Talk:Mathematical proof

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A vital article.

Do you consider anything, or are there limitations? The reason I ask is that I have been working in an old field that would be consuidered new, as it has not been explored to any extent that I am aware of. It does things like taking the 3, 4, 5 triangle theorem and advances it to the realm of possibly being able to create matter through the use of the numbers generated as frequencies of atomics. Within this it also seems to eliminate all but 3 through 8, using these combnations to illucidate the apparent tetrahedral as the basis for all matter structure in a solid light format. If these things are what this page is all about, then I have come to the right place. Bear in mind that many of the usual laws of both mathematics and physics including constants become subject to a different paradigm that might seem to negate what is now comonly accepted by most of accademia. It has been found that thee seems to be three sets of individual laws or sets of constants depending on exactly what is being dealt with. All that is available to date is a very general philosiphy of this new math, though I will make that available upon request. DKBailey contact me at Drakedkb8848@aol.com

Your material seems to be original research and is therefore not suitable for Wikipedia, which attempts to be an encyclopedia of established knowledge. Once your theories have been published and well received, we will cover them here. AxelBoldt 23:03 Jan 4, 2003 (UTC)

I removed two paragraphs because they contain numerous statements that I can not verify. For example, the second sentence in the first paragraph I removed: "In the context of proof theory, where purely formal proofs are considered, such not entirely formal demonstrations in mathematics are often called "social proofs"". I've never heard of "social proofs" and when I clicked on the link "proof theory" in this sentence, it didn't even mention "social proofs". So apparantly the phrase "social proofs" is not used as often as this sentence suggested. The rest of these two paragraphs was of poor quality as well. Jan 12, 2005. —The preceding unsigned comment was added by 68.35.253.247 (talk • contribs) .

I'm sorry, but I think it is bad manners to delete material because you cannot verify it without asking about it first. Similarly, poor quality should be rewritten instead of deleted. For instance, the first sentence you deleted read "Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity." What is wrong about that? Hence, I reverted your removal with the exception about the phrase mentioning "social proofs", where you at least gave some evidence. -- Jitse Niesen (talk) 13:34, 12 January 2006 (UTC)

While it is true that most proofs do use natural language, it is also true that the only proofs that mathematicians will accept are those that (at least in principle) can be reduced to logic + ZFC. The ambiguity of natural language must be clearly resolvable from the given context, if not, the argument will not be accepted by mathematicians as a valid proof. Thus to say that ambiguity is allowed in mathematical proofs is misleading, every statement made during the proof must have (given the context) only one possible meaning. MvH Jan 12, 2005.

Firstly, welcome at Wikipedia!

I understood the sentence "Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity" as meaning that proofs usually include some amount of natural language and that natural language admits some ambiguity without implying that proof may admit ambiguity, but I can see your point. Saying that proofs must be reducable to logic + ZFC is a theoretical construct. In practice, which proofs are deemed acceptable is not an easy thing to determine. Standards differ; at least historically (some proofs accepted in say Euler's time wouldn't be accepted nowadays) and I think also across areas within mathematics.

I think there is a social aspect here, proofs are partially means to convince others that certain results are true. This aspect arises sometimes in conversations between mathematicians; I assume that there is also scholarly work in this direction but this is really not my speciality. I think that this aspect should be mentioned in this article. This is why I disagreed with your removal, though I now regret the violence with which I disagreed, sorry about it.

I do agree that these two paragraphs can be improved a lot and I hope you will be able to do so. If you still think that it would be better to remove the fragment, please say so and I'll ask some logicians what they think about it. -- Jitse Niesen (talk) 22:23, 12 January 2006 (UTC)

I think that in concrete examples of proofs, practically all mathematicians have the same opinion about when a proof is valid and when it is not (although they might ignore some of the philosophical issues and foundational issues studied by logicians and set theorists). But that's just an observation about mathematical proofs, and not a definition. In principle, a complete proof is something that uses only logic, ZFC, and previously proven results. However, it would be too cumbersome to spell everything out in terms of logic; details (especially those that don't help the understanding) have to be skipped because otherwise most proofs would become so long that they would be of no value to anyone (except to a computer). This raises the question: which details may be omitted? The answer is a practical one: For a mathematician, a proof is valid when a reader can reasonably be expected to fill in the details up to any level of detail that the reader desires. For instance, if you send a paper to the Journal of SomeAreaInMath, you may assume that your reader masters that SomeAreaInMath at PhD level and is familiar with common notation in that part of math. If such readers can check the proof and fill in the details with reasonable effort, then the proof is considered to be complete. While this might not sound very precise, in practice there is a near unanimous agreement about what constitutes a valid proof and what does not. A good proof must not contain any statement that could be misinterpreted due to ambiguity. If there are ambiguous statements that are not easily resolved then the reader is justified in rejecting the proof as unreadable. MvH 12 Jan 2006.

