So You Want to Study Mathematics…
Introduction
March 6th, 2022
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.
Anyone can understand mathematics. The joy of learning to speak the language of humanity and of the universe is accessible to anyone who has (1) the desire to think about things a little more deeply and (2) the curiosity and patience to discover what’s possible.
Godspeed!
Before You Begin
Popular Math Books
Before you dive into the more formal and difficult coursework, you might find it helpful or fun to read some books about mathematics and mathematicians that are a little more accessible than many of the textbooks you will find in the curriculum that follows.
Here are a handful of my favorite popular mathematics books, ranked in order of difficulty:
e: The Story of a Number by Eli Maor (Level: Easy). A fun, accessible book that will get you excited about mathematics.
The Joy Of X: A Guided Tour of Math, from One to Infinity by Steven H. Strogatz (Level: Easy). A lot of fun to read, but make sure you get the paperback or hardcover version for readability purposes.
Fermat’s Enigma by Simon Singh (Level: Easy). A beautifully-written book about Fermat’s Last Theorem.
The Man Who Loved Only Numbers by Paul Hoffman (Level: Easy). A compulsively-readable biography of Paul Erdős.
The Man Who Knew Infinity by Robert Kanigel (which was also made into a film) (Level: Easy). A wonderful biography of Srinivasa Ramanujan.
Flatland by Edwin A. Abbott (Level: Easy). A classic. I highly recommend the annotated version, which adds extra joy to the reading experience.
A Mathematician’s Apology by G.H. Hardy (Level: Medium). One of the most beautiful things ever written about mathematics, by one of the greatest mathematicians of all time.
Fearless Symmetry by Avner Ash and Robert Gross (Level: Difficult). One of my all-time favorites.
Proofs from THE BOOK by Martin Aigner and Günter M. Ziegler (Level: Difficult). This book is an absolute joy to read in small bits and pieces. The more math you learn, the more you will fall in love with it.
Prerequisites
Before you begin working through the curriculum below, you need to be familiar with some basic mathematics:
High school mathematics: A high school education — which should include pre-algebra, algebra 1, geometry, algebra 2, and trigonometry — is sufficient. If you need a refresher or if you are unfamiliar with the material, I recommend either working through the Khan Academy math courses (https://www.khanacademy.org/) or the book Why Math? by R.D. Driver.
Precalculus: Precalculus with Calculus Previews by Dennis G. Zill and Jacqueline M. Dewar is a very comprehensive, well-written book (like every one of Zill’s wonderful textbooks!). If you haven’t taken a precalculus course or studied precalculus before, I recommend working through the entire book and getting comfortable with the concepts before moving onto the calculus course that kicks off our math curriculum. You can supplement this textbook with the Khan Academy Precalculus course if needed.
How to Study
Every person learns in their own special way, and knowing your learning style is important: do you learn by reading, by taking notes, by talking, by watching, by doing, or by a combination of some or all of these? For example, I learn by reading and by note-taking, so I read through textbooks very carefully, take copious notes, and summarize each concept in my own words before moving on to something new. Think about this before you begin so that you'll know how to structure your studies.
Regardless of your learning style, you'll still need to solve the problems in each textbook. Just like in physics, solving problems is the only way to understand mathematics. There's no way around it.
One tough thing about learning on your own is that you may not know whether you are solving the problems correctly. A number of the textbooks listed below have answers to selected exercises in the back of the book, but these aren’t always adequate for two reasons: (1) they often only show the solutions to the problems, and not the steps taken to get there; and (2) it’s much better to do all of the exercises rather than just a select few. The good news is that many of the solutions (and step-by-step ways to solve them) can be found online with a simple google search. If you are going to google the answers, however, please first try to solve the problems on your own, and try multiple times (you’re not in school trying to get a perfect grade — you’re trying to learn and understand).
Further Reading and Misc. Fun Stuff
If you’re enjoying with these courