I am throwing various net.kooks into the mix? David Petry 22:43, 13 March 2006 (UTC)
    • I state without hesitation that they're wrong because they're wrong, not because I don't understand them. Set theory does have a connection with the real world. It makes falsifiable predictions about real-world observations, that have not been falsified. The simplest explanation of these observations is that the objects of discourse of set theory really exist, as real, though non-physical, objects.
    • As for the net.kooks, I'm talking about people like Nathan, whom you tried to rehabilitate in a noble but doomed and thankless undertaking. He really genuinely never did get it, and I don't think you've come to terms with that. --Trovatore 22:49, 13 March 2006 (UTC)
      • Oh. But that's absurd. At first, when I thought there was a chance that he really was 11 years old, I did try to explain the issues to him, but after just a few emails, he admitted he wasn't just a kid, and I've never attempted any more "rehabilitation". David Petry
        • Fair enough. Maybe you weren't counting him, or AP, or any of the psychoceramics who (on Usenet) massively outnumber any Sohas or Zeilerbergers who might drop by. But then who are you counting? Take each of those paragraphs that starts "anti-Cantorians believe" or "from the anti-Cantorian point of view"–is the individual claim in that paragraph the view of Soha? Zeilerberger? Weyl? Gauss? Who exactly? Do you have any evidence that all these people subscribe to all your claimed anti-Cantorian positions? If your evidence is your own compilation of their writings, then that's still OR, so a better question is, can you find some writer who's abstracted these ideas from the writings of the people you mention?
        • I'm not saying it's always forbidden to do a certain amount of analysis of other people's thought for the reader. But I think you go way way way beyond that, attributing a specific set of positions to a school, when it's not clear that this school as you conceive it even exists. There's no doubt that there are various people holding various of these positions, but that doesn't constitute a school.
          • I have stated several times already that there is no *school* that I'm aware of, but rather, the article is based on comments made by various people whose views should be taken seriously. To spend much time arguing against Cantor's theory is a fool's task, and serious people seldom do more than make passing comments to the effect that Cantorian set theory has gone beyond what can be considered reality, and that the advent of the computer has given us a better idea of what the reality is. I believe that is enough to warrant a few paragraphs in the Wikipedia. David Petry 02:52, 15 March 2006 (UTC)
        • If it does exist, someone else will have noticed it. Poke around in the journals and find some cites. That would make a good article. The preface as it stands clearly violates NOR and NPOV. --Trovatore 02:56, 14 March 2006 (UTC)
          • In (many) years past, I have seen articles in, for example, the Mathematical Intelligencer, where a discussion of the reality of Cantorian set theory takes place. I doubt that such a discussion would be considered a suitable topic for the vast majority of mathematical journals. I think you're trying to apply NOR and NPOV in a very strict and self-serving way, and not the way in which such policies were intended. David Petry 02:52, 15 March 2006 (UTC)
            • There are lots and lots of discussions of the reality or otherwise of set theory. You'll find most of them in philosophy journals, I think (your insistence on distinguishing your argument from "philosophical" ones is frankly bizarre). You may well be able to find something in those sources that resembles the point you're making. I hope you'll make the effort, because right now what you've got is a documentation of a social phenomenon that only you claim to have noticed (at least in the form in which you present it). I don't think it's at all a stretch to say that that's outside policy.
              • First of all, I personally am not interested in philosophical arguments, and I would much prefer to keep them separate from (practical) mathematical arguments. Let me remind you about how I became interested in this topic. When I was in graduate school, the idea which I thought was really worth pursuing was the idea of teaching computers (artificial intelligences) to really and truly understand mathematics. This lead to the idea that the connection between mathematics and computation should be explored within the foundations of mathematics, and once I started exploring that idea, it became quite clear that conventional set theory contains elements of fantasy. Then, with a little more research, I came to realize that at least a few other mathematicians have been saying essentially the same thing I have been saying, although not in exactly the same words that I use. I believe that if you truly did understand what I am saying, you would agree that I am saying almost exactly what others are also saying, but I don't think you do understand what I am saying. The situation is similar to a devout religious person being unable to understand criticisms of his religion. I think you have been blinded by too much study of a fantasy world. You know how to reason about the fantasy world, and then conclude that since you can reason about it consistently, it must be real. But it's still a fantasy in the sense that it cannot be relevant to the task of helping us understand the world in which we live. It cannot be used to create models of the physical world. David Petry 21:20, 15 March 2006 (UTC)
                • Your argument above is philosophical; it's really strange that you won't admit that. "Practical mathematical arguments" do not discuss what is real and what isn't; as soon as you bring that question up, you're on the philosopher's turf. Maybe you have too limited an idea of what philosophy is?
                  • When a scientists asserts that religion is distinct from science, and explains exactly what he means by that, is he engaging in philosophy? I don't think so, and I think most people agree with me. All I'm saying is that the mathematics that is relevant to understanding the world we live in, is itself scientific in nature, and set theory introduces what is essentially an element of theology (i.e. a mythology about a world beyond what exists from a scientific point of view) into mathematics (and hence changes the definition of mathematics) David Petry 23:35, 15 March 2006 (UTC)
                    • Yes, of course, such a scientist is engaging in philosophy. It's not even a close call. He's discussing the boundaries of science; science itself can't do that. About the rest of it, as you know, I agree that mathematics is relevant to the world and is scientific, but disagree that that excludes set theory. When we discuss these points we are doing philosophy, not mathematics. --Trovatore 01:01, 16 March 2006 (UTC)
                • But yes, in fact, set theory can be relevant to the task of understanding the world in which we live, and can be and is used to give an infrastructure for models of the physical world. However that's all beside the point here. Even if you were completely right about all this, you still couldn't write a personal essay about it and post it here. But you may well be able to find good, reviewed sources that explain your philosophical viewpoint, and then you can report what they say. I hope you will do that. It really would be an addition to WP. --Trovatore 21:52, 15 March 2006 (UTC)
            • I'm really not against you here, Dave. I'd like to see you bring this up to standard; I think there really could be an interesting article on this subject. But it has to be much more grounded in published material, and much less personal. The reader really shouldn't be able to tell who wrote an encyclopedia article, whereas to anyone familiar with your writings, you have your byline in every paragraph. --Trovatore 04:25, 15 March 2006 (UTC)
              • I can't say you're wrong, but I'm not the only one responsible for writing this article. If you see a better way to write the article without removing the few ideas that I want to keep in the article, then please do so. David Petry 21:20, 15 March 2006 (UTC)
            • I assume you are still talking about the preface and not the complete article. Here's a definition from Wikipedia: "A preface (Med. Latin prefatia, for classical praefatio, praefari, to speak beforehand) is an introduction to a book, also any preliminary or introductory statement." This strikes me as SUBJECTIVE content. karl m
  • Further comment You can look up Karl Malbrain on Google Groups. I'd love to hear what, if anything, you think he actually believes. I've never been able to decide if he's serious or not, nor whether he has any point of view of his own, or simply enjoys sarcasm for its own sake. --Trovatore 03:19, 11 March 2006 (UTC)
• Support Barnaby dawson 10:10, 13 March 2006 (UTC)

