Talk:Mathematical proof

message: What is an effective proof?  — Preceding unsigned comment added by 2001:628:2120:601:7447:9B9B:6C18:BF6E (talk) 11:49, 9 December 2015 (UTC)
thread_title:  effective proof 

message: The intro currently says that in the great body of math, ZFC is the standard foundation.  I think this statement gives a ludicrously wrong impression.  It's like saying that for most nonfiction authors, the Dewey Decimal system is the standard method of organizing knowledge.  --Jorend 15:32, 14 December 2006 (UTC)
thread_title: Informal proofs, continued

message: I made a link to the section at Proof theory which mentions that formal proofs can be automatically checked but are harder to find (although even the latter is computable, if I understand Godel correctly; it's just that you can't necessarily tell if a proof exists to find). It would be good to say a bit more about this, either here or there: specifically (a) how "hard" it is to convert informal into formal proofs, and (b) the implications for the extent to which it is really "known" that most "theorems" are indeed true. A distinction is often made between absolute truth in mathematics and empirical truth in science, but the predominance of informality calls this into question. (Presumably the answer to (a) is therefore "very"; can anything more precise be said?)  —Preceding unsigned comment added by 194.81.223.66 (talk) 10:39, 19 October 2007 (UTC)
thread_title:  Formal and informal proof 

message: 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?  --anonymous comment
: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.  --anonymous comment
::To the original poster:  You're right.  But math is not this sort of grand project to build everything on ZFC.  Don't get me wrong, such things exist.  I'm a huge fan of Metamath, for example.  But proofs serve many, many purposes, not just one.
::Say you're working on group theory.  You don't really care about the details of predicate calculus and ZFC.  You just want certain things to work, like "if A = B then B = A".  You don't care how, and you certainly aren't going to cite a proof of "the reflexive property of equality" explicitly in your work.
::In short, you're imagining a cetain level of rigour, that's way beyond what your average mathematician needs.  Because his purpose in writing proofs isn't what you imagine.  You really should check out Metamath.  Here's a giant library of proofs that all explicitly link backward exactly as you suggest.  But Metamath proofs don't fulfill the other purposes of proofs particularly well... purposes like communicating the mathematician's line of thought.  --Jorend 15:10, 14 December 2006 (UTC)
thread_title:  Prove everything 

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

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Proof by Transposition? 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Examples? 

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

text: Fits in both echastion and computer sections, as well as in in aeshtetics section, all for different reasons, so mention it thrice.  EricDiesel (talk) 19:25, 23 September 2008 (UTC)
thread_title:  Four colour theorem 

message: 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)
thread_title:  Methods of proof. 

message: 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)
thread_title:  Bijection? 

message: 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)
thread_title:  Second proof by contradiction example 

message: * 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)
thread_title: Proof archive - Do we need one?

message: Am I going ga-ga, or is the example in "proof by construction" just nuts?
If AD is a median and G is the centroid then BG extended is another median and therefore X is the midpoint of AC.  Where's this 1:5 coming from???
Anyway, even if this example were correct, it wouldn't be especially helpful in explaining what proof by construction is.  The main article on constructive proofs is a far clearer explanation, and gives the example of transcendental numbers, which provide a good example of the distinction between non-constructive existence proofs and constructive ones.  Could someone who's more closely involved with this project have a look at this issue and fix it up?
Hugh McManus 08:50, 1 June 2007 (UTC)
replies: text: Geometric example in "Proof by construction" didn't make sense to me either, so I have replaced it with the example that is used in the proof by construction article. Gandalf61 12:48, 1 June 2007 (UTC)
thread_title: Proof by construction example - huh?

message: If a mathematical proof is not a type of formal proof, but rather are usually informal, would it not at least be appropriate to say that they intend to mirror some formal proof, even if they are not themselves fully rigorous or formalized the same way.
Perhaps the last sentence of the lead paragraph can explicate the proper relationship? Pontiff Greg Bard (talk) 23:22, 15 May 2008 (UTC)
thread_title:  Relation to formal proof 

