In what order does mathematics build upon itself?

Question

I have many times come across mathematical proofs, theorems, conjectures, and generally interesting things. For example:

• Twin primes conjecture
• Millennium problems
• Ramsey theory
• Fermat's last
• Reimann Hypothesis
• Reimann Zeta function
• Galois theory / Galois groups

And I would of course very much like to understand these as much as possible. When I go to the Wikipedia pages of these topics, for example, I come across a lot unknown terminology and concepts.

My question would be, where should I begin, to try to understand, say, Andrew Wiles's proof?

I have just very successfully completed Calculus AB, and though I know it may be a long way before I can completely understand these advanced things, I'm eager to begin somewhere. I have watched a series of college lectures on "Higher Math 101", but I really want to get into the meat of things.

(By the way, I didn't know what to tag this)

Answer 1

Mastering AB level Calculus is not sufficient at all to be able to understand some very complex work in number theory, abstract algebra or other fields.

My best advice, is to follow by order the books listed here to understand every topic from the basics building up to the complex stuff.

To keep motivated, don't forget to read the Princeton Companion to mathematics from time to time.

Answer 2

The Riemann $\zeta$ function for real arguments $>1$ is $\zeta(s)=\sum_{n=1}^\infty 1/n^s$. So how does one define it if $s$ is not real or if the real part of $s$ is less than $1$? Notice that you can probably at least understand that question without knowing anything beyond first-year calculus. If you know that $n^s= e^{s\log_e n}$ and that $e^{i\theta}=\cos\theta+i\sin\theta$ then that's a step in that direction. As to why $e^{i\theta}=\cos\theta+i\sin\theta$, the beginnings of an understanding of that are at least hinted at by the fact that multiplying by $i$ is the same as rotating $90^\circ$ counterclockwise.

Often the way topics are presented is logically rigorous, and that in effect means that one describes an argument that would be approved by a sound proof-checking algorithm. Normally one doesn't write down enough to hand to such an algorithm, but one describes it in such a way that would enable a mathematician reading the argument to do that.

With many topics one can learn quite a lot before one reaches the point where one can understand a logically rigorous account. But the math curriculum is in many respects not designed to facilitate that. For example, I can give a quick intuitive argument for the proposition that $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$ and similarly for sine and cosine, and thereby "prove" the assertion above about $e^{i\theta}$. If one doesn't insist on logical rigor in the initial encounter with a subject, I can say something like this: Define the $\zeta$ function for $s>1$ as above, then say that for real numbers $s$ near $2$, one has $\zeta(s) = \sum_{n=0}^\infty c_n(s-2)^n$, then apply that series when $s$ is near $2$ but not real, then say that's $\zeta(s)$ for non-real $s$. Not the least of the concerns resulting from the deficiency of logical rigor in this argument is whether I start with $2$ and you start with $1.9$, we both get the same value of $\zeta(1.8+0.1i)$? I think learning stuff like that, with the realization that it is not rigorous, can make you less confused when you later reach the point where you can learn the rigorous version of it.