I believe the question is premature and support the wikipedia definition of consensus -- a GENERAL AGREEMENT among participants. This does not mean an agreement generally among most participants (which is majority rule), but a GENERAL AGREEMENT among all participants. -- karl m

David Petry has just put the preface back. He stated in his edit comment that the result of the vote was 2 Yes 2 No and 1 abstain. This is not correct we have 2 Yes 1 No and 1 abstain (we have in addition 1 comment on the voting system). Which is as close to a consensus as possible (given the number of voters). 1 Vote against is to be expected in any disputed vote.

Of course we should work towards the goal of consensus. But we cannot argue that we should choose option X above option Y because Y has no consensus in favor. This is not a good decision procedure because it may be that neither X nor Y have a consensus in favor (probably the case here (if we define consensus as overwealming support)).

But, we DON'T define consensus this way here. See definition from Wikipedia above. karl m

I shall delete the preface but I shall also expand the introduction and response section to include a little of the former preface. Hopefully this will be regarded as a reasonable compromise. Barnaby dawson 12:37, 19 March 2006 (UTC)

Well, I did my part, but if other people have basically lost interest in discussing this (as I suspect they have) or in participating in this poll, then we'll just have to say basically there is no consensus for anything. What's strange is that from reading the prior discussions I'm sure there is a kind of consensus, but if the majority of editors refuse to participate, then there's nothing to be done. I'm got other things to do than take on this crusade against OR, and I don't care enough to keep this little part of Wikipedia from turning into crap, as my general sentiment is that it takes way too much effort to stop that kind of thing from happening. I'm no longer watching this page. Whatever happens is fine with me. --C S (Talk) 20:26, 19 March 2006 (UTC)

Original research

My general impression to date has been that David Petry is attempting to push a certain brand of original research as if it were gospel truth. Wikipedia is not the place for these types of activities. I strongly urge David to write up an article for American Mathematical Monthly, and see if he can stir up a debate there. If there is some success in that formal setting, then perhaps a synopsis could be warrented here. But right now, my general impression is that David doesn't quite understand the theory, and is converting his energies into original research and POV pushing. This is not an acceptable behaviour pattern on WP. linas 05:17, 20 March 2006 (UTC)

• As I see it, the only "original research" I am doing is finding new ways to explain the concepts so that people who don't know the technical language can understand them. I don't believe that is the kind of "original research" that the Wikipedia disallows.

