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What is an effective proof? — Preceding unsigned comment added by 2001:628:2120:601:7447:9B9B:6C18:BF6E ( talk) 11:49, 9 December 2015 (UTC)
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) .
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)
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)
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
IMHO this could be moved to proof (mathematics). What do you think? googl t 19:27, 15 August 2006 (UTC)
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)
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)
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)
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)
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)
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)
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.
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)
I have to disagree with this:
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"?
This is arbitrary:
-- Pokipsy76 ( talk) 10:30, 1 January 2008 (UTC)
I'll propose a rewrite and see if that gets the changes started:
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)
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)
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)
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)
I have removed the following nonsense, added twice by an anonymous editor:
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)
—Preceding unsigned comment added by 220.233.197.84 ( talk) 20:31, 16 August 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)
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.
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)
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)
There are several things that I think this article is missing:
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)
Gandalf61 rewrite is a big improvement on my original scribbled note jot (my other edits could be similarly improved).
Some Homeless (proofs) are subjects of derision by all-
Oh, well. Tautologist ( talk) 19:56, 27 September 2008 (UTC)
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:
By the way, does anyone mind if I clean up the references to be like the ones in Group (mathematics) or the ones in Mathematical logic? — Carl ( CBM · talk) 21:09, 5 October 2008 (UTC)
The article says a proof is "a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true". I think few mathematicians would recognize this as a description of proof. Eric Weisstein's "A rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition" (on MathWorld, http://mathworld.wolfram.com/Proof.html) is much closer to the mark. The fact that a proof is rigorous is key to the concept of mathematical proof and is a distinguishing feature of mathematics.
The text in a math book or math journal labeled "proof" is intended to convey or communicate the existence of a proof to the reader. The proof itself is better thought of as existing in the minds of the author and reader. The author checks that there is a rigorous proof by checking that each step logically follows from those that come before and filling in all details needed to verify this. However, the author doesn't include all these details in what they write. Instead, they decide how best to communicate this structure to a reader who has the background and knowledge that they are assuming for their readers. The reader, when reading the proof, is expected to check that a rigorous proof exists by checking that each step logically follows from those that come before and by filling in all details needed to verify this.
This underlying logical structure/scaffolding actually permeates a math book or journal article. Every sentence or group of sentences in a math book/article can be classified as a definition, theorem, proof, or remark. (Some sentences might have elements of more than one of these categories.) Math books/articles try to guide the reader by providing explicit labels, e.g., "Theorem", "Proof", for the more important or lengthier such items. However, it is expected that the reader will figure out the function of the unlabeled parts by their content. One convention that helps the reader is that sentences containing a word or phrase in italics are definitions of the italicized word or phrase (some authors use bold). If a sentence in an unlabeled part is functioning as part of a proof, the reader is expected to check that the proof is correct/rigorous. For some declarative sentences, the author may not give any justification. For these mini theorems, the reader is expected to provide the proof without help from the author. When I first realized, in talking with colleagues who were not mathematicians, that most non-mathematicians are unaware of this structure in math books/articles, I was rather surprised.
To understand the relationship between mathematical proofs and formal proofs, it helps to understand the purpose of the latter.
One purpose of formal logical systems and formal proofs is as a mathematical model of mathematics and mathematical proof. This mathematical model allows the mathematical study of mathematics. For this purpose it is important that a formal proof capture the key features of a mathematical proof. However, as with any mathematical model, the model is not the object. A formal proof is a mathematical proof, but most mathematical proofs are not formal proofs.
Another purpose of formal logical systems is to provide a foundation for mathematics. For example, ZFC is an adequate foundation for almost all of modern mathematics. When used for this purpose, it suffices to prove that the mathematics in question could be translated into ZFC. It isn't necessary to actually do the translation. Of course, the proof that such a translation could be done is a mathematical proof.
