Talk:Constructivism (mathematics)

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Suggestion for renaming (2004)[edit]

I think a better name for this page is constructivism (mathematics). Mathematicians and philosophers of mathematics do not normally call it "mathematical contstructivism". The point of the word "mathematics" in the title is only to distinguish it from what is called "constructivism" in other fields such as architecture or educational psychology. Michael Hardy 02:55, 6 Sep 2004 (UTC)

Ditto on Michael's comment. Also, this may seem like a silly question, but I know almost nothing about this. Why do constructivists accept an "algorithm that takes any positive integer n and spits out two rational numbers, etc., etc." Doesn't this necessarily involve the use of infinite sets? (The algorithm itself appears to be a SEQUENCE, isn't it? It's a function from the natural numbers into Q x Q.) Or is there something more subtle going on here? This is a serious question to ask. I'm a math person and the example of Cauchy sequences fails to explain to me what's different. Both traditional and constructivist formulations of the reals appear identical to me the way it's explained here. Revolver 06:51, 30 Sep 2004 (UTC)
I guess my question just boils down to this: what exactly is an algorithm?? The answer to this question seems to be at the heart of what constructivism is, yet this question is completely ignored. Revolver 06:56, 30 Sep 2004 (UTC)
Good question. An algorithm should be thought of as an element of a set inductively defined by composition of very primitive algorithmic operations, NOT as an arbitrary function. The functions produced by such algorithms will not usually include all "classicaly possible" functions. This article could be improved, maybe I will (later). 65.50.26.149 02:19, 9 August 2005 (UTC)

The text claims that there exists a bijection between the reals produced by an algorithm and naturals. This is of course true classically, but is it true constructively? To construct such a bijection (in constructive mathematics), one should be able to recognize (algorithmically) whether two instruction sets produce the same real. I simply cannot see how this could be accomplished.

Correct. That statement is false. You cannot decide equality of two algorithms, and in fact you cannot event decide whether a finite string of symbols IS a valid algorithm. 65.50.26.149 02:19, 9 August 2005 (UTC)
By the way, does some know the source of the information constructivism here (algorithm and so on). There is a hot diskussion in german wiki, because the german constructive mathematics do use Chauchy-squences (a real number is an abstraction of two Chauchy-sequences which difference is a zero-sequence; Paul Lorenzen). But there is a person who translated this english article saying: Its english, so it must be true. Paul Conradi 12:00, 9 August 2005 (UTC)
Hah. How does he know the english article wasn't written by a german-speaking person anyway? I didn't write the original bit, I just did some corrections, but I believe the definition used there is essentially Errett Bishop's. There can be other constructions of the real numbers, for instance using choice sequences (Brouwer). This article could still use some improvement obviously. 192.75.48.150 17:33, 9 August 2005 (UTC)

I have no idea what "But by enumerating algorithms, we can only construct a partial function from the naturals onto the reals. And even though there is also of course a trivial injection from the naturals to the reals, still one cannot construct a bijection out of these; non-constructive parts of classical set theory are required." means. What are "these"? Jim Apple 11:18, 18 August 2005 (UTC)

Good question. That whole bit was somewhat opaque. I've separated out the whole cardinality thing and hopefully clarified it. -Dan 15:38, 23 August 2005 (UTC)


I'm still baffled. Why do we enumarate algorithms to construct a function T rather than just enumerating functions T? Wy can't we be sure to enumerate only the full (rather than also the partial) ones? What does "the algorithm may fail to satisfy the constraints" mean? What constraints? Didn't we create it in a constructive system? How could it them fail to be constructive? Jim Apple 17:50, 7 September 2005 (UTC)
We can't just will away the non-terminating algos / partial functions because we actually have a theorem that basically says not all functions are either total or partial, essentially by the diagonal argument. This theorem isn't valid in all variants of constructivism, but the question "but aren't the computable numbers countable?" only really comes up in the context where functions are algorithms (recursive constructive mathematics). I will try to clarify again. -Dan 02:30, 20 September 2005 (UTC)

Rejects infinite sequences?[edit]

There seems to be a contradiction in the article. At one point, we see that, "Constructivism also rejects the use of infinite objects, such infinite sets and sequences." In the very next section, we are told that a real number is a function f:N --> Q x Q satisfying certain properties.