PS. Standards about what is a valid proof have indeed changed historically. But that does not mean that there still is a debate among mathematicians about what constitutes a valid proof. I think that "may use logic, ZFC, previously proven results, and may omit details that a reader can be expected to fill in" are the generally accepted criteria for judging validity of a proof. MVH Jan 12, 2006.

I remember attending a lecture given by Graham Higman in which, in response to a question, said something like (I can't remember the exact words - it was about 30 years ago!) "A proof is a form of words that convinces the mathamatical community of the truth of a proposition". Interestingly no one challenged this although there were some emminent mathematicians in the audience. How many mathematicians could quote the axioms of ZFC (Even if they were once required to attend a course on the subject)? Most could quote some version of the Axiom of Choice (AC) and perhaps the Axiom of Infinity, but I don't think most mathematicians have ever really considered the philosophical (not sure if that is the right word to use here!) status of the axioms. For instance most mathematicians would take acceptance or non-acceptance of AC mark the borderline between constructivist and non-constructivist versions of mathematics however this is not the case. Take AC in the form - I am being deliberately informal here - "Given an infinite collection of non-empty sets there is a set which contains precisely one element of each". To a platonist, one who believes in the literal eternal existence of mathematical objects this is non-controversial. However an intuitionist the claim to be able to exhibit an infinite collection of sets precisely is the claim that one can exhibit an element of each set so again the axiom is again uncontroversial. It is only when one adopts a "half-hearted" version of constructivism, somewhere between platonism and intuitionism that the axiom of choice becomes controversial. I would suggest (this is only an opinion, I have not made a scientific survey!) that the majority of mathematicians, if pushed, would subscribe to some version of "formalism" (i.e. the notion that it is the business of mathematics to contruct axiomatic systems and prove things inside them.). On this view one could accept ZFC on Mondays, Wednesdays and Fridays, and reject it on Tuesdays, Thursdays and Saturdays. I believe mathematicians tend find the idea that there is some sort of objective and universally accepted standard of mathematical proof psycologically comforting and for this subscribe to the idea - there is often a certain amount of resistance to even holding a sensible discussion of the issues - but I don't think there actually is such a standard, or perhaps I should say that the Higman quote is about the nearest one could get to such a standard. Bernard Hurley 09:21, 1 October 2006 (UTC)

Contents 1 Prove everything 2 Moving 3 Proof by Transposition? 4 Examples? 5 Visual for Page 6 Four colour theorem 7 Methods of proof. 8 Bijection? 9 Second proof by contradiction example 10 Proof archive - Do we need one?

[edit] Prove everything

If the purpose of mathematical proof is to prove everything starting from a set of axioms [say, ZFC], shouldn't all mathematical proofs provide links to what comes previously, so that we could trace every proof back to the axioms?

Within proof theory, quite a lot of proofs do quote any 'standard results' used, which in turn can be used to further trace back results to the axiom set used. For most practical purposes, however, it is enough to know that a result has been widely established as correct. Having said that, I'm not sure what you're suggesting here as to updating the article.

[edit] Moving

IMHO this could be moved to proof (mathematics). What do you think? googl t 19:27, 15 August 2006 (UTC)

Seems reasonable, although I don't really know if it's worth the trouble. People are more likely to search Mathematical proof than Proof (mathematics). Meekohi 17:05, 17 August 2006 (UTC)

[edit] Proof by Transposition?

