Talk:Constructivism (mathematics)

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[edit] Suggestion for renaming (2004)

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)

[edit] Rejects infinite sequences?

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)

[edit] Variants of Constructive Computability

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)

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

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)

[edit] Inconsistent

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)

[edit] Game semantics?

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)

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

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)

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

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)

[edit] Attitude of mathematicians POV?

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).