Talk:Theorem

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?

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)

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)

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)

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, an algorithm is a procedure, and a theorem is a (provable) assertion -- orthogonal concepts, except that we often prove theorems about algorithms. For instance, the theorem often referred to as "division algorithm" is actually a theorem about the division algorithm, asserting that it always terminates in finite time, that its outputs (the quotient and remainder) have certain properties, and that its outputs are the only integers having these properties. Calling this theorem "division algorithm" rather than "theorem on the division algorithm" is just normal human sloppiness sanctioned by long usage.Hippasus the Younger 04:12, 17 April 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)

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

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)

classification of finite simple groups

Imagine my surprise, after reading in the lede about how a proof should not be confused with the theorem it proves, that the section on trivia explains that the classification of finite simple groups is the "longest theorem"! The theorem actually is not so long; even if you went into some details about the groups it would still be not so long. I suppose if you started explaining about all the sporadic groups like the Monster, it would start getting longer but that's true of any complex theorem. --C S (Talk) 00:38, 22 April 2007 (UTC)

Lead too long and other problems

This article is still a mess, despite being WP:MATHCOTW. There seems to be a lack of boldness in improving the article, which really needs a lot of reorganisation. I would like to help, but cannot do it tonight, so please bug me on my talk page if I forget. Geometry guy 21:51, 28 April 2007 (UTC)

I've now written a new lead and moved some of the previous lead material into sections. These sections still need reorganisation and expansion, but it would be more fun to do it in collaboration. Anyone? ;) Geometry guy 14:39, 5 May 2007 (UTC)

Well, okay, the weekend is not the best time to find collaborators. Anyway, I've added a few pictures and controversial points, which I hope begin to answer Salix alba's question: "Is there anything more which could be said which is not covered by proof?" Geometry guy 19:59, 5 May 2007 (UTC)

Excellent edits! One thing -- there are in fact many more particles in the observable universe than 1.59*10^40 (see googol, for instance), so I removed that claim from the article. Kier07 16:52, 6 May 2007 (UTC)

Thanks for the ref, but the Mertens bound involves the exponential of 1.59*10^40, which is vastly larger, cf. googolplex! Geometry guy 19:17, 6 May 2007 (UTC)

Woops -- sorry about that! Kier07 00:00, 7 May 2007 (UTC)

By the way, what does it mean in the lead: it can be shown (indeed proven) that there are mathematical statements which are true but not formally provable? I know about statements such as the continuum hypothesis which are not disprovable (and hence could be called "true"), but which are also not provable. Is this what we're referring to, or is it something else? We should say somewhere in the article what we mean by this. It was my understanding that the only "truth" a theorem has is that it follows from axioms and other theorems. Kier07 17:54, 6 May 2007 (UTC)

This is a vague reference to Godel's first incompleteness theorem, which should be discussed in the body of the article, but is not at present. It states that in any axiomatic system strong enough to contain arithmetic, there are true statements which are not provable within the system. I'm glad you pointed this out! Geometry guy 19:17, 6 May 2007 (UTC)

It is certainly the case that one can talk about mathematical statements being true, even if they are not provable. Usually, if a mathematician, and indeed probably you or anybody else, were to say that the statement "every even number greater than two is the sum of two primes" were true, s/he would mean that every even number greater than two is the sum of two primes. To pick the classical example, "snow is white" is true if and only if snow is white. Note all this makes perfect sense. It is not a mathematical statement's fault, however, if it is in fact not a theorem, i.e. provable within your formal system. --C S (Talk) 19:51, 6 May 2007 (UTC)

The lead is looking great. I'm still confused by the statement that "The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical." I'd like to raise the maximum power theorem again (see above). This "theorem" is related to "empirical phenomenon" of electronic circuits - even though the theorem is provable, it's truth is dependent on empirical observation and verification. Hence theorem does not seem to be fundamentally deductive... Sholto Maud 01:31, 7 May 2007 (UTC)

Thanks. I have had a look at maximum power theorem and clarified it in a couple of places. From a mathematical point of view, there seem to be two theorems here, but one is a generalization of the other. To state the theorem as a mathematical theorem, one simply needs to add the hypothesis "For an electrical circuit satisfying Ohm's law and Joule's law". However, in physics and engineering, it is common to omit such hypotheses, since everyone knows (empirically) that electrical circuits satisfy Ohm's law and Joule's law under reasonable physical assumptions.

The mathematical formulation of the theorem is purely deductive, independent of any empirical observation and verification (as the proof shows). It proves, for example, that the empirically observed "maximum power principle" (that you maximize power when the load resistance equals the internal resistence) is a logical consequence of the empirical observations of Ohm and Joule.

I just want to break this down a little - "..the empirically observed X is a logical consequence of the empirical observations made by Y." Is this saying that an emprical observation is a logical consequence of an empirical observation? Sholto Maud 13:21, 7 May 2007 (UTC)

Yes, although your wording is open to misinterpretation, so let me give a precise example. Suppose you vary the voltage in a resistive electrical circuit and measure the current and power output. From your measurements you make the following empirical observations.

1. The current is proportional to the voltage.
2. The power output is proportional to the product of the current and the voltage.
3. The power output is proportional to the square of the voltage.

Then, for example, the third of these empirical observations is a logical consequence of the first two. Geometry guy 13:56, 7 May 2007 (UTC)

But I would apply the same method with Pythag theorem - from length measurements make empirical observations - so then what is the difference between mathematical and physical theorems? Sholto Maud 21:55, 7 May 2007 (UTC)

In principle there should be no difference! This is exactly the point I made below. Geometry guy 22:05, 7 May 2007 (UTC)

Hence if you find a circuit for which the maximum power theorem does not hold, you can conclude either that Ohm's law, or Joule's law (or both) is not valid for the given circuit.

All that has happened here is a blurring of the distinction between a principle and the theorem which may be used to derive it from other principles. This is no different in spirit from the blurring of the distinction between the division algorithm and the theorem which proves that it works. Geometry guy 12:21, 7 May 2007 (UTC)

Copyediting and physical theorems

I did some minor copyediting this evening. The only controversial point is whether theorems "should" be expressed as symbolic statements, and are not for convenience, or whether there is no need to worry about formalization. Different people take different positions, so I toned down the lede to take less of a stand about this. Also there was some confusion about Godel's theorem. CMummert · talk 02:56, 7 May 2007 (UTC)

Re: copyediting. "Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the way that such evidence is used to support scientific theories."

At issue is the separation of theorem (purely abstract) from theory (empirical/concrete reality). It seems to be the case that the maximum power theorem is a statement that cannot be shown to be true by mathematical proof only, but that the truth of the statement requires empirical evidence.

a) are there theorems of mathematical-physics that are not purely abstract?

b) does empirical evidence (e.g. of electronic circuits) influence the truth value of a proof?