-- David Marcus ( talk) 19:55, 24 December 2008 (UTC)
This is not distinct from a direct proof. Any reason not to remove it from the article? Bongo matic 13:28, 24 March 2009 (UTC)
I don't understand the words: "(or sometimes just called reasons)". What are called reasons? The lines? Myrvin ( talk) 09:03, 15 October 2009 (UTC)
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'm also interested in this new revision - see my comments on your talk page. -- Joth ( talk) 09:27, 1 June 2009 (UTC)
The reference for Buss, 1997 is missing. What is the publication? Myrvin ( talk) 11:20, 12 July 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)
I recently came across the article on statistical proof. When when I first arrived at the page [19] - it was a mess. There is a bit of debate going on in the discussion page on the merit of statistical proof having its own page and even if such a thing exists. There are obvious logical ties to mathematical proof, so I thought I would come here and ask others who know about mathematical proof to share their thoughts. Is statistical proof distinct from a mathematical proof? Is this something that could turn into a small section or Wikilink in this article? Thought I would raise the issue and see what comes of it. Thanks. Thompsma ( talk) 21:25, 11 November 2011 (UTC)
First, many apologies. I know I typed something like this when I made a small change a few days ago. Somehow it did not remain so I am typing this first. I have no interest in a reversion conflict. Just wanted to make a change and state the reason. The change is to return a few sentences relating to "the first proof by contradiction" I have no opinion about the value one way or the other about these few sentences. My problem is that they were removed with the total justification being something like "Euclid's proof of the infinitude of primes was the first proof by contradiction" (not a direct quote.) As Wikipedia itself correctly says (Euclid's Proof) "Euclid is often erroneously reported to have proved this result by contradiction" (and other parts of Wikipedia do say that) That is based on the following common misconception that Euclid book IX prop 20 says something like
Prop: There are infinitely many primes
1. Suppose that there were finitely many and list them p1,p2,..pk
2. Use the list to create the number P=p1*p2*...*pk+1
3. There is a prime q dividing P (maybe q=P) and it is not in the list CONTRADICTION
It is true that the proof is nowadays often presented this way and that there are claims (including some parts of Wikipedia) that this is essentially what Euclid wrote. However what was actually written was more like, there are always more primes (than in any finite list)
Prop: No finite list of primes includes all primes
1. Given a list p1,p2,..pk (here is how to get something else)
2. Use the list to create the number P=(p1*p2*...*pk)+1
3. There is a prime q dividing P (maybe q=P) and it is not in the list (there, something else)
SO I am happy with some other reason to remove the claim I reverted, but not the given one. Gentlemath ( talk) 02:31, 16 February 2015 (UTC)
Inductive inference and deductive inference are both regularly used in mathematical proof.
The article "Mathematical induction" declares the method of mathematical induction as deductive inference, but, it's inductive inference.
It neither clear nor unambiguous (nor non-controversial) to call "inductive inference" (exhaustive inductive inference, as of proof by induction) instead "deductive inference", because it's not.
It is clear and unambiguous that "exhaustive" deductive or inductive inference (i.e., covering all cases) does maintain derivability of conclusion from premise and otherwise maintains the grounds for mathematical proof.
It is very widely understood that "proof by induction" is "mathematical proof by mathematical induction" in any context of mathematical proof. — Preceding unsigned comment added by 75.172.122.39 ( talk) 07:10, 18 March 2017 (UTC)
This has that the disambiguation of mathematical inductive inference and the inference of reasonable expectations would go into the article on induction, to leave clearly in the main article of mathematical proof the deductive and mainly inductive inferential arguments as relevant to the derivability via inference of conclusion from premise. "Induction inference of expectations" ("common sense") should be disambiguated from exhaustive "inductive inference by cases" (mathematical induction), instead of overloading the definition of deductive inference (the contrapositive, that syllogism is the inductive).
This involves a rather significant difference in definitions of overloaded primary terms.
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From basic mathematical study I have come across the notion of story proofs. Some are used in a book I am reading (Introduction to Probability by Blitzstein and Hwang). It seems like its own category of proof, so perhaps a section should be added here, or a new Wikipedia page created. I would do it, but I'm not confident enough in my mathematical knowledge to consider editing this page. So I figured I would provide a suggestion instead. Proxyma ( talk) 23:44, 28 June 2017 (UTC)
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What is an effective proof? — Preceding unsigned comment added by 2001:628:2120:601:7447:9B9B:6C18:BF6E ( talk) 11:49, 9 December 2015 (UTC)
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) .