I've never heard this term before; I've always called this technique "Proving the Contrapositive." Is it possible to put both terms in that heading, or at least a note in the section that it's talking about the contrapositive here? I'm eager to assist with this project (including the overall WikiProject: Mathematics), so please let me know how I can help. Feel free to leave a message on my Talk page to do so. Thanks, JaimeLesMaths 05:39, 28 September 2006 (UTC)

Contraposition and transposition are different (but related) concepts - see their articles. The method referred to under the heading "Transposition" depends on the rule (P → Q) ↔ (~Q → ~P), which is the rule of transposition, hence it is properly called "Proof by transposition". Gandalf61 13:05, 28 September 2006 (UTC)

OK, so transposition is the rule that states that a statement is logically equivalent to its contrapositive. Then I'm for keeping the heading the same and just adding that the formal name for the referenced equivalent statement is the contrapositive. Or maybe just even a See Also: Contrapositive would do. --JaimeLesMaths 06:07, 30 September 2006 (UTC)

[edit] Examples?

I added an example to the Proof by Contradiction sub-section in an effort to beef up the content and make the concepts more understandable. I think it would be great to add such examples to all such sub-sections. Thoughts? --JaimeLesMaths 06:25, 28 September 2006 (UTC)

Nice idea, but this article is not the right place. Mathematical proof is meant to be a short survey article with links to more detailed articles. The "Types of proof" section started out as an annotated list of proof types. Then someone added sub-headings. Now you are proposing examples of every method of proof that it refers to. With all these additions, the article will become too long, and new readers will not be able to see the wood for the trees. Examples belong in the detailed articles themselves - so the example of proof by contradiction that you have added should be moved to the proof by contradiction article. Gandalf61 11:42, 28 September 2006 (UTC)

I like the idea of having an example proof. Prefreably the simplest one about. One example would help readers understand the concept of proof in general as distinct from particular methods of proof. In general this artice needs considerable expansion to get it above its Start-status. This is particularly important as the article is one of Wikipedia 1.0 core topics suplements so is of high visability. --Salix alba (talk) 12:39, 28 September 2006 (UTC)

But my fear is that this article then becomes bloated with multiple examples, and before long someone slaps a "too technical" tag on it, and someone else start to write a simpler introductory article, and the cycle starts all over again ! Gandalf61 09:42, 29 September 2006 (UTC)

I certainly don't want the article to get too bloated, and I see now that the proof I gave is also provided in the reductio ad absurdum article. Perhaps instead of a full example in each one, a direct link to an example in the appropriate article with a descriptive title of what is being proved? But at least one full example of a proof (perhaps annotated in detail?) is needed, I think, though not necessarily the one I provided. Other than that, even though this is a survey article, it needs more meat/eggplant to it. I don't even know if I support this idea, but would a section of "common mistakes" in proof be useful? Or at least a link to logical fallacies. --JaimeLesMaths 06:19, 30 September 2006 (UTC)

[edit] Visual for Page

Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem by Simon Singh has a picture of the first page of the proof of Fermat's Last Theorem. Might be a nice visual for the page, though I'm not 100% sure about policy for putting it in here. I guess my question is if the proof itself is considered "public domain" or not. In any case, to get this up to featured article standards, some visual would be nice. Any other suggestions? --JaimeLesMaths 06:25, 28 September 2006 (UTC)

Bill Casselman has a beautiful image of a fragment of Elements which can be found at http://www.math.ubc.ca/~cass/Euclid/papyrus/tha.jpg. It'd make for a great picture if there were a "history" section in this article. I've never uploaded an image before, so I'm not sure how to go about it. I think he took the photo himself; I could email to see what sort of copyright he claims on it. shotwell 16:03, 8 October 2006 (UTC)

I think it makes a great picture regardless of where it was placed in the article. If it's available, let's use it. I'm not sure of the picture uploading process either, but I'm sure someone here can help us out. --JaimeLesMaths 00:10, 9 October 2006 (UTC)

Proposition II.5 from Euclid's Elements

Uploading a picture is not that hard (but not completely straightforward either). It's explained at Wikipedia:Uploading images and commons:Commons:First step. Anyway, I uploaded the picture (see at the right). However, I'm not sure it's a good illustration for this pape as it only contains the theorem, not the proof. -- Jitse Niesen (talk) 07:19, 9 October 2006 (UTC)

I have not yet clarified the copyright on that picture with its author. shotwell 12:39, 9 October 2006 (UTC)