If it is the case that maximum power in electronic circuits is the domain of empirical inquiry & hence theory, does maximum power theory generate the maximum power theorem and associated hypotheses?Sholto Maud 04:12, 7 May 2007 (UTC)

When I went through, I added the adjective "mathematical" several times, and changed the lede to start with "in mathematics," to make it clear that this article is currently about mathematical theorems only, not theorems in physics and engineering. The proof given in the maximum power theorem article does appear to be rigorous, however. I am not quite sure what you re asking. CMummert · talk 11:48, 7 May 2007 (UTC)

I think it is appropriate that this article should be primarily, but not exclusively, about the mathematical notion of a theorem. However, more could certainly be said about the use of the term theorem in physics and engineering. I already introduced a sentence at the end of the section on "Relation with theories" and would encourage other editors, such as Sholto Maud, to expand it. From my point of view, many physical theorems are closely related to mathematical ones, except that certain physical assumptions are not explicitly mentioned. I have explained this for maximum power theorem above, but it can also be seen at equipartition theorem: despite having worked extensively on the latter article, I still do not know what the hypotheses of this theorem are! The lack of explicit hypotheses makes physical theorems more flexible than mathematical ones: for instance one says that "the equipartition theorem applies to the canonical ensemble and to the microcanonical ensemble, but it does not apply when quantum effects are significant". From a mathematical point of view, this language sounds odd, because mathematical theorems always apply. More could be said about this in the present article. Geometry guy 12:38, 7 May 2007 (UTC)

Perhaps the article should acknowledge the ambiguity at the outset. E.g., "In mathematics, a theorem is a statement that can be shown to be true by a mathematical proof on the basis of explicitly stated or previously agreed assumptions. ... In physics, a theorem is ... In mathematical physics a theorem is understood to be..." If there are three different interpretations there should be an argument for why any one should have priority over the others. Sholto Maud 13:11, 7 May 2007 (UTC)

I think that goes too far. I have already explained how the use of the word "theorem" in the physical sciences is closely related to (and indeed derives from) the mathematical notion. It is not a different idea; it is simply that different standards of preciseness and rigour are applied in different areas (this is true even within mathematics). When you claim that a physical theorem can state an empirical truth, I think you are confusing the conclusions of the theorem with the theorem itself, and I have demonstrated that the maximum power theorem is deductive. In any case, the lead of an article should reflect its content, so this discussion needs to be more fully developed in the body of the article. Geometry guy 14:12, 7 May 2007 (UTC)

Yes I think the max pow theorem is deductive as you say, but I haven't seen the suggested derivation of the physical science theorem from mathematical theorem. I'm also a little unclear on the role of hypotheses: When one predicts that a conclusion is emprically observable as a logical consequence of prior empirical observations (are these the premises?) is this predicted conclusion called the hypothesis (which would seem to be the same meaning as employed in science)? Sholto Maud 21:55, 7 May 2007 (UTC)

Outside of mathematics hypothesis has too many meanings. A better word is premise. The prior emperical observations in the mathematically-phrased maximum power theorem are only premises in the sense that they are assumed to hold, not in the sense that they have been observed experimentally. The mathematically-phrased maximum power theorem would be true even if Ohm's law were never true. Geometry guy 22:15, 7 May 2007 (UTC)

Ok. I think I'm following. So, most theorems have two components, called the premises and the conclusions. In principle, there is no difference between physical and mathematical theorems. However in practice there is a difference, and this difference depends on how the theorem is "phrased". A theorem may be phrased in many different ways, it can be mathematically-phrased, physically-phrased, biologically-phrased, financially-phrased etc. As Geometry guy stated above, a mathematically-phrased maximum power theorem would be true even if Ohm's law were never true. This means that there is no truth test required for the assumptions made in the premises (which makes it close to a mathematical-logic-phrasing). However a physically-phrased maximum power theorem can only be true when Ohm's law is also true - a truth test is required for the premises. And these are the different standards of preciseness and rigour that are applied. Have I understood? Sholto Maud 02:15, 8 May 2007 (UTC)

Partly, but I still think you are over-egging the pudding. Probably I took the discussion in the wrong direction with "mathematically-phrased". The only real difference between different fields is which assumptions are explicitly stated. In practice, in any domain of mathematics or physics or whatever, the premise of a theorem will include implicit assumptions which are not written down in the statement. These take various forms.

• Logical foundations. See Carroll's paradox for the difficulties which arise when you try to make assumptions about the nature of logical argument explicit! This is not a trivial point: proof by contradiction uses the law of the excluded middle; this is seldom mentioned in the premise, despite the fact that it can be useful to work without this law. In contrast, the axiom of choice is quite often mentioned when it is used, but again this depends on the context. More generally...
• Axioms e.g., any natural number n has a successor n+1. When talking about foundations, such axioms are mentioned explicitly, but usually they are just implicitly assumed. Alternatively they can be viewed as...
• Definitions. In the premise "Let G be a group" there is an implicit assumption that a group is a set with certain operations and properties. Whether such definitions are given depends on the context. Even the intended meaning varies from context to context, e.g. a ring sometimes has a multiplicative identity, sometimes not.

Thus even within mathematics the use of implicit assumptions varies depending on the context. So if you start talking about "mathematically-phrased" theorems, you might as well also say "set-theoretically phrased", "number-theoretically phrased", "geometrically-phrased", and so on.

Physics and engineering are no different. For instance the implicit assumption that Ohm's law holds is not a "truth test": it is part of the definition of a "resistive electrical circuit" in this context! The only difference between the two formulations is which assumptions are implicit, and which are explicit. This is not fundamental to the nature of a theorem, just a context-dependent convenience. Geometry guy 15:12, 8 May 2007 (UTC)

I appreciate your efforts, but perhaps we're talking about a different pudding. I think I'm questioning whether the conclusions of the maximum power theorem are true or not, since this will affect the truth of the assumptions - agreed that the implicit assumption that Ohm's law holds is not a "truth test". But the truth of the assumption can be tested against a real circuit (in fact many mathematicians refuted Ohm's reasoning and it was, apparently, empirical repeatable reality that gave support to Jacobi's law). We know that the theorem is proven, but the truth seems stronger than the proof in this instance. Truth makes a phenomenological claim about the measureable properties of any specific electronic circuit. I can only confirm the truth value of the theorem if I make an empirical electronic circuit and measure kW with different load resistance. If this is right then the truth of some theorems seems to be linked to empirical content and the strict separation between theorem and scientific theory is brought into question: if the electronic circuit does not give the predicted results, then the theorem, while proven, is not true and falsifiable. Sholto Maud 21:44, 8 May 2007 (UTC)

Questioning whether the conclusions of a theorem are true or not is different than saying a theorem is not true. The theorem "if 0 = 1, then every prime number is composite" is clearly true. You can question the conclusion all you want, and indeed the conclusion is false. But the theorem isn't. The maximum power theorem is true, regardless of whether a person finds the conclusion objectionable or not. Just because a theorem isn't phrased in an "if then" format doesn't mean there aren't implicit assumptions that fit into the "if" portion. --C S (Talk) 22:04, 8 May 2007 (UTC)

