This is an archive of past discussions about Mathematical proof. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.
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 by68.35.253.247 (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 Hurley09:21, 1 October 2006 (UTC)
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. --Jorend15:32, 14 December 2006 (UTC)
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)
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. --Jorend15:10, 14 December 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). Meekohi17:05, 17 August 2006 (UTC)
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, JaimeLesMaths05:39, 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". Gandalf6113:05, 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. --JaimeLesMaths06:07, 30 September 2006 (UTC)
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? --JaimeLesMaths06:25, 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. Gandalf6111:42, 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 ! Gandalf6109:42, 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. --JaimeLesMaths06:19, 30 September 2006 (UTC)
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? --JaimeLesMaths06:25, 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. shotwell16:03, 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. --JaimeLesMaths00:10, 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)
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)
It's been 7 years without a new picture; I think this one is great, and the diagram is the beginning of the proof, so I do think it is appropriate.
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)
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 Thomas01:01, 26 October 2006 (UTC)
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)
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.
Proof archive - Do we need one?
The "Proof Archive" from the German Wikipedia. (Is this something we should get in the English Wikipedia?) -- Johnbibby
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 McManus08:50, 1 June 2007 (UTC)
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:
"Assumed" by whom?
"Unless indicated otherwise" according to what?
Why ZFC and not NBG or New Foundation?
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
What about Synthetic geometry or visual geometric proofs of algebraic equalities? What have they to do with ZFC?
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.
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)
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)
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.
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)
It seems to me that often their are many mathematical proofs which are not on the internet, or when they can be found are haphazardly located in unrelated topic web pages.
It would be nice to have a mathematical proofs wiki, where people can add as many alternative methods for proving a given thing as seem generally distinct. Often wikipedia doesn't provide
proofs at all. I think the trick would be to have links to
proofs through category headings like linear algebra --> matrices --> invertible matrices equivalence statement proofs.
--> just found out that there is such a thing called plantmath, will add a link
--> wrote a small description of planetmath and wikiproof.
Sorry, but I removed your descriptions - including these sites under external links is fine, but writing descriptions of them here gives them undue weight and looks like advertising. Planetmath has an article in its own right, and the other site could have its own article too as long as it is notable enough. Gandalf61 (talk)
Etymology
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)
I am suspicious of the etymology in the article. Hacking does not seem to derive the word from probare, although he does refer to the meaning of test. We need a lexicographical citation. I don't find it in the OED or Chambers for Mathematical proof. Myrvin (talk) 10:52, 12 July 2009 (UTC)
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)
"Statistical proof" Section
There is an error in the deletion explanation, "Statistical proof" - already covered in "Probabilistic proof", in the article edit history. Therefore, I restored the "Statistical proof" section.
1. Material in the "Statistical proof" section is not contained in the "Probabilistic proof" section. More strongly, there is no overlap.
2. A probabilistic proof is nonconstructive. Neither of the terms, "constructive" or "nonconstructive", would be used to describe a statistical proof. (A “proof in statistics” might or might not be a “probabilistic proof”, however, as explained in the deleted section, a “statistical proof” differs from a “proof in statistics”, the former being data analysis pertaining to data, the later to mathematics pertaining to variables abstractly representing data.)
3. The two kinds of proof do not use “proof” with the same meaning. As explained in the deleted section, "statistical proof" has two meanings, and both are different from "proof in statistics". Both meanings are very different from "probabilistic proof", which may or may not be a kind of "proof in statistics", but never a “statistical proof”.
4. A statistical proof produces statements like “OK, you can sell this drug in the US, having established a p value less than .05”. A probabilistic proof is a method of proof to conclude things like, “if there is such and such kind or graph, then a so and so kind of subgraph exists”. A "statistical proof" is either the mass of deduction and methods used to arrive at a probability statement about specific “data”, or an evaluation of that statement, as more fully explained in the deleted section. A Statistical proof produces conclusions like “Cigarettes are associated with cancer in this way, based on this data.”
5. A probabilistic proof is a method of proving existence theorems, which does not construct the thing proved to exist. For example, suppose you can prove the Expectation of something is less than 0, then you have proved that there exists an element less than 0. Probabilistic proof has more meaningful and classic examples.
