Talk:Theorem

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Contents 1 How Tos 2 How does 3 Comments 4 References 5 Badly placed text 6 Definition of 'hypothesis" 7 Corollary

[edit] How Tos

I would like to learn more on how to create my own basic theorems and proofs. Are there any good sites covering this subject?

[edit] How does

Could someone please provide a reference or statement of how a theorem, like Clausius' entropy theorem, evolves into a 'principle', and how a 'principle' evolves into a physical law, like entropy the second law of thermodynamics. Sholto Maud 09:20, 13 December 2005 (UTC)

You're confusing "theorem" with theory. -- llywrch 20:51, 13 December 2005 (UTC)

• Ok thanks. But could you then please clarify what the term "theorem" refers to on the maximum power theorem entry? I'd really appreciate it. Sholto Maud 22:23, 13 December 2005 (UTC)

Looking at the material on this page, the theory page & the Maximum power theorem, I would conclude the following:

• A theorem is a statement which we can prove is true by at least one argument based on other theorems & axioms. (ISTR Gauss once creating several proofs for one theorem, in a quest to find the simplest & most elegant proof for that statement -- so a theorem can have more than one proof.)
• A theory is a statement which we can't prove is true -- but we can prove is false -- based on experimentation, & to some degree on arguments based on other theories. For example, no one really knows if the Theory of Thermodynamics is true, but experiments designed to verify it have failed to shown it to be false so many times that many people have for convenience assumed it is true. (And a theory that has become so enshrined as true or correct often is renamed as a Law, e.g., the Law of Thermodynamics.)

Let's stop for a moment & review the differences here. In one case, we can prove a statement true; in another, we can only prove it false. These are not the same thing, unless we also assume that a statement can only be true or false: & experience shows us that statements are often partly true or partly false. Thus, no matter how many times we prove a theory is not false, we can never be 100% sure that it is true.

• The maximum power theorem. Here it gets a little confusing: theorems are usually associated with mathematics, & theories with science. However, in a science-related field (electrical engineering), we find a statement labelled as a theorem. Reading the article, I noticed that there is a section labelled "Proof of theorem for resistive circuits": because this statement's truth rests on an argument based on other theorems & axioms, we can conclude this statement is a theorem.

Now it may happen that someone encounters a case for which this theorem is not true. What would happen is that one would need to review the truth of all of the theorems & axioms this particular theorem depends on, & reformulate the statement that made this statement false. (This would be the same procedure a mathematician would need to follow -- although such an event would shatter the entire structure of this discipline, as it did with the discovery of non-Euclidean geometry. But I understand that as of this writing there are few such surprises remaining to be found.)

Does this help? -- llywrch 18:14, 14 December 2005 (UTC)

Thank you for your considered contribution lywrch. It does help a little. I like the distinction between theorem as provable as true, and theory as not so and only falsifiable. This interpretation seems to me to make a theorem a more powerful statement qua epistemic truth, than a theory. But as with life, this also beggs more questions.

• Firstly, the transition between theory and law does not seem adequate for the rigor demanded by most systems of science. For instance, there is no specification of how many times we need to fail to show a theory false in order for it to be renamed a law, and thus considered true, as a "pseudo theorem". "Failiing to show" seems to be a measurable phenomenon, but there is no specification of what measure will change the status of a theory.
• Secondly, what happens in transdiscipline known as "mathematical physics"? I mean if theorem → mathematics, and theory → science & physics, then is mathematical physics, "theorem theory"? Such that we have a statement or proposition that is both falsifiable and provable? When you say ""Proof of theorem for resistive circuits": because this statement's truth rests on an argument based on other theorems & axioms, we can conclude this statement is a theorem." is it not also the case that the statement's truth rests on the actual measurable properties of the electromagnetic system, and so it is both a theory and theorem?
• Thirdly, if a theorem-theory can evolve into the status of a theorem-principle, and then theorem-law, by a process of repeated observations, then this suggests that we may be able to generate new laws, of thermodynamics for instance, over time. But when at what critical point does the theory become law?

Sholto Maud 21:48, 14 December 2005 (UTC)

Sholto, you're now asking questions that a philosopher of science would be better prepared to answer. I'm just a guy who adds articles to Wikipedia, & while I'm willing to share my opinions, I doubt that they may be as insightful as someone who has studied these issues would be; your thoughts are likely just as valid as my own. But I'll offer a few points for you to ponder further:

