Talk:Brouwer–Hilbert controversy

deletions, speedy deletions, AfD

Most of the material here is copied from the pages for Brouwer and Hilbert. This important foundational controversy deserves a page of its own in my opinion. If you differ, please express yourself here before jumping the gun with the above. Katzmik (talk) 13:37, 9 September 2008 (UTC)

How does this differ from the general topic of intuitionism in the early 20th century? That is, why is it named 'Brouwer-Hilbert' instead of 'Intuitionism-formalism'? It seems to me that the use of the two names personalizes the issue in a way that isn't accurate, as there were numerous other mathematicians supporting each side. — Carl (CBM · talk) 13:49, 10 September 2008 (UTC)

This certainly sounds like a valid objection. Perhaps we can get a few people to comment on this, perhaps including Bill Wvbailey. It seems to me that B and H were those who spearheaded the respective camps, and in fact the confrontation did turn personal. If a majority of editors disagree this page should be deleted. Katzmik (talk) 14:03, 10 September 2008 (UTC)

I looked through the intuitionism page again. It contains very few details on Brouwer-Hilbert. Katzmik (talk) 14:12, 10 September 2008 (UTC)

There were certainly some personal aspects (as will all human endeavors), but I expect the main purpose of this article is to cover the overall controversy, not just the personal aspects. From my point of view as a mathematical logician the controversy was just a manifestation of the debate about foundations of mathematics in the early 20th century. So I tend to view this article as a chance for an expanded version of Foundations_of_mathematics#Foundational_crisis. I'll probably be able to contribute quite a bit to the article, actually.

I think that a title like Foundational crisis in mathematics or Controversy on foundations of mathematics in the early 20th century would be better, since it conveys the overall subject of the article more accurately. By naming the article "Hilbert-Brouwer", we implicitly limit the scope to just these two people. — Carl (CBM · talk) 14:13, 10 September 2008 (UTC)

I think the second title would be fine if it were not so long. By mentioning the two main protagonists we certainly specify the period. Everybody readily recognizes that we are talking about foundations, as well. Nobody will mind if Brouwer-Hilbert controversy mentions other people, as well :) At any rate I am certainly looking forward to reading your contributions. Katzmik (talk) 14:20, 10 September 2008 (UTC)

I think the first (Foundational crisis in mathematics) is actually better; the time period can be clarified in the text, and there really was only one such crisis to date. — Carl (CBM · talk) 14:35, 10 September 2008 (UTC)

I think there were many crises, actually. Some of them were resolved my mutual agreement and are no longer considered a crisis, but certainly at the time they were. To consider one of the earliest examples I know, consider the controversy over the complex numbers a few centuries ago. Once it was realized that complex numbers are helpful in solving cubic and quartic equations, people got very upset over using such imaginary quantities. It does not really matter that today we all agree upon a solution. At the time it certainly seemed like a crisis. More recently, there was a crisis over the notion of infinity (I am sure you can place this better than I do historically). People got very upset about straightforward talk about infinity as if one knew what it was. There is a bit of a relic of this crisis even today as when people prefer to write "increases without bound" instead of "goes to infinity". The soul-searching discussions over the AC documented in Wagon's book (including Borel) are another example. What is a crisis? Katzmik (talk) 14:44, 10 September 2008 (UTC)

I don't disagree, but "the" foundational crisis was the debate in the early 20th century, spurred by the discovery of paradoxes (and by some bad proofs) in the 19th century.

By the way, regarding speedy deletion, I have have found that as long as you give one or two references, even the most esoteric topic is unlikely to be speedy deleted. Even one external link, like in Godel-Gentzen negative translation, can be enough. The key is just to make it patently obvious to a random viewer that the subject has already been covered somewhere else. This article shouldn't have any problems.

It will be a little while before I can try to add more content here; I need to get some books from the library, but more importantly I have to wait until I have time. Maybe this weekend. — Carl (CBM · talk) 15:03, 10 September 2008 (UTC)

[Written before an edit conflict]: Hmm. I see merit to both sides. I personally like the name "Brouwer-Hilbert controversy": (1) It's catchy and engaging, (2) All aspects -- the personal and the professional and the foundational -- could be covered in one place.

