Talk:Theorem/Archive 2

Squares and triangles
GregBard has reverted my change of all the $$\triangle \,$$'s and $$\square \,$$'s to A and B. Who actually believes triangles and squares make the article more accessible? Plus do we really need $$\mathcal{FS} \,.$$ repeated in practically every single sentence after the context has been established? Dmcq (talk) 15:17, 16 October 2010 (UTC)
 * The example had solid squares and triangles which were very readable, I would prefer we bring that whole section back for reasons I explain above. Perhaps we could use solid symbols instead of outlined ones.Greg Bard (talk) 15:21, 16 October 2010 (UTC)
 * Perhaps we go back to A and B thanks. And I actually have a certificate in doing some pedagogy thank you so don't say you're better than me at that. Dmcq (talk) 15:25, 16 October 2010 (UTC)
 * Certificate or not. Using letters doesn't help in terms of pedagogy. The point is that the principle is sound no matter what symbols you use. You insist on using what is familiar to you, why is that? It doesn't help people understand that the principle applies to the unfamiliar as well. If you want to remain in-the-box with your thinking, that is fine. However imposing that our readers must remain in-the-box is a disservice.
 * You have reinserted a lot of redundant matierial, which I took the time and effort to incorporate thoughtfully. The "in logic" section is not exactly popular among your bretheren, so I have no idea why that is back. "Lore" Really? Would someone other than me get rid of some of that garbage please. Please take the time to edit thoughtfully, don't just be reactionary. Greg Bard (talk) 16:17, 16 October 2010 (UTC) Greg Bard (talk) 16:10, 16 October 2010 (UTC)
 * I reinserted the stuff you removed with no justification other than you wanted to go off and edit it yourself. So you were saying it was worth keeping but you went and removed it. And please don't go stuffing me into a 'your brethren' box. Dmcq (talk) 16:26, 16 October 2010 (UTC)
 * Re the 'in-box'. The whole basis of Wikipedia is that it should be in-box. We are supposed to be summarizing what others have done. Which reminds me, more citations would be a very good idea. Dmcq (talk) 16:31, 16 October 2010 (UTC)
 * Yeah, you have missed the point here. Yes everyone knows that Wikipedia is an encyclopedia and the content is within an objective box. However the reader should, on their own take what they have learned and be able to apply it out side the box. Squares and triangle do help in that regard. The same old Xs and Ys do not.
 * Most of that content could be left out entirtely as duplicated or irrelevant. I refactored it to a sub page for the sake of making it easier for everyone to see the mess and use it as a source of readding content thoughtfully. Adding it back does not improve this article AT ALL. "Lore?" Seriously, what are you thinking? Greg Bard (talk) 21:43, 16 October 2010 (UTC)
 * In regard the symbols; I think it's important that we note that a "formal language" and "formal theory" use arbitrary symbols, not just roman letters, but I would like to point out the Greg's "pedagogy" argument is severely flawed. That is one of the things which made "New Math" inaccessible to the teachers, and hence to the students.  — Arthur Rubin  (talk) 18:06, 16 October 2010 (UTC)
 * That is off the deep end again Arthur. We don't need to take it to the extreme of "New Math," I'm just saying that squares and triangles are a neutral (NPOV) way to communicate without math jargon. You also seem inconsistent with your above statement. What exactly are you fair-mindedly agreeing with me on? Greg Bard (talk) 21:43, 16 October 2010 (UTC)
 * I agree that there is a point to using symbols not normally associated with "formulas" or "mathematics"; but that it is not a good pedagogical tool. Fortunately, Wikipedia's purpose is not pedagogical, but to be informative.
 * As for "Lore", I would use something like "in popular culture" or "in popular mathematical culture", but it seems an appropriate subheading.
 * Rather than deleting sections which you think may need to be readded, may I suggest tagging them with relevance-section, or some other appropriate tag. That could be dealt with in a more sensible manner, as we could all see the topic of discussion.  — Arthur Rubin  (talk) 22:06, 16 October 2010 (UTC)

Draft shortened lead
In mathematics, a theorem is a statement proven on the basis of previously accepted or established statements. In mathematical logic, theorems are modeled as formulas that can be derived from axioms according to the inference rules of a fixed formal system without any additional assumptions.

The expression that results from a derivation is a syntactic consequence of all the expressions that precede it, regardless of semantics. In mathematics, the derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. Although they can be written in a completely symbolic form using, for example, propositional calculus, theorems and their proofs are often expressed in a natural language such as English.

The statements of theorems have two components, called the hypotheses and the conclusions. The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.

A theorem which may be simply stated but with a proof that involves surprising and subtle connections may be referred to as "deep", for example Fermat's Last Theorem.