6. The two kinds of proof are related only in a relation of two terms in their titles, i.e., statistics is related to probability theory in that the former is used to prove theorems in the latter. However, a "probabilistic proof" could never be called a "statistical proof", nor vice versa. A "proof in statistics" may, however, use a "probabilistic proof", to establish existence. However, a "proof in statistics" differs from a "statistical proof", as explained in the deleted section. The two sections, "probabilistic proof" and "statistical proof" should not appear next to each other to avoid confusion because of apparent relation through terms in their titles. EricDiesel (talk) 01:16, 21 September 2008 (UTC)
If I understand you correctly, what you are calling a "statistical proof" is actually a statistical test - a method of reaching statistical conclusions (such as "cigarettes are associated with cancer" or "x is true with a probability of greater than 99%") using experimental data. I don't believe any mathematician would call a statistical test a proof, and it is confusing to include them in this article, which focusses on mathematical methods of proof. If this is just your own usage of the term "statistical proof" then it is original research, and should be removed from the article. To show that this section is not original research, you need to provide a reliable source that uses the term "statistical proof" with the meaning that you have assigned to it. Gandalf61 (talk) 08:30, 21 September 2008 (UTC)
Re your first sentence, agreded as to "proof" in mathematics (I used to be a probabilist, and I would never call a "statistical proof" a "proof" in a talk in a math department, though "proofs" in the latter usage may occur in the former, if you read and track down all the footnotes). A "statistical test", may or may not be part of a statistical proof. "Staistical Test" has, over time, become a technical expression referring to the Neyman-Person classical theoretical framework, further put in a desicion theoretic framework as in, e.g., T. Ferguson's text. With the advent of computers and the subsequent ability to use nonparametric methods, this classic "statistical test" framework is used less in "statistical proofs". (See B. Efron's Scientific American article predicting this would occur.) An example of a "statistical proof" without a "statistical test" was the establishment of menstural synchrony using data from sororities, done by the former director of the Center for Research on Women at Stanford, which used a circular distance measure (akin but not identical to reasoning in group theory), but contained no formal "statistical test", (e.g., independence assumptions could not be established). "Statistical proofs" can have dazzling complexity, and debatable methodology, (while lacking in mathematical "depth", as I use the term), as in the current "factors in autism" debates. EricDiesel (talk) 17:29, 21 September 2008 (UTC)
All irrelevant. Provide a reliable source that uses "statistical proof" with the meaning that you have assigned to it. If you cannot, I will remove your mini-essay again, on the grounds that it is original research. Gandalf61 (talk) 18:24, 21 September 2008 (UTC)
Re "reliable source that uses "statistical proof" with the meaning that I have assigned to it"
1 I have addressed your initial deletion reason, by demonstrating that a "statistical proof" is unrelated to a "probabilistic proof".)
2 I have addressed your next objection that "proof" is used, sometimes but not always, differently from "statistical proof.
3 I address your "statistical test" objection with a number of sourced examples below, not using a "statistical test". (The random number generation and number theory example below is unrelated to test, both uses are consistent with the deleted section, fully dispositive by themselves, but are a minority use for a more restricted definition of statistical proof.)
4 Here are some diverse uses of “statistical proof”, all sourced, all but one in conventional academic contexts, and all consistent with the deleted section. (I note that none of the other sections are sourced, but this one will be in excess.) None but one of the following uses of “statistical proof” uses a “statistical test”, but all are consistent with the deleted article section. In only one use below, a statistical proof is a “proof” in mathematics, but it is not a “test”. ---
Diverse reliable source, all that use "statistical proof" consistent with the meaning in deleted section -
Physics context –“ the Large Hadron Collider could start to indicate exotic new things like the Higgs boson within months - though it could take a year or two to have the statistical proof needed to confirm its presence” [3]
Sports context – “Southern Cal’s defensive line dominated Ohio State last weekend. Here’s statistical proof” [4]; [5]; Stein's paradox in statistics, Bradley Efron, SCIENTIFIC AMERICAN 236: 119 (1977)
Pharmeceutical industry context – “supported by statistical proof both from the providers” [6]
Educational policy context – “I don’t have any statistical proof of this, but I firmly believe that cheating and plagiarism have increased in recent years,” Brown, who is the history department chair, said” [7]
Marketing context – “based on clear statistical proof about which combination of design works most successfully in converting sales and delivering customers” [8]
Cryptography, number theory, chaotic series, random number generation, etc. pure mathematics context – “a statistical proof can just mean ‘most’ cases, which may mean that some cases which fit under the guarantee are not really secure” [9], “in number theory and commutative algebra... in particular the statistical proof of the lemma.” [10], “Whether constant π (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for some statistical proof” (This is a derogatory use, but still goes under this bulletin heading as pertaining to a chaotic series, and it does not use “statistical test” in the use of the Wiki article.) [11], “these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E” [12]
IT context use – “Maxymiser's technology allows them to test different approaches to content and provides clear statistical proof of what works [13]
Bayesian analysis context – “(I leave this citation to someone else, but the article section is at least consistent with some Bayesian uses.)”
