Talk:Mathematical proof

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[edit] Informal proofs

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)

[edit] Informal proofs, continued

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)

[edit] Formal and informal proof

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)

[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? --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)

[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
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?

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)

[edit] Proof by construction example - huh?

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)

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)

[edit] Disagree with the first paragraph

I have to disagree with this:

In mathematics, a proof is a demonstration that, assuming certain axioms and rules of inference, some statement is necessarily true.

I don't think that this has ever been a standard definition of "mathematical proof". For example Euclids' proof that there are infinitely many primes couldn't be expressed in the "axiom->theorem" form until Peano axiomatized arithmetic, and have been called "a proof" for centuries (togheter with many others that didn't have the "axiom->theorem" form until Zermelo&C. axiomatized set theory or Robinson axiomatized Non-Standard Analysis).

Moreover: Why a proof of the irrationality of e should be "a demostration that assuming certain axioms and rules of inference e is necessarily irrational" and not just "a demostration that e is irrational"?

In virtually all branches of mathematics, the assumed axioms are ZFC (Zermelo–Fraenkel set theory, with the axiom of choice), unless indicated otherwise. ZFC formalizes mathematical intuition about set theory, and set theory suffices to describe contemporary algebra and analysis.

This is arbitrary:

  1. "Assumed" by whom?
  2. "Unless indicated otherwise" according to what?
  3. Why ZFC and not NBG or New Foundation?
  4. Virtually all mathetaticians make proofs without ever knowing exactly what ZFC (NBG or whatsover) is and having no idea of how the ZFC axioms would work in their (informal) proof
  5. What about Synthetic geometry or visual geometric proofs of algebraic equalities? What have they to do with ZFC?

--Pokipsy76 (talk) 10:30, 1 January 2008 (UTC)

I'll propose a rewrite and see if that gets the changes started:

In mathematics, a proof is a rigorous demonstration that, assuming certain axioms and rules of inference, some statement is necessarily true, within the accepted standards of the field. A proof is a logical argument, not an empirical one. That is, one must demonstrate that a proposition is true in all relevant cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence believed (but not proven) to be true is known as a conjecture. In virtually all branches of mathematics, the assumed axioms are ZFC (Zermelo–Fraenkel set theory, with the axiom of choice), unless indicated otherwise. ZFC formalizes mathematical intuition about set theory, and set theory suffices to describe contemporary algebra and analysis.

This is a really rough set of changes, but I thought I'd try to get the ball rolling. The ZFC stuff might be rephrased and relocated to another part of the article (as opposed to just struck entirely). Also, I didn't wiki-link anything, although there's alot of that missing I think from this paragraph (as it is now, and in this rough draft). --Cheeser1 (talk) 22:39, 3 January 2008 (UTC)

This is an improvement. Some suggestions for modifications to this replacement text:
  • Replace statement by mathematical statement (and hyperlink).
  • Replace rigorous by: convincing. Not all demonstrations that are commonly accepted as a valid proof within "the accepted standards of the field" are rigorous. We may instead wish to add some paragraph or sentence somewhere stating that informal proofs are generally considered acceptable when it is obvious to experts in the field how to make them formal and rigorous.
  • Replace one must demonstrate by: the proof must demonstrate.
  • Replace all relevant cases by: all cases to which it applies, without a single exception
  • Add after believed: or strongly suspected.
  • The part (but not proven) is redundant and can be deleted.
Applying this results in the following text:
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true, within the accepted standards of the field. A proof is a logical argument, not an empirical one. That is, the proof must demonstrate that a proposition is true in all cases to which it applies, without a single exception. An unproven proposition believed or strongly suspected to be true is known as a conjecture.
 --Lambiam 10:20, 4 January 2008 (UTC)
It seems OK, let's edit the paragraph.--Pokipsy76 (talk) 17:38, 8 January 2008 (UTC)
Y Done  --Lambiam 22:08, 8 January 2008 (UTC)
I like the changes you made, and I agree that it definitely sounds better now. I'm glad we fixed it up. --Cheeser1 (talk) 22:54, 8 January 2008 (UTC)

[edit] formal proofs

I removed a statement that mathematical proofs are formal proofs. Certainly there is a formal character of mathematical proofs. But it is not the sense described in the article formal proof. Mathematical proofs are almost always expressed in natural language, not in a formal language. While it is commonly assumed that mathematical proofs could be recast as formal proofs, that doesn't mean that they are formal proofs to begin with. — Carl (CBM · talk) 02:22, 12 May 2008 (UTC)

