[ycombinete]

The problem is that the definitions of in such footnotes would be full of terms that one also needs defined.

For example here's the first few sentences from the Wikipedia page on Compact Sets:

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.[1] The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact.

[nimih]

Another way of stating this, which I think is rather non-obvious to people who have not studied some amount of pure math, is that technical mathematical definitions are almost never helpful absent a substantial amount of additional context, because their purpose is generally to capture and formalize a much more intuitively-understood idea or motivating phenomena. A professor of mine liked to joke that MathWorld was a website devoted to collecting definitions which were as technically correct as possible while also being useful to nobody[1], and I think the quoted wikipedia passage follows in that tradition beautifully.

[1] I don't know if this is still the case, but it certainly was in the late '00s.

[tzs]

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| <= 1,

> 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...

[trueismywork]

You would probably end up writing copies of such books, since same proof will be included multiple times. And more copies written by humans means more errors.

Automated theorem proverbs can probably solve this problem though..

I had a professor who derived Riemann metric from just area of triangle and limits over the course of a semester. So I see what you're getting at, but such books will probably he too long for most people to read anyway.