Comment by alnar | Hacker News Reader

Comment by alnar | Hacker News Reader

SOME ADVICE BEFORE YOU DO A DEEP DIVE INTO WHATEVER YOU END UP STUDYING MATH-WISE:

My Background: Current Undergraduate in CS and I recently added Mathematics

The most difficult part for a person who hasn't done a lot of math to become a person who does a lot of math is to read and understand rigorous proofs. You will encounter countless difficult proofs in any mathematical topic you try to study. Read a few books on mathematical thinking and proof techniques before/during/after reading any other dense math book.

Like you, I realize the value of having a mathematical mindset and want to have a deep understanding. When I added math as a major, I had a very hard time jumping from computational courses (typical math courses, geared towards any major) to theoretical and conceptual courses (proof-based courses that use all the fun and interesting math books everyone has linked here). These books helped:

https://www.amazon.com/How-Read-Proofs-Introduction-Mathemat...

https://www.amazon.com/How-Think-Like-Mathematician-Undergra...

https://press.princeton.edu/titles/669.html

<3 this is a great book, obvi since its george polya

> The most difficult part for a person who hasn't done a lot of math to become a person who does a lot of math is to read and understand rigorous proofs.

I’m afraid you haven’t delved into any advanced topics. That’s actually about the easiest part, and could be mastered by ten year olds (certainly myself when I was ten).

alnar is likely referring to the US system, or something like it: unless you are tutored externally (rich), an autodidact outlier (gifted), or selected for honors courses, you basically take computational courses for 14 years (with one cursory stop for euclidean geometry) and are then thrown into proofs at the age of 19-20, if at all. it's widely recognized as a problem in the math pipeline, which is why many US universities have "transition" courses for non-honors students.

so it's not that the basics are intellectually difficult as much as practically difficult (unfamiliar, disorienting) for many students. many "transition" books talk about the difficulty in adjusting from talent being redefined from perfectionist "plug and chug" (APs, SATs) to reasoning and creativity.

btw, i'm impressed that you could master college-level proofs at 10. i have a kid about that age who is pretty good at logical reasoning, but i'm not sure what topic (at that level) he could do a rigorous proof about; maybe numbers, as in landau? can you say more about the materials you used?

This has little to do with education systems, I was talking about objective difficulty. Try to tell any mathematician that the hardest part of math is rigorous proofs; if they don’t laugh in your face, they are just being polite. One may think the transition is “hard”, until one actually gets into more advanced topics. As you said, it’s just unfamiliar to the uninitiated, at best.

Reading and presenting rigorous proofs in elementary number theory, Euclidean geometry, etc. is easy for gifted ten year olds and definitely manageable for a lot of fifteen year olds. You asked about material — it doesn’t actually matter, and I don’t recall specifics; any entry level treatment of elementary number theory should do (really beautiful subject with a very low barrier of entry). For kids who have eyes on IMO, it’s very common to be throwing around perfectly rigorous proofs at young ages.

Reply on news.ycombinator.com