Okay, I think I understand what you're saying, although I misunderstood before (my earlier response is left above since Geometry guy commented on it). But I think what you are talking about isn't too relevant to the main thrust of the article, but may be very nice in a small section on applicability of mathematics to science. As a mathematical theorem, the maximum power theorem is true; however, you are right that in order for it to be useful it should agree with and predict real world results. I think the confusion here is that when we talk about truth we are really talking about mathematical truth. In science, there really is no equivalent. Even when you say the theorem would be true if you tested it and found it agreed with the results, I don't believe it's actually correct to say it's true. It just hasn't been falsified. A scientific theory isn't "true", it's either "useful" or not, falsified within a certain domain or not. Newton's theory isn't true or false. It's useful in certain contexts, less useful or completely falsified in others. --C S (Talk) 22:17, 8 May 2007 (UTC)

Chan-Ho Suh pre-empted my response in fine form, but here it is anyway (you/your = Sholto Maud, not Chan-Ho Suh)...

That is a completely independent (though quite reasonable) question. Whether the conclusions of a theorem are true has no logical implication for the validity of the premise. For example, "If 0 = 1 then 1 = 1" is a theorem, but the fact that the conclusion is true does not imply that the premise is! Nevertheless, in practice, conclusions are used to provide evidence for hypotheses. For instance the Riemann hypothesis has many plausible consequences, and these support (but do not prove) the idea that it may be true. Similarly, the empirical observation of the maximum power principle provides evidence for Ohm's law. Such reasoning, however, has nothing to do with the notion of a theorem. You don't confirm the truth value of a theorem by making a measurement, you do it by giving a proof! You have understood nothing if you do not understand this! In particular, your last sentence is pure nonsense (sorry to be so blunt!). Geometry guy 22:10, 8 May 2007 (UTC)

Thankyou Chan-Ho Suh, and Geometry guy. I agree that the last sentence reads poorly - I have no excuse - and agree that a conclusion being true does not imply that the premise is. Despite both efforts I feel there is a small bit missing that needs to be locked down.

1. Useful: Chan-Ho Suh introduced the concept of 'usefulness' - this needs definition. It assumes a distinction between useful and non-useful theorems, but this has not been previously discussed. Is the proposition that a useful theorem should both agree with and predict real world results, and a non-useful theorem should not agree with or predict real world results?

2. Mathematical truth: again introduced by Chan-Ho Suh. I think this is used by Geometry guy in the statement, "You don't confirm the truth value of a theorem by making a measurement..." - a mathematically true theorem is one that is proven, if it is not proven it is not mathematically true. Note that Geometry guy says we may empirically observe the maximum power principle, but Geometry guy does not use the term maximum power theorem (Wikipedia has 2 different entries for these terms). Is the proposition that one cannot empirically observe a theorem?

3. So it seems that a useful theorem which agrees with and predicts real world results should be empirically observable. But this is outlawed in 2, hence no theorem is useful (or perhaps this should read no theorem is empirically useful as opposed to mathematically useful) Sholto Maud 01:44, 9 May 2007 (UTC)

This is pure nonsense again. There is no need at all to define "useful" or categorize theorems by "usefulness", and the rest of the logic is fallacious. The answer to your two questions are "no" and "no", and your conclusion is false.

• The maximum power theorem is empirically observable in the sense that one can observe empirically that circuits which satisfy Ohm's law and Joule's law satisfy the conclusions of the maximum power theorem (which is what I meant by the maximum power principle: please reread my comment in the previous section instead of confusing the issue by linking to its philosophical meaning). Such empirical observations do not prove the theorem, however.
• The maximum power theorem is useful because it tells you that you don't actually need to check empirically whether a circuit satisfies the conclusions, as long as you know it obeys Ohm's law and Joule's law. The contrapositive could be useful as well, as I have already remarked.

Geometry guy 11:01, 9 May 2007 (UTC)

Actually I'm not the one that introduced usefulness into this discussion, Sholto Maud...you are! You were confusing what everyone else would call usefulness with "truth", so I merely pointed it out.

"As a mathematical theorem, the maximum power theorem is true; however, you are right that in order for it to be useful it should agree with and predict real world results." Could you explain how you are using useful in this sentence Sholto Maud 22:17, 9 May 2007 (UTC)

But as I explained, whether a theorem is useful, applicable, or whatnot, in the real world, is not very relevant to whether a theorem is true. I think the prime source of confusion here is that you don't realize that scientific theories are not true or false.

No, no. I'm comfortable with this notion. Sholto Maud 22:17, 9 May 2007 (UTC)

They are useful or not in certain domains. Scientific theories are also stated in a manner called "falsifiable", but that doesn't have to do with "truth value". --C S (Talk) 17:24, 9 May 2007 (UTC)

It's probably overkill, but I just can't resist! Here's one of my favorite quotations from Albert Einstein, as quoted by Ludwig von Mises in Human Action.

How can mathematics, a product of human reason that does not depend on any experience, so exquisitely fit the objects of reality? Is human reason able to discover, unaided by experience, through pure reasoning, the features of real things? ... As far as the theorems of mathematics refer to reality they are not certain, and as far as they are certain, they do not refer to reality.

So if Einstein couldn't figure out why some "theorems" fit the world of our experience so neatly, how can we possibly hope to do so?  ;^> DavidCBryant 17:02, 15 May 2007 (UTC)

Pure nonsense (continuation)

Please excuse my reasoning and thankyou both for your persistence. I act from good intention, but I'm confused again.

• From the article: "Mathematical theorems ... are purely abstract formal statements"
• From the article: "...the proof of a theorem cannot involve experiments or other empirical evidence in the way that such evidence is used to support scientific theories."
• Geometry guy: "You don't confirm the truth value of a theorem by making a measurement, you do it by giving a proof!"

I think I'm ok with this much: proving a theorem makes it true. But I have an impedance mismatch with the above and the below...

• Geometry guy: "The maximum power theorem is empirically observable in the sense that one can observe empirically that circuits which satisfy Ohm's law".

Please quote/reread the full sentence and note the use of the words "in the sense that". It is not empirically observable in the sense that its truth follows from empirical observation. My statement was quite clear: you are just confusing the issue by using the slipperiness of language to tie yourself up in knots. Geometry guy 12:25, 9 May 2007 (UTC)

Full sentence reads: "The maximum power theorem is empirically observable in the sense that one can observe empirically that circuits which satisfy Ohm's law and Joule's law satisfy the conclusions of the maximum power theorem (which is what I meant by the maximum power principle: please reread my comment in the previous section instead of confusing the issue by linking to its philosophical meaning)." - sorry I'm a little confused by the distinction between maximum power principle_Wikipedia and maximum power principle_{Geometry guy} and maximum power theorem_Wikipedia, so I left out the parenthesis. As for "...and Joule's law satisfy the conclusions of the maximum power theorem..." I didn't see the second meaning of "observe empirically" I think you may be referring to. hmmmm.... Sholto Maud 12:56, 9 May 2007 (UTC)

Perhaps I shouldn't have shifted the meaning, but this situation is no different from observing that two oranges and three oranges together give five oranges. This observation illustrates the theorem that 2+3=5 but does not establish its truth. Geometry guy 13:23, 9 May 2007 (UTC)

If a theorem is empirically observable then can't it be faslified? For example:

• Chan-Ho Suh: "Even when you say the theorem would be true if you tested it and found it agreed with the results, I don't believe it's actually correct to say it's true. It just hasn't been falsified."