• Consider, if you will, the following quote from Harvey Friedman (who is widely regarded as a leader in the foundations of mathematics)

An absolutely crucial issue for the foundations of mathematics is whether there is any significant use of set theoretic methods for "real mathematics". [...] In particular, mathematics is really Pi01. YES, Pi01. When it is Pi02 or Pi03, it BEGS to become Pi01 by placing upper bounds on the existential quantifiers. Look at Fields medals, prize winning work, million dollar problems, etcetera. Sure, sometimes there are real numbers and continuous functions, but usually it is clear by approximation that what is going on is very finitary. E.g., finite simplicial complexes. And when that is not clear, time and time again when the problems are solved, there is a Pi01 essence that is the hard part that easily implies the full result. So the CONTENT is Pi01. And even when that is perhaps debatable, the cases people are really interested in push us down to Pi01. http://www.cs.nyu.edu/pipermail/fom/2006-January/009526.html

• What I am doing is replacing "Pi01" with something like "has clear computational meaning", and then explaining what that means. Again, what I am doing is not "original research" in the sense that Wikipedia disallows.

• Anyway, the article as it stands is rubbish. It is of no use to anyone at all. It is a propaganda piece for the pro-Cantorian view. It has been written by people who do not understand the issues. I intend to do what I can to change it. David Petry 19:58, 20 March 2006 (UTC)

I have reverted the intro because it is imho biased and the list of quotations excessive. We already have a quotes section. This section wants a few well selected quotes. The other quotes may be used to back up an argument but they will not enhance the article itself. Let me explain in what ways I regard it as biased:

While axiomatic set theory is almost universally accepted as the gold standard of mathematical proof, there are, nevertheless, numerous pure mathematicians, logicians, and applied mathematics who point out that Cantorian set theory goes far beyond what is needed for "real" mathematics, or "scientifically applicable" mathematics, or "core" mathematics. That is, "real" mathematics has close connections to computation, but the world of sets going far beyond that is a fantasy world. Some mathematicians argue it should be expunged.

• Here I think the the term "real" as applied to mathematics, the concept of "scientically applicable", the concept of "core" mathematics, the precise definition of computation and the idea that set theory is a fantasy world are all subjective and therefore the use of "point out", which implies truth, is inappropriate. Furthermore I think that there is no coherent group of people who hold all (or even most) of the views presented in this paragraph.

First of all, the words "real", "core", "scientifically applicable" are words that leading researchers such as Friedman and Feferman actually use. They believe they are "pointing out" something, even if you don't. You may not see any coherence to the group of people in the quotes I have presented, but I do. Why is your view superior to mine? David Petry 01:36, 27 March 2006 (UTC)

I'm not saying my view is preferable. However NPOV requires that you present views as views and not as facts. Some people do not think that any subset of sentences of ZF is the core of mathematics, the only real mathematics or the only scientifically applicable mathematics. You're "seeing" a coherent group of people holding a point of view" does not a make such a group appropriate material for an encylopedia article. It seems to me that the closest approximation to such a group might be the finitists. If finitists are not the subject of this article the burden of proof therefore lies on you to demonstrate that a group exists distinct from them and hold a coherent set of beliefs. Barnaby dawson 17:33, 29 March 2006 (UTC)

I'm claiming that there is a great deal of "coherence" about what the problem is. There is in fact very little agreement among the people who agree about the problem about what the solution is. But the point of the article is merely to point out that some people (i.e. leading researchers) see a problem, without taking a stand on what the solution is. David Petry 01:06, 30 March 2006 (UTC)

So this is your belief. I do not see an identifiable group holding opinions dinstinct from the finitists. I certainly do not think we should have an encylopedia article about such a group without some scholarship to back it up. We could have an encyclopedia article deeling with general objections to ZFC (unlike Trovatore I do regard ZFC as being the de facto foundation of mathematics) or even objections to the concept of set as foundational (perhaps closer to what Trovatore would want). However, I still feel you need to justify the existence of this page as encyclopedic. Right now this article is essentially an analysis of objections to cantor's diagonal argument. Right now the article has a split personality between finitist like ideas and objections to cantor's diagonal argument (some of them certainly not finitist arguments). Barnaby dawson 08:03, 30 March 2006 (UTC)

The search for a foundation for mathematics which does not include the ethereal sets of Cantorian set theory is an area of active research. Unfortunately, there do not seem to be any articles in the mainstream peer-reviewed literature on this topic that present the topic in way appropriate for an encyclopedia article, and on this basis, a majority of the editors of the Wikipedia are arguing that any attempt to explain the topic in terms appropriate for an encyclopedia amounts to original research, which is not allowed. Hence, for now, this article will consist of quotes from various mathematicians illustrating their rejection of Cantor's set theory, to be followed by an essay from the Cantorian point of view arguing that all rejection of Cantor's set theory is based on a misuderstanding os the diagonal argument.