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)
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)
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
IMHO this could be moved to proof (mathematics). What do you think? googl t 19:27, 15 August 2006 (UTC)
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)
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)
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)
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)
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)
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)
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.
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)
I have to disagree with this:
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"?
This is arbitrary:
-- Pokipsy76 ( talk) 10:30, 1 January 2008 (UTC)
I'll propose a rewrite and see if that gets the changes started:
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)
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)
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)
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)
I have removed the following nonsense, added twice by an anonymous editor:
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)
—Preceding unsigned comment added by 220.233.197.84 ( talk) 20:31, 16 August 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)
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.
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)
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)
There are several things that I think this article is missing:
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)
Gandalf61 rewrite is a big improvement on my original scribbled note jot (my other edits could be similarly improved).
Some Homeless (proofs) are subjects of derision by all-
Oh, well. Tautologist ( talk) 19:56, 27 September 2008 (UTC)
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:
By the way, does anyone mind if I clean up the references to be like the ones in Group (mathematics) or the ones in Mathematical logic? — Carl ( CBM · talk) 21:09, 5 October 2008 (UTC)
The article says a proof is "a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true". I think few mathematicians would recognize this as a description of proof. Eric Weisstein's "A rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition" (on MathWorld, http://mathworld.wolfram.com/Proof.html) is much closer to the mark. The fact that a proof is rigorous is key to the concept of mathematical proof and is a distinguishing feature of mathematics.
The text in a math book or math journal labeled "proof" is intended to convey or communicate the existence of a proof to the reader. The proof itself is better thought of as existing in the minds of the author and reader. The author checks that there is a rigorous proof by checking that each step logically follows from those that come before and filling in all details needed to verify this. However, the author doesn't include all these details in what they write. Instead, they decide how best to communicate this structure to a reader who has the background and knowledge that they are assuming for their readers. The reader, when reading the proof, is expected to check that a rigorous proof exists by checking that each step logically follows from those that come before and by filling in all details needed to verify this.
This underlying logical structure/scaffolding actually permeates a math book or journal article. Every sentence or group of sentences in a math book/article can be classified as a definition, theorem, proof, or remark. (Some sentences might have elements of more than one of these categories.) Math books/articles try to guide the reader by providing explicit labels, e.g., "Theorem", "Proof", for the more important or lengthier such items. However, it is expected that the reader will figure out the function of the unlabeled parts by their content. One convention that helps the reader is that sentences containing a word or phrase in italics are definitions of the italicized word or phrase (some authors use bold). If a sentence in an unlabeled part is functioning as part of a proof, the reader is expected to check that the proof is correct/rigorous. For some declarative sentences, the author may not give any justification. For these mini theorems, the reader is expected to provide the proof without help from the author. When I first realized, in talking with colleagues who were not mathematicians, that most non-mathematicians are unaware of this structure in math books/articles, I was rather surprised.
To understand the relationship between mathematical proofs and formal proofs, it helps to understand the purpose of the latter.
One purpose of formal logical systems and formal proofs is as a mathematical model of mathematics and mathematical proof. This mathematical model allows the mathematical study of mathematics. For this purpose it is important that a formal proof capture the key features of a mathematical proof. However, as with any mathematical model, the model is not the object. A formal proof is a mathematical proof, but most mathematical proofs are not formal proofs.
Another purpose of formal logical systems is to provide a foundation for mathematics. For example, ZFC is an adequate foundation for almost all of modern mathematics. When used for this purpose, it suffices to prove that the mathematics in question could be translated into ZFC. It isn't necessary to actually do the translation. Of course, the proof that such a translation could be done is a mathematical proof.
