What Is Mathematics, Really?

A nice, dense philosophy book, written with lots of wit. This was NOT a book that you can zone out on, and - indeed - it required Full Attention to reap its benefits. This meant that I read it in tiny bursts and, if before bed, would basically immediately fall asleep. This would be a good book to sneak into a meditation retreat, or to have on a long flight/train ride, or anywhere else it's easy to focus and be alert.

I was looking for something that talked about the kinda default way everyone thinks about math as being inherently Platonic - that is, that we think of math as something True, beyond human fallibility and human society even. This book argues firmly against that: Hersh establishes the “mainstream” philosophy of math as being Platonist, but then discusses his “humanist” belief in math as a socio-cultural construct. The book is a combination of sly jokes/author personality (which I usually dislike, but here really enjoyed - he was funny!), a philosophical discussion using impressively simple examples, and then a survey of philosophers - from Plato onwards - and where they fell on the mainstream/”magical math from the eternal cosmos” vs. humanist/”math is just a game and shorthand” spectrum. He also interestingly framed the mainstream vs. humanist philosophy of math as correlating strongly with the right-wing vs. left-wing beliefs of the philosophers themselves (!) and - even more surprisingly - with the fixed mindset/elitism vs. growth mindset/math education is shitty paradigm. That last dichotomy was the most surprising - and the most eye-opening for me! It definitely explained my own fraught relationship with math (brief bio below).

So I came into the book firmly on the Platonist side - I'm not Platonist on anything else, but I had subconsciously swallowed the mainstream attitudes whole: math is magical truth from the cosmos, something we discover rather than invent, and something that you can either see or not. Or, as my favorite Leonardo da Vinci quote says: there are those who see, there are those who see when they are shown, and there are those that do not see. Magical math!

Hersh dismantles this argument by noting the ways that math is a social convention, collaboratively and imperfectly invented using impossible-to-totally-logically-underpin proofs written by real, living, fallible mathematicians who mostly take it on faith, and that many previously-held mathematical truths (Euclidean space) have been dismantled or revised. I think this book would have been a bunch more enjoyable for actual pure/academic mathematicians, since he drops a bunch of stuff I had absolutely zero knowledge of - very exotic stuff! But his general points he makes concisely and with great wit. I also really enjoyed the survey of philosophers: I loved the short bios with sarcastic, lively interjections (boy, he really has it in for Wittgenstein, haha!), and I came away from this book with a list of people I keep meaning to read more about: John Stuart Mill (u r a hero, jsm), John von Neumann (also such a hero, wow), Karl Popper (world 3!!!! WORLD 3!! what a brilliant idea), Bertrand Russell.

Okay, small feminist aside on the surprising way this philosophical argument impacts the mindset/math education stuff: part of the mainstream “magical Platonic math” mentality is a strong elitism. If math is something that's baked into reality's DNA (and, again, not a tool we invented to understand reality's DNA), but if it's the actual - ahem - SOURCE CODE - then there are people who can read this code/see these truths, and those that can't. It matters less how you teach it, since there's just One True Math and “smart” people eventually get there one way or another. This, of course, perpetuates all manner of bad things: like how only white dudes have the True (Math) Sight. If you permit me the indulgence, I realized recently that my “math bio” is like a textbook case of stereotype threat:
- In high school, I considered myself “bad at math”, but scored a higher math SAT score than my (male) friend - who went on to major in mechanical engineering. :/
- In college, I minored in math and adored every class. When I asked my calc 2 prof if I should apply to an applied math master's programs (WHAT FUN!), he discouraged me. ://
- In grad school, I struggled through those goddamn Lagrangians and assumed the struggle was because I was “bad at math” (and not because grad school is grad school).
- In one of my first jobs, I figured I'd never learn to use Stata (the statistical programming language we used) well, because I was “bad at coding” and “bad at math” and didn't “think logically”. (I ended up writing the Stata tutorials for staff training, lecturing on it and TAing it.)
- Now I'm a data scientist, and I math it up for ~40 hrs/week with GREAT PLEASURE and yet I STILL constantly have imposter syndrome and assume I “have no natural talent” for math and will “never be that good at it”, but am grateful that my joy/passion for doing it at least keeps me from totally sucking.
I mean, LOOK AT THAT. It's INSANE. I have literally mathed with pleasure near-constantly since I was 15, and have meanwhile never recognized it as a core part of my identity, but instead felt always like an interloper. I mean, DAMN.

Anyway, so there you go. Definitely recommended, especially if you like epistemology.