But what is a sequence if not a function with domain N? The article does comment that the issue is which functions are allowed, but this doesn't really avert the problem. A sequence just is a function with domain N and if we allow any such function, then we contradict the claim that constructivism disallows sequences.

Alternatively, perhaps the author has some other definition of sequence in mind, but I surely don't know what.

Phiwum 12:31, 11 September 2005 (UTC)

That is odd. I'm not sure what was meant! Maybe the author was referring to actual infinity and intuitionism. In any case, I think the sentence doesn't really belong here. -Dan 02:30, 20 September 2005 (UTC)

Variants of Constructive Computability[edit]

The present text says: "This then opens the question as to what sort of function from a countable set to a countable set, such as f and g above, can actually be constructed. Different versions of constructivism diverge on this point. Constructions can be defined as broadly as free choice sequences, which is the intuitionistic view, or as narrowly as algorithms (or more technically, the computable functions), or even left unspecified. If, for instance, the algorithmic view is taken, then the reals as constructed here are essentially what classically would be called the computable numbers."

It should be pointed out, however, that there are various degrees of computability. One traditional view says: a symbol sequence is computable if it can be generated by a halting program on a universal Turing machine. This excludes infinite sequences. A more relaxed variant is computability in the limit: a symbol sequence is computable in the limit if there is a finite, possibly non-halting program that incrementally outputs every symbol of the sequence. This includes the dyadic expansion of pi but still excludes most of the real numbers, because most do not have a finite program. One has to be careful though: there is a difference between traditional Turing machines that cannot edit their previous outputs, and generalized Turing machines, which can. According to Jürgen Schmidhuber, the constructively describable symbol sequences are those that have a finite program running on a generalized Turing machine, such that any output symbol eventually converges, that is, it does not change any more. Due to limitations first exhibited by Kurt Gödel it may be impossible to predict the convergence time itself by a halting program, otherwise the halting problem could be solved. Schmidhuber uses this approach to define the set of formally describable or constructively computable universes for a constructive theory of everything. References: 1. Algorithmic Theories of Everything http://arxiv.org/abs/quant-ph/0011122 (2000) 2. Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit. International Journal of Foundations of Computer Science 13(4):587-612, 2002 http://www.idsia.ch/~juergen/kolmogorov.html . 3. Overview site: http://www.idsia.ch/~juergen/computeruniverse.html 4. Kurt Gödel, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme," Monatshefte für Mathematik und Physik 38: 173-98. Discrepancy (talk) 19:53, 29 March 2008 (UTC)

I think the text was trying to get at the main three branches of constructivism, namely intuitionism (choice sequences), russian constructivism (algorithms), or Bishop-style (general, unspecified). I have no idea where this "generalised computable" stuff comes from. May I ask what counts as a constructive proof that "any output symbol eventually converges"? it seems to me the answer would have to be "a generalised turing machine that, given a symbol position, outputs the time at which it converges." Only now the time is a natural number, therefore expressed by a finite number of symbols. So for that generalised turing machine, since any of its output symbols converge, and there are only finitely many of them, there would have to be a finite time after which all its output symbols converged. In other words, it halts. So it is actually a traditional Turing machine. But if the termination time of a generalised Turing machine can be given by a traditional Turing machine, then it also is a traditional Turing machine.
Now I could very well be missing something, but to be honest, the "theory of everything" bit makes it sound rather dodgy to begin with. So I suspect this bit ought to be moved to a different article about "generalised computability", or even removed altogether. --Unzerlegbarkeit (talk) 21:38, 25 June 2008 (UTC)
Moved. --Unzerlegbarkeit (talk) 17:44, 27 June 2008 (UTC)

It is NECESSARY in this case to make the disambiguation notice LONG[edit]

Joriki has again revised the disambiguation notice, making it read as follows:

This article is about constructivism in the philosophy of mathematics, not in mathematics education. For that and other meanings of the word, see constructivism.