-- David Marcus ( talk) 19:55, 24 December 2008 (UTC)
This is not distinct from a direct proof. Any reason not to remove it from the article? Bongo matic 13:28, 24 March 2009 (UTC)
I don't understand the words: "(or sometimes just called reasons)". What are called reasons? The lines? Myrvin ( talk) 09:03, 15 October 2009 (UTC)
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'm also interested in this new revision - see my comments on your talk page. -- Joth ( talk) 09:27, 1 June 2009 (UTC)
The reference for Buss, 1997 is missing. What is the publication? Myrvin ( talk) 11:20, 12 July 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)
I recently came across the article on statistical proof. When when I first arrived at the page [19] - it was a mess. There is a bit of debate going on in the discussion page on the merit of statistical proof having its own page and even if such a thing exists. There are obvious logical ties to mathematical proof, so I thought I would come here and ask others who know about mathematical proof to share their thoughts. Is statistical proof distinct from a mathematical proof? Is this something that could turn into a small section or Wikilink in this article? Thought I would raise the issue and see what comes of it. Thanks. Thompsma ( talk) 21:25, 11 November 2011 (UTC)
First, many apologies. I know I typed something like this when I made a small change a few days ago. Somehow it did not remain so I am typing this first. I have no interest in a reversion conflict. Just wanted to make a change and state the reason. The change is to return a few sentences relating to "the first proof by contradiction" I have no opinion about the value one way or the other about these few sentences. My problem is that they were removed with the total justification being something like "Euclid's proof of the infinitude of primes was the first proof by contradiction" (not a direct quote.) As Wikipedia itself correctly says (Euclid's Proof) "Euclid is often erroneously reported to have proved this result by contradiction" (and other parts of Wikipedia do say that) That is based on the following common misconception that Euclid book IX prop 20 says something like
Prop: There are infinitely many primes
1. Suppose that there were finitely many and list them p1,p2,..pk
2. Use the list to create the number P=p1*p2*...*pk+1
3. There is a prime q dividing P (maybe q=P) and it is not in the list CONTRADICTION
It is true that the proof is nowadays often presented this way and that there are claims (including some parts of Wikipedia) that this is essentially what Euclid wrote. However what was actually written was more like, there are always more primes (than in any finite list)
Prop: No finite list of primes includes all primes
1. Given a list p1,p2,..pk (here is how to get something else)
2. Use the list to create the number P=(p1*p2*...*pk)+1
3. There is a prime q dividing P (maybe q=P) and it is not in the list (there, something else)
SO I am happy with some other reason to remove the claim I reverted, but not the given one. Gentlemath ( talk) 02:31, 16 February 2015 (UTC)
Inductive inference and deductive inference are both regularly used in mathematical proof.
The article "Mathematical induction" declares the method of mathematical induction as deductive inference, but, it's inductive inference.
It neither clear nor unambiguous (nor non-controversial) to call "inductive inference" (exhaustive inductive inference, as of proof by induction) instead "deductive inference", because it's not.
It is clear and unambiguous that "exhaustive" deductive or inductive inference (i.e., covering all cases) does maintain derivability of conclusion from premise and otherwise maintains the grounds for mathematical proof.
It is very widely understood that "proof by induction" is "mathematical proof by mathematical induction" in any context of mathematical proof. — Preceding unsigned comment added by 75.172.122.39 ( talk) 07:10, 18 March 2017 (UTC)
This has that the disambiguation of mathematical inductive inference and the inference of reasonable expectations would go into the article on induction, to leave clearly in the main article of mathematical proof the deductive and mainly inductive inferential arguments as relevant to the derivability via inference of conclusion from premise. "Induction inference of expectations" ("common sense") should be disambiguated from exhaustive "inductive inference by cases" (mathematical induction), instead of overloading the definition of deductive inference (the contrapositive, that syllogism is the inductive).
This involves a rather significant difference in definitions of overloaded primary terms.
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From basic mathematical study I have come across the notion of story proofs. Some are used in a book I am reading (Introduction to Probability by Blitzstein and Hwang). It seems like its own category of proof, so perhaps a section should be added here, or a new Wikipedia page created. I would do it, but I'm not confident enough in my mathematical knowledge to consider editing this page. So I figured I would provide a suggestion instead. Proxyma ( talk) 23:44, 28 June 2017 (UTC)