I'd like to see a series of sites or interactive ebooks that work as described below, which would address that problem. I can't actually build such a site or ebook because (1) I'm not sufficiently mathematically sophisticated and (2) my front end skills aren't up to it.
Each one would be devoted to one interesting major mathematical theorem, such as the prime number theorem. The initial view would be a presentation of the theorem as it would be presented if it were a new discover being published in an appropriate journal for the relevant field.
At any point in that view you can pick a term or a step in the proof and ask for it to be expanded. There are two expansion options.
1. More detail. You use this when you understand what is being said but just don't see how they got from A to B. It adds in the missing details.
2. Background. You use a background expansion when you need to know more about some term or theorem that is being used.
I'll give an example of how these might work later.
The details or background you get from either of those expansions can themselves be expanded. Background expansion should work all the way down to pre-college mathematics.
Ideally if you started at the initial view of the prime number theorem site without having had any college mathematics and did background expansions all the way down you would end up learning the necessary calculus and complex analysis and number theory to understand the initial journal-level proof.
You would not learn all of the calculus or complex analysis or number theory one would normally learn in those courses. You would just learn the parts necessary for the top level proof.
I think it would be possible to choose a selection of interesting major theorems to do these sites for such that their combined background expansions would include everything that you'd get in a normal college mathematics degree program.
I think many people would find that a more interesting way to learn the material from a college mathematics degree program than the normal series of courses that each covers one subject. By getting it via background expansions everything you are learning you are learning for a specific application which can be more motivating.
Here's an example of what expansions might do.
There's a theorem from Liouville that says that if a is a real root of an irreducible polynomial with integer coefficiants of degree v >= 2, and p and q are any integers (q != 0) then there is a constant C > 0 which does not depend on p and q such that
|a-p/q| > C/q^v
In Gelfond's book "Transcendental and Algebraic Numbers" he says (I'm changing the names of some of his variables because I don't want to deal with typing greek letters and some of his terminology for clarity):
> The proof of this is quite straightforward. Suppose a is a real root of an irreducible equation
f(x) = a0 x^v + ... + av = 0,
> where all of the ai (i = 0, 1, ..., v) are integers. Then, using the mean value theorem, we get
|f(p/q)| = |a-p/q| |f'(z)| >= 1/q^v; z = a + t(p/q-a),
|t|
> from which the Liouville theorem follows directly.
Directly for Gelfond maybe. It certainly does not follow directly for me! I understand everything he's saying, but the steps between some of the things is a little too big for me. I'd need to use a details expansion (maybe more than once). What that should give is something along the lines of how Wikipedia proves that theorem, which is the first lemma in the "Liouville numbers and transcendence" section of their article on Liouville numbers [1].
If I didn't know what the mean value theorem is (or the extreme value theorem, which shows up after expansion to a Wikipedia level of detail) that would be time for a background expansion.
[1] https://en.wikipedia.org/wiki/Liouville_number#Liouville_num...