# Theories, Theorems, Lemmas, and Corollaries

I've been getting so many requests for "basics" posts that I'm having trouble keeping up! There are so many basic things in math that non-mathematicians are confused about. I'm doing my best to keep up: if you've requested a "basics" topic and I haven't gotten around to it, rest assured, I'm doing my best, and I will get to it eventually!

One of the things that multiple people have written to be about is confusion about what a mathematician means by a theory; and what the difference is between a theory and a theorem?

Math folks do use the term "theory" in a very different way than most scientists. For a good explanation of the scientific use of the word theory, check out John's post at Evolving Thoughts. For mathematicians, the word "theory" has come to mean something more like "independent field of study" - set theory is the study of mathematics starting from the basic idea of sets; category theory is the study of the basic idea of function or mapping; homology theory is the study of mathematical structures in terms of chain complexes, and so on. Each independent sub-branch of mathematics is called a mathematical theory.

A theorem is a statement which is proven by valid logical inference within a mathematical theory from the fundamental axioms of that theory. So, for example, the pythagorean theorem is a proven statement within the mathematical theory of geometry: given the basic axioms of euclidean geometry, you can prove the pythagorean theorem. In some sense, a theorem is similar to what most scientists call a theory, although it's not the same thing. A scientific theory is a statement inferred from a set of observations or facts, and which is consistent with all of the observations related to whatever phenomenon the theory explains. But a theory is never proven to be true: a new observation can always invalidate a scientific theory. A theorem is a statement which is proven from known facts: if it's really a theorem, then it's a theorem forever: no "new facts" can come along and cause a theorem to become invalid. In Euclidean geometry, the square of the length of the hypotenuse of a right triangle will always be the sum of the squares of the other two sides. No new observation can ever change that: within the realm of Euclidean geometry, that is an absolute, unchangeable fact: it's a theorem.

If you read math, you'll also see references to a bunch of terms related to the idea of
a theorem:

Lemma
A lemma is really just another word for a theorem. The idea behind the distinction
is that a lemma is a proven statement which is not interesting in and of itself, but which
is proven as a step in the proof of some more interesting statement. In a proof of a complex
theorem, we often break it down into steps - smaller theorems which can be combined to prove the
complex theorem. When those smaller theorems don't have any particular interest to the author
of the proof except as stepping stones towards the proof the main theorem, they're called
lemmas.
Corollary
A Corollary is a theorem which so obviously follows from the truth of some other theorem that
it doesn't require a proof of its own. Corollaries come up in two main contexts in math. First,
given a complicated theorem, it's often helpful for readers to understand what the theorem
means by showing several corollaries of the theorem that are concrete enough to be
easily understood. By understanding those corollaries, the reader gains insight into the
meaning of the theorem from which they derive. Secondly, often when we want to prove some
specific statement, it turns out to be easier to prove a more generic statement, and then
show that the specific statement obviously follows from the more general. For example, if
I wanted to prove that a triangle with sides of lengths 3, 4, and 5 is a right triangle,
I'd just point at the proof of the pythagorean theorem, and then say that since
32+42=52, the fact that it's a right triangle is
a simple corollary of the Pythogorean theorem.
Proposition
This is a nasty one, because I've seen it used in two very different ways, and I've yet to figure out any community/subject area cue to use for figuring out which meaning a given writer
is using. In some contexts, proposition means "a basic, fundamental statement for which no proof needs to be presented." (That is, it's a provable statement, but the authors believe that it isn't necessary to present the proof, either because the proof is so trivially simple that you should be able to immediately see how you'd prove it; or because it's something so fundamental to the subject area that all of the readers must have already seen the proof.) The other meaning that I've seen used for proposition is "A statement which is being put forward as something to be proven or disproven", as in "Consider the proposition that X", which leads into either a proof or disproof of "X".
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For those of you who read my comment in the "Axioms" basics post, a theorem is a statement that is true in all models of the axiom system. This is equivalent to what Mark says; the equivalence follows from the properties of soundness and completeness.

I would suggest that Mark do a basics on these, but I wouldn't consider these "basic".

As for the pecking order of lemma/corollary/proposition, I have nothing to add; I agree with what Mark has. I typically use the first instance of proposition. I use conjecture for the second usage.

In practice, a proposition tends to be less momentous than a theorem but more generally useful than a lemma. When there are some things called "theorems" and some called "propositions", the theorems tend to be natural stopping points.

Think of it like composing a musical track. The propositions are the patterns, made up of licks, riffs, and fills. Some of those riffs are interesting enough in their own right to be considered separately -- the lemmas. A series of patterns builds up and culminates in the overall track -- a theorem. I think a good track to use is Brainbug's "Nightmare" (Sinister Strings mix).

