Soft question regarding math theses

Question

I'm currently in high school and I am wondering about how math papers a formated. In standard literature studies essays, we are told to "come up with" a thesis statement, present it in the introduction then and prove it by giving arguments and examples from a given text.

Is this the same for math papers? Do the mathematicians already know what the result will be before the paper is written?

For example, if a math paper encompasses the proving of a certain conjecture, will the result of the entire paper be stated in the intro of the paper similar to that of a paper regarding literature study?

Answer 1

This depends on the style of the author and/or the journal and/or the specific field of study. Of course, mathematical proofs consist of a single logically rigourous argument which demonstrates a particular result, not a series of arguments, examples and pieces of evidence like, perhaps, in literature. A possible format for a mathematical paper might be:

Introduction. Wiles [1] showed that the Diophantine equation $$x^n+y^n=z^n$$ has no solution in integers for $n\geq3$. We generalise this problem by considering the equation when $x,y,z$ are $2\times 2$ matrices with integer entries. The main result is the following.

Theorem 1. stuff

Furthermore, we show another result relating to ....

Theorem 2. more stuff

(Please don't scrutinise the mathematical content of the above, it is merely for example.) And the sections following this might be named "proof of theorem 1", "proof of theorem 2", for example.

But this is far from the only structure a paper can take. Another natural (and thus common) approach is to give some brief background and ask an important question in the introduction, and then work through the steps in a systematic manner before ending up at the final result (instead of stating the result in the introduction up front). This is somewhat the way Wiles' famous paper Modular elliptic curves and Fermat's Last Theorem was written.

Really, the specifics of the formatting don't matter so much as long as the content can be conveyed in a clear way. The only purpose to formatting is to serve to aid understanding. As long as understanding is clear (for the readers), the paper is well-written.

Answer 2

I have attached a link to a short paper written by Paul Erdos in 1946 which I believe is a great little example of a good piece of mathematics (cited 595 times which is a lot for a 3-page math paper):

https://www.cs.umd.edu/~gasarch/TOPICS/erdos_dist/erdos.pdf

This little paper came with a big conjecture that was only "completely" solved recently by Larry Guth and Nets Katz:

https://annals.math.princeton.edu/2015/181-1/p02.

In short, mathematicians do not necessarily have all the answers to their problems when they write their papers. They present whatever they discover after a period of tackling the problem. In Erdos' case, he wrote in the introduction of the paper

"The following theorem establishes rough bounds for arbitrary $n$. Though I have sought to improve this result for many years, I have not been able to do so."

Despite the fact that he did not solve the problem, he pointed to a direction that allowed others to continue working which I believe is one of the most important aspect of a good math paper.