• This is an encyclopedia. The pages ought not to comment on the process used to create them (with very few obvious exceptions such as wikipedia). To represent that something is active research and also that there is very little literature on the subject also seems strange to me. An article should not engage in self criticism.

The preface is intended to be temporary until this debate is settled. I only said there is very little literature that discusses the issues in a way appropriate for an encyclopedia article (i.e. I don't know of any review articles discussing and comparing a wide variety of related but disinct views) Certainly there are many highly technical articles taking a specific focus. David Petry 01:36, 27 March 2006 (UTC)

The important sentence is this one "a majority of the editors of the Wikipedia are arguing that any attempt to explain the topic in terms appropriate for an encyclopedia amounts to original research". Note the box we have at the top of the article. This is sufficient warning that the article is under development (furthermore it is carefully designed not to be in the actual text of the article). Barnaby dawson 17:33, 29 March 2006 (UTC)

The comments in the box do not give any hint about the nature of the dispute. David Petry 00:56, 30 March 2006 (UTC)

Thats why the box refers you to the article. Barnaby dawson 08:03, 30 March 2006 (UTC)

In conclusion I do not think this article needs a preface (it already has an introduction) and this current preface certainly shouldn't be allowed to stand. Barnaby dawson 18:25, 26 March 2006 (UTC)

what a mess

It occurs to me that there isn't anyone, including David, who actually likes the article. David would like it if allowed to present his own view unopposed, by which I don't necessarily mean his own view of mathematics, but his own view of how various classes of thinkers see it. That obviously isn't going to happen. (And, of course, shouldn't happen.)

Not "unopposed", but undistorted by people who don't understand the issues. David Petry 01:28, 27 March 2006 (UTC)

So is there any point in keeping the thing? Maybe we should just delete it and be done with it; I gather that even Dave would prefer that outcome to a non-"prefaced" version. Then, as someone suggested way back when, a new article could be written on various positions as to whether set theory is a good foundations for mathematics, which would provide space for significant "anti" arguments, including, if it can be sourced, the view that what comes out of set theory is lacking in practical importance.

(By the way, someone needs to take a look at the "philosophical objections" page; don't remember the name. Is it still around? We really should clean up the detritus from this little war.) --Trovatore 18:48, 26 March 2006 (UTC)

On the last point, never mind. The article is philosophical objections to Cantor's theory; it seems to have been deleted and then redirected here. --Trovatore 19:51, 26 March 2006 (UTC)

I concur with Trovatore. I think this article has a biased focus. However, there should certainly be an article on criticisms of ZFC as the foundation of mathematics. Let us see. We should have:

• Intuitionism
• Objections to AC, maybe some stuff on Solovay's model.
• Objections to Power set
• Objections to the axiom of infinity
• NF/NFU
• Objections to the concept of set as foundational
• Finitism and Ultrafinitism
• Other alternative logics such as brazilian logic
• Category theory as an alternative but compatible foundation

What should it be called? alternative foundations to mathematics? Barnaby dawson 21:28, 26 March 2006 (UTC)

But what Dawson proposes would definitely have a biased focus. David Petry 01:28, 27 March 2006 (UTC)

In what way might I ask? Barnaby dawson 17:25, 27 March 2006 (UTC)

It focuses on philosophical argumentation. The single idea I want to express in this article is that the ethereal world of Cantorian infinite sets goes far beyond what is needed for "real" mathematics. That is not a philosophical idea. David Petry 00:24, 28 March 2006 (UTC)

I think you will find very few people to agree with you that the question of what is and is not "real mathematics" can be anything but a philosophical question.

Mathematics plays a vital role in our society. It underlies our technology. It is used by engineers, physicists, economists, and computer scientists. "Real" mathematics is the mathematics used by such people. It is an observation in the real world that leads to the conclusion that Cantor's world of infinite sets has no role to play in "real" mathematics. That is not philosophy. David Petry 23:01, 28 March 2006 (UTC)

First, the claim that real mathematics is the sort used by such people, is likely to be viewed as a philosophical claim by others writing about it, whether you so categorize it or not. As for your claim about "an observation in the real world", this is not in fact what we observe.

