Talk:Nonconstructive proof

message: To make this into a proper article, we need some examples of nonconstructive proofs, and discussion thereof.
: See Votes for deletion for some discussion about the article too :) Dysprosia 10:52, 22 Aug 2003 (UTC)

message: I removed "or an existence proof" from the opening sentence, because as I see it, constructive proofs are also existence proofs! The existence theorem article distinguishes "pure existence theorems" which are proven by nonconstructive methods; in this terminology, wouldn't a nonconstructive proof just be a "pure existence proof"? -- Oliver P. 11:25, 22 Aug 2003 (UTC)
replies: text: I put it back - Sorry, I missed the talk. But it is also known as an existence proof as well, but I'll add your point. Dysprosia 11:27, 22 Aug 2003 (UTC)
replies: text:  Okay, fair enough. Thanks. :) -- Oliver P. 11:33, 22 Aug 2003 (UTC)
replies: text:  No problem-o :) Dysprosia 11:34, 22 Aug 2003 (UTC)
thread_title:  Same as existence proof? 

message: Also, I'm not sure how far we have to fo to accommodate supporters of mathematical constructivism. If they say that nonconstructive proofs are invalid, do we have to say that they only purport to prove things? -- Oliver P. 11:25, 22 Aug 2003 (UTC)
replies: text: Nonconstructive proofs always prove something, it just may not be what they purport to prove: any nonconstructive proof of a proposition P can be thought of as a constructive proof of not (not P), that is, a constructive refutation of the impossibility of P.  Constructively, it is not the case that not (not P) implies P. William Lovas 04:12, 3 February 2006 (UTC)
thread_title:  Accommodating mathematical constructivism 

message: I think that in linear algebra class we first proved that there existed a function that satiesifed the properties we wanted for the determinant, before showing what it actually was -- Tarquin 11:40, 22 Aug 2003 (UTC)
thread_title:  LA example 

message: * Nonconstructive
** Dictionary definition. -- Oliver P. 10:34, 22 Aug 2003 (UTC)
thread_title:  From VfD 

message: I'm fairly sure the example with x^2 and sqrt(x) is not really an example of a true nonconstructive proof. What is the thing whose existence is proved but not constructed? The sqrt(x) function seems to me to be a specific thing, so it can't be that. What is it then? But, I'd prefer someone more expert in logic to confirm/deny this. Revolver 18:47, 2 September 2005 (UTC)
replies: text: I agree this is not an example; I noticed this while reading the article and went to the talk page to leave a note to this effect. Also the Hex example is weak, since Hex is finite. Lambiam 11:53, 14 January 2006 (UTC)
thread_title:  "easier" example 

message: My understanding of an existence proof (or, to be more precise, a pure existence proof) is a proof that goes like this:
:Either P or not P. If not P then P. Therefore P.
and so proves P without showing how to find or construct an explicit case of P. For example, Cantor's diagonal argument shows that that no countable list can contain all real numbers; the set of algebraic numbers is countable; therefore there are real numbers that are not algebraic. This is a proof of the existence of transcendental numbers without giving us a clue of how to find or construct one.
Now our axiom of choice article says: "A proof requiring the axiom of choice is always nonconstructive". So the proof in our Banach–Tarski paradox article is nonconstructive since it uses the axiom of choice. Yet that proof gives a very detailed construction for an appropriate dissection of a 3-d ball. The only non-explicit step in that proof is using the axiom of choice to pick exactly one point from every orbit of the group of rotations H. It seems to me that the Banach-Tarski proof does not follow the pattern of a pure existence proof - it says "If you grant me a tool called axiom of choice, I give you an explicit dissection of a 3-d ball with certain non-intuitive properties".
This makes me question whether pure existence proof and nonconstructive proof really are synonomous. It seems to me that pure existence proof has a narrower definition that nonconstructive proof - the Banach-Tarski proof is nonconstructive but is not just a pure existence proof. And so the line in this article that says: "The term pure existence proof is often used as a synonym for nonconstructive proof" should be qualified by explaining that the terms are not in fact synonomous (despite what MathWorld says here). Or maybe nonconstructive proof is used in two slightly different ways, only one of which is synonomous with pure existence proof. Thoughts ? Gandalf61 15:59, 18 June 2007 (UTC)
thread_title:  Same as existence proof ? (redux) 

message: A constructive proof is just one that shows the existence of something by explicitly constructing it. These proofs -- as far as most mathematicians are concerned -- often employ the law of the excluded middle in their reasoning (contrary to what the article states). 
On the other hand, a constructivist proof is one that uses a different form of logic, which indeed disallows the law of the excluded middle.
This distinction needs to be made clear in the article.Daqu (talk) 20:44, 23 November 2007 (UTC)
thread_title: Need to distinguish between "constructive proof" and "constructivist proof"