• The scientific disciplines extend in a continuum from the "hard" sciences (which are most like mathematics like physics or astronomy) to the "soft" sciences (like sociology or anthropology). Those at the one end best lend themselves to a rigorous approach like mathematics, & offer some basis for arguing the truth of theorems; those at the other at best offer theories, which sometimes do not lend themselves to being proven false. So none of the sciences are really as rigorous as we might think.
• The difference between "theory" & "law" is a fitting philosophical problem -- & I also suspect that a certain amount of subjectivity enters into promoting a theory to a law. In other words, I don't have a concise, clear answer for determining the difference -- but an academic who specializes in the philosophy of science might.
• I don't think that the statements described by "theory" & "theorem" are disjunctive groups: a statement that is true is also not false. If both approaches point to a statement being correct, then how would they conflict?
• Lastly, theorems depend on axioms, which by definition are assumed to be true; as I suggested above, experience may show that an axiom is indeed false. (This was the case with Euclid's famous axiom about parallel lines: doubt about this axiom led to the discovery of non-Euclidean geometries, thus demonstrating the underlying natures of logical proof & geometry.) Despite the certainty that logical reasoning gives us, we don't know if our conclusions are true until we encounter something that clearly proves that they are not.

I sincerely believe you are struggling with a worthy problem. However, I don't think I can provide you the help you need to be successful with this search. -- llywrch 04:42, 15 December 2005 (UTC)

[edit] Comments

While this article is useful as an introduction or definition of this term, it would improve this article if it answered questions like:

• What is the relationship of theorems in mathematics? Are they similar to experiments in the empirical sciences?
• How are the theorems of Euclid's Elements different from today's more rigorous theorems?
• What form did theorems have before Euclid?
• Do the concepts "theory" & "theorem" have more in common than a similar name?

This article could cover a lot more points. -- llywrch 20:51, 13 December 2005 (UTC)

• Seconded. Sholto Maud 19:16, 11 February 2006 (UTC)

[edit] References

This page needs references. Some parts seem correct, but others are illogical (incorrect typological order, among other grammatical issues). I've made a few corrections. Fuzzform 00:16, 31 March 2006 (UTC)

[edit] Badly placed text

This text should be in an article called "Mathematical Terminology". There's no structural reason to put all these definitions together in the same article called "Theorem".

And there are some fundamental errors in the definitions, we need to find external sources. But I think that first this name problem should be corrected.

And just one more thing: there is NO difference between mathematical algorithms and the ones in Computer Science! Arthur Gabriel de Santana a.k.a. Rox 11:38, 28 December 2006 (UTC)

What have mathematical algorithms got to do with theorems? Sholto Maud 21:54, 1 March 2007 (UTC)

I think the point is that the Division algorithm is really a theorem, despite it's name. But it's certainly rather unclear at the moment. Algebraist 02:03, 4 March 2007 (UTC)

Well, I edited it some. What would be really helpful would be a guide to mathematical writing that we could cite for some of this stuff. CMummert · talk 04:29, 4 March 2007 (UTC)

[edit] Definition of 'hypothesis"

This article (Theorem) says "In this case A is called the Hypothesis" but hypothesis in wiki has a definition that does not seem to accord with this usage of the term "hypothesis". So there needs to be some good disambiguation.Nznancy 22:17, 9 January 2007 (UTC) nznancy, 10 Jan 2007

[edit] Corollary

In the Terminology section, Corollary links back to Theorem. What IS this, some sort of a Moebius article? Lou Sander 01:18, 22 January 2007 (UTC)

I noticed the same thing just now, why doesn't corollary have its own article, if proposition and lemma do? -Dmz5*Edits**Talk* 03:39, 22 January 2007 (UTC) (i keep getting signed out)

It once had its own very short article, but somebody merged it. You can see the old stuff by clicking the link in the "redirected from" line of the Theorem article when you get to it through Corollary. Once you are there, look at History.

I just checked out Proposition and Lemma. The Lemma article definitely pertains to math. The Proposition article pertains mostly to philosophy and logic, though Proposition (disambiguation) mentions its mathematical meaning. Somebody needs to write articles on Corollary and on Proposition (mathematics). I don't have enough subject matter knowledge to do it myself. Lou Sander 05:17, 22 January 2007 (UTC)

It seems to me that the article Lemma (mathematics) is just a definition; I would rather see all these definitions gathered into one article where they can be properly compared and contrasted rather than in separate articles. And WP:NOT#DICT says that articles should not be created just to give definitions. CMummert · talk 13:19, 22 January 2007 (UTC)

That seems like a good idea to me. Proposition merits the same treatment, as do maybe some others. Lou Sander 18:23, 22 January 2007 (UTC)

I would also like to see all definitions gathered into an article, with discussion of how they are related and why they are useful with examples. Sholto Maud 21:56, 1 March 2007 (UTC)

I just noticed the same thing when trying to search for Corollary. I think it should have its own article, I'll bring it up with the people at the Mathematics wikiproject.--Jersey Devil 00:24, 29 March 2007 (UTC)

This article is perpetually under-referenced; I don't see how we are going to find enough references to make TWO articles that are more than just dictdefs. CMummert · talk 00:39, 29 March 2007 (UTC)