However, I'm sure you can find non-crackpot people who claim it is. If they write about it systematically, as opposed to just making don't-bother-me-about-it kind of statements, it's likely to be in philosophical publications. If what you want to document is precisely the don't-bother-me-about-it reaction, then you probably have to find some sort of sociology-of-math type work. There are plenty of mathematicians who prefer not to think about philosophy at all; you could call them pragmatic without being pragmatists, but almost by definition they can't do any serious exposition of their own view of the matter, as they would then be moving into philosophy. (As a corollary—you personally are not one of these people.) --Trovatore 23:14, 28 March 2006 (UTC)

You've just identified a bias and weakness in the Wikipedia. You're claiming that people with the "don't bother me with rubbish" attitude should have no say in the Wikipedia, and I suspect most editors agree with you, even if not explicitly. But there's something wrong with that. What can be done about it? David Petry 00:07, 29 March 2006 (UTC)

Of course you're entitled, for your own use, to classify knowledge in whatever categories you like, so I'm not inclined to argue with you on this point. I do suggest, though, that other people who work seriously on the issues you discuss, probably think they're doing philosophy. If you want to find their work, the place to look would be philosophy journals. --Trovatore 02:12, 28 March 2006 (UTC)

What name should the new article have?

Any ideas on what name a general article on alternatives to (and criticisms of) ZFC should have? Barnaby dawson 11:33, 28 March 2006 (UTC)

So first of all I wouldn't focus on ZFC per se. I think it's a misconception that ZFC is used as foundations currently. It's really more the iterative hierarchy. Almost no one does formal axiomatic proofs, in any theory; the intended interpretation is the primary thing rather than the axiom set. --Trovatore 12:31, 28 March 2006 (UTC)

How about a general article on alternative foundations to mathematics? Do we already have such an article? Incidentally I would disagree with you on the ZFC point but we can have an article which allows us to discuss other foundations such as category theory and other first order set theories so we can disagree without danger of disagreeing on article content :) Barnaby dawson 08:08, 30 March 2006 (UTC)

My idea of the ideal article

Here's my idea of what the ideal article on this topic would be.

First, we start with the quote from Harvey Friedman, but rewrite it in non-technical language that would be accessible to, say, a bright high school student.

Then, we include a little bit of the history behind this topic, mentioning the quotes from Herman Weyl and Kronecker, and mentioning intuitionism and constructivism.

Then, we let the philosophers insert their take on the subject, but *clearly* labeling it as philosophy (and insisting they keep their comments short and simple). If they want to mention crank objections to Cantor's ideas, that would be fine, as long as they keep it short and simple.

And that's all. Any discussion of alternate foundations belongs in a separate article. Likewise, any serious philosophical discussions about objections to the axiom of infinity etc. (other than briefly mentioning that such arguments appear in the literature) belong in separate article. David Petry 23:21, 28 March 2006 (UTC)

Do you by any chance think Harvey Friedman is on your side on this? If so you've severely misinterpreted his work. He may very well believe that the core content of mathematics consists of Π⁰₁ statements. But he also believes that set theory has a key role in deriving such statements. A big part of the motivation of reverse mathematics is to quantify exactly how much such theory you need for what, thereby "deriving the axioms from the theorems". His more recent work extends this to a project called "Boolean relation theory" where he attempts to justify large cardinal axioms from such consequences. --Trovatore 23:29, 28 March 2006 (UTC)

No, I do not think Friedman is "on my side". But he states the case against Cantorian set theory so clearly that it's clear that he understands the issue. And, once again, it's the assertion that set theory (and large cardinal axioms, etc.) is consistent that leads to the Pi01 statements. A person could easily believe that Cantor introduced a fantasy world into mathematics, and also believe that Friedman is doing useful work. David Petry 23:56, 28 March 2006 (UTC)

Harvey believes very passionately in good guys and bad guys, and it's completely clear which one you would be. To adduce his work in support of your position, without mentioning that, seems seriously misleading. --Trovatore 00:35, 29 March 2006 (UTC)

Preface

A preface has been switching back and forth between existence and non-existence. I have been removing it and David Petry has been adding it. Neither of us has violated the 3RR due perhaps to our slow response times :)

David Petry argues that the preface shouldn't be removed without a consensus in favor of removing it. I could if I wished maintain that it shouldn't be added without a consensus in favor. We could decide this matter in several ways:

• Talk it over till one side convinces the other (already attempted)
• Try to find wikipedia policy resolving the matter (I would say NPOV David Petry would disagree)
• Resolve it on the basis of absolute consensus (Not possible as there is still argument)
• Resolve it on the basis of consensus (I suspect there is a consensus for removal but again DP would disagree)
• Resolve it on the basis of experience of wikipedia (Certainly would imply removal of preface but Don't think we should go there)
• Resolve it on the basis of experience outside of wikipedia (Don't think we should go there)
• Resolve it on the basis of the majority opinion (Would imply removal of the preface)