I deleted the sentence "Theorems have two components, called the hypotheses and the conclusions" from the lede, for four reasons. First, the point is better addressed in "Informal accounts". Second, it was interrupting a paragraph primarily about the nature of proof, and so was out of place. Third, it is grammatically strained since "Theorems", "hypotheses", and "conclusions" are all plural, but the latter two must be counted as singular if they compose "two components". Finally, many natural language theorems do not cleave so tidily into two easily identified halves. (Howald (talk) 02:25, 28 November 2011 (UTC))

Theorem vs. theory
Someone please add a little bit about what the difference is. I am sure many will ask the question. Thanks. 82.43.199.163 (talk) 20:21, 7 March 2013 (UTC)

removing POV tag with no active discussion per Template:POV
I've removed an old neutrality tag from this page that appears to have no active discussion per the instructions at Template:POV:
 * This template is not meant to be a permanent resident on any article. Remove this template whenever:
 * There is consensus on the talkpage or the NPOV Noticeboard that the issue has been resolved
 * It is not clear what the neutrality issue is, and no satisfactory explanation has been given
 * In the absence of any discussion, or if the discussion has become dormant.

Since there's no evidence of ongoing discussion, I'm removing the tag for now. If discussion is continuing and I've failed to see it, however, please feel free to restore the template and continue to address the issues. Thanks to everybody working on this one! -- Khazar2 (talk) 22:11, 21 July 2013 (UTC)

Classification
objected to my classifying a theorem as a mathematical proof and undid an edit of mine that had several other changes. Since there was no rationale for undoing them, I will reinstate the edit and argue for my classification here. Any dictionary definition for "theorem" includes two parts, "statement" and "proof", or their equivalents. These are the defining characteristics, and not much else. We can add Theorem to Category:Statements, but there is no Category:Statements that are proved or the equivalent, and if there were such a category I'm not sure what would be in there besides theorem and lemma. But this article contains 59 uses of the word "proof", so Category:Mathematical proofs seems like the category most likely to contain articles that readers of this article would want to find. RockMagnetist(talk) 23:19, 3 October 2015 (UTC)
 * A theorem is only the last line of some proof. A theorem *has* a proof, but is not itself a proof. Youknowwhatimsayin (talk) 23:37, 3 October 2015 (UTC)
 * I didn't think it was necessary for me to state something so obvious. Yes, it is not itself a proof, but it is inseparable from proof; a statement without a proof is just a statement. Hence the rest of my reasoning. RockMagnetist(talk) 00:55, 4 October 2015 (UTC)
 * Categories aren't for merely related things. They are classification of what the things are. You have removed some appropriate categories (syntax, for instance) and are now putting inappropriate categories in. Youknowwhatimsayin (talk) 01:07, 4 October 2015 (UTC)
 * So you're claiming that a theorem is syntax but isn't proof? RockMagnetist(talk) 01:16, 4 October 2015 (UTC)
 * I looked a little closer at Category:Mathematical proofs, and it looks like a highly appropriate category. First, the preamble says "This category includes articles on basic topics related to mathematical proofs, including terminology and proof techniques." In other words, not just examples of proofs. Second, theorems are mentioned throughout the article Mathematical proof, 30 times in all. Compare that to Category:Syntax and Syntax, which have zero mentions. RockMagnetist(talk) 21:47, 4 October 2015 (UTC)
 * What's written on the category page is of secondary importance. The important thing is whether the categorization is appropriate, given the category's name.
 * The notion of theorem is related to the notion of proof; no one disputes that. But a theorem is not a proof.  Since the "proofs" category is a plural count noun, one expects things categorized by it to be instances of that noun, so it's confusing.  The "syntax" category doesn't have the same problem, because "syntax" is not a plural count noun. --Trovatore (talk) 21:57, 4 October 2015 (UTC)
 * I think an appropriate solution is to rename it Category:Mathematical proof (singular). That sort of thing is often done by speedy renaming to align the names of categories with their topic categories. RockMagnetist(talk) 22:08, 4 October 2015 (UTC)
 * That's not a bad idea. We could then have a "mathematical proofs" category that's really for articles about individual proofs. --Trovatore (talk) 22:21, 4 October 2015 (UTC)
 * Indeed, it already exists: Category:Article proofs; and there is also Category:Articles containing proofs. All the more reason to rename their parent category! RockMagnetist(talk) 22:24, 4 October 2015 (UTC)
 * Huh. "Article proofs" is an even worse name in my estimation (it sounds like galley proofs of articles).  "Mathematical proofs" would be a much better name. --Trovatore (talk) 22:31, 4 October 2015 (UTC)
 * I agree, they are a bit strange. Category:Article proofs is actually for "all pages which provide mathematical proofs of adjunct mathematics and physics articles"; and Category:Articles containing proofs is a hidden maintenance category. It doesn't say how something gets into this category. RockMagnetist(talk) 22:41, 4 October 2015 (UTC)
 * I have started a discussion of the latter category at WikiProject Mathematics. RockMagnetist(talk) 23:17, 4 October 2015 (UTC)