If a theorem can be falsified then this would seem to imply that an empirical observation has the capacity to adjust the truth value of the theorem, regardless of the proof. But this result doesn't seem to agree with the article. Sholto Maud 12:07, 9 May 2007 (UTC)

This disagreement arises simply because you have misquoted me and interpreted "empirically observable" in two subtly different ways. The beauty of having a proof is that if someone happens to observe one day a circuit which satisfies Ohm's law and Joule's law but not the conclusions of the maximum power theorem, then this does not falsify the theorem — instead it shows that something went wrong during the experiment! Geometry guy 12:25, 9 May 2007 (UTC)

So would you say then that theorems can't actually be falsified? Sholto Maud 12:45, 9 May 2007 (UTC)

Yes. However, people are fallible, and so a mathematical statement and argument which claims to be a theorem and its proof might not be. The claim to theoremhood can be denied either by finding an error in the proof (so the statement is unproven) or by finding a counterexample (which shows that the statement is false). The latter process is similar to the scientific notion of falsifiability, but using the term here only generates confusion. In particular, a statement for which there is a counterexample is not a theorem. Geometry guy 13:23, 9 May 2007 (UTC)

Chan-Ho Suh were you implying that theorems can be falsified with the sentence ""Even when you say the theorem would be true if you tested it and found it agreed with the results, I don't believe it's actually correct to say it's true. It just hasn't been falsified."  ? Sholto Maud 22:20, 9 May 2007 (UTC)

• I was thinking about this last night, and perhaps my confusion is a result of the term "maximum power theorem", and that it should rather be "maximal theorem_power", so that we know that the theorem does not have to be specifically about kiloWatts. Rather the maximal theorem is generic and could be applied anywhere, and electrical engineers have just applied it to electric circuits. Would that explain it? Sholto Maud 23:36, 9 May 2007 (UTC)

Circuit theorem

I noticed that the maximum power theorem has recently been reclassified as "circuit theorem" - but there is no Wikipedia entry for circuit theorem, and theorem doesn't state the criteria for mathematical theorem, circuit theorem etc. Sholto Maud 04:47, 12 May 2007 (UTC)

"I think the definition of a mathematical theorem ("a theorem is a statement that can be shown to be true by a mathematical proof on the basis of explicitly stated or previously agreed assumptions") works equally well for circuit and physics theorems. There could very well be a difference, but I can't think of one, sorry. Roger 02:46, 11 May 2007 (UTC)"

This has been done to death already, but since I like to beat a dead horse I'll try to clarify two or three things for Sholto Maud.

When physicists speak of theorems they're not using the word the same way mathematicians do. Natural languages are inherently ambiguous. The word "theorem" has two different meanings (for mathematicians, and for physicists).

When mathematicians speak of "assumptions" and "axioms" they are speaking of purely logical propositions that do not have any necessary connection to the "real world". The ideas of mathematics, including its theorems, exist only in our minds.

Physical laws would (in the view of most physicists) exist whether there were human beings to observe them, or not. So the "assumptions" the physicists make are not the same as mathematical axioms, because the physicist is making an assumption about a world that exists outside of his mind.

The logical thought processes that connect a physical theorem with the physicist's assumptions are indeed the same as the deductive logic of mathematics. The difference is in the nature of the assumptions. If the physicist discovers that his assumptions were incorrect, he'll have to start all over again, because there is only one "real world" to which his theories can apply. The mathematician doesn't care about that – rather than proclaim an axiom "wrong", he'll just invent a new branch of mathematics by choosing a new set of assumptions. For example, the discovery of non-Euclidean geometries did not invalidate Euclidean geometry; it merely broadened the field of geometry.

If none of this seems very helpful, try chewing on the article about David Hume and then the one about the problem of incomplete induction. DavidCBryant 16:13, 14 May 2007 (UTC)

Maybe the article should start with, "The word "theorem" has two different distinct meanings, a mathematic meaning, and a physical meaning. When physicists speak of theorems they're not using the word the same way mathematicians do. The difference appears to be that mathematical theorems are not applied to the real world, whereas physical assumptions are applied. In mathematics, assumptions exist only in our minds. In contrast when a the physicist discovers that their assumptions are incorrect, they have to start their system (see Mind-Body_Dualism)." is this right? Sholto Maud 21:40, 14 May 2007 (UTC)

I'm not sure such a distinction belongs in the article. This article is about mathematics, and it says so in the very first sentence.

I'm not convinced. In strict terms the article is about the word 'theorem', and the very first sentance implies ambiguity, or at least the need for greater clarification. The qualification "In mathematics..." says that one might also have started the article, "In physics...". Sholto Maud 05:22, 16 May 2007 (UTC)

Depending on one's philosophical point of view, the difference between "arbitrary axioms" and "axioms that agree with the real world" is not necessarily as great as I've tried to paint it. See, for instance, Birkhoff's theorem. That theorem can be understood as a purely mathematical statement about the solutions to Maxwell's equations. It's only when one starts to think of electromagnetic fields as actually existing in the real world that the more subtle point about the nature of one's assumptions becomes significant and Birkhoff's theorem becomes a physical (and not merely mathematical) theorem. DavidCBryant 10:30, 15 May 2007 (UTC)

Circuit theorem just means "theorem in the theory of circuits". The same is true, modulo variations of grammar, for pretty much any X in the phrase X theorem. These derived concepts do not need to be defined and the criteria are obvious (e.g., "the theorem is about electrical circuits").

The concept of a theorem is fundamentally mathematical, and other uses of the term are variations on the same basic theme. Geometry guy 12:18, 25 May 2007 (UTC)

Substantive concerns

I think the article is fairly good, stylistically speaking. Here are a few more substantive concerns I have.