In short every sensible option has either been tried or would result in the removal of the preface (as it currently stands). Therefore I regard it as being disruptive behaviour to continue to add the preface. For one thing it necessitates our repeatedly removing it when we could be updating other articles. I have identified many problems with the preface. I can't really sensibly rewrite it as I don't see a coherent subject matter for it. However, I will not allow it to remain on the article without being edited by someone. I will support the creation of a new article. And I would support the changing of this article to one on "computationalism" if I can be convinced that such a group exists outside of David Petry's mind. Otherwise I think this article should morph into one on objections to cantors diagonal argument (which it would probably be quite good as). Barnaby dawson 08:12, 30 March 2006 (UTC)

Situation as I see it

There's very little interest in this article, but a few fanatics have decided that they simply are not going to allow a simple idea, clearly supported by quotes of prominent authorities, to appear in the Wikipedia. I am thoroughly disgusted by the behavior of those fanatics, but I don't see how to deal with them. It seems that they can win by the rules of the Wikipedia. I hate to see garbage in the Wikipedia, which the current article is, but for now, I give up. David Petry 02:38, 31 March 2006 (UTC)

Very bad intro

The first sentence is:

The pure mathematicians and applied mathematicians who object to Cantor's theory of sets claim that Cantor introduced into mathematics an element of fantasy which should be expunged.

How bad! This is definitely not the way to introduce a topic!!--Pokipsy76 13:09, 24 May 2006 (UTC)

Proposal for deletion

I can't see why this article exists. There is no particular movement disputing Cantor's theory — whatever "Cantor's theory" means. There are some cranks that don't like the proof |X| < |P(X)| and there are some folks (philosophically credible and otherwise) that don't like the axiom of infinity. There was a historical controversy involving Kronecker and others.

But there is not "a" controversy. In fact, there is not a clearly definable theory under discussion here (though the article suggests ZF). This is a collection of snippets (many of them not objecting to Cantor's theorem) that have no coherent theme at all.

I just don't see the point of this page. (Full disclosure: I am familiar with the original creator, David Petry, and I have some idea of his aim with this page. But that has to do with original "research" and so is irrelevant here.) Phiwum 14:44, 24 May 2006 (UTC)

I agree that the present article really doesn't belong in the Wikipedia. I don't see how anyone could get anything out of it.

I had a fairly specific purpose for writing the article in the first place. It is very clear that lots of people come to the conclusion that there is something wrong with the ideas introduced into mathematics by Cantor. Time and time again, people (generally not pure mathematicians, but often technically oriented, intelligent people) come into newsgroups like sci.math and sci.logic to complain that there is something fishy about Cantor's ideas about the transfinite. The purpose of the article was to provide such people with useful information that might help them understand what is going on, and hence, the endless and worthless battles in the newsgroups might come to an end.

Anyway, no article would be better than the current article, in my opinion. David Petry 00:51, 27 May 2006 (UTC)

I disagree. Such an article is usefull and the "anti-Cantor" position is an interesting piece of mathematical *history*. The use of AC [Axiom of Choice] may have some mathematical controversy, Cantor's ordinals and cardinals have *none*. The article should mention the initial objections of Wittgenstein et. al. which were subsequently abandonned once it was realised how rich and usefull Cantor's construction was, and how any "anti-Cantorians" would lose vast areas of mathematics and logic while gaining nothing in return. Endomorphic 00:05, 8 December 2006 (UTC)

I have nothing against a historical article of the sort you mention and I would not suggest deleting the article in its current form. When I proposed deletion earlier, it was for an utterly different kind of article, an unabashed collection of original half-thoughts from one particular self-described "anti-Cantorian". Phiwum 16:15, 8 December 2006 (UTC)

I have only seen the current state of the article, and I still saw it implicitly claim that some active serious mathematicians were 'anti-Cantorians'. I claim the article should be written with arguments against Cantor written in the past tense as such views are not held today. My main beef lies with statements like "From the start, Cantor's Theory was controversial among mathematicians and (later) philosophers", which seems to imply that there is still professional controversy, as oposed to something like "Cantor's radical new construction was initially met with scepticism by other prominent mathematicians and philosophers" or something to that effect. We could even throw in Cantor's quote "I see it, but I don't believe it!". There used the exist 'anti-Cantorians'; there aren't any now. Wittgenstein died in 1951; the 1982 reference from Kline seems only to be used to supply a quote from Poncaire, who died in 1912. The other sources are all pretty old. We should either provide some references to *current* researchers who strongly challenge the legitimacy of Cantor's construction, or rewrite the article to reflect the modern consensus. Endomorphic 21:04, 13 December 2006 (UTC)

I have heard rumors that some modern philosophers still dispute modern set theory. I have not come across any reputable criticisms in my own studies of philosophy of mathematics — aside from those who want to reject the axiom of infinity, which is a perfectly respectable position, but not particularly "anti-Cantorian".