"Unproven theorems"
In the first line of the article, it says "a theorem is a statement that has been proven ". But, for instance, the conjecture known as "Fermat' Last Theorem" was called a theorem long before it was proven. Shouldn't this widespread but less than stringent use of the word be discussed in the article? Or have I missed it? Wdanbae (talk) 14:06, 5 February 2016 (UTC):
 * The case of Fermat's last theorem is very specific, because Fermat claimed to have a proof, and the statement has been known as "Fermat's last theorem" a long time before the common use of the word "conjecture". This is briefly discussed in section Terminology. D.Lazard (talk) 15:52, 5 February 2016 (UTC)
 * I thought there were a few other misnomers around, but I am no pro. Wdanbae (talk) 18:56, 5 February 2016 (UTC)
 * Just because a statement is called a theorem, doesn't make it a theorem. So although Fermat's last theorem had the word "theorem" as part of its name, it was not in fact a theorem until it had been proven. I doubt there are any other examples of a statement being named a theorem, without there being an accepted proof. Paul August &#9742; 19:16, 5 February 2016 (UTC)
 * Well, at least arguably, it's always been a theorem. Before Wiles, it wasn't known to be a theorem.
 * As a separate point, I am not sure the word "theorem" has always been used this strictly. The English translation of Hilbert's remarks on his first problem has it:
 * The investigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.
 * That's Mary Newson's translation; German sources seem to use Satz, which in a non-mathematical context means "sentence", so it's possible that she just chose a less-than-optimal word here.  Still, the word "theorem" suggests an element of a "theory"; that doesn't suggest in and of itself that it must be provable.  I would be curious to know how long the word has been generally restricted to statements that have proofs. --Trovatore (talk) 22:17, 5 February 2016 (UTC)
 * The OED defines theorem as follows: "1a. Chiefly Logic and Math. A statement which has been proved to be true, or asserted as true and capable of being proved." An example of each meaning:
 * 1806 C. Hutton Course Math. (ed. 5) I. 2  A Theorem is a demonstrative proposition; in which some property is asserted, and the truth of it required to be proved.
 * 2010  Nature 11 Mar. 165/3   Until 2001, the Poincaré conjecture was one of the most famous open problems in maths; now it is one more theorem.
 * Of course, that is consistent with a change in the meaning over time. RockMagnetist(talk) 23:42, 5 February 2016 (UTC)

Strong and Weak Theorems
I tried to make an edit, and it (rightfully) got undone due to lack of references. I was attempting to define the terms strong theorem and weak theorem, which have multiple definitions and have been used in those different contexts in various other Wikipedia pages. So, I think it would be reasonable to try to work together to find the necessary references to make it all cohesive. The three usages I know of, with examples:


 * when neither theorem implies the other, because the "stronger" one has both a stronger hypothesis and a stronger conclusion, so it proves more restrictions are true in a more restrictive special case, as with the law of large numbers.
 * when a theorem appears to have a stronger hypothesis or conclusion, but actually is equivalent, as with strong induction.
 * when two theorems are similar enough so that one forces the other to be true (but not the other way around), in which the forcing theorem is said to be "stronger" than the other "weak" theorem, as with... well, I struggle to find an example at the moment on Wikipedia itself, but I can find a Math StackExchange reference that supports it. (Note, for instance, that the strong law of large numbers would actually not be a stronger theorem by the accepted definition there. The weak law wouldn't be stronger, either, though; both have propositions which only one of them prove, as neither implies the other.) — Preceding unsigned comment added by Jandew (talk • contribs) 23:12, 23 November 2016 (UTC)
 * It is true that "strong" and "weak" are sometimes used for qualifying theorems. I may add Hilbert's Nullstellensatz and Goldbach's weak conjecture to your examples. However, these are traditional qualifications, which, as far as I know, have never been formalized in any reliable source. Personally, I doubt that a general definition of these terms is useful, and I consider that it could be confusing as giving a mathematical label to terms that have not any mathematical content. In any case, Wikipedia is not the place for defining terms that are not defined in the literature.
 * By the way (and this confirms that defining "strong" and "weak" may be confusing) your post above contains several errors: two theorems may not be qualified as equivalent nor non-equivalent (your second item); they are true or wrong. Similarly, a theorem cannot "force" another theorem to be true (third item). D.Lazard (talk) 09:04, 24 November 2016 (UTC)

Four color theorem
Four color theorem has human readable proof.

So shouldn't the picture caption be changed or the example removed all together as pointless? Or it could be replaced with something like: "For years this theorem had only compter derived proof but since ..."?Linkato1 (talk) 16:33, 4 October 2017 (UTC)