• In the section Formal and informal notions, the form of proof involving logical equivalency (if and only if theorems) isn't mentioned. Such theorems are very common, and probably ought to be discussed.
• The same section might also benefit by mentioning the difference between existence theorems and constructive theorems. I guess the discussion of Merten's conjecture in the next section hints at this (existence of a counterexample, without actually producing it), but the distinction ought perhaps to be more explicitly drawn. I guess we don't want to drag in a lot of stuff about Brouwer, but it might be good to mention him.
• I know that this article is about theorems, and not proofs, but the two concepts are pretty closely connected. One or two more examples of theorems might be worked into the discussion somehow, to illustrate subtle differences between different methods of proof, or how the same theorem might be stated (and proved) in two slightly different ways. My favorite example along these lines is Euclid's proof that there are an infinity of prime numbers, versus Euler's demonstration that the sum of the reciprocals of the primes diverges. Another good example is the demonstration that √2 is irrational, either by contradiction, or by actually constructing the sequences of Pell numbers that approach the limit from above and from below.
• In the section Terminology many of the distinctions drawn are fairly fuzzy, and the wiki-links aren't as helpful as they might be. I tried to rectify some of that by avoiding links through dab pages, but more work is needed. One thing that occurs to me is that certain phrases (e.g. "Law of Large Numbers") have assumed a precise meaning in mathematics, so they serve as a sort of shorthand for mathematicians. Can that idea be used to help explain the distinctions between identities, rules, principles, and laws?
• I'm not sure exactly where it ought to fit, but the structure of axioms and theorems in Euclid's Elements is commonly held up as a model of well-organized logic. Without getting into all the pros and cons, I wonder if a nod in that direction might improve the article. Maybe a short section on why mathematicians construct theorems, building up from "self-evident" truths or axioms through very simple theorems and ultimately arriving at statements that are fairly hard to prove would benefit some readers. DavidCBryant 18:01, 15 May 2007 (UTC)

Misses the point of being a theorem

"In mathematics, a theorem is a statement that can be proved on the basis of explicitly stated or previously agreed assumptions."

The ideas of "provability" and "truth" come into the picture only as interpretations of theorems. A theorem is a "derivation" from the alphabet and rules of the language.

The example I have seen is the language FS (stands for 'Formal System' from Benson Mates) whose alphabet consists of stars and daggers *, †, and whose formation rule for wffs is:

'Any string of symbols of FS which is at least 3 symbols long, and which is not infinitely long, is a formula of FS. Nothing else is a formula of FS.'

The single axiom of FS is: '†*†'

The transformation rule for FS is:

'Any occurrence of '†' in a formula of FS may be replaced by an occurance of the string '†*' and the result is a formula (wff) of FS.'

Theorems in FS are defined as those formulae in FS of which a derivation can be constructed, the last line of which is that formula.

• 1) †*† (Given as axiom)
• 2) †**† (transformation rule)
• 3) †**†* (transformation rule)

Therefore '†**†*' is a theorem of FS. Yet no one would claim it as "true" or "proved." It is merely derived.

Two metatheorems of FS are:

• Every theorem of FS begins with '†'
• Every theorem of FS has exactly two daggers.

This article would seem to neglect this more fundamental definition of a theorem. Gregbard 21:54, 5 July 2007 (UTC)

The definition of a "theorem" as any derivable expression in a given formal system is not completely uncommon, but it isn't the meaning that mathematicians in general attach to the word. They mean a particular type of mathematical statement expressed in natural language. The more general definition does belong in this article, but in a new section on more general meanings of the term. Would you be interested in writing that section? — Carl (CBM · talk) 23:01, 5 July 2007 (UTC)

I kept the content that User:Gregbard added, but I moved it around a little. The lede section, per WP:LEDE, is meant to be a summary of the overall article, a mini-article that can stand alone, so too much detail isn't right. I think it's important to keep the link to mathematics near the top, as well as the link to logic. I made a whole section on theorems in logic, which could be longer than it is right now. By having a section, we can give a much more complete picture than is possible in the lede. — Carl (CBM · talk) 05:14, 6 July 2007 (UTC)

Hmm. How silly is a NPOV box going to look on the top of an article about 'theorem?' "A mere general sense of the term is used in logic, where a theorem is..." The wikipedia is math-centric rather than logic-centric. This phenomenon that has just occurred I find fascinating.

I'm pretty sure an encyclopedia article should be organized from general to specific. The opening paragraph is dumbed down so as to totally ignore the one essential property of theoremhood. The interpretation as "truth" or "proof" is a "mere" interpretation that mathematicians use.

The key is derivability not truth or proof. The logical aspect of it should come first, then the math aspect. The same organization applies to math articles with computer science applications. The CS part comes after the math part. Gregbard 08:43, 6 July 2007 (UTC)

Wikipedia is reader-centric. We proceed from the general to the specific only in the sense of proceeding from the broad interest and general use to minority interest and specialist use. A theorem is a statement with a proof based on previously agreed assumptions. This already includes the idea that the statement might be a wff in a formal system, and that a proof is a derivation based on the axioms and transformation rules of that system. It also includes the notion of a theorem used in physics, in which some of the assumptions are implicit, or "physical".

I agree there is a distinction between "truth" and "theorem" and this article already makes it. There is no such distinction between a derivation and a proof except in a very specialist arena. This article is supposed to encompass the notion of a theorem in physics, logic, mathematics, engineering, computer science... You may regard the other uses as mere interpretations of the logical use, but that is a niche view. It should be expressed for encyclopedic completeness, but it should not drive the article. Geometry guy 12:26, 6 July 2007 (UTC)

If it is indeed true that the distinction is limited to a "very specialist arena" like you say, is that supposed to be a good thing? Do you think we're better off with it staying that way? It isn't my "niche view" that makes these other uses interpretations of the logical principle. The existence of an interpretation means something to a logician. The use of theorem by mathematicians to mean truth is an interpretation. It is to confuse the reality for the interpretation of the reality. You can go around saying it's true all day long. That model works well for mathematicians. But please don't portray theorems to the general public as 'the reality' rather than as a model of reality. I think they can take it. Although I think some mathematicians who are attracted to the field by it's apparent certainty may get their hearts broken if they faced the truth. Gregbard 13:11, 6 July 2007 (UTC)

The distinction between truth and theorem is widely appreciated, even by mathematicians! And even physicists understand the distinction between reality and a model of reality. So I do not see where you are coming from here. For instance, most mathematicians I know are delighted, not disheartened, by results like Godel's incompleteness theorems.

As for portrayal: this is an encyclopedia. There is no original research here. If proof and derivation are widely regarded as synonyms, then that is what we write. It makes no difference whatsoever whether we are better off staying that way or not. If you want to change the world, this is not the place to do it! Geometry guy 13:35, 6 July 2007 (UTC)

Indeed, the usage that calls a string generated by a term rewriting system a "theorem" is muchless common than the general mathematical usage. I have tried, in the lede, to clearly distinguish the notion of derivability from the notion of truth; the second paragraph, written by Gregbard, stresses this. — Carl (CBM · talk) 15:33, 6 July 2007 (UTC)

Gregbard, please think about WP:NPOV. I just consulted the Oxford English Dictionary, in which a theorem is defined to be "A universal or general proposition or statement, not self-evident (thus distinguished from AXIOM), but demonstrable by argument (in the strict sense, by necessary reasoning)". Just to be sure that this reflects a majoritarian view, I also consulted Webster's Third New International Dictionary, which gives as the first definition "A statement in mathematics that has been proved". So it appears that the ideas of "proof" and by implication "truth" are closely associated with the word theorem, in popular usage.

