Talk:Constructivism (philosophy of mathematics)

message: 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)
replies: text: 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)

text: 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)
replies: text: 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)

text: 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)
replies: text: 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)
replies: text: 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)

text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Suggestion for renaming (2004) 

message: 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)
replies: text: 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)
thread_title:  Rejects infinite sequences? 

message: 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)
thread_title: It is NECESSARY in this case to make the disambiguation notice LONG

message: 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)
replies: text: 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)
replies: text: My Mathematical Encyclopedic Dictionary says constructivism is a kind of intuitionism.--Nixer 18:31, 11 June 2006 (UTC)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Inconsistent  

message: 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)
replies: text: Because of Paul Lorenzen, perhaps. -Dan 00:15, 30 December 2005 (UTC)
thread_title:  Game semantics? 

message: 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)
replies: text: Try asking at the Reference desk. CMummert 13:40, 10 December 2006 (UTC)
thread_title:  what is the implication of intuitionism in teaching and learning of mathematics? 

message: 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)
replies: text: 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)
thread_title:  Attitude of mathematicians POV? 

message: Hi,
Does anyone know whether Bieberbach was a contstructivist?  Katzmik (talk) 10:24, 23 October 2008 (UTC)
replies: text: 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)
replies: text:  He'd be 123 years old this year.  He died in 1982.  See Ludwig Bieberbach. Michael Hardy (talk) 19:37, 20 March 2009 (UTC)
thread_title:  Bieberbach 

message: Troelstra's 1991 history  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)
thread_title:  Bishop won 

message: 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)
replies: text: 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)
thread_title:  Cardinality section 

message: 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)
thread_title:  Boundaries 

message: 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)
replies: text: 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 . — Carl (CBM · talk) 13:23, 27 February 2012 (UTC)
thread_title:  plagarism? 

message: 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)
replies: text: 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)
thread_title:  Confusion of philosophy and mathematics 

message: 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)
thread_title:  Confused section about classical measure theory and the axiom of choice 

message: 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 (talk • contribs) 19:20, 24 August 2017 (UTC)
thread_title:  Fool question