This still misses the point. If you say "constructivism in the philosophy of mathematics, not in mathematics education", it makes it sound as if constructivism is the name of something that can apply either to the philosophy of mathematics or to (mathematical or other) education, and there are separate articles about these. That is wrong! The point is that the word "constructivism" refers to two entirely different things, not to one thing that is applied differently to philosophy of mathematics and to education. One is about the sense in which mathematical objects such as numbers, functions, sets, structures, etc. may be said to exist, and how that existence can be known. The other is about how people learn by constructing knowledge. Michael Hardy 19:29, 6 October 2005 (UTC)

Inconsistent[edit]

This seems to be contradictory: Constructivism is often confused with intuitionism, but in fact, intuitionism is only one kind of constructivism. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Constructivism does not, and is entirely consonant with an objective view of mathematics. If "intuitionism is only one kind of constructivism" then it cannot be that constructivism asserts the objective view of mathematics while intuitionism takes a contrary view. I'm not up on the terminology, so I don't know what the correct usage is. Perhaps "Intuitionism is related to constructivism"? --66.82.9.35 19:15, 10 November 2005 (UTC)

That would be contradictory, but what the author probably meant to say is that constructivism (in general) does not assert that mathematics is subjective, which is different from saying that constructivism asserts that mathematics is not subjective. -Dan 19:48, 10 November 2005 (UTC)
My Mathematical Encyclopedic Dictionary says constructivism is a kind of intuitionism.--Nixer 18:31, 11 June 2006 (UTC)
Really? Which encyclopedia is this? Does it say that there are intuitionists who are not constructivists? It seems to me that SEP disagrees. As do I, since I think it is possible (and common) to subscribe to constructivist methods, without subscribing to the intuitionist philosophy. -Dan 19:22, 11 June 2006 (UTC)
The encyclopedia's article is written by Markov. And yes, it says constructivism is a branch of intuitionism. It was inspired with intuitionist philosophy, but rjects non-constructive objects (for example, actual infinity). For example, Brouwer was intuitionist, but not constructivist (constructivism developed later). The article on constructivist logic says it differs from intuitionist one, for example in that it accepts Chirch thesis and Markov's principle. But both use the same Heyting formal system.--Nixer 06:42, 12 June 2006 (UTC)
Ah. I see. Yes, that is correct for Markov's branch of constructivism, also called "recursive" constructivism, or "Russian" constructivism. When the article was written by Markov, he probably just called it "constructivism". But Markov died almost 30 years ago, so the article must be even older. "Constructivism" refers to something more general now, and it may or may not assume Church's thesis and Markov's principle. That said, I think there should be at least a section in this article on Markov's programme! -Dan 14:12, 12 June 2006 (UTC)
The encyclopedy is printed in 1988 and I suppose other articles were written after the death of Markov. For example the articles on constructivist logic (which describes defference between constructivist and intuitionist logic) and on constructivist analysis are written by Kushner (by the way, both dont have their pages in Wikipedia).--Nixer 14:49, 12 June 2006 (UTC)
Kushner is also an authority. But at the same time Bishop's constructivism does not assume Church's thesis or Markov's principle, nor does Martin-Löf's. I can't really explain it, maybe the article is from an older edition and just wasn't updated. (It is unfortunate that Markov and Kushner have no article yet). -Dan 15:37, 12 June 2006 (UTC)

Game semantics?[edit]

Would someone care to explain why there is a link to game semantics at the bottom of the page? Game semantics has no more to do with constructivism than any other kind of semantics related to computation and programming languages. I am inclined to delete the reference to game sematics. Frege 22:43, 17 December 2005 (UTC)

Because of Paul Lorenzen, perhaps. -Dan 00:15, 30 December 2005 (UTC)

this article makes no mention of mathematics without axiom of choice[edit]

why is the axiom of choice mentioned nowhere in this article? -lethe talk + 09:29, 15 March 2006 (UTC)