message: I have removed the following nonsense, added twice by an anonymous editor:
:Proof by intuitive lemma
: 
:Often problems of mathematical proof can be reduced to a much simpler form by merely considering an appropriate lemma. 
:For example, to prove that π is irrational you could start by assuming the obvious result, "π2 is irrational". Then the result follows.
This is incorrect because the result "π2 is irrational" is clearly not obvious - otherwise the irrationality of π would have been proved in antiquity, not in the 18th century. The term "Proof by intuitive lemma" is not notable. And the red-link also indicates that this is not a serious contribution. Gandalf61 (talk) 15:23, 27 May 2008 (UTC)
replies: text: I agree with you entirely, both about the irrationality of π2 not being obvious, and that that added text is not helpful. I'm going to remind the IP editor to avoid edit warring, and remove the text again. — Carl (CBM · talk) 15:36, 27 May 2008 (UTC)
replies: text: Thank you for your help :) Gandalf61 (talk) 15:42, 27 May 2008 (UTC)
thread_title:  Removed nonsense 

message: *1. On introducing Shakespeare's, Macbeth, Macbeth rode out upon the "heath" (a plane), "line" against line (a line of Kerns, a military line of men) , "point against point" (a point is a spear or javelin) , "lapp'd [covered] in proof [covered in armour]".  An emeritus Shakespeare scholar at UCLA claimed Shakespeare was talking about geometry, and the metaphor was armor against the false, developed further in the story.  Does anyone have anything to contribute as a section of this article regarding the etymology or early history of "proof", armour against false propositions, other than use in law? 
*2. Hacking discusses "probity" in relation to "probability", with ironic etymology.  But more interesting is that "probity" comes from God without empirical test, like proofs were once thought to.  Does anyone know more about this relationship?  The parallel would agrue against coincidence.  I included this fact in the article without stating the inference from the fact.  Tautologist (talk) 19:48, 3 October 2008 (UTC)
replies: text: Websters 1913 edition has same etymology. as does the The Century Dictionary, 1911. and wikitionary wikt:proof.--Salix (talk): 17:58, 12 July 2009 (UTC)

text: Yes I also found it in the New Shorter OED. I'll cite that since its new. I wonder why the OED doen't - maybe I misread it. Myrvin (talk) 21:02, 12 July 2009 (UTC)
thread_title:  Etymology 

message: Gandalf61 rewrite is a big improvement on my original scribbled note jot (my other edits could be similarly improved).  
*1. Maybe append clause on loss of apriori certainty, re "math is not empirical", and related debates, e.g., post- computational neuroscience "neo-psychologism" as to certainty never being there in the first place, written with spin of new functionalist phil of mind perspective, etc.  
*2. Also maybe could be added semi-technical brief sentence or two explaining foundatoin theory tranlation of proofs via to symbolic proof structures allowing computers are able to "think" and "prove" things, without reader having headache of linking to main article; this would both preserve mathematical appearance of article in methods section and clarify things w/o tracking down via link. Tautologist (talk) 17:05, 27 September 2008 (UTC)
thread_title:  Computer proofs 

message: Some Homeless (proofs) are subjects of derision by all- 
* In recent math journal article- "There is only a statistical proof of this."
* From Wiki Statistics article "edit summary" for reason to delete brief mention of same "only a statistical proof" from Wiki Statistics article - 
:"(→Use of Statistical Methods Outside the Framework of Inference: Remove sections - these refers to uses of mathematics, not statistics)" 
Oh, well.  Tautologist (talk) 19:56, 27 September 2008 (UTC)
thread_title:  statistical proof - "Homelessness" not dealt with in ''this'' election cycle 