If I were writing the Elements, for example, I would call I.18 ("In any triangle the angle opposite the greater side is greater") a proposition. I would also call I.16 ("In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles") a proposition, and I.17 ("In any triangle the sum of any two angles is less than two right angles") its corollary. The constructions like I.1 ("To construct an equilateral triangle on a given finite straight line") are clearly lemmas. Everything meshes together, culminating in I.47, the Pythagorean Theorem, and I.48 its corollary-coda.

MarkCC wrote:

For example, if I wanted to prove that a triangle with sides of lengths 3, 4, and 5 is a right triangle, I'd just point at the proof of the pythagorean theorem, and then say that since 32+42=52, the fact that it's a right triangle is a simple corollary of the Pythogorean theorem.

Isn't this a simple corollary of the converse of the Pythagorean theorem? The theorem itself (Euclid 1:47) goes from "right triangle" to "sum of squares equals square of hypotenuse"; to prove the statement quoted above, you have to go in the opposite direction (Euclid 1:48).

Thanks for addressing the request (I was one of those asking for some mention of "theory").

And, in the pecking order, don't forget the lowly "Claim" (for the general public: claims are usually something like "sub-theorems" inside of long proofs; they get their own proof and end-of-proof marker, and help to put some order in the long argument).

But the converse of the Pythagorean Theorem is almost a trivial corollary of the theorem itself. The only additional element needed is SSS.

Also, in the traditional translation of Euclid's terminology, "proposition" means the same thing as "theorem."

There really is no uniformity in the use of the word "Proposition". When I was teaching, I would often use it for a statement worthy of setting by itself in a heading before proving it; but I also used it in casual but careful speech to refer to any statement that was grammatical enough to have a truth value, whether true or false. So, the proposition that the Moon is made of green cheese has been known to be false since July 1969.

Excellent, and clear! Funny, we never hear a President or a Creationist dismiss something as "just a theorem." Mark has helped to explain why.

But where the Art of Mathematics begins is somehow beyond all that (which is part of what Godel was getting at).

"... a very great deal more truth can become known than can be proven."
Richard P. Feynman - Nobel Lecture
nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html

One common way that I've seen "proposition" used is as an adjunct to a definition. For example, if you were introducing group theory, you would first define a group (in terms of its operations and axioms), and then list a bunch of propositions, which are mini-theorems that are straightforwardly proven from the definition, and then can effectively be used as part of the definition from then on.

By Pseudonym (not verified) on 13 Mar 2007 #permalink

I agree that there is no uniform usage of "Proposition" in the math literature. I tend to use it or "Fact" for important theorems from other sources which I am stating and using, but not proving my own paper. I have also seen it for everything else under the sun: things that should be lemmas, theorems, or corollaries. I am pretty sure I have even seen the term used for conjectures.

I think "Proposition" exists for long math papers, when the writer gets tired of everything being a "Theorem." I think I threw in a Proposition or two in my thesis for precisely that reason.

In my university undergraduate maths course, "proposition" tended to be used for a more routine theorem. "Theorem" tended to be reserved for something a bit more spectacular. It's all too far back in the past for me to cite examples. ;-)

Either could be preceded by "lemmas" or followed by "corollaries".

But the proportion would maybe be three or four propositions to each theorem. You'd encounter one or two theorems in each lecture, and the rest would be propositions. (Or so I was told: my lecture attendance record wasn't all that great...)

Jonathan, here in Italy politicians use the word "Teorema" with the meaning of "something which is built up by enemies of mine, without any evidence". That's really sad.

(this idea of writing about basics is great! I just started something similar on my own blog, of course in Italian)

Everytime I've asked a mathematician about the meaning of "theorem," I've gotten an answer like your "a statement which is proven by valid logical inference within a mathematical theory from the fundamental axioms of that theory." That's fine and understood, but it implies that 23+44=67 is a theorem since it is a statement that can be proven from the axioms of the number system. In the normal usage of mathematics, it's obvious that a provable result has to be central or important or noteworthy in some other sense in order to count as a theorem. Has anybody tried to formalize or at least thematize what it takes to count as a theorem? Would such a criterion be part of mathematics, or part of something else such as metamathematics, the philosophy of mathematics, or maybe the ethnography of the mathematical tribe?

"23+44=67 is a theorem" -- yes, it is.

Dr. Eric W. Weisstein's definition at MathWorld.com (and see the references he gives):

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.

Although not absolutely standard, the Greeks distinguished between "problems" (roughly, the construction of various figures) and "theorems" (establishing the properties of said figures; Heath 1956, pp. 252, 262, and 264).

According to the Nobel Prize-winning physicist Richard Feynman (1985), any theorem, no matter how difficult to prove in the first place, is viewed as "trivial" by mathematicians once it has been proven. Therefore, there are exactly two types of mathematical objects: trivial ones, and those which have not yet been proven.