So now I'm wondering how it works into the article. The issue you describe seems to me like content for the lede. If that is the way people see it. The truth is that a mathematical proof itself is an abstract object. (However, I am not interested in pushing that issue here.) It is that abstraction that manifests in natural or formal language. We don't need to harp on that, however I would like to provide a path for those who want some fundamentals. Pontiff Greg Bard (talk) 03:39, 12 May 2008 (UTC)
No, the proof is not an abstract object, though it may, and likely does, contain various types of abstractions. Tparameter (talk) 04:07, 12 May 2008 (UTC)
No, Tparameter, I am absolutely clear: the proof itself is an abstract object. When you see chalk lines on a board, or ink marks on a page representing in language a proof, really what you are talking about is a token of the type of abstract object that the proof is. The many and varied tokens of the same proof are the ones, for instance, in a French mathematics class, and and American one... they talk about the SAME proof do they not? Pontiff Greg Bard (talk) 06:08, 12 May 2008 (UTC)
This is why these things need to be in the article.Pontiff Greg Bard (talk) 06:09, 12 May 2008 (UTC)
What you're saying is not necessarily true, and is therefore not a rule, and thus should not be discussed in the article. An analogy to what you're trying to say might be a vector compared to an instance of that vector - the abstraction compared to a particular instance of it. However, you assume there is some abstract proof floating in some layer of abstraction above where the mathematician/logician lays out his proof. Not necessarily true. More importantly, not the appropriate place to discuss this type of abstract concept. It will only add confusion, and take away from the substance of the article. You might try this sort of thing in happy, pointing out that anyone being happy is only a token representation of the abstract concept of "happiness". Of course, I'm being facetious - but, hopefully you get my point. This token distinction is a discussion of its own, and should not be introduced in every conceivable place it can be imagined. Better to defer to mathematicians (not referring to myself) in math articles, IMO. All the best. Tparameter (talk) 07:34, 12 May 2008 (UTC)
Yes, it is, in fact, necessarily true that a mathematical proof is an abstract object. I don't seek to insert this kind of stuff in any old article like happy. There are a few topics for which it is important, and for the most part they are in the template:logic, not everywhere. I do not see the need to insert "Abraham Lincoln was a human..." into every biography. That is the type of thing that is obvious and not helpful. But for these topics which people other than mathematicians study we should respect a wider audience. There are clearly reasonable cases and clearly reasonable cases (this is why I am not pushing the issue for math proof) where these things should be discussed: set, theorem, and really only a few more. I hope you think about what I have said, and as a matter of conscience relent in the view that we can never never never talk about this abstract distracting stuff. I'm serious. There are some people are looking for exactly this kind of stuff when they are looking at these articles. You are denying them. It's really pretty little to ask in the scheme of things.
By the way, if we explore this whole "distracting" justification critically, we find that it really isn't a very good principle for our editing. Clarify, don't duck your head in the sand. My goodness. Be well, Pontiff Greg Bard (talk) 08:11, 12 May 2008 (UTC)

Gregbard, you said "It is that abstraction that manifests in natural or formal language. ". I'm sure that a more common viewpoint among mathematicians is that the proof "is" the natural language expression, not an abstract idea behind that expression. Mathematicians use the word "same" to indicate lots of equivalence relations.

I'm certain people have written a lot about the nature of mathematical proof (although I know of only a little bit of their work). Are you taking your ideas here from a published source? I'd be glad to look through it to get a sense of what's going on. — Carl (CBM · talk) 11:55, 12 May 2008 (UTC)

I'm going to have to look for some sources. What you say may be ture. However, if mathematicians generally believe that the proof, or the set or the theorem is the chalk on the board, then all of the logical consequences of that belief go along with it. Really we have identified a misimpression which could stand some clarification. You will find in general that my edits seek to clarify. One can express sets theorems and proofs in all kinds of languages, however we still say it is the same theorem, etc. To some it may seem a silly point, whereas other study this aspect of it specifically. Pontiff Greg Bard (talk) 00:57, 14 May 2008 (UTC)
There are lots of equivalence relations that people describe as "same". In the most literal, intensional sense, a proof is not the "same" proof when translated into a different natural language. — Carl (CBM · talk) 01:00, 14 May 2008 (UTC)
Carl, now we are just fudging on what it means to be the same. Please (all I can do is beg at this point), let me correct you. First of all a lot of people don't use the word "literally" correctly. Literally speaking, they are the same proof (a proof of the validity of modus ponens for instance), and it is only the tokens (the French, American, natural deduction method, and axiomatic versions) of it that are different. This distinction, I believe is well known, and accepted. If you and others have have a hard time accepting this, then it really proves my point that this kind of material needs to be covered. Pontiff Greg Bard (talk) 23:10, 15 May 2008 (UTC)

[edit] Relation to formal proof

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)

[edit] Removed nonsense

I have removed the following nonsense, added twice by an anonymous editor:

Proof by intuitive lemma
Main article: 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)

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)
Thank you for your help :) Gandalf61 (talk) 15:42, 27 May 2008 (UTC)