When Hilbert lost Weyl to the "dark side", he was pretty upset. There really was a personal animosity between the two . . . for which Brouwer as an old man ultimately paid a price in personal/professional isolation. As a young man Brouwer was nasty and weird and to make his mark came out in full cry against the aged Great Man -- this come at the end of Hilbert's career -- and these assaults by an young upstart (on his sacred Formalism) and the loss of disciples was upsetting. There may be an interesting tie-in with the Platonist Goedel who points out that his proofs are "constructive" -- so you know that this controversy was very much on his mind . . . and his gently-delivered assault on Hilbert's "programme" was fatal. Poor Hilbert: first he is pestered by Brouwer's assault and the loss of Weyl, then his "programme" is demolished by the constructivist/Platonist Goedel, and then the Nazis finish off his faculty. (I am confused/uncertain about the relationship between constructivism and Intuitionism -- does "constructivism" exist without Intuitionism at its core?). There may also be an interesting tie-in with the Cantorian notion of "completed infinites" -- Hilbert accepted this at least tacitly in his acceptance of non-constructive proof (double-negation extended over the infinite -> reductio ad absurdum extended over the infinite), and with the various "paradoxes", and with various failed attempts such as Finsler's. (To figure out what these tie-ins are, if they really exist, and refresh my failing memory I would have to go away and study the various Brouwer papers and commentary in van Heijenoort. This would be a big undertaking. ). BTW, does anyone know of any good books on the "foundational crisis"? Bill Wvbailey (talk) 15:22, 10 September 2008 (UTC)

Yes, constructivism is quite independent of Brouwer's intuitionism, although the motivations of the two programs are closely related. Bishop's work on constructive analysis, for example, does not carry the metaphysical connotations that Brouwer's intuitionism does. In contemporary formal work, people often use the term "intuitionistic" to mean "without excluded middle", but for Brouwer it meant far more than that.

As for books, I took the following from the library this morning:

• From Brouwer to Hilbert, Mancoso, ed.
• Gnomes in the Fog, Hesseling
• Oxford Handbook of Philosophy of Mathematics and Logic, Shapiro, ed.
• Mathematics: A concise history and philosophy, Anglin.

— Carl (CBM · talk) 16:18, 10 September 2008 (UTC)

Thanks. I've got Anglin, and I've seen references here at Wikipedia to Mancoso and Hesseling. I'll look into Bishop and Shipiro, too. This gives me a good excuse to bike up to the library on the hill. BillWvbailey (talk) 18:13, 10 September 2008 (UTC)

Added side-bar: I'm no fan of completed infinities. I'd prefer "increases without bound". Here's why: From the physicist's point of view there is a completed, countable boundary -- all the potential energy states of the universe. Of course, if there are more than one universe and they are accessible then there might be an infinite number, but not a completed infinity. As far as I'm concerned the issue is still with us. Bill Wvbailey (talk) 15:36, 10 September 2008 (UTC)

Apart from a vanishing number of finitists, the issue is completely settled within the mathematics community. — Carl (CBM · talk) 16:18, 10 September 2008 (UTC)

I'm probably a retro-radical finist/constructivist. But I'm not a mathematician, so I guess I can be such a thing without too-dire consequences to myself. Bill Wvbailey (talk) 18:13, 10 September 2008 (UTC)

tone of article

Hi, Thanks for your edits. I am a little concerned about the tone of the article. Hilbert is one of the most revered figures in mathematics. Any page about him should be more circumspect that the current version of the article. Katzmik (talk) 12:57, 14 September 2008 (UTC)

I don't think that awe of Hilbert is the main issue, but I agree there are serious issues with tone. We're looking for a professional presentation, and I don't think that section titles like "Conflict!" meet that goal. I'm planning to work on the article this afternoon, and I'll try to change the tone some at that time. — Carl (CBM · talk) 13:17, 14 September 2008 (UTC)

I agree the awe is not the main issue. What I was trying to point out, as I am sure you agree, is that wikipedia tends to represent the mainstream rather than searching for any kind of quixotic truth. Even if Hilbert were dead wrong and Brouwer completely right, such a viewpoint would not represent the mainstream and thus amount to POV pushing. Katzmik (talk) 13:47, 14 September 2008 (UTC)

There's no POV-pushing at all -- when I went into it I had no idea what I would find. I merely relied on the available sources to guide me, van H in particular. The issues at hand were quite complicated, and that comes out in both the primary and secondary references. Basically what happened is all documented (you sure can't complain that there aren't any sources). As for tone: oh well, The tone I used is certainly less formal that that of wikipedia, but c'est la vie -- so much for the notion of "compelling prose"; the tonemeisters can solve this vexing problem. What I wrote needs vetting, in particular the very long footnote which is in part redundant, and some of the stuff around axiomatic induction versus "primitive induction" and confusion around Goedel's proof V. I have no particular stake in this except I did enjoy doing the research and finding out what I found -- it was pretty interesting. And I think it's pretty damned thorough piece of work. (There is a more having to do with Logicists, but that might be off-topic a bit). (BTW: I'd suggest the vetters read the van H sources before doing too much vetting other than for tone and glaring inaccuracies). Bill Wvbailey (talk) 16:22, 14 September 2008 (UTC)