The distinction between an "interpretation" of a logical principle and the principle itself is vacuous. You can say that a "well-formed formula" is "derived" and "valid" – I can say with equal force that it has been "proved" and is therefore "true". The distinction between the two forms of expression is purely semantic. But the second way of talking about it is more widely accepted by people who speak English, and the majoritarian point of view is the one this article ought to stress.

One other thing. The dictionary does not define the word "theoremhood", so I am excising it from the article, on the ground that it is a neologism. DavidCBryant 15:51, 6 July 2007 (UTC)

I am completely outside this discussion, but I just wanted to mention that it seems odd to me that "theorem" would be defined as "a statement that has been proved". I was taught back in school that a proven statement is a postulate, while a theorem is a statement that is accepted as true, but which has not been fully proven or for which no full proof exists or can exist, due to the nature of the statement. Your thoughts? — KieferSkunk (talk) — 21:32, 28 July 2007 (UTC)

It's the opposite. The postulate is accepted (like an axiom), a theorem has a derivation associated with it, although not necessarily a "proof" (This I think, is the point these guys are getting lost on.) If it has a "proof" there is a derivation to which the proof corresponds. Be well, Gregbard 23:07, 28 July 2007 (UTC)

Okay, thanks for clarifying. :) — KieferSkunk (talk) — 16:04, 31 July 2007 (UTC)

lede

I did intend to say "more general", thanks for fixing that.

Geometry Guy mentioned in an edit summary that the lede is too long. I don't think it's so bad - the first four paragraphs are a reasonable summary of the material in the article, which is desirable for WP:LEDE. The last paragraph, perhaps, could move down to the section on terminology. — Carl (CBM · talk) 15:21, 6 July 2007 (UTC)

I'm not satisfied with the lede. It needs a statement of what a theorem is. The qualification "in mathematics" is against the rules, unless there are equivalent statements, "in logic", "in metaphysics", "in physics" etc... But then it needs to be explained why there are different interpretations in different fields - perhaps with some history as to how the different uses came about. If theorems are about proofs or derivations as argued above, then the rest of the article needs an commonly accepted good example of a proof and a derivation. Sholto Maud 06:07, 31 July 2007 (UTC)

All theorems have a derivation associated with them, that is why it is my view that that should be first. I'm indifferent to a having the "in logic" or "in math," etcetera, although that should be somewhere. The theorem as derivation is more a more fundamental definition because it does not presume truth or justification etc. We should always begin articles with a more general or fundamental overview. This is so that people learning from them are not prejudiced forever. Gregbard 07:03, 31 July 2007 (UTC)

In its ordinary sense in mathematical logic, the word "proof" doesn't assume truth either - it refers to a derivation. The idea that a derivation is not a "proof" without some semantical interpretation is not found in mathematical logic. The fact that the deduction rules used in most mathematical proofs are sound is of course important, but not part of the definition of a proof.

I have already pointed out to Gregbard that Samuel Buss's article in the Handbook of Proof Theory claims exactly the opposite of what Gregbard claims - Buss claims that the idea of a proof as a logical derivation is less general than the concept of a natural language proof. Buss says,

"There are two distinct viewpoints about what a mathematical proof is. The first view is that proofs are social constructions by which mathematicians convince one another of the truth of theorems. ... Of course, it is impossible to precisely define what constitutes a valid proof in this social sense, and the standards for valid proofs may vary with the audience and over time. ... The second view of proofs is more narrow in scope: in this view, a proof consists of a string of symbols which satisfy some precisely stated set of rules and which prove a theorem, which must also be expressed as a string of symbols. ... Proofs of the latter kind are called "formal" proofs to distinguish them from "social" proofs. (p. 2)"

This viewpoint is not uncommon. — Carl (CBM · talk) 04:52, 4 August 2007 (UTC)

I think if you read this quote carefully, you will realize that brother Buss is talking about comparing "social" theorems with "formal" theorems. He is not comparing "logical" theorems to "mathematical" ones. The formal proof is more narrow than the social proof because social proofs range from "aw c'mon" to "This here fact A implies that there fact B, and fact A is undisputed, therefore B is true." In that regard, what consists of a social proof is more broad than a formal proof whose requirements for proofhood are more narrow.

The mathematical use of proof assumes that justification goes along with the derivation. We are supposed to be compelled to believe something by such a proof. That is a presumption. A goal of philosophy and logic is to strip away the presumptions.

The description of the formal proof as "more narrow" refers to the fact that there are requirements of a formal proof that there are not for a social one. That's true. But that is not what is being discussed. The sense of "more broad" and "more narrow" that matters as it concerns the organization of an encyclopedia is the one definition includes all of the cases of the other.

"Buss claims that the idea of a proof as a logical derivation is less general than the concept of a natural language proof."

The word "proof" which is a form of the word "prove" can hardly be thought not to imply justification. That is baggage. The goal is to put forward a definition that does not have baggage. Then talk about the different kinds baggage in the article. In this regard theorem as derivation is a more general definition than theorem as proof (or proved).

It seems we are refering to different pairs of things when we are saying A is more general then B. The use of the word theorem in logic is more general then the usage in mathematics. Perhaps brother Buss would agree in this. Every theorem of mathematics corresponds to a logical derivation, not the other way around.

It is also of note that Buss says "...that the IDEA of a proof as logical derivation is less general ... " The derivation itself is in fact more general. Be well, Gregbard 13:38, 6 August 2007 (UTC)

Ok so if we step back from the task making a definition for a moment, what we can say is that there is no commonly agreed definition of the "theorem" concept. This is the most general statement one can deduce from all the above comments. I propose we start the lede saying that there is no commonly agreed definition of the "theorem" concept, and then say what the various different definitions are and how they differ. Sholto Maud 02:30, 6 August 2007 (UTC)

(←) It's not true that there is no commonly accepted definition; what's true is that there are two commonly accepted definitions. While you are free to try to rephrase the lede more clearly, it already explains what the two definitions are and how they relate to each other. — Carl (CBM · talk) 12:02, 6 August 2007 (UTC)

Token revert

Greg, I reverted because I think what you added was OR. Of course, I could be wrong - so, please give some sources that verify what you are saying about tokens, abstract objects, and so forth - because it sounds just like what you were trying to add to some other articles. Thanks. Tparameter (talk) 01:52, 13 May 2008 (UTC)

Until I get some good material on this, I have placed the tag for lacking interdisciplinary content. Be well, Pontiff Greg Bard (talk) 18:24, 14 May 2008 (UTC)

I still don't agree with this addition:

It is universally acknowledged that numbers and the other objects of pure mathematics are abstract. In essence, a theorem is a type of abstract object. We only experience them as tokens of that abstraction. For instance, one token of a theorem is the formula of a formal language which is derived in a formal system; another token of which is the equivalent statement in natural language, which can be proved in a mathematical proof; and another of which may be chalk marks on a board representing that theorem.