There is no reason why it should not be mentioned, just that nobody mentioned it yet... -Dan 19:25, 11 June 2006 (UTC)
If you have to use the axiom of choice for instance in a proof, then you are sure, that you don't make constructiv mathematics. But anyway ... continue being happy. --Paul Conradi 23:16, 12 June 2006 (UTC)
Thanks, I will. But I think you misunderstood what I wrote. Too many negatives, I guess. -Dan 23:34, 12 June 2006 (UTC)
Intuitionists in the school of Brouwer and Bishop typically accept the axiom of choice in very strong forms. This is because of the way the interpret provability. If they can prove, for example, that every element of a particular set is nonempty, the limits they place on method of proof will require them to produce a choice function for the set to prove that each element is nonempty (read nonempty to mean there exists an element of the set). So the axiom of choice adds no strength. The axiom of choice is only strong in the presence of the law of the excluded middle.
A serious flaw is that the article as it stands doesn't make any effort to distinguish various schools of constructivism. All of the following have been called constructive at some point or anouther:
  • Intutionionism (Brouwer)
  • Bishop's constructive analysis
  • Russian style constructive mathematics
  • Finitism and Ultrafinitism
  • Constructive Type theory as led by Martin Löf
  • Constructive ZF and Intuitionistic ZF (current research topics)
  • Rejection of the axiom of choice by working mathematicians
The article ought to mention all of these. It needs some serious help. CMummert 12:33, 25 June 2006 (UTC)
Well, "set" isn't the foundation of any of these approaches, except CZF/IZF/ZF. We might say, for Bishop and Löf, very roughly, if a "set" is constructed, rather than separated, then the axiom of choice works. If you apply choice to arbitrarily separated sets, you run into the Goodman Myhill theorem. But it's hard to do justice to the issue in a few sentences. I really think it needs its own article. 72.137.20.109 18:14, 25 June 2006 (UTC)
I think that the issue raised by Lethe above is that the article could include a sentence which says that the issue of whether the axiom of choice is “constructive” in the presence of classical logic is not an issue that constructivists are typically interested in. It is an issue that some mathematicians who practice nonconctructive mathematics have been interested in. This is a common confusion because so much is made of the contentious history of the axiom of choice.
By the way, thanks for the reminder about Goodman/Myhill. I am more used to finite type Heyting Arithmetic, where there is the full choice scheme and schemes to define functions (and thus sets) of finite type by primitive recursion but no other comprehension/separation at all. You're right that the axiom of choice does imply LEM in the presence of comprehension. CMummert 19:32, 25 June 2006 (UTC)

My english is not so good. So I don't like to edit this article. Maybe someone would like to add, that the constructive mathematicans typically don't make use of axiom of choice in a proof, because while using it there is no concrete construction (operation) given, that leads to the lemma to be proofed. --Paul Conradi 13:14, 14 June 2006 (UTC)

I made a section stub. Certainly the whole article needs work. -Dan 14:05, 14 June 2006 (UTC)
Thanks :) --Paul Conradi 16:18, 17 June 2006 (UTC)

what is the implication of intuitionism in teaching and learning of mathematics?[edit]

as i'm asking on the headline can u send me on my e-mail(abelo121@yahoo.com) thank u abel —The preceding unsigned comment was added by 213.55.95.4 (talk) 07:50, 10 December 2006 (UTC).

Try asking at the Reference desk. CMummert 13:40, 10 December 2006 (UTC)

Attitude of mathematicians POV?[edit]

The attitude of mathematicians article does not read as being very NPOV. Perhaps more citations are needed, rather that just quoting one mathematician's POV. —The preceding unsigned comment was added by 165.228.227.39 (talk) 04:39, 17 January 2007 (UTC).


The sentence

More recently, the formalism of constructivist mathematics has been gaining increased credibility since natural applications for it have been found in typed lambda calculi, topos theory and categorical logic, which are extremely notable subjects in foundational mathematics and computer science.

also seems to me to be bogus. I'd say the technical concepts developed by constructivists have found applications in computer science. That says nothing about the acceptance of constructivist philosophy, any more than use of the Pythagorean theorem in geometry says something about the acceptance of mathematical Platonism. Computer scientists generally also accept non-constructive proofs (e.g. graph minor theorem) and complexity theory (e.g. the quest for P!=NP) is largely a search for proofs of non-existence of certain classes of algorithms.