message: The article says a proof is "a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true". I think few mathematicians would recognize this as a description of proof. Eric Weisstein's "A rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition" (on MathWorld, http://mathworld.wolfram.com/Proof.html) is much closer to the mark. The fact that a proof is rigorous is key to the concept of mathematical proof and is a distinguishing feature of mathematics.
The text in a math book or math journal labeled "proof" is intended to convey or communicate the existence of a proof to the reader. The proof itself is better thought of as existing in the minds of the author and reader. The author checks that there is a rigorous proof by checking that each step logically follows from those that come before and filling in all details needed to verify this. However, the author doesn't include all these details in what they write. Instead, they decide how best to communicate this structure to a reader who has the background and knowledge that they are assuming for their readers. The reader, when reading the proof, is expected to check that a rigorous proof exists by checking that each step logically follows from those that come before and by filling in all details needed to verify this.
This underlying logical structure/scaffolding actually permeates a math book or journal article. Every sentence or group of sentences in a math book/article can be classified as a definition, theorem, proof, or remark. (Some sentences might have elements of more than one of these categories.) Math books/articles try to guide the reader by providing explicit labels, e.g., "Theorem", "Proof", for the more important or lengthier such items. However, it is expected that the reader will figure out the function of the unlabeled parts by their content. One convention that helps the reader is that sentences containing a word or phrase in italics are definitions of the italicized word or phrase (some authors use bold). If a sentence in an unlabeled part is functioning as part of a proof, the reader is expected to check that the proof is correct/rigorous. For some declarative sentences, the author may not give any justification. For these mini theorems, the reader is expected to provide the proof without help from the author. When I first realized, in talking with colleagues who were not mathematicians, that most non-mathematicians are unaware of this structure in math books/articles, I was rather surprised.
To understand the relationship between mathematical proofs and formal proofs, it helps to understand the purpose of the latter.
One purpose of formal logical systems and formal proofs is as a mathematical model of mathematics and mathematical proof. This mathematical model allows the mathematical study of mathematics. For this purpose it is important that a formal proof capture the key features of a mathematical proof. However, as with any mathematical model, the model is not the object. A formal proof is a mathematical proof, but most mathematical proofs are not formal proofs.
Another purpose of formal logical systems is to provide a foundation for mathematics. For example, ZFC is an adequate foundation for almost all of modern mathematics. When used for this purpose, it suffices to prove that the mathematics in question could be translated into ZFC. It isn't necessary to actually do the translation. Of course, the proof that such a translation could be done is a mathematical proof.
--David Marcus (talk) 19:55, 24 December 2008 (UTC)
thread_title:  Rigor's role, communicating proofs, purpose of formal proofs 

message: This is not distinct from a direct proof. Any reason not to remove it from the article? Bongomatic 13:28, 24 March 2009 (UTC)
replies: text: These particular types of direct proofs are very commonly taught in grade school in the U.S., to the point where many students immediately think of a two-column proof when the hear the word "proof", instead of thinking of the natural-language proofs that mathematicians would think of. So I think that these are of enough interest to the general reader to include (because they may well wonder why the thing that they learned to call a "proof" is not included). There are several other sections in the article about types of direct proofs: elementary proofs, combinatorial proofs, etc. Pretty much every type of proof listed in the "Proof methods" section is a direct proof. Maybe this could be clarified some. — Carl (CBM · talk) 14:02, 24 March 2009 (UTC)

text: It just means that the right-hand column in a two column proof is often headed "Reasons". I have rewritten that section of the article to make it clearer. Gandalf61 (talk) 09:28, 15 October 2009 (UTC)
thread_title:  Two-column proof 

message: As a student of mathematics having assisted with proofs classes, I believe this article is dismally organized. As a summary of the concept of mathematical proofs, it reads fairly well in sections (1) and (2), but section (3) on the methods of proof (which is probably the most important section to someone looking for specific information) is a mess. The main categories of proof techniques are lumped right in with subcategories, field-specific techniques, and several that are not methods of proof at all such as "visual proofs" and computer-assisted proofs (i.e., demonstrations and tools). First of all, the techniques listed can at least be divided into direct proofs, proofs by contrapositive (or transposition), and proofs by contradiction. Arguably, existence proofs and proofs by induction could also be top-level divisions. More specific proof categories such as combinatorial proofs - which may themselves refer to several different methods of proof - could be listed in a final, separate sub-section. Secondly, the two mentioned earlier, visual proofs and computer-assisted proofs, should probably be presented in an "Other" section with a header explaining that they are not, themselves, proof techniques. The latter would be a tool for proving by cases, which is a type of direct proof.
I do not pretend to have the supreme layout which will present the material most clearly to all audiences - I just think it needs to be done much, much better than it has been. I've given a few ideas already. Does any disagree or have other ideas for organizing the "Methods of proof" section? Kiyura (talk) 03:36, 2 May 2009 (UTC)
replies: text: Sounds good to me. — Carl (CBM · talk) 15:23, 19 May 2009 (UTC)
replies: text: When you have it up (I assume in a sub-page), can you please post a notification and wikilink here? Bongomatic 16:23, 19 May 2009 (UTC)

text: I have finished my initial revision, here it is: User:Kiyura/Mathematical_Proof. Please see the associated Talk page as well. Since you two, at least, have replied (thank you), I will not apply anything to the main article until some consensus is reached. Kiyura (talk) 20:13, 23 May 2009 (UTC)
thread_title:  Very messy! 

