In highschool I was terrible at trig and then avoided maths a long time.
Months ago pseudonym73 in the handmade hero chat mentioned that there's an easy-ish method of integration that is somewhat little known because calculus classes focus so much on integration by parts and so on. He recommended a book on it, manuel bronstein's symbolic integration, which is about how to write integration solvers. So immediately two things struck me. The first was that the very first definition
, a terse definition of group
meant for people who already knew what groups were, was the first definition of a group that made any sense to me. And you know a couple years ago I read edward frenkels popular math book 'love and math' and it is all about group theory and I tried to understand groups in that book and I learned nothing. And all those paragraphs in the wikipedia page trying to explain it all in english language terms and motivations and I learned nothing. But I hope everyone can see from that book that a group, in computing science terms, is a type
This has really changed mathematics for me hugely: I used to think I needed these long-winded rationale-giving explanations before I understood something, but in reality it's that terse notation that makes mathematics work for me. Everything you need to know to understand a mathematical object on an area the size of a business card (with an example on the back). I understand this is problematic if you've been scared off maths for a long time and you don't know what the hell an upside down A is supposed to mean. In my opinion any maths not aimed at, say, upper level math undergrads and above, uses a symbol without first defining it should be launched into the sun. As a nice counter-example, there's a great (free) textbook on the discrete fourier transform by Julius O Smith that assumes no mathematical background--since its discrete, you get away without any integrals, and he's free to focus on defining complex numbers and the complex exponential function without getting into calculus
The second thing was that I really had too poor a grasp on even basic calculus to be reading a book on symbolic integration so I went off to read up on calculus instead. In particular I found a book that comes highly recommended from mathematicians, Spivak's Calculus, that is more or a less a series of terse definitions. It's not so dry as that, but it kind of is, and it works. It's really great to read a maths textbook that first begins by rigorously defining what a number is and what hell addition is supposed to be, then steamrolls over everything you didn't learn in highschool, and I recommend it to everyone who has never done so
I guess maths is like a presentation: if you're standing there just restating everything on a slide it's not a very good presentation. I've found when reading maths my eyes often flow over the equations and skip to the text as if I'm going to find what is really 'meant' there, and this is a terrible
habit. If anything, i'd be better served trying to guess what the text would say from the equations
Since then I've always had a current maths textbook I've been reading. Recently I finished a book, approximation theory and approximation practice
, where I found myself so surprised by some of the theorems I found myself implementing them to experiment a little and to find out if they were really saying what I thought they were saying. It's a nice feeling
edit: after writing this post I ended up clicking around the ATAP site and it turns out there are lectures on it available from the author. It's really a genuinely good book: https://people.maths.ox.ac.uk/trefethen/atapvideos.html
In Sept-Oct 2015 Nick Trefethen, Professor of Numerical Analysis at Oxford and Global Distinguished Professor at NYU, gave an 18-hour series of lectures at NYU presenting all 28 chapters of this book. The lectures are full of unusual mathematical, philosophical, and historical perspectives, and everything is illustrated by Chebfun demos on the computer. This is a unique resource for those who want a deeper understanding of the foundations of numerical computation.