A Mathematician’s Lament (2002) [pdf]

I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it.

I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break into theoretical math classes but I've found the bar for entry pretty high (especially when I only have room for one or two courses), with most classes and even peers asking for years of experience and "mathematical maturity." Have any of you ever succeeded in learning some math outside formal curricula?

This is how I feel about doing most coding tutorials versus the MIT intro to CS through python, or project euler.

When coding is presented as: - here is a problem - how would you solve this problem? - here are some hints to get you started

it is incredibly fun for me. when it is presented as:

- follow along - look what you did!

it can be a bit dry.

> The difference between a euclidean domain and a principal ideal domain

A Euclidean domain is an integral domain where the Euclidean algorithm works. For the Euclidean algorithm to work, you need to be able to divide two elements and produce a remainder that's smaller than the divisor.

So an ED requires a notion of "smallness" (the Euclidean norm) which interacts with division in a way that makes the Euclidean algorithm work (remainder is always smaller than divisor).

A principal ideal domain is an integral domain where every ideal is principal (can be generated by one element). It can be useful for you to know that certain situations cannot happen, e.g. in a PID you can say, "Let I be an ideal of D, then I = ..." and do something with the generating element g. It lets you pass from an ideal to a single generating element in a proof, which may be a useful capability. The PID concept is also part of a taxonomy, since Euclidean domains ⊆ principal ideal domains.

We really need to teach people _how_ to teach induction, which is only done right when you put quotes around your Boolean statements; the "implies" symbol gets jumbled up with everything else otherwise, and not using it at all is passing up on a great tool. One can do simple proofs-by-induction without a single English word, completely symbolically, and have it be understood easily, if one uses quotes and correct LaTeX formatting (or good handwriting)

Induction doesn't just involve numbers and equality signs, it involves _statements_ with variables inside of them, and non-programmers need to be made well-aware of this (and taught Boolean logic early, PLEASE)

> (...) while reading Noam Chomsky's book Syntactic Structures on his honeymoon in 1961

Which makes you wonder, what kind of person reads this stuff on the honeymoon.