Constructability and Mathematical Existence by Charles Chihara may be a good reference for this article. 67.122.210.149 (talk) 17:14, 17 November 2008 (UTC)

Bieberbach[edit]

Hi,

Does anyone know whether Bieberbach was a contstructivist? Katzmik (talk) 10:24, 23 October 2008 (UTC)

I wouldn't have thought so, but he is still around, you could e-mail him and ask. 67.122.210.149 (talk) 17:16, 17 November 2008 (UTC)
He'd be 123 years old this year. He died in 1982. See Ludwig Bieberbach. Michael Hardy (talk) 19:37, 20 March 2009 (UTC)

Idea: document divergences[edit]

I have an idea that the article would benefit immensely from a more hands-on account of where classical and constructive mathematics diverge, organised on a thematic basis. The kind of thing I have in mind is to say that, say, in measure theory, how the accepting or rejecting AC gives different worlds, where classical measure theory is interested in the complexity of constructions of non-measurable sets, and constructive measure theory is concerned with things like integration of functions over notions of the computable real line. Doing this properly is well beyond my mathematical-recreations pay grade, though. I'd appreciate some help in coming up with a good set of topics. My initial ideas are:

  1. The above bit on measure theory and Lebesgue integration;
  2. Analysis and Specker's theorem;
  3. Maybe there's something interesting in ideal theory and Buchberger's computable algebra?

Thanks! — Charles Stewart (talk) 14:36, 20 March 2009 (UTC)

I can help some with this. The main difficulty is keeping straight what you mean by "constructive mathematics". There are many meanings to the term, and they diverge differently from classical mathematics. So you cannot just say, "Differences in algebra" because on the one hand there are results in computable algebra about the effectiveness of various operations on computable structures, and on the other hand there are independence results over ZF or ZFC, and the character of these results is radically different. — Carl (CBM · talk) 16:03, 20 March 2009 (UTC)
I think there are two kinds of constructive mathematics that are more important than the others, firstly the account from Errett Bishop and formalised by Martin-Löf, which represents a very purist reading of the BHK interpretation, and second, classical recursive mathematics, which is the sort of constructivism promoted by Markov, and which doesn't seem to be well reported on Wikipedia at all. I'm interested in documenting controversies close to core mathematics; I see set theory as being relevant here only through things like Borel algebras. — Charles Stewart (talk) 19:24, 20 March 2009 (UTC)
The received viewpoint of Bridges and Richman (1987) is there are three programs of constructive analysis: Bishop's program, the intuitionistic program, and the Russian constructivist program in which everything is ultimately a natural number. You're right that Bishop's program has been very influential, but the approach via intuitionistic subsystems of arithmetic (e.g. Metamathematical investigation) is equally important, and is more closely related to contemporary work in proof theory.
The issue with set theory is that (1) most mathematicians immediately think about the axiom of choice when they hear the word "constructive" and (2) the article here already tried to talk about constructive set theory (I improved that section some this morning).
This article can certainly also discuss classical recursive mathematics, as in the Handbook of Recursive Mathematics, but I might quibble over whether that is ordinarily considered to be a school of "constructivism".
My main point, in any case, is that we have to be sure to qualify statements like "In constructive mathematics, you cannot prove X" by saying which school of constructive mathematics we are referring to, because they disagree. — Carl (CBM · talk) 22:24, 20 March 2009 (UTC)
I agree with all of this. Three points to make for the sake of precision: (i) I think that classical recursive mathematics and Markov's constructivism are essentially the same, and I think that this is the constructivism that the Russian constructivist school came to hold, although I feel some doubt that I am right in this, (ii) The most important theory of constructive set theory, CZF, is a more or less successful attempt (ie. modulo hand-waving) to clothe Martin-Loef's type theory in the language of ZF, and (iii) I don't think Brouwerian intuitionism was really faithfully followed by anyone else: Heyting's approach is essentially the same sort of thing as Bishop--Martin-Loef, and who else really tried to take up Brouwer? Kolmogorov?
And: excellent! I am pleased. I think that documenting the three schools is a good place to start, and then get started on the real content. It sounds like I have most to say on measure theory, and you have most to say on analysis? If so, there's a possible division of labour. — Charles Stewart (talk) 20:12, 21 March 2009 (UTC)
That sounds good to me. — Carl (CBM · talk) 00:50, 22 March 2009 (UTC)
Made a start, not very well written, haven't fact checked anything or provided refs, but I am hopeful that something good will grow out of this. — Charles Stewart (talk) 09:55, 23 March 2009 (UTC)
Rah rah rah! linas (talk) 15:56, 8 May 2010 (UTC)

Bishop won[edit]

Troelstra's 1991 history [1] asserts repeatedly in section 8 that Bishop's constructive mathematics won, in the sense that that has become the focus of constructive mathematics, displacing Brouwer-Heyting style mathamtics, and Markov-style CRM on the other, and for the reason that Bishop's approach to constructive mathematics was more focussed on mathematically interesting results.

