Back in 2016, I typed up a little guide to studying physics called “So You Want to Learn Physics.” It ended up being pretty popular, so I started working on other guides, including a guide to studying philosophy (“So You Want to Study Philosophy”), which I published in 2021, and this long-awaited guide to studying mathematics, which I am sharing with you today.
I absolutely love mathematics. I think it is the purest and most beautiful of all the intellectual disciplines. It is the universal language, both of human beings and of the universe itself. Sadly, there is all sorts of baggage around learning it (at least in the US educational system) that is completely unnecessary and awful and prevents many people from experiencing the pure joy of mathematics. One of the lies I have heard so many people repeat is that everyone is either a “math person” or a “language person” — such a profoundly ignorant and damaging statement. Here is the truth: if you can understand the structure of literature, if you can understand the basic grammar of the English language or any other language, then you can understand the basics of the language of the universe. That doesn’t mean it’s easy — no, mathematics is an incredibly challenging discipline, and there is nothing easy or straightforward about it — but, honestly, I have yet to find a single topic, discipline, or intellectual pursuit that is easy or straightforward to learn at any advanced level.
The secret to learning math is this: accept that it is a difficult subject and that understanding it is going to be hard, study it in small manageable pieces (like the curriculum I’ve put together here), be patient with yourself and with your study, and work diligently to understand it. I promise you that it is worth every moment, every effort, every precious bit of energy.
My goal here is to provide a roadmap for anyone interested in understanding mathematics at an advanced level. Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics. This guide only covers an undergraduate mathematics curriculum, because, unlike the fields of physics and philosophy (both of which I have studied at the graduate level), that’s where my math knowledge ends. While I have taken a few graduate courses in mathematics and have studied a handful of topics in mathematics (including differential geometry and logic) at the graduate level, I don’t have enough experience or knowledge to feel comfortable evaluating graduate-level mathematics textbooks, and, as a matter of principle, I won’t recommend or include a textbook in one of my guides that I haven’t studied (whether in full or in part) either on my own or for a course. I’m always learning new things, so if/when that ever changes, I’ll update this guide.