The late mathematician P. Erdos has often been associated with the observation that "a mathematician is a machine for converting coffee into theorems" (e.g., Hoffman 1998, p. 7). However, this characterization appears to be due to his friend, Alfred RÃ©nyi (MacTutor, Malkevitch). This thought was developed further by Erdos' friend and Hungarian mathematician Paul TurÃ¡n), who suggested that weak coffee was suitable "only for lemmas" (MacTutor, Malkevitch).

R. Graham has estimated that upwards of 250000 mathematical theorems are published each year (Hoffman 1998, p. 204).

A101273 Theorems from propositional calculus, translated into decimal digits.
http://www.research.att.com/~njas/sequences/A101273

and one of my responses (which has a few errors and needs to be updated):
A100200 Decimal Goedelization of antitheorems from propositional calculus, in Richard Schroeppel's metatheory of A101273.
http://www.research.att.com/~njas/sequences/A100200

For mathematicians, the word "theory" has come to mean something more like "independent field of study"

Personally I still see the parallel here. In science theories are connected knowledge justified by observations and in math they are justified in the theorems.

Personally I still see the parallel here. In science theories are connected knowledge justified by observations and in math they are justified in the theorems.

Sure, the usage of the word in both cases is not totally unrelated (that would be a surprising case of linguistic divergence). The important points, though, are the differences. Math theories are defined by their objects; in science, you can have two or three theories dealing with the same objects and data, and giving alternative explanations for them. On the other hand, a scientific theory may be shown to be wrong; in math, the worst that can happen to a theory is that it becomes obsolete or dÃ©modÃ©. And math theories are not expected to be predictive, just useful (or fun :-) ). Etc.

Trying to ask a mathematician what makes a provable proposition count as a theorem is like trying to change the subject with a bunch of dolphins who just want to go on talking about how much they like to eat fish. I have heard the line about turning coffee into theorems decades ago and I know the Feynman bit as well. I don't need to hear about proofs and axioms. I want to ask a different question: what is it about a result that makes it significant enough to qualify as a theorem since in practice propositions such as "23+44=67," are not counted as theorems even though they qualify according to the formal definition. If such results did count as theorems, after all, it would be mighty easy to get a publication in the math business.

Before the announcement of the Fermat proof, I had the recurrent fantasy of finding the proof myself and publishing it as a lemma in a paper entitled "A new proof that the cube of 3 plus the cube of 4 doesn't equal the cube of 5."

I just want to give my own answer to Jim's question above, from what I know personally.

A statement such as "23+44=67" is a theorem, and one could certainly prove it in some theory such as Peano Arithmetic. However, no-one would attempt to publish such a theorem. The criteria for publication is, roughly, how interesting the result is to the referees. So "23+44=67" wouldn't be published on the grounds that everyone already knows it and so it isn't interesting, not on the grounds that it isn't a theorem.

As far as I'm concerned, Lemma, Proposition, Corollary, etc. are all just synonyms for Theorem, but are used as relative importance indicators within some particular piece of mathematical writing, like a paper, book, course, etc.

That is, Lemmas, Propositions, Corollaries, Results, Claims and anything else are all Theorems in their own right in the widest picture, but are frequently viewed as less important in a particular context, say a book on Group Theory, or a paper proving the Four Colour Theorem, or whatever.

I'm not sure after all that I've explained myself terribly well, but hopefully someone might get what I'm trying to say.

By Joseph Cooper (not verified) on 14 Mar 2007 #permalink

I don't think my question is about terminology. It's about the criterion that mathematicians use, consciously or unconsciously, to rate a result seriously enough to call it a theorem, to matter in mathematics. The question is parallel to one that literary theorists sometimes ask about what makes a sequence of narrated events a plot. Telling how one damn thing happened after another is a necessary but apparently not sufficient condition.

I assume some of you guys referee papers. What makes you decide a result is important enough to publish? It may well be that there is no interesting answer to my question. I'm not trying to be a pest about it, though; and I promise to go away after you tell me again that a theorem is a result that follows validly from premises, etc. and I realize that the question is inaudible to mathematicians.

Math theories are defined by their objects; in science, you can have two or three theories dealing with the same objects and data, and giving alternative explanations for them.

I think this happens in math too. For example, manifolds have topology, metrics, geometry, differentiability, et cetera.

On the other hand, a scientific theory may be shown to be wrong;

Again, I don't see a qualitative difference.

Theorems, even fundamental ones, can be found wrong later. Scientific theories may have areas of applicability and be superseded but perhaps not abandoned. (For example, different theories of gravitation.) Some non-predictive model physics aren't only useful to learn methods in the full theories, they also raise fun discussions. Etc. :-)

Obviously, taste differs.

