Resources for the kind of math learning project I’ve recently undertaken have never been more readily available. This post offers summaries of some of the resources I’m using for the initial phase of the project, which I’m calling “intro to math” as a shorthand for a breadth-first overview and introduction to proof. I make specific reference to resources that support my personal goals and strategies for math learning, including academic math, contest math, computational math, and special topics. I plan to make more detailed standalone posts on several of these introductory resources, and I also expect to share information about more specialized and advanced resources as I progress in my program. Stay tuned for more!
An extraordinary variety of resources on math learning have become available in recent years. This includes books, which are an old format but are more accessible now than ever before. Newer formats include videos and online apps. In order to create a sort of “math immersion” environment, I’m using a bunch of different such resources, but my primary sources are textbooks.
One great thing about learning math (or really any subject) nowadays is that it’s easier than ever to find first-rate resources. Instead of using some random textbook selected by an apathetic faculty committee, you can use one that hundreds or thousands of real students have studied and enjoyed. My general strategy for finding such books, especially ones that align with my math learning goals, involves googling the general subject and then reading suggestions at places like Math Stack Exchange and Quora, as well as consulting reviews at Amazon. Examining the table of contents, reading the preface, and taking a look at the first couple of chapters is usually enough for me to reach an informed conclusion about whether the book meets my needs.
As previously noted, I have a strong preference for PDFs, which are often difficult to find for purchase but are clearly the best format for math ebooks.1 In addition to using highly recommended books, I also prefer to use books in their second edition or later. Among other reasons, using a later edition means that many of the inevitable first-edition typos will have been fixed. (Typos in math textbooks are the worst—with natural language, I can generally error-correct, but with math it’s hard to know if the problem is me or the text.) Finally, because I’m mostly learning alone, I generally give precedence to books that have full solutions manuals available.2 This is a key part of my strategy for using books to learn math without having to rely too much on outside help. And I find that it greatly lowers the psychological barrier to attempting to solve problems myself—an essential part of math learning—without worrying that I might get permanently stuck.
For other resource types, I typically use a similar process. For videos, I pay especially close attention to what the YouTube algorithm surfaces. For example, I found Start Learning Mathematics (Section 2.2) by googling for videos on constructing the real numbers, and YouTube quickly started recommending other useful videos as well.
To support this goal, I’m using mainly books, with some videos. The techniques above led to a large number of possibilities, and there’s no one right answer that works for everyone. After an extensive winnowing process, these are the main sources I decided to use for an introduction to pure mathematics, with a focus on breadth and proof-writing.
These are just brief summaries. I’m planning standalone posts on at least two of the following titles.
Pure Mathematics for Beginners by Steve Warner (solutions, full post). This is my primary text for pure math since it covers an amazing breadth of subjects while still focusing on mathematical proof. Note that I am using the original, shorter version of the text and not the expanded version available at Amazon.3
How to Prove It: A Structured Approach (3rd Edition) by Daniel Velleman (solutions). I am also studying this book intensely and doing many of the problems. How to Prove It has some overlap with Pure Mathematics for Beginners while having less breadth overall but more depth on how to write proofs.
Discrete Mathematics with Applications (Fifth Edition) by Susanna Epp. I’m mainly using this as a supplement for now, but it’s certainly good enough to be a primary text, especially for students in computer science and related fields.4 Discrete Mathematics with Applications covers several topics (such as graph theory and algorithms) not covered by the other introductory texts listed.
There are literally jillions of math videos out there. Here are just a couple of ones I’ve found particularly useful as beginner-level introductions to rigorous math.
Start Learning Mathematics from The Bright Side of Mathematics. This remarkable course builds up the various number systems of mathematics from scratch, starting with logic and set theory, moving through the definition of the natural numbers and constructions of the integers, rationals, and reals, and ending with the extension to complex numbers. I’ll have a lot more to say about this amazing YouTube channel in the future.
Introduction to Higher Mathematics by Bill Shillito. The production values of Introduction to Higher Mathematics are a little dated—as Bill Shillito himself notes, the course is “ages old at this point”—but the videos provide a great overview of the subject and the material is hard to find all in one place like this.
I mentioned before that I’m using Ph.D. qualifying exams to provide general guidance about which direction to go. There are actually a lot of options to choose from; I’m currently broadly aiming toward the Ph.D. quals used by the Harvard University math department. In addition to coming from one of the top math graduate programs, Harvard’s quals have significantly more breadth compared to most of the other quals I’ve found online. Having taken several undergraduate courses in the Harvard math department back in the day,5 there’s some sentimental value here as well.
In the spirit of “as easy as possible (but no easier)”, I’m initially pursuing this goal using books aimed at high-school students, namely The Art of Problem Solving (a.k.a. AoPS). These books are not to be trifled with, though; even as a trained theoretical physicist, I still find many of the problems contained in these books quite challenging. They also have remarkable breadth, including algebra, Euclidean geometry, analytic geometry, combinatorics, probability and statistics, number theory, and graph theory.
The Art of Problem Solving, Volume 1: the Basics by Sandor Lehoczky and Richard Rusczyk (solutions). Great coverage of both standard high-school topics (logarithms, algebra, geometry, trigonometry) and more advanced topics (probability and statistics, number theory).
The Art of Problem Solving, Volume 2: and Beyond by Richard Rusczyk and Sandor Lehoczky (solutions). Even more advanced topics, including vectors and matrices, analytic geometry, combinatorics, graph theory, and number theory (linear and quadratic congruences, Euler’s \( \phi \) function, Diophantine equations, and more—I told you these “high school” books weren’t to be trifled with).
These are the only contest resources I’m using for now (I’m currently about halfway through Volume 1 of AoPS), but many other resources are available. There are books for Math Olympiad (extremely challenging “high school” math) and the Putnam exam (brutal college-level math), and the Art of Problem Solving website has tons of great content (including an archive of problems from the AMC 12).
Pursuing this goal is fairly straightforward. I’m taking a lightweight approach, reading just enough to get started (one might say, learning enough to be dangerous) and focusing on applying the techniques to resources for the other goals.
As outlined above, my hands are quite full learning the basics, but I have taken some tentative steps toward one of my preferred special topics: geometric (Clifford) algebra.
A Swift Introduction to Geometric Algebra by sudgylacmoe. This entertaining and impressively popular video has helped get a lot of people interested in geometric algebra, including me. (I had the good fortune to collaborate with sudgy last year on a video version of The Tau Manifesto.)
A New Language for Physics by Arthur Lasenby. A thorough two-hour talk from a conference on geometric algebra given by a professor of astrophysics and cosmology (emeritus) at Cambridge University. Lasenby also coauthored Geometric Algebra for Physicists, which is a little advanced but is definitely on my reading list.
Because of the huge variety of great resources available, it’s important to have strategies for maintaining focus. One technique I’m using involves thinking of myself as having some small number of “slots” (corresponding roughly to a typical undergraduate or graduate course load) and moving back and forth between them. For this early intro-to-math phase, I’m using four slots, corresponding to four primary resources:
Pure Mathematics for Beginners by Steve Warner
How to Prove It by Daniel Velleman
The Art of Problem Solving, Volume 1 by Lehoczky and Rusczyk
Sage for Undergraduates by Gregory Bard