As far as van H goes, the dates 1979-1931 may be appropriate for an article about mathematical paradoxes, but in this case I think there may be a misprint :) Katzmik (talk) 08:55, 15 September 2008 (UTC)

It did appear as though van H had solved the problem of travel backwards in time. He was a marvelous mathematician and writer, but I don't think he was that clever. Bill Wvbailey (talk) 15:07, 15 September 2008 (UTC)

footnote 2

Can this be changed to a subsection? Katzmik (talk) 16:05, 17 September 2008 (UTC)

Yes, perhaps ... I wrote this footnote first, then found myself kind of re-writing it and expanding it into separate paragraphs. In a way the footnote is a summary of the "Deeper philosophic differences" section.

Or perhaps ... Because some of it is repetitive re the "Deeper philosophic differences" section, rather than giving it its own separate section, maybe parts of it could be distributed into the other paragraphs where its sub-topics more-naturally should be placed.

Or, if there is nothing being added by it, then the redundant pieces could be cut. Any approach would work as far as I'm concerned. Bill Wvbailey (talk) 18:45, 17 September 2008 (UTC)

The point missed by Hilbert

Somehow "the point missed by Hilbert and pointed out by Poincare" needs to be explained better. Katzmik (talk) 14:58, 23 September 2008 (UTC)

Informal tone, solecisms, etc

Katzmik, you wrote: "The section starting with a philosophical defeat, etc is somewhat informal in style. Could you try to eliminate solecisms when you get a chance?"

I'm not sure what you mean by "solecism". I had to look up the word. At least in wikiland's dictionary it means "errors in syntax". I don't see any errors in syntax. I'll remove the "I" and replace it with "one" to abstract the tone a tad, but I'm not sure what this stiffening is going to gain the reader. If you have specific areas that bother you please edit away, as you've been doing.

Tone-editing may be gilding lilies at this point. I'm more concerned about getting the facts, and accurate communication of what really happened, which is not so easy in this particular case, IMHO. For example, the writing of Anglin in his 1994 Mathematics: A Concise History and Philosphy seems off the mark; unfortunately we've used that reference in other articles (in general it's too simplistic). And Kleene 1952, while quite useful, does not discuss much other than application of the LoEM over the infinite. The best broad-brush, secondary-source writing that I've seen is the tiny summary paragraph by Dawson.

The following I'm writing to clear my own thinking as address issues in this article ... sorry ... plus I found some new stuff. This thing is growing hugely ... it's complicated ... can't seem to get my arms around it ...

There are several currents flowing in the "dispute". The first current was the broadest issue -- the philosophy (and I'm not sure what fancy name to give it, we need a philosopher here... "idealism?") in support of Hilbert's "axiomatic systems". Hilbert had achieved great success with his "axiomatization" of geometry, and he was pushing it further with his "problems for the 20th century -- into number theory and physics. It was here that he began to meet resistance. This breaks into four sub-issues: (1a) the notion of "formal axiomatic system" in general, and (1b) the notion of "formal axiom" (i.e. symbols on a page) versus "informal axiom" -- the intuitive idea, (1c) the specific choice of formal axioms to build the "formal axiomatic system" -- empirically-derived (Russell) versus "intuitive" (Brouwer) versus ?? (Hilbert), and (1d) as was beginning to be the case in physics (see more below), the abstraction of axioms and the notion of testing only the consequences of "the axiomatic structure" as a whole (e.g. Hilbert's notion of axiomatization and Weyl's complaint).

The second current had to do with Hilbert's adoption of the "completed infinity" in his analysis-proofs (and its use in set theory -- I guess... see the Goedel quote below), with constructivism and finitism coming in conflict with Cantor's philosophy of the infinite -- Cantor's philosphy was Catholic religion-based, which is interesting in its own right. I don't know the history well enough here, but apparently the finitism/constructivism came (via Aristotle, see quote below) from British Empiricism through Gauss through Kronecker through Russell, Poincare, Brouwer, etc. Whereas the "completed infinite" comes from ?? (Anglin points to Anaximander (610-540 B.C.), cf p. 14), it definitely does not come from Aristotle, who opposed the completed infinite (Anglin p. 64: "Aristotle was a staunch finitist"). So given this murky history, we have Hilbert adopting the "completed infinite" because it allowed him to come up with "clever proofs" that were based on the notion of "non-constructive existence proof" (cf Kleene p. 50).