If you want to edit the article to make it clear that the debate is long past, then I have no objections. If someone else wants to cite recent evidence that the debate is ongoing, then they are welcome to. Until then, I think your criticism makes sense. Phiwum 01:08, 14 December 2006 (UTC)

Finitism

Sorry to opine on the subject and not the article here, but I cannot resist. That a finite universe should necessitate a finite mathematics is preposterous! Not only would a finite mathematics desecrate the most primitive and simplest structure of mathematics, (i.e., Peano's Axioms), and make the whole thing terribly clumsy and awkward, but would also run the risk of total obsolescence in the face of empirical, environmental discoveries (e.g., the finite universe somehow expands and contains now a larger number of primitive units).

Also the argument that if it cannot be varified on a computer it is irrelevant to our world is equally preposterous. Again, mathematics deals with a realm that cannot be dealt with by the limitations of ANY SINGLE computer, THEREFORE can make predicitions (theorems) that will hold up against the experiments of century after century of more and more powerful computers.

Mathematics is not subordinate to the experimental results tomorrow will bring, or to the supply of resources available, is not a science but an art, and yet certainly is undeniably invaluable to our understanding of the physical universe. This infinite paradise that upsets a few antagonists and entertains many mathematicians explains so much of our universe, be it finite or infinite. And many others as well. The objections have no basis. MotherFunctor 07:33, 16 January 2007 (UTC)

True, there are a lot of those who do not see the difference between the physical constraints of reality and the unlimited nature of abstract mathematical concepts. Just because you can't actually count up to a googol doesn't mean the arithmetic concept of that number is meaningless or nonexistent. It's a "real" number in the abstract sense, but not in the physical sense. Numbers, sets, and other notions exist only as ideas in our minds, not as tangible objects of reality, so obviously they are not constrained to abide by the laws of physics. Whether or not our universe is infinite in size is utterly irrelevant to the mathematical concepts of infinity and transfinite cardinalities. I'm not sure if the article points out this distinction or not. — Loadmaster 20:51, 8 May 2007 (UTC)

Missing the point?

It seems to me that this isn't so much a disagreement over Cantor's "theory" as it is over the axioms that give rise to the theory's consistency in mainstream mathematics. Cantor's theorem is consistent if certain axioms are accepted; it's not if they aren't. In the end it's all an arbitrary choice, whichever axioms are most useful to your purpose. The system will always be incomplete no matter which ones you choose, so does it really matter? Incrediblub 09:56, 8 March 2007 (UTC)

Well, yes, it does matter. If you take Cantor's theory, you get a huge amount of structure and are able to do things like measure theory, topology, and functional analysis. If you reject Cantor, you end up shrugging and looking silly any time an infinity pops up. How many natural numbers are there? Cantor: Aleph zero. Wittgenstein: I can't count them. How many real numbers are there? Cantor: Aleph one. It's strictly more than Aleph zero. Wittgenstein: *blushes* Endomorphic 20:53, 8 March 2007 (UTC)

Um, just to keep things straight here — Cantor believed the cardinality of the reals was aleph-one, but he did not consider the question settled; he actively sought a proof and did not find it. Now we know that a particular widely-used axiomatic system, ZFC, will never settle the question—but that does not stop it from being an active area of inquiry, because the independence of CH from ZFC tells us more about ZFC than it tells us about CH. --Trovatore 21:14, 8 March 2007 (UTC)

It seems to me that Incrediblub must be confused when he says that Cantor's theorem is consistent if certain axioms are excepted and is not they aren't. You can't make a theory (I suppose incrediblub actually means a "theory", not a "theorem", as a theorem is just something provable from a theory, but never mind) inconsistent by taking axioms away. You can only make a theory inconsistent by adding new axioms. MichealT 20:13, 29 March 2007 (UTC)

I think Incrediblub is simply echoing a common complaint, that the "meaning" of Cantor's (or anyone else's) theorem is only as meaningful as the axioms it is based upon. Which is of course true, and other, just as "meaningful" theorems can be developed under different sets of axioms (different theories). But assuming you accept the basic axioms of set theory (i.e., you accept that they are "true"), Cantor's theorem is correct. You can't accept the axioms without also accepting the conclusions (theorems) drawn from them. So to reject Cantor's theorem is to ultimately reject the axioms it is derived from. Which is the point (I think) that Incrediblub was trying to make.

And yes, it does matter what we choose as axioms and the theorems resulting from those axioms, as Endomorphic said. It's not useful to ignore useful theorems. After all, Gödel showed that arithmetic is an incomplete system, yet we don't abandon it because of this. — Loadmaster 15:41, 9 April 2007 (UTC)

Guys, be a little careful with all this stuff about axiomatics, as it relates to Cantor. Cantor did not propose an axiomatic system at all, at least as ordinarily understood. --Trovatore 22:28, 9 April 2007 (UTC)

Strange use of Axiom of Infinity

The current article, under heading "Cantor's Argument, contains a bullet point that reads

"That there exists at least one infinite set of things, usually identified with the set of all finite whole numbers or "natural numbers". This assumption (not formally specified by Cantor) is captured in formal set theory by the axiom of infinity. This assumption allows us to prove, together with Cantor's theorem, that there exists at least one set that cannot be correlated one-to-one with all its subsets."