First, whether a number is an abstract object doesn't reflect on whether a theorem is. But more importantly, I don't think there is any support demonstrated in the literature for the idea that a theorem consists of both a formula in a formula language and a natural language statement. Not only is no support demonstrated, I don't think there is support in the literature for it. I'll see if I can't find some actual references on a theorems in those "introduction to higher mathematics" books. — Carl (CBM · talk) 22:24, 16 May 2008 (UTC)

Under what interpretation is a theorem not an "object of pure mathematics." I am really just shocked at this point. Pontiff Greg Bard (talk) 22:38, 16 May 2008 (UTC)

You seem to be completely ignoring the actual use of the word. We refer to each of these things as, for instance, a modus ponens theorem: the chalk on the board, the idea of it. However, if I asked you if the theorem written in yellow chalk is a "yellow theorem" you would think I'm nuts. It an abstract object, and that is obviously how we treat it. Pontiff Greg Bard (talk) 22:47, 16 May 2008 (UTC)

Please see the quotes below, which each support my contention that a theorem is simply a statement. What I disagree with isn't particularly the "abstract" part, which I think only misses the point. The thing I find doubtful is the claim about tokens that represent the same theorem. I'm going to rephrase your sentences some while leaving the abstract object part. — Carl (CBM · talk) 23:11, 16 May 2008 (UTC)

Some quotes

Each of the following supports the general viewpoint that a theorem is a statement that has been (can be) proved.

• Smith, Eggen, Andre, A transition to advanced mathematics, p. 26

  "A proof is a justification of a statement called a theorem."

• Encyclopedia Britannica, "theorem (logic and mathematics)":

  "in mathematics and logic, a proposition or statement that is demonstrated. ... The statement “If two lines intersect, each pair of vertical angles is equal,” for example, is a theorem."

• Tarski, Introduction to logic, the very first paragraph:

  "Every scientific theory is a system of sentences which are accepted as true and which may be called laws or asserted statements of, for short, simply statements. In mathematics these statements follow one another in a definite order which will be discussed in detail in Chapter VI, and they are, as a rule, accompanied by considerations intended to establish their validity. Considerations of this kind are called proofs, and the statements established by them are called theorems."

• Enderton, A mathematical introduction to logic, p. 117

  "Notice that we use the word "theorem" on two different levels. We say that α is a theorem of Γ if $\Gamma \vdash \alpha$. We also make numerous statements in English, each called a theorem, such as the one below. ..."

— Carl (CBM · talk) 23:11, 16 May 2008 (UTC)

Carl, it will shock and amaze you to learn that a statement is also an abstract object. (I am not really very interested in inserting that fact in that article, although ...) Furthermore, logicians always deal with the idealized version of a statement. I will try to find where I read that recently. If the tokens of the type of statement that it is do not properly reflect the idea, then it is not a token of that type of thing. It's the token that's wrong, not the idea. Logicians act accordingly. The following is false:

"In each of these settings, a theorem is an abstract expression of a proposition that can be logically deduced; the difference between the settings is, essentially, whether the statement is in natural language or in a formal language."

Actually, in each of those settings, we are talking about only the tokens, not the type. Therefore those are not "abstract expressions" (a term from somewhere?? or OR), those are the concrete ones. You identify a difference in culture, however certainly it makes no difference as far as the theorem is concerned whether it is in natural, or formal language. Pontiff Greg Bard (talk) 23:29, 16 May 2008 (UTC)

It's not clear to me that the theorems are not themselves tokens, that is, expressions in a natural or formal language. What they are tokens of is a question that I cannot answer.

I disagree that it makes no difference whether a theorem is expressed in a natural or formal language. A formal statement of a mathematical theorem can be said to formalize the theorem, or to represent the theorem, but is not itself the same as the natural language theorem. At least, that is a reasonably common viewpoint, in my estimation; I am not claiming it is a unanimous opinion. I am certain is it more common for texts to say that a natural language theorem can be turned into a formalized one than for them to say that a natural language theorem is the same as the corresponding formal theorem.

I think it's your turn to provide some sources, since I have found four above that seem to simply identify a theorem with a statement. Is there any source that describes a theorem as a type with various tokens? Note that, even if there is such a source, I'm not convinced that we should discuss that here, since it may be a very nonstandard text, or just mention it in passing. But if there isn't any mainstream text that mentions type/token issues in the context of theorem, that would be a sign that we don't need to do so in this article. — Carl (CBM · talk) 00:27, 17 May 2008 (UTC)

Obviously, there must not be an easy verification of the type/token claims, which means that most likely it's trivial information or wrong information. Either way, shouldn't be in the article. Tparameter (talk) 04:29, 17 May 2008 (UTC)

Where did you get these ideas about what is trivial, wrong, or obvious. 0 for 3 mister. The fact is, if you must know, that I have limited time and resources. I think I have explained it all sufficiently. Philosophers of language study theorems because they are statements that tell us about truths of the world. Logicians care about the type-token distinction because failure to care may result in ambiguity. This is all straightforward. I think you had compared apples to theorems at one point. I'm sorry, but there really is much for you guys to learn here. Pontiff Greg Bard (talk) 10:30, 17 May 2008 (UTC)

(←) One of the reasons I enjoy contributing to WP is because I learn a lot of things from other people. There is much more that I don't know than I do know. On the other hand, there are a few areas I am very familiar with, particularly within mathematical logic.

If you asked me for an off-the-cuff, unresearched opinion, I would agree that there is a universal called theoremhood of which individual theorems are tokens. I would also agree that there is a universal called "Fermat's-last-theoremhood" that includes all theorems that are different expressions of the underlying idea of Fermat's last theorem. But Fermat's-last-theoremhood is not itself a theorem, it's a type of theorems (it corresponds to something like red-chairhood vis-a-vis chairhood).

The claim I believe you are making is different - that Fermat's-last-theoremhood is the theorem, and that the things I call theorems are merely tokens of it. In the way I ordinarily use the language, the theorem itself is a token, not a type. (In a particular sense; of course each theorem has a corresponding type of similar theorems, just as Fermat's last theorem has Fermat's-last-theoremhood).