For example, I just found this in Kolmogorov 1925:

§5. An excellent example of a proposition unprovable without the help of an illegitmate use of the principle of excluded middle is given by Brouwer (1920); he shows that it cannot be considered proved that every real number has an infinite decimal expansion. He even exhib its a definite number for which it is not know whether it has a first digit in its decimal expansion." ([He gives more examples, very interesting ones, cf Kolmogorov 1925 in van Heijenoort:436]

It is this adoption of "completed infinity" as applied to Hilbert's 2nd problem that Brouwer complains about. The exact mechanism of the use of this is embedded in Hilbert's axioms. Kolmogorov discusses in detail Hilbert's "first axiom of negation" and his "second axiom of negation"

"Thus, from the intuitionistic point of view neither of Hilbert's two axioms of negation can be taken as an axiom of the general logic of judgments" (van H:421).

Thirdly, there was something major going on in physics (it was becoming more abstract, and Newtonian physics was being put "on its head"). Just the fact that in 1900 Hilbert had proposed, along with his problem #2, his axiomatization-of-physics problem hints that he felt he was hot onto something big. With regards to physics, the mathematician Poincare was involved deeply (I was just reading last night in Einstein's bio about Poincare's physics). Both Poincare and Lorentz were very close to discovering Einstein's version of relativity, but neither could give up the aether, nor did either apparently understand what Einstein had done (he posed a counter-intuitive hypothesis or "axiom" and ran with it...). Poincare was, in particular, very very close. There's also the niggling controversy re Hilbert's mathematical contribution to Einstein's general relativity. How this plays in exactly to the Brouwer-Hilbert controversy I'm not quite sure (I'm pretty sure its there, though excepting we see this tie-in via Poincare, and this (apparently-) Hilbertian notion of axiomizing every intellectual endeavor of mankind with abstract axioms (not intuitive ones).

Fourthly, there was the little bitty problem with "the Russell paradox" that the young Russell discovered in Frege's axiomatization of logic (cf van Heinjenoort: 124, 126). (As to what broad philosophical school Russell fell in -- he eschews Empricism but he also eschews the 'rationalists' Descartes, Leibniz, etc cf his 1912 p. 74-75 and other stuff scattered throughout. He seems to admire Kant.) And the whole philosophy represented by Whitehead-Russell's axiomatization of logic, and its apparent failure. Kleene observes here that

"the logistic thesis can be questioned finally on the ground that logic presupposes mathematical ideas in its formulation. In the intuitionistic view, an essential mathematical kernal is contained in the idea of iteration, which must be used e.g. in describing the hierarcy of types or the notion of a deduction from given premises." (Kleene 1952/1971:46).

Lastly, and connected to "fourthly", there's also the not-so-little matter of "impredicative defintions" (cf Kleene p. 42 -- 44) that hadn't gone away when Goedel commented on them in his 1933 On Intuitionistic Arithmetic and Number theory:

"... intuitionistic arithmetic and number theory are only apparently narrower than the classical versions, and in fact contain them (using a somewhat deviant interpretation). The reason for this lies in the fact that the intuitionistic prohibition against negating universal propositions to form purely existential propositions is made ineffective by permitting the predicate of absurditiy to be applied to universal propositions, which leads formally to exactly the same propositions as are asserted in classical mathematics. Intuitionism would seem to result in genuine restrictions only for analysis and set theory, and these restrictions are the result, not of the denial of tertium non datur, but rather of the prohibition of impredicative concepts. The above considerations, of course, yield a consistency proof for classical arithmetic and number theory. However, this proof is certainly not "finitary" in the sense given by Herbrand, following Hilbert" (boldface added added, Davis 1965:80)

So what Goedel is implying is: analysis and set theory at their base rest on "impredicative concepts" (! I've seen this notion elsewhere, too. Have modern mathematicians finally conquered this beast?)