Politics context – “Evolution got a cameo role in Philippine politics a few days back … alleged statistical proof of his invincibility. De Castro may wilt in presidential debates” [14]
Classical testing context – “(insert any textbook cite over twenty years old containing ‘statistical test’, here)”
Medical practitioner context – “all doctors can do is to point that scientifically, there is no statistical proof” [15]
Economic context – “Claiming they have ‘solid, statistical proof’ the two say that data actually suggests the economy is not heading into a recession”” [16]
Despite the wide, consistent, but not well-defined usages, except for non-comprehensive isolated articles, like I. J. Good’s exposition [17], which is antiquated, hyper-classical, pre-computer methods, anti- free wheeling data analysis, and parametric assumption laden, there is no “collected uses” literature. Further, none but two of the above quoted usages are consistent with Good’s discussion. One reason for absence of a “collected uses” is disdain for data analysis by “real” mathematician, Bayesian vs. NonBayesian encampments, and most importantly, if a student inquires into the history of statistics, there is usually a shudder and change of topic, in recollection of the Galton then Hitler path, or the overt racism of Fisher and his circle.
PS, just a guess, but if we actually know each other, and this is another clean the Augean Stables practical joke, I remind you of my last repartee to such a joke, my for the moment repartee is that looking at sources for all these uses, all over the map, has been fun. EricDiesel (talk) 21:12, 21 September 2008 (UTC)
Thank you for the references. FYI, in-line citations don't work well in talk pages, as there is no sensible place to put the <references/> tag (which you omitted anyway), so I have converted your in-line citations to external links for clarity. I will add one or two of the more useful references to the article. I will also attempt to simplify and clarify your rather opaque style - this article is meant to be an overview article linking to more detaild articles, and the "Statistical proof" section is, at present, far too long.
As regards your strange remarks about a "practical joke" and your "last repartee", they make even less sense to me than the rest of your writing, but I suggest you review WP:AGF before throwing out accusations and veiled threats at other editors. Gandalf61 (talk) 08:37, 22 September 2008 (UTC)
One more, I happened to have been reading The Singularity is Near, and on returning to it after writing this, I noticed that it is chalk full (especially for a popular science book; recall hawking's remark that sales are halved for each included equation) of scatter plots of logarithms) of scatter plots as statistical proofs. These might be the most common of all. EricDiesel (talk) 22:44, 21 September 2008 (UTC)
Throw that book in the bin right now! Its a great big bunch of hokum. Rays central argument falls apart if you recalculate the graphs assuming we are 1000 years in the future, the log-log plots would look almost identical pointing to a singularity at which ever point to calculate from. --Salix alba (talk) 23:21, 21 September 2008 (UTC)
Ha, I guess they weren't very good "visual statistical proofs" then, eh? I am reading his book on an Amazon Kindle, which does not yet have readable pictures. So I not only can't see the scatter plots, I can't throw the book in the bin 'either. Still, it should be an interesting next thirty years. All of us have singularities that are near, as we all gotta go sometime. Gotta go. EricDiesel (talk) 23:29, 21 September 2008 (UTC)
Comment - After perusing several shelves of Statistics texts, and poring over the net, I have found that, interestingly, the expression “statistical proof”, appears to be used in almost every field... from Number theory to Baseball. Every field, that is, with the exception of Statistics!! EricDiesel (talk) 19:06, 22 September 2008 (UTC)
.