Jim:

The criteria for deciding whether to call something a theorem, a lemma, a corollary, etc., are really purely subjective. Technically, they're all theorems. But for clarity of exposition, we use a variety of terms to differentiate
between the different theorems that come up.

In general, when writing about some topic, there's some central focus of the writing. Things that are major points that demonstrate an important fact about the central focus, and which (for the purposes of the text) requires an independent proof, we call theorems. Things that we only introduce because they're useful steps in some proof, but which don't demonstrate a central, important fact in the context of the text get called lemmas; etc.

It's not purely a matter of "Statement X is profound, so it's a theorem, but statement Y is obvious, so it isn't". It's contextual - something which is a theorem in one text can be a lemma, or a proposition, or even an un-named statement in another text. Specifically naming something a theorem is a statement that *for that text*, it's illustrating something important.

I'd give some examples, but too much of my stuff is packed in boxes at the moment. I know this is vague, but there's a lot of overlap between algebraic topology and category theory; and when I was writing the category theory posts, I remember that there are some things that are theorems with elaborate proofs in one of my topology texts which didn't even rate as lemmas in my category theory book.

So, to back to an example from earlier in the thread: "23+44=67" is a theorem. But in most texts, it's not one of central importance. So it would almost never be labelled as a theorem in a text, because for most texts, the result of an arbitrary simple addition just isn't a demonstration of anything fundamental.

But you can find places where similar things do rate as named theorems. In Russell and Whitehead's principia, the theorem "1+1=2" does end up rating as a first-class named theorem - they take several hundred pages to work up to the point where they can prove that.

Torbjön:

The qualitative difference between a mathematical theorem and a scientific theory is that a theory can be wrong. A theory is derived from observations, but further observations can show that it's completely wrong, and it gets discarded as false. A theorem is always true - it may fall out of favor, so that it isn't widely used, because it's easier to use some other theorem or approach in proofs - but the theorem can never be invalidated if it is, in fact, a theorem.

Mark: What if we believe something to be a theorem, then later discover an error in the proof - possibly even one profound enough to enable a proof of the negation of the claim in the supposed theorem?

By Michael Ralston (not verified) on 15 Mar 2007 #permalink

Michael:

Then it was never a theorem in the first place. A theorem is, by definition, a statement which is provably true via valid inference. If there's an error in the proof, then we may have thought that it was a theorem, but we were wrong - because it didn't have a valid proof. If the proof is correct, then you can't "untheorem" or invalidate it.

It's one of the ways that math is sort of cleaner than physical/applied sciences: math has genuine proofs; most physical sciences have inferences from observations, but it's rare that you have anything like a conclusive mathematical proof. Most sciences can make "proofs" based on inferences from interpretations of observations - and that's where the uncertainty comes in. A scientific theory always ultimately rests on observations and interpretations of observations - if those turn out be wrong, or even just incomplete, it can utterly destroy the validity of the proof of the theory. But in math, we start from axioms, and given a set of axioms and a set of inference rules, if you generate a proof today, that proof will be valid forever.

Which leads to the other big difference between scientific theories and mathematical theorems. A mathematical theorem is dependent on some set of axioms. The axioms and acceptable inference rulus are chosen by the mathematician for some domain. A statement that is a theorem under one set of axioms and inference rules is not necessarily a theorem under some other one. There are plenty of cases of things that are theorems under classical first order predicate logic, but which are not theorems under first order intuitionistic logic.

A fascinating historico-philosophical essay explores what is meant by Definitions, Theorems, and Proofs on p. 10:

http://arxiv.org/PS_cache/math/pdf/0703/0703427.pdf

Date: Wed, 14 Mar 2007 16:17:56 GMT (32kb)

Mathematical knowledge: internal, social and cultural aspects
Authors: Yu. I. Manin
Comments: Commissioned for vol. 2 of ``Mathematics and Culture'', ed. by C. Bartocci and P. Odifreddi
Subj-class: History and Overview; Algebraic Geometry

"I discuss some general aspects of the creation, interpretation, and reception of mathematics as a part of civilization and culture."

He gets to Shannon and Kolmogorov on p.20, and ends with von Neumann and Hiroshima.

but the theorem can never be invalidated if it is, in fact, a theorem.

I appreciate that. But it can be initially wrong before someone finds the problem. Witness Wiles work on Fermat-Wiles theorem, or the problems with older theories of differentials.

In principle, it can even be by observation. (From calculation a disproof, by computer for example.) But these similarities are rather tenuous or even specious.

most physical sciences have inferences from observations, but it's rare that you have anything like a conclusive mathematical proof

Ah, but I use testing. Once a theory has been tested on its predictions, it would also be valid forever for its areas of application. It can be superseded, but will remain valid and perhaps useful. Witness classical mechanics.

Again, these similarities are rather tenuous or even specious.