I don't know of anyone who disagrees with this assessment. Is it fair to structure the article assuming in the lede that constructive mathematics means Bishop style (perhaps with a footnote), and relegate talk of other approaches to a later section? — Charles Stewart (talk) 13:46, 15 July 2009 (UTC)

Cardinality section[edit]

In the cardinality section of the article, there is a discussion of how it is difficult to prove that the real numbers are countable using only constructivist logic. There is a big problem with the reasoning put forth in this section: The real numbers are NOT countable, as anyone who has taken a basic course in set theory can tell you. However, right in the middle of the last paragraph of this section, the claim is made that "They are, therefore exactly countable."

The claim is also made at the end of the same paragraph that "cardinality of sets fails to be totally ordered". This statement is also blatantly untrue; in fact, cardinal numbers are used as an example of totally ordered sets in the Total Order. Am I missing something important? Because neither of these statements are in any way true, so of course we have difficulty proving them using constructivist logic. 128.120.218.130 (talk) 23:58, 5 November 2009 (UTC)

That section is written in a very confusing way. It is referring to a certain interpretation of constructive mathematics in which "real number" is redefined to mean "computable real number". Then it tries to explain how this would affect cardinality results. Really that whole section should be cut and then rewritten from scratch; I think it is likely to be confusing to anyone who does not already know what is going on. — Carl (CBM · talk) 00:21, 6 November 2009 (UTC)

Boundaries[edit]

I'm having basic problems understanding things; some trouble determining what is accepted by constructivists, and what is not. For example, I presume that CH is rejected, and also Martin's axiom MA. Where does V=L fit in? Suslin conjecture? Axiom of determinacy? Also, there's plenty of sentiment that "Statement X might not be true, because it requires AC" (and zillions of examples of such X's), but are there any results along the lines of "Statement X is unprovable without AC"? So, for example, we have "Assume AC then the Vitali set exists." Under what circumstances is it possible to say "the Vitali set is not constructible, and could never be"? or that "It is impossible to make the Vitali set unless one assumes AC"? (how about "Statement X is unprovable without CH, or without MA, etc."?) Or are these questions off-topic? If so, why? linas (talk) 15:56, 8 May 2010 (UTC)

Hilbert and constructivism[edit]

In reference to the edit I made [2], quickly reverted: the article currently seems to be confused on Hilbert's attitude to constructivism (proponent or opponent?). Perhaps part of the problem is the finitism article, which doesn't seem to be H's finitism. I don't have the time right now to research and rewrite the article(s) to reflect H's different responses to constructivist criticism; hopefully someone else does. Hibbleton (talk) 22:05, 9 March 2011 (UTC)