I added bullet points to different inconsitent uses of "statistical proof", clarified when it is a mathematical proof and when it is not, and referenced sources using the expression. But Gandalf61's comment that this section is too long has not been adequately addressed. EricDiesel (talk) 16:21, 23 September 2008 (UTC)
(Thread moved from other talk page.) Please can I ask you yo stop writing your mini-essays in the mathematical proof article. That article is meant to be an overview article that provides a summary of the topic and links to more detailed articles. Your essays are by far the longest sections in the article, and they are making it unbalanced and unreadable. If you think we should have an article about all the different uses of the term statistical proof, for example, then by all means start a statistical proof article. But please, please stop overloading the mathematical proof article. Gandalf61 (talk) 16:13, 23 September 2008 (UTC)
Recall that your original objection and deletion of the entire section was that "'statistical proof' is covered in the 'probabilistic proof' section." The first contribution I put in was short and consistent with all of the bullet points, but you correctly objected that my formulation would be "original research". I have been working up a Statistical proof article for the past couple of days to lin to. I have tried to address all of your concerns in a systematic manner, but the latest is not specific enough, except for a legitimate length objection. but I can not figure out which information to delete without deleting essential distinctions, so please specify which information does not belong here, if it is clear to you. EricDiesel (talk) 16:38, 23 September 2008 (UTC)
I removed a claim that fractal theory developed without proofs. To the extent that fractal theory is a subfield of dynamical systems, formal proofs are employed. For example, Lyapunov exponents and Hausdorff dimension are formalized to the usual standard of mathematical rigor. — Carl (CBM • talk) 14:16, 22 September 2008 (UTC)
That's why I put the "citation needed" tag. Mandelbrot was my best friend's uncle when I was a mid teen, and a dinner table "fact" was that he was having problems with others who were asserting he was not really doing mathematics, in that he was developing pictures and not proving theorems. I also read something similar regarding the history of fractals using some kind of non rigorous method, different from the conventional theorem-proof framework, with some examples to understand what was meant, but which I can not recall. I also recall, as a beginning grad student, meeting another grad student in physics at a film, who said he was working with fractals, then having my hands slapped when I expressed an interest in fractals, being told "that's not really mathematics, its just a way of making pictures". There may still be a grain of truth in the deleted claim, and I will try to track it down with a source. I presume that the purpose of the "citation needed" tag is a request to others to fact check an questionable assertion, so let me know if I am using it incorrectly. EricDiesel (talk) 14:26, 22 September 2008 (UTC)
No, what you did is perfectly fine. I do it myself, if there is something that I believe to be true but don't have a good reference for (but only very occasionally). Of course, if someone else disagrees, they may remove it and leave their concerns on the talk page, which is what I did.
In this case, the issue may be the wording. The only things I know about the early development of chaos theory I learned from the Chaos book, which I read too long ago. I also have the impression that Mandelbrot did quite a bit of experimental work before he and other mathematicians could prove rigorous results. But I don't think that's the same as saying the fractal theory developed "without proofs". For example, I looked through Mandelbrot's publications on Mathscinet, and MR0719054 caught my eye. It is for a 1983 paper where Mandelbrot uses illustrations to discuss conjectures on the Mandelbrot set - but the reviewer points to papers in 1982 by Douady/Hubbard and Ruelle covering similar topics in a rigorous way. So while Mandelbrot might not have always used proofs, other mathematicians were studying the same topics rigorously at the same time. — Carl (CBM • talk) 14:53, 22 September 2008 (UTC)
My knowledge of Mandelbrot comes from mid 1970's family dinners. I was a run away, and usually ate dinner at his nephew's home. In the later 70-s or early 80's, in the math department at UCLA, I brought up the fantastic pictures and his Fractal Geometry of Nature book, which I was unable to read, to my runaway-kid-math-mentor, Ernst Strauss, a geometer. He said it was not mathematics, "just pictures". This remark was repeated by Kai Lai Chung in 1982, who terrified me and made me feel dumb for even bringing it up. (Chung was the guy who boasted of a "perfect", "elementary" probability text, having eliminated all pictures, an attitude the opposite of the computere driven, data analytic oriented, Statistics department next door. Chung was also very nasty to grad students (and others), deriding their work as "trivial", making one of the very smartest grad students (much smarter than me) cry and leave the department. Following this, I did some stuff Chung liked, Chung asked me to be his student following that, and I told him, no, as I could only think in pictures, and he was not doing anything interesting anyway. Interestingly though, we ended up friends, shared a friendship with Alonzo Church, and Chung had great Church stories from Princeton, where Chung was a student of Church in some courses, when church was very young, and had the same beautiful and bizzare idiosynchratic personality traits he had in old age.) Ok, back to the issue. I was abuot to suggest a "terminology" change too, as mandelbrot's fractals have been embeded in the very rigor he rebelled against. Perhaps a better expression would be "fractal geeometry for Mandelbrot", for the following reasons. -
Regarding “just making pictures” and calling the activity “doing mathematics”, “Mandelbrot, working at the IBM Research Laboratory, did some computer simulations for these sets on the reasonable assumption that, if you wanted to prove something, it might be helpful to know the answer ahead of time.”[18] A sort of exploratory mathematics, cf. exploratory data analysis, derided only twenty years ago. Here is one comment regarding Mandelbrot and “proof” of theorems, “… brought home again to Benoit [Mandelbrot] that there was a ‘mathematics of the eye’, that visualization of a problem was as valid a method as any for finding a solution. Amazingly, he found himself alone with this conjecture. The teaching of mathematics in France was dominated by a handful of dogmatic mathematicians hiding behind the pseudonym ‘Bourabki’… ” Of course, this quote is from what is essentially a comic book, but this only makes the very point of the “conjecture” Mandelbrot was said to be making!. The irony of the context of the quoted passage has the elements of a political cartoon.