So my doubt that the type/token distinction is relevant in the context of theorems comes both from my own mental analysis of what's going on, and from my doubt that there are texts in mathematical logic or mathematics that discuss theorems in those terms. I can certainly be wrong, but the best way to demonstrate that is with some sources to back up your position. — Carl (CBM · talk) 12:21, 17 May 2008 (UTC)

Hi guys, can I join in? Not to be outdone quotation-wise:-

• [in the language L] "A sentence Φ is a theorem in logic (or theorem, for short) if an only if is derivable from the empty set of sentences." (Mates, 1972, p. 127)
• [in the axiomatic system for L1] "A sentence Φ is a theorem if an only if Φ is the last line of a proof. We shall write
  |- Φ
  as an abbreviation for
  all closures of Φ are theorems" (Mates, ibid, p. 166)

It is clear that the term theorem in the above two quotations is referring to a sentence in a language and as such a theorem is, being a sentence, a string of symbols. To allow the possibility that there are other meaning of the (English) word theorem lets give it the name theorem1, so we will say a theorem1 is a string of symbols which &c. We would then say e.g. that Fa is a string of symbols which is a sentence (in some languages) but is not a theorem1 (in a consistent theory) but Fa v ~Fa is a string of symbols, and is a sentence and is a theorem &c. And so is (x)(Fx v ~Fx) In other words in one sense of theorem, i.e. theorem1, a theorem is just string of symbols (marks on paper, verbal utterances &c., and Fa v ~Fa and (x)(Fx v ~Fx) are examples of same.

Does the term, or could the term theorem have any other meaning? I turn to my copy of Euclid, The Thirteen books of the elements Dover 1956, in which Proclus (ed. Friedein) is quoted p. 124 as distinguishing between problems and theorems the latter exhibiting the essential attributes of the generation, division &c. of figures and again, page 126 "...when any one enunciates that' [emphasis added] ‘‘In isosceles triangles the angles at the base are equal we must say that he enunciates a theorem." The words in isosceles triangles the angles at the base are equal is a string of symbols. On my reading it is not the string of symbols that is being describes as a theorem but what is asserted’’’ when the words are used to make an enunciation. On that reading the theorem that isosceles triangles the angles at the base are equal for which Euclid provides a proof (Book 1, proposition 5) can be enunciated by the string of symbols isosceles triangles the angles at the base are equal but IS not the string of symbols isosceles triangles the angles at the base are equal. To suggest that the theorem proved by Euclid was the string of symbols in isosceles triangles the angles at the base are equal would be to suggest that Euclid wrote or uttered that string of symbols meaning then in the sense understood in English speaking countries several centuries later. Lets us call that sense of the word theorem theorem2. Far be if for me to say that Proclus was right to think that there is something apart from the strings of symbols he read, wrote, heard and said, something which he calls a theroem2, and which he believes Euclid proved. Nevertheless he does, rightly or wrongly, appear to use the term theorem in a different sense, a sense which does NOT denote a string of symbols.

I put it to you that:

Hans is saying that

a theorem1 is a string of symbols, "tokens, that is, expressions in a natural or formal language"

and

Gregbard is saying that

a theorem2 is "an abstract object" and, if I understand him, a theorem2 is an "idealized version of" a corresponding theorem1.

Perhaps Gregbard would say that the sense of a theorem1 is a theorem2, that many theorems1 (e.g. isosceles triangles the angles at the base are equal and the same translated into German, or written in the original Greek) have the same theorem2 as their sense. Being theorems, i.e. proven to be true, the reference (bedeuten) are the same, the True.

I may be quite wrong in surmising what Hans and Greg mean by the word theorem. If they mean the same thing then they definitely disagree about its status and proprieties. If they mean different things by the word then it is not obvious that there is any disagreement of that kind at all. I suggest both parties define just what they DO mean by the word "theorem" before they continue their debate about. --Philogo 03:34, 18 May 2008 (UTC)

I think you have mistaken me (Carl) with Hans. There is the same issue here as in many areas of mathematics: two statements can be intensionally different theorems (as in the example of translation from Greek to English you provided), but still be equivalent in some way. Similarly, two knots can be intensionally different (tied according to different instructions, say) but be equivalent ("the same") knots. In the case of knots, there is a precise definition in knot theory of when two knots are equivalent. In the case of theorems, there is no good definition I know of that expresses when two theorems express the same abstract assertion.

My goal in presenting the references above was to demonstrate the common viewpoint that a theorem is a statement, sentence, or proposition. The difference between a theorem qua statement and theorem qua proposition is not particularly germane in most areas of mathematics, so there is little written about it in mathematics texts as far as I can tell.

My specific concerns with text I have removed from the article are:

• I don't agree that a statement in a formal language, and a different statement in a natural language, should be described as tokens of the same "theorem". This is a way people could look at things, but I don't know of any evidence that this way of looking at it is common in the literature. Rather, they would say that each of the two statements is a theorem (compare Enderton's quote above), but that they express the same idea.
• Upon reflection, I believe I identify a theorem with a token, not with a type. I don't believe there is a well-established word for the type of things of which theorems are tokens (for example, there are many seemingly independent statements equivalent to the Riemann hypothesis. Do these express the "same" theorem as the Riemann hypothesis does? In many cases the equivalence is very subtle, and the statements were not intended to be equivalent.)

I don't see that these issues warrant discussion in the article, because they seem to be mostly ignored in the literature. But I am open to being proved wrong. — Carl (CBM · talk) 12:17, 18 May 2008 (UTC)

What you are describing is a formulation. There are many formulations of the RH, however we still say that they are "of the RH." We don't call each formulation a new hypothesis. People in their vernacular, call a theorem the chalk on the board, and it works pragmatically just fine. It's serviceable. However, strickly speaking there is a distinction. I am not so interested in saying the abstract part is the theorem and the chalk is merely a token of it. They are both referred to correctly as the theorem just fine. There is a sense in which the theorem of chalk on the board is a theorem, and then there is the sense in which it is a type of abstract object. However, as I stated earlier, that logicians always deal with the idealized version. This is what we are talking about that someone asserted this theorem. Nobody thinks that logicians assert the chalk on the board. However, there is also a sense in which a theorem written in yellow chalk is a "yellow theorem." If one made such a claim, they would need to address the type token distinction in order to be clear. I believe it is a WP goal to cover all senses of a term. Under that justification alone, this distinction belongs in.

Also, it does not matter if, for instance, Fred believes that theorems "float in the air," and Joe believes that a theorem reflects a "structure in God's brain," etc. These metaphysical issues have nothing to do with what I am talking about. Logicians with such beliefs can perfectly well work together on the same theorems just fine. So if Carl believes in "theoremhood" that is fine. However, the type-token distinction is intended to follow reason (its not metaphysics). In that regard it is not merely a matter of looking at it differently as a "way of understanding." The point is to identify a legitimate distinction, so therefore it is not a POV issue in the article. Pontiff Greg Bard (talk) 13:28, 18 May 2008 (UTC)

As I said above, I think the burden is on you here to provide some sort of published source to show that not only are your ideas plausible (of course they are plausible), but that they are of sufficient interest to the mathematics or logic community to include here. As I also pointed out, it seems to me that theorems are tokens, not types, so in particular I would appreciate any source to the contrary, so that I can see the argument that is being presented there. My guess is that such a source would be using the word theorem in a different sense than mathematicians do. — Carl (CBM · talk) 13:46, 18 May 2008 (UTC)