What are the requirements of a mathematical theorem?

Question

Most important, I would be interested on the fact that the theorem must be expressed with the least amount of words or requirements in order to be "elegant". I dont know how to describe, but I know it has a name :D

PS. Tags might not be correct, I accept suggestions.

Edit: I will try to explain what I mean. It is agreed that when defining a new concept, there are 2 parts involved: the base term and the specifics. For example: a square is a rectangle where all sides are equals. The base term is rectangle, the specifics are "all sides equal". Then you can move further with "what is a rectangle", "what is a parallelogram" etc.

What is important here is that When defining a term, you try to (must?) take the closest possible term and add the minimum number of details. So a square is not defined as a quadrilateral with only right angles and all sides equal. This definition would not be "elegant", or optimal. Or .

For theorem, it is somewhat similar. Some examples:

• Instead of "If a number is a multiple of 10, then it ends in 0; and conversely, if a number ends in 0, then it is a multiple of 10", it is customary to say "A number is a multiple of 10 if and only if it ends in 0".
• The Pythagorean theorem is very simple: "the square of the hypotenuse is equal to the sum of the squares of the other two sides". But it could be made more complex, like "in a right triangle, the square of the side opposing the right angle is equal to the sum of the square of the other 2 sides", and here is already some redundancy. If we take the all know image with 4 right triangle and a square in the middle, then the mathematical formula would be (a+b)^2 - 4*a*b/2 = c^2. However, it is simplified to a^2+b^2=c^2.

What I highlight here is that there's a willing of the author to formulate the theorem as simple and "elegant" as possible.

Answer 1

A mathematical result, in order to be considered 'okay' by the mathematical community, only needs to meet three criteria:

1. It's correct. Well, obviously, the result needs to be right. By that, I mean it cannot contain any inexplicable leaps of faith or any gaps in the logic. Any false conclusions in the proof can lead to its downfall, but in some cases can still be remedied (for example, Andrew Wiles' proof of Fermat's Last Theorem had a flaw in the first version, but was eventually fixed after a year of hard work). Anyhow, if you publish a result, other mathematicians are bound to referee your paper, and they need to be able to ensure that the result you have is perfectly flawless.
2. It's readable. The key here is that a (or a few) mathematicians needs to be able to read your paper, understand your proof, and then judge whether it is correct. If your results are correct but hidden behind a wall of weirdly phrased language, or with obscure mathematical symbols you made up just for your paper, or 1000+ pages of tough work, then it is almost confirmed that no one will be able to understand you. Then you will have failed to produce a readable and valid proof of your result, even if your idea is right. After all, communication is key in math. An example is Mochizuki's supposed proof of the ABC Conjecture, which is hidden behind thousands of pages of work in a brand new field of math Mochizuki claims to have invented. (Though he might be wrong after all; see here.)
3. It's useful. Imagine that I wrote a paper which defined the notation $\{\dots\}_\Omega$ to mean an omega set, but the definition of an omega set is just identical to that of a set, but its written in a fancy way. Sure it is 'correct': I most certainly did not make any logical errors in the definition. Sure it is readable, but this definition is clearly utterly useless. Now I can prove that all omega sets are sets and vice versa, and that the real numbers are an omega set, but all of these results are useless and I most certainly won't be able to publish them. This is of course an extreme example, but the moral of the story is that any proof or result you give must be somehow related to some other field which people are already interested in. It must make a valuable contribution to understanding a certain field, or solving a certain problem. Otherwise, the proof might as well not exist. See this answer on Academia.SE.

Now, for 'smaller' proofs, there are also a bunch of smaller stylistic concerns that mathematicians like to see observed. However these are purely aesthetic and don't really matter that much to mathematical content. The most significant one is perhaps:

• Don't use proof by contradiction. A proof by contradiction is not logically flawed at all, of course (at least in most cases). However, it is considered an undesirable approach if another 'better' proof exists. See here for a more detailed explanation why.