One last thing (Joe Biden look out, for long-windedness competition), as you probably inferred by now, I am a picture person, and unfamiliar with the theory of fractal geometry beyond a very cursory knowledge, although people are deceived into thinking the opposite as I can talk on and on for hours on the subject in my lectures during ecology field trips. I can only understand a proof and remember it if I can draw a picture of the content. I was just utterly humiliated (my favorite emotion) for this by my new friend, User:Stan Shebs. I was invited to contribute to the Wiki plant-pages. I was referred to User:Stan Shebs as fellow desert enthusiast (finatic, in my case). When I saw his personal photo on his user page, I thought I could boast by pinpointing the exact location, date, and time, of the photo, all from the coloration of the plants in the photo's distant background. He responded something like, “very close, but it says where and when the picture was taken in the text below the picture”! (I consider humiliation, along with eavesdropping and plagiarism, to be the highest form of flattery, so fear not in pointing out errors in my contributions, but supplement your criticism with a picture so I can understand it.) EricDiesel (talk) 18:08, 22 September 2008 (UTC)
I think that, rather than making claims about fractal geometry itself, it may be better to just mention people, such as Mandelbrot of Doron Zeilberger. There are some things that can be said about the contemporary "experimental math" movement, although it does not seem to be gaining much ground among mathematicians overall. — Carl (CBM · talk) 19:00, 22 September 2008 (UTC)
What to do with the pictures? Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were coins of the realm ant the conventions of that day dictated that journals only published theorems" Indra's Pearls p. viii.
This whole "Mathematics without proofs" section does not belong in an article about mathematical proofs. It is a poorly written mini-essay, unencyclopedic in style and POV. I propose it should be removed. Gandalf61 (talk) 23:27, 22 September 2008 (UTC)
Ok to remove. I will rewrite with just one or two sentences (or you can sum it down). I was just trying to be responsive to carl, and including quotes to avoid source onjections that I have seen in other articles. I am new to Wikipedia, wrote my first articles on Palin's churches and Pastors, and am noticing wildly divergent standards in the different article topics. Your rewording of statistical proof is nice and short and easy to read. It misses a couple of things, but a sentence or two will fix it. EricDiesel (talk) 00:07, 23 September 2008 (UTC)
Missing content
There are several things that I think this article is missing:
The role of mathematical proof in contemporary mathematics
The tension between natural language proofs and formal proofs
The much-discussed status of proofs that are too long or too esoteric to understand, including:
the four-color theorem
Fermat's last theorem (isn't there stuff by vos Savant we could mention?)
the Poincare conjecture
The spectrum of meanings of "constructive proof"
The "experimental mathematics" program
The first four of these ought to have sections; the last one maybe just a paragraph. As it stands, the article is a good start, but it reflects a "Proof 101" viewpoint rather than the broader scholarly viewpoint. — Carl (CBM · talk) 13:48, 23 September 2008 (UTC)
We can certainly link to those articles (although I don't think e.g. proof theory covers any of the points I mentioned), and write some things here in summary style. From my point of view, this article is like a hypothetical article on derivatives that consists primarily of derivative rules rather than discussion of the overall concept of derivative. It's a start, but doesn't give a good overall perspective on the concept of mathematical proof. Anyway, if I have time to work on it, I'll try to add a little content so you can see what I'm thinking. At 20k in length, this article is not at all too long at the moment. — Carl (CBM · talk) 14:52, 23 September 2008 (UTC)
This overview article already contains links to most of those articles. If you think that proof theory, for example, needs expanding then by all means add content in that article - but not here. EricDiesel's mini essays have already made this article unbalanced (I have asked him to put them somewhere else) and it is in danger of becoming completely unreadable. Gandalf61 (talk) 16:22, 23 September 2008 (UTC)
Now rewritten to remove "essay" quality, at least in "statistical proof" and "history" sections, and moved bulky stuff to links. EricDiesel (talk)
I do agree the article has readability problems; I think that some rearranging of topics will help with that. It will be a little while before I can work on this one, but I think the topic is sufficiently important to warrant a slightly deeper analysis than this version (for example) provides. — Carl (CBM · talk) 16:31, 23 September 2008 (UTC)
deeper topic "Proof 201" stuff should go at end, after "Proof 101" and not early on, then 301 depth in pages of the links, and 401 in the linked pages as references articles.
