Sunday, August 27, 2023

Theorems and Lemmas and Proofs, Oh My!

I was recently asked by a non-mathematician about the difference between the terms Theorem, Lemma, etc. My first reaction was I probably have a blog post on that. Actually, I looked and I don't seem to. Since I have, according to Ken Reagan, over 1000 posts (see here and here) I can easily confuse things I meant to write a post on with things I wrote a post on. My next thought was Lance probably has a post on that. I asked him, and he also thought he had, but also had not. So now I will!

Open Question: A well defined question that you don't know the answer to and may not even have a guess as to which way it goes. The above is not quite right: sometimes an open question is not that well defined (e.g., Hilbert's 6th problem: Makes Physics Rigorous) and sometimes you have some idea, or a rooting interest, in how it goes. I tried to find some open questions in Mathematics where people in the know don't have a consensus opinion.  I can think of two off hand: Is Graph Isom in P? and is Factoring in P. Maybe the Unique Game Conjecture, though I think people in the know think it's true. Here is a website of open questions, but I think for all of them people in the know  think we know how they go: here.

Conjecture: A statement that you think is true, and may even have some evidence that its true, but have not proven yet. I am used to using this term in math, and hence I hope someone will PROVE the conjecture. Are there conjectures in empirical sciences? If so, then how do they finally decide it's true? Also note- I blogged about conjectures and how once they are proven the conjecturer is forgotten here. EXAMPLES OF CONJECTURES: The same link as in open problems above. 

True story but I will leave out names: There was a conjecture which I will call B's Conjecture. C & S solved it AS WRITTEN but clearly NOT AS INTENDED.  Even so, C & S got a published paper out of it. This paper made M so mad that he wrote a GREAT paper that solved the conjecture as intended (and in the opposite direction). That paper also got published. So one conjecture lead to two opposite solutions and two papers. 

Wild-Ass Guess: You can take a wild-ass guess what this is. 

Hypothesis: An assumption that you may not think is true but are curious what may be derived from it. The Continuum Hypothesis is one. For some reason Riemann's problem is called The Riemann Hypothesis even thought it's really a conjecture.  So is my notion of Hypothesis wrong? In any case, if you know other things that are called Hypothesis then please leave a comment. 

Lemma: A statement that is proven but only of interest in the service of proving a theorem. There are exceptions where a Lemma ends up being very important, see here. The word is also used in English, see here, but I've never heard of the word being used that way.

Theorem: A statement that has been proven. Usually it is somewhat general. There are a few exceptions: Fermat's Last Theorem was called that before it was a Theorem. If you know other things that were called theorems but weren't,  please comment.  EXAMPLES OF THEOREMS: The Fundamental Theorem of X (fill in the X), Ramsey's Theorem, VDW's theorem, Cook-Levin Theorem, The Governor's theorem (see here). There are many more theorems that have names and many that do not. 

Corollary: A statement that follows directly from a Theorem. Perhaps an interesting subcase of a Theorem. Often this is what you really care about. When trying to find a famous corollary I instead found The Roosevelt Corollary to the Monroe Doctrine,  Corollaries of the Pythagorean theorem, and Uses of the word Corollary in English. Are there any famous corollaries in mathematics that have names?

Claim: I do the following though I do not know if its common: During a proof I have something that I need for it, but it is  tied-to-the-proof-of-the-theorem so it would be hard to make a lemma. So I prove it inside the proof of the theorem and call it a claim. I use Claim, Proof of Claim, End of Proof of Claim to delimit it. 

Porism: A statement that you can get from a theorem by a minor adjustment of the proof. I've also heard the phrase Corollary of the proof of Theorem X. I first saw this in Jefferey Hirst's Phd Thesis which is here, on Reverse Mathematics. I liked the notion so much I've used it a few times. It does not seem to have caught on; however,  there is a Wikipedia entry for the term here which also gives two examples of its use, which are not from Hirst's  thesis or my papers. 

Proposition: I see this so rarely that I looked up what the difference is between a Proposition and a Theorem. From what I read a Proposition is either of lesser importance, or is easy enough to not need to give a prove, as opposed to a Theorem which is important and needs a proof.

Axiom: A statement that one assume is true and usually they are self-evident and true. Exceptions are The Axiom of Choice which some people reject since it is non-constructive. Also some people do not thing The Axiom of Determinacy is self-evident. Same for Large Cardinal Axioms. But really, most axioms are self-evident. Note that all branches of math use Axioms.

Postulate: Euclid used the term Postulate instead of Axiom. Actually, Euclid wrote in Ancient Greek so to say he used the term Postulate is probably not quite right. However, the term Postulate seems to mean an axiom of Euclid's, or perhaps an axiom in Geometry. One exception: Bertrand's Postulate which was a conjecture but is now a theorem. The link is to a math-stacks where there is some explanation for the weird name. 

Paradox:  A Paradox is a statement that is  paradoxical. Hmmm. that last sentence is self-referential, so its not enlightening. A paradox is supposed to be a statement that seems absurd or self contradictory, though under closer examination may make sense. Russell's Paradox shows that Frege's definition of a set does not work. The Monty Hall paradox, and the Banach-Tarski Paradox are just theorems that at first glance seem to be absurd. The Monty Hall Paradox is not absurd. Darling thinks the BT-paradox means that math is broken, see this post here for more on that.


  1. You left out Observation: a lemma with a very easy proof

  2. There is also "thesis" as in "Church's thesis". Is there another such use of "thesis"?

  3. (Bill G) How would define `Thesis' in this context? A statement that most people accept?

    1. In this context, it is something we (e.g., the community) are going to accept, but we can't prove it because it refers to something that does not have a precise definition.

      I don't know if I'd include self-evident as being a usual property of an axiom. Axioms are things that we assume are true, so we don't have to list them in the hypotheses of a theorem. Some are more self-evident than others.

  4. A "thesis," as in the "Church-Turing Thesis," is a statement which most people in the community agree to (have to) accept, for else they are considered part of some other community. Even anachronistically, then, Alan Turing arguably did not belong to the computer science community as we have come to understand it:

  5. Since you're asking for other hypotheses, ETH (Exponential Time Hypothesis) would be one :)

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  7. There are also laws: for example the law of large numbers and Muphry's Law.

  8. Also missing is "definition", this may sound self evident but I actually once had a student ask me what the difference between a theorem and a definition is.

    The two can get quite close two each other, for example:

    A function $f(x)$ is continuous if for all $a$ in it's domain $\lim_{x \to a} f(x) = f(a)$.

    All polynomials are continuous.

    In the student's mind, both cases defined something to be "continuous" and so he wondered why I called one a definition and the other a theorem.

    1. Yes! Many non-mathematicians don't know what a definition is (even though the math meaning is one of the English meanings).

  9. It will be nice if you show an example how to use these mathematical terms. For example, mathematical article written without those terms (see Exact solution for the nonlinear pendulum
    December 2006Revista de ensino de física 29(4):645-648
    DOI: 10.1590/S1806-11172007000400024) and how it would be written with mathematical language. I am still confused with the term theorem. To my understanding, If I want to develop a solution to the differential equation (a proposition) , I write axioms or lemmas representing formulas I will be using when doing proof. Then make a claim of my solution and develop proof using these axioms. In results, I write this claim as theorem which I proved previously. Is that a proper way of doing it? P.S. - I have never seen an article with the term " Wild Ass Guess".