message: The reference for Buss, 1997 is missing. What is the publication? Myrvin (talk) 11:20, 12 July 2009 (UTC)
replies: text: I think it must be Samuel R. Buss.  His website only lists four 1997 publications, none of which seems promising: http://www.math.ucsd.edu/~sbuss/ResearchWeb/index.html.  I think that that reference should be cut or improved. Colin Rowat (talk) 18:35, 22 December 2015 (UTC)
replies: text: The year is wrong; I've found a 1998 book that matches the claim and page number. I'll use it to fix the reference. —David Eppstein (talk) 18:55, 22 December 2015 (UTC)
thread_title:  Buss 

message: is one of the main contributors to Wikipedia (over 67,000 edits; he's ranked 198 in the number of edits), and practically all of his edits have to do with Islamic science, technology and philosophy. This editor has persistently misused sources here over several years. This editor's contributions are always well provided with citations, but examination of these sources often reveals either a blatant misrepresentation of those sources or a selective interpretation, going beyond any reasonable interpretation of the authors' intent. Please see: Wikipedia:Requests for comment/Jagged 85. I searched the page history, and found 5 edits by Jagged 85 in March 2010. Tobby72 (talk) 14:03, 13 June 2010 (UTC)
replies: text: Fixed. Gandalf61 (talk) 15:36, 13 June 2010 (UTC)
thread_title:  Misuse of sources 

message: I recently came across the article on statistical proof. When when I first arrived at the page - it was a mess. There is a bit of debate going on in the discussion page on the merit of statistical proof having its own page and even if such a thing exists. There are obvious logical ties to mathematical proof, so I thought I would come here and ask others who know about mathematical proof to share their thoughts. Is statistical proof distinct from a mathematical proof? Is this something that could turn into a small section or Wikilink in this article? Thought I would raise the issue and see what comes of it. Thanks.Thompsma (talk) 21:25, 11 November 2011 (UTC)
thread_title:  Statistical proof 

message: First, many apologies. I know I typed something like this when I made a small change a few days ago. Somehow it did not remain so I am typing this first. I have no interest in a reversion conflict. Just wanted to make a change and state the reason. The change is to return a few sentences relating to "the first proof by contradiction" I have no opinion about the value one way or the other about these few sentences. My problem is that they were removed with the total justification being something like "Euclid's proof of the infinitude of primes was the first proof by contradiction" (not a direct quote.) As Wikipedia itself correctly says (Euclid's Proof) "Euclid is often erroneously reported to have proved this result by contradiction" (and other parts of Wikipedia do say that) That is based on the following common misconception that Euclid book IX prop 20 says something like
Prop: There are infinitely many primes
1. Suppose that there were finitely many and list them p1,p2,..pk
2. Use the list to create the number P=p1*p2*...*pk+1
3. There is a prime q dividing P (maybe q=P) and it is not in the list CONTRADICTION
It is true that the proof is nowadays often presented this way and that there are claims (including some parts of Wikipedia) that this is essentially what Euclid wrote. However what was actually written was more like, there are always more primes (than in any finite list)
Prop: No finite list of primes includes all primes
1. Given a list p1,p2,..pk (here is how to get something else)
2. Use the list to create the number P=(p1*p2*...*pk)+1
3. There is a prime q dividing P (maybe q=P) and it is not in the list (there, something else)
SO I am happy with some other reason to remove the claim I reverted, but not the given one. Gentlemath (talk) 02:31, 16 February 2015 (UTC)
thread_title:  Euclid and Indirect Proof 

message: Inductive inference and deductive inference are both regularly used in mathematical proof.  
The article "Mathematical induction" declares the method of mathematical induction as deductive inference, but, it's inductive inference.
It neither clear nor unambiguous (nor non-controversial) to call "inductive inference" (exhaustive inductive inference, as of proof by induction) instead "deductive inference", because it's not.
It is clear and unambiguous that "exhaustive" deductive or inductive inference (i.e., covering all cases) does maintain derivability of conclusion from premise and otherwise maintains the grounds for mathematical proof.
It is very widely understood that "proof by induction" is "mathematical proof by mathematical induction" in any context of mathematical proof.  — Preceding unsigned comment added by 75.172.122.39 (talk) 07:10, 18 March 2017 (UTC)
thread_title:  Induction and Deduction as methods of inference 

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