I'm glad you posted here. I think your list of types of constructivism is an improvement, independent of whether Hilbert should be included there, so I have put it back in the article.
I generally agree Hilbert is not viewed as a finitist in the way that, say, Kronecker or Doron Zeilberger are. Hilbert was interested in finitism only in the sense that he was looking to show that at least some mathematics could be done finitisicaly, but Hilbert was a staunch advocate of infinitary methods at the same time (e.g. the "paradise Cantor has created" quote). — Carl (CBM · talk) 02:55, 10 March 2011 (UTC)
This may be true, but removing references to Hilbert-Bernays makes about as much sense as removing all mention of Edward Nelson's contribution to constructivism just because he is interested in non-standard analysis. The real problem with Hilbert-Bernays is that it does not fit neatly into the fantasy tale of the Brouwer-Hilbert controversy which would like to line everybody up on one of two sides. We shouldn't necessarily encourage that. Hilbert-Bernays contribution to finitism may in the end have been more valuable than Kronecker-Brouwer. Tkuvho (talk) 12:58, 10 March 2011 (UTC)
Could you expand on what you're thinking of for Hilbert and Bernays' contributions to finitism? From my POV it's certainly accurate to say that the work of Hilbert was relevant to finitism; it's something else to say Hilbert was a finitist or, more generally, a constructivist. Stephen Simpson has described Hilbert's work in this area as "finitistic reductionism" rather than finitism, and he has some papers on that, which could be useful as sources. — Carl (CBM · talk) 13:09, 10 March 2011 (UTC)
Are you familiar with a version of finitism that is not at the same time finitistic reductionism? I would like to hear about the Hahn-Banach theorem in a finitist framework. Tkuvho (talk) 13:49, 10 March 2011 (UTC)
If I use "finitist" to refer to a person who feels that things such as the Hahn-Banach theorem are simply meaningless, because they refer to infinite sets that don't exist, then I wouldn't call Hilbert that sort of finitist. But there is no arguing that what we now think of as finitism was very strongly influenced by Hilbert's program, and Hilbert argued at times for a very formalistic account of mathematics. I went back and checked Troelstra's article, which is what we cite on that sentence, and Troelstra explicitly lists Hilbert and Bernays. All this convinces me you're correct that we should include Hilbert and Bernays in the sentence, so I edited it again. — Carl (CBM · talk) 14:55, 10 March 2011 (UTC)
Thanks. Tkuvho (talk) 20:48, 10 March 2011 (UTC)

plagarism?[edit]

Large chunks of this article seem to come word for word from a book Philosophy of Mathematics John Francis 2008 http://www.amazon.com/gp/search?index=books&linkCode=qs&keywords=8182202671 Of course for all I know that book is a rip off from Wikipedia..... Gentlemath (talk) 07:37, 27 February 2012 (UTC)

I looked at it on Google Books. The book is simply a bunch of Wikipedia articles with a fake author. The publisher's website is at [3]. — Carl (CBM · talk) 13:23, 27 February 2012 (UTC)

Confusion of philosophy and mathematics[edit]

This article seems to make the mistake of combining constructivism the philosophical notion applied to mathematics (i.e. math is a social construct) and constructivism as a mathematical philosophy (i.e. math should avoid using certain techniques like the axiom of choice and proof by contradiction). The finitism of Hilbert is much more the second than the first, as are most of the examples. In particular, computer science operates almost exclusively on constructive grounds, but that doesn't preclude computer scientists from being philosophically Platonic idealist or mathematical monists.

I think this article needs to be drastically rewritten to reflect that distinction, if not split into two separate articles. — Preceding unsigned comment added by 76.123.34.94 (talk) 03:49, 18 October 2012 (UTC)

Which part of the article would you say is about mathematics as a social construct? I just read through it again and it all appears (to me) to be about a single topic. — Carl (CBM · talk) 11:22, 18 October 2012 (UTC)

Confused section about classical measure theory and the axiom of choice[edit]

I am removing this very confused paragraph: "Classical measure theory makes deep usage of the axiom of choice, which is fundamental to, first, distinction between measurable and non-measurable sets, the existence of the latter being behind such famous results as the Banach–Tarski paradox, and secondly the hierarchies of notions of measure captured by notions such as Borel algebras, which are an important source of intuitions in set theory. Measure theory provides the foundation for the modern notion of integral, the Lebesgue integral."

You don't need the axiom of choice to define measurability of sets, the Borel hierarchy, or the Lebesgue integral. It's true that the axiom of choice is used to create non-measurable sets, but these are a pathology to be avoided rather than a significant part of the theory.

It's true that the axiom of choice is usually used freely in the development of classical measure theory, but I think this usage is mostly for trivial things. I don't think it would be that hard to develop classical measure theory without the axiom of choice, although it would be a little annoying. It is much harder to develop a truly constructive measure theory, so I am emphasizing that in the article. David9550 (talk) 21:06, 9 March 2015 (UTC)

Fool question[edit]

Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics.

Is it because you limit yourself to construct proofs by contradiction? — Preceding unsigned comment added by Ore4444 (talkcontribs) 19:20, 24 August 2017 (UTC)