(With regards to impredicative definitions, There's another Goedel quote I've read but haven't found yet, re all the paradoxes arising from impredicative defintions. Kleene attributes this notion to Poincare and Russell (cf Kleene p. 42)). With respect to this assertion re the following paradoxes (cf Kleene in his §11 p. 36-40): (A) The Burali-Forti paradox 1897, (B) "in the theory of transfinite cardinals, particularly Cantor's paradox (found by him in 1899)", (C) The Russell paradox 1902-3, (D) The Richard Paradox 1905, Dixon 1906, (E) the Epimenides (or Liar) paradox (6th C. B.C.) Kleene has this to say:

"Each of the antinomies of §11 involves an impredicative definition". (and cf Kleene p. 42 his section IMPREDICATIVE DEFINITION)

He goes on to agree with Goedel that analysis contains impredicative definitions, and he gives the example of Weyl in his "Das Kontinuum" (1918) attempting to find out how much of analysis could be retained, given restrictions on the use of "impredicative definitions". Weyl "was able to obtain a fair part of analysis, but not the theorem that an arbitrary non-empty set M of real numbers having an upper bound has a least upper bound (CF. also Weyl 1919)" (Kleene p. 43).

Kleene in his 1952 is unable to resolve the difficulty. "But the same argument [a proposed definition of least upper bound] can be used to uphold the impredicative definitions in the paradoxes." (Kleene 43)

Maybe there's some stuff here that could be transported over to the "foundations" article, too. Bill Wvbailey (talk) 16:51, 23 September 2008 (UTC)

hilbert's basis theorem and boxer's fists

A quick comparison of dates: basis theorem, 1888; boxer remark, 1927; shows there is something incoherent about the presentation here, which seeks to suggest that Hilbert argued against Brouwer because he wanted to defend his basis theorem. Tkuvho (talk) 11:53, 6 May 2010 (UTC)

Not incoherent at all. He was defending the notion of non-constructive proof, of which his basis theorem was but one example. As noted in the basis-theorem article the Hilbert proof is non-constructive (it does not produce an algorithm, it does not produce an object); presumably it uses reductio ad absurdum which requires either a double negative ~(~p) together with a "for all p" operator, or p V ~p with a "for all p" operator over an infinite set p; to achieve this he needed to "complete" the infinite and thus his proof offended the constructivists and intuitionists and finitists. It's the non-constructive form of the proof that started the hullabaloo. BillWvbailey (talk) 20:33, 6 May 2010 (UTC)

Hi Wv, I look forward to constructive work on the article. Your description of LEM as "completing the infinite" is somewhat simplistic. Thus, constructivists accept the validity of LEM over Z, even though it is a completed infinity. I still think the section I marked as "synthesis" needs work. You claim that it is all over the literature, but so far I have seen only that it is all over Reid's book which falls somewhat short of scholarly history. One does not resolve a disagreement by removing the "synthesis" flag, as I am sure you will agree upon reflection. Tkuvho (talk) 04:36, 7 May 2010 (UTC)

I don't understand your complaint. But rather than hang a banner on it, go ahead and fix it. Then after we see what your fixes are, we can react to them. BillWvbailey (talk) 00:41, 8 May 2010 (UTC)

Let's start at the beginning, namely the section title: "A philosophical defeat in the quest for "truth" in the choice of axioms". Now the "truth" in question is not sourced. Who is reaching for truth here? It appears to be the wikieditor who wrote the section. The Weyl quote says nothing about "truth". Furthermore, the article presents Weyl as siding with Brouwer. This may have been the case in the 20s, but it is well-known that Weyl reversed himself, and came to view Brouwer's approach as destructive of mathematics. This is not made clear in the article at all, on the contrary. Tkuvho (talk) 09:40, 11 May 2010 (UTC)

While I probably wrote it, I agree with your delete re "Brouwer's last years". Don't end there, keep on editing. Make the article better. Bill Wvbailey (talk) 02:24, 15 May 2010 (UTC)

Hmm... The problem is that some of the most basic premises here are flawed. The idea that Hilbert abandoned the search for "truth" whereas Brouwer championed the latter, is a caricature of the controversy. The premise that Hilbert's axiomatisation of geometry "set the stage" for the controversy has no semblance to reality. What set the stage for this foundational controversy was the emergence of the so-called antinomies. What does axiomatisation of geometry have to do with it?? Tkuvho (talk) 16:35, 16 May 2010 (UTC)

Gödel's incompleteness theorems?

I have to admit I didn't understand everything in this article, but in the article on Gödel's incompleteness theorems, it says right at the start that they put an end to Hilbert's aspirations that feature prominently in this article. Maybe the incompleteness theorems should be mentioned here as well? --84.63.203.226 (talk) 01:16, 12 March 2016 (UTC)