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 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
- 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.
- 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”.