And yeah, I suppose you're right about its suitability. shotwell 12:45, 9 October 2006 (UTC)

Regarding copyright, as I understand it, the papyrus is written centuries ago and hence in the public domain. Taking a picture is not a creative act and hence does not make the photographer eligible to claim copyright. See Wikipedia:Public domain. -- Jitse Niesen (talk) 03:58, 10 October 2006 (UTC)

[edit] Four colour theorem

I'm wondering if its worth including something about the four colour theorem and other proofs which have only been mechnacially verified. --Salix alba (talk) 09:37, 1 October 2006 (UTC)

I think that's a good, commonly-known example for proof by exhaustion. --JaimeLesMaths 21:33, 3 October 2006 (UTC)

[edit] Methods of proof.

Hi everyone. I was looking over the "Methods of proof" section and it feels very verbose, "probabilistic proof" for example feels more like an application of probability theory than much of a different approach to proof, similarly "combinatorial proof". "Direct proof" clearly deserves mention as does "contradiction" or "contrapostion" or "transposition" or any one of the names it seems to be listed under. "Induction" too is a well known method and so could be listed. As for the others, how about moving them to a quick list of other common methods of proof and just keep short descriptions of the key ones? Richard Thomas 01:01, 26 October 2006 (UTC)

[edit] Bijection?

The text from Bijection section of the article reads: "Usually a bijection is used to show that the two interpretations give the same result." Was this meant to be part of the Combinatorial proof section? I checked the Bijection article, but I didn't see anything there that made sense in the context of this article except for its relationship to Combinatorial proof. I'm going to remove the heading for now, but feel free to put it back if the text is expanded and made clearer. --JaimeLesMaths ^(talk!edits) 22:04, 28 October 2006 (UTC)

[edit] Second proof by contradiction example

I'm moving this example here because, at the least, it needs formatting cleanup. However, I don't think that this example is best for the article. It's not mentioned in the main reductio ad absurdum article, and it's not easily comprehensible to non-mathematicians. I want to be clear that I'm not wedded to the example I've added staying either (not trying to WP:OWN the article), but simply that this example needs some work/discussion before being re-added to the main article. (See also discussion above about whether we should have any examples in article.) --JaimeLesMaths ^(talk!edits) 22:16, 28 October 2006 (UTC)

Another little problem in Number Theory can be proved using proof by contradiction. The DIVISION ALGORITHM states that : Given any integers a and b with a not equal to 0, there exist unique integers q and r such that b=qa+r, 0<_r<|a|. If b is indivisible by a , then r satisfies the stronger inequality 0<r<|a| LEMMA 1. If an integer u divides an integer v, v not equal to 0,then v=up, p not equal to 0. hence |v| = |up| = |u| |p|. As p is not equal to 0 and |q| is either greater than or equal to 1 , thus |v| is greater than or equal to |u| Proof:Consider S = { b-ak | b-ak >_0, k belongs to Z,the set of integers } Clearly, b + |ab| belongs to S. Thus, S is non-empty. By the well-ordering principle, S has a least element, say b-'aq = r. If r>_ |'a| , then 0<_r-|a|<r ; and r-|a| belongs to S : which is a contradiction! Thus, 0<_r<|a| Now, to prove the uniqueness of q and r, let b= am+n and also b = ak+l' with 0<_n<|a| and 0<_l<|a|. If n is not equal to l , let l>n. then 0<l-n<|a|. But l-n=a(m-k). Thus a divides (l-n). But this contradicts lemma 1. Thus, m=k and n=l.

[edit] Proof archive - Do we need one?

• The "Proof Archive" from the German Wikipedia. (Is this something we should get in the English Wikipedia?) -- Johnbibby

That is not the German Wikipedia, but the German wikibooks. Wikipedia is an encyclopaedia, Wikibooks are text books. On the general point on whether we should have proofs in an encyclopaedia, see Wikipedia:Manual of Style (mathematics)#Proofs for what's probably the current feeling, Wikipedia:WikiProject Mathematics/Proofs for some discussion, and Category:Article proofs for something similar to the Beweisarchiv. I don't want to shoot down your idea immediately, but only to make you aware that the place of proofs here has been discussed before. -- Jitse Niesen (talk) 02:12, 4 November 2006 (UTC)