4-Color fits illustratively in all of "exhaustion", "computers", and "aesthetics" sections.
Article sections best proceed from a dictionary like section ("Proof 001") - then to general ecyclopedia stuff ideas and history- then to specialized encyclopedia content ("Proof 201") with links, else general users will give up at the beginning without finding what they came for.
"Experimental" rewritten per your previous comments and Ganalf61's, but still needs revision.
But seriously, the current version, while not as pretty as Gandolf61’s terseity, serves as a quick reference by an undergrad given an exercise to do both a “probabilistic proof” and a “constructive proof” of something, just what an encyclopedia is for. It’s also good as a survey of conversational use. But, I came across this article following links, after a referral from User:Katzman, and would like to see a general, readable not terse, but short expositions of the stuff Carl is talking about, just out of curiosity. Tautologist (talk) 23:30, 25 September 2008 (UTC)
@CBM: I think there should be a section about termination analysis stating that for any formal system there exists a statement that is provable but not provable in that system. People don't generally invent a formal system until they figure out a proof that everything provable in that system is true. For any formal system, it's possible for somebody to think of a way to count all algorithms that can be proven in that system to generate a number. Using Cantor's diagonal argument, you can find an algorithm that generates a number that none of those algorithms generate. Although you can't prove in that system that that algorithm generates a number, you know it's true, which means a proof for it exists and therefore a proof is an abstract object. Blackbombchu (talk) 01:37, 13 June 2015 (UTC)
Computer proofs
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)
Since i already ruined the conciseness of this article, we might as well print out the 4 "color" proof to give the reader the "flavor" Tautologist (talk) 20:29, 29 September 2008 (UTC)
statistical proof - "Homelessness" not dealt with in this election cycle
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)"
I edited and rearranged a lot of content today. I started a section on "nature and purpose" of mathematical proof, although it's only a stub right now. There are still many topics the article doesn't discuss, or only mentions in passing:
The role of mathematical proof in mathematics
The concept of proof in intuitionistic mathematics; link to the article on the foundational debates of the 1920s.
Proof in mathematics education, esp. at the pre-college level.
More source stuff - FYI all, Steven Krantz at Washington in St Lous has a new book in draft, The Proof is in the Pudding, A Look at the Changing Nature of Mathematical Proof.
More possible section topics - In Time Love Memory (about the fly-room "dirt-under-the fingernails" molecular bio crowd at Ctech, Jonathan Wiener talks about an "association" gene (a la Hume induction). Recent dinner talk with the same crowd there indicates there may be a "proof" gene about to be inferred (circularly using proof concepts for the tools to make the inferences, of course). Lots of (perhaps justified) hyper-psychologism from molec bio and neuroscience recent events, too. Tautologist (talk) 00:40, 6 October 2008 (UTC)
My take on Quine, from Two Dogmas and later, is very different than User:Carl's (else what does "two", "dogmas" and "empiricism" mean in the title? Also "web" of belief for Q, and of associations in neurosci, contradicts) The new sentence sounds more like a description of Frege than Quine on this. Tautologist (talk) 01:04, 6 October 2008 (UTC)
The sentence using Quine as a source for the claim that mathematical proofs are analytic was already there, worded slightly differently. Thanks for pointing it out, though. That paragraph (the whole section, really) is pretty vague at the moment. I commented out the Quine sentence for now. My thought is that some of the references from the history section of Tautology (logic) may apply here as well.
Rigor's role, communicating proofs, purpose of formal proofs
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.
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)
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)
Very messy!
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)
Well then, since no one has anything at all to say, I'll take it upon myself. I will post a major revision of the article, both content and organization, on my user page by this Friday May 22. If there are still no comments, I will apply the revision on June 1. Kiyura (talk) 15:18, 19 May 2009 (UTC)
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)
Jagged 85 (talk·contribs) 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)