Well, we don't need that axiom/assumption to prove the conclusion, indeed it is easy to prove that there is no finite set that can be corelated one to one with the set of its subsets, so all the finite sets are examples of sets that can't (basically, a finite number is not equal any number that is more than it, and number of subsets of a set is at least one more than the number of elements of teh set). So why claim that the fairly heavy axiom of infinity allows us to prove the trivial point that at least one set can not be correlated with its subsets, when we have a hole pile of (trivila) example to hand with finite sets? MichealT 20:26, 29 March 2007 (UTC)

No. You say "it is easy to prove that there is no finite set that can be corelated one to one with the set of its subsets. Of course - Cantor's Theorem proves that. It proves that no set can be correlated one-one with its subsets, therefore no finite set can be correlated one-one with its subsets. But Cantor's Theorem does not prove that any set exists. The axiom of infinity (which asserts the existence of a null set and an infinite set) is necessary to assert the existence of any set at all. edward (buckner) 17:51, 5 April 2007 (UTC)

As far as I recall, standard first-order logic proves there exists a set x such that x = x. We do not need the axiom of infinity to prove that some set is smaller than its powerset. Phiwum 16:57, 9 April 2007 (UTC)

Discussions about ZFC aside, the "Cantor's Argument" section remains poorly organised. It could be made shorter, more concise, and more straighfoward. For instance, the axiom of infinity is not required to prove that any set is strictly smaller than the power set, but it's needed in this context to introduce an infinite set, whose power set is then demonstratobly a strictly "larger infinity". The article fails to communicate this. The line "There are a number of steps implicit in his argument" is awful; being a mathematician, Cantor's steps are quite explicit. Endomorphic 21:12, 9 April 2007 (UTC)

Comment

Comment: Why are the authors of Cantor's diagonal argument afraid to mention that many mathematicians, past and present, object to the diagonal argument and to Cantor's theory in general. Why is there no cross-link in that article to the "Controversy over Cantor's theory"? Intellectual honesty demands that this be fixed.--66.67.96.142 (talk) 17:24, 29 May 2008 (UTC) —Preceding unsigned comment added by 66.67.96.142 (talk)

Can you really cite notable present mathematicians objecting the diagonal argument?--Pokipsy76 (talk) 17:46, 29 May 2008 (UTC)

Pokipsy76: Elsewhere I find the assertion that "...something on the order of 90% or so of working mathematicians accept Cantorian set theory both in theory and in practice, to some extent. (Source: The Mathematical Experience, Davis/Hersh)". I haven't checked this assertion for accuracy, but if true, that's not very encouraging for Cantor. Suppose the Pythagorean theorem enjoyed the same level of support! —Preceding unsigned comment added by 66.67.96.142 (talk) 22:44, 29 May 2008 (UTC)

Wittgenstein

I never had much time for W, so I may be biased, but Wittgenstein explicitly denies Hume's principle, arguing that our concept of number depends essentially on counting. "Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one". makes no sense. If W explicitly denies Hume's principle, no-one has managed to find a quote saying so. Thinking of large as being closer to infinity is rather besides the point, here William M. Connolley (talk) 21:41, 29 May 2008 (UTC)

Nonsensical objections

This page sort of makes me laugh. The thing is, it's not that the philosophical objections to Cantor's theory are right or wrong, they are *nonsensical*. One problem is that sentences ending in "... exists." or "... does not exist" are nonsensical. It makes no sense to say "the set of all natural numbers exists" or "the set of all natural numbers does not exist". It does make sense to say "the set of all natural numbers exists in the set of all sets of natural numbers".

It also makes no sense to say "completed infinity" with respect to the set of natural numbers, for example (or any other infinite set in mathematics). The standard meaning of the phrases "natural number" and "set of natural numbers" is meant to be absent of spatiotemporality. That is the intended meaning. It makes no sense to talk of spatial or temporal properties of the things these (standardly interpreted) phrases refer to because they have none. However as soon as one says the word "completed", one is talking about the temporal, and in this context it makes no sense, being a conflation of the spatiotemporal and the non-spatiotemporal.

The only objection one could raise to this consideration is to say "the very notion of the non-spatiotemporal is incoherent", an objection which can be raised to absolutely any notion.

Assuming this objection is not raised, we have two main reasons why this sort of "philosophical objection" is itself incoherent, corresponding to the first and second paragraphs respectively. —Preceding unsigned comment added by 86.9.9.235 (talk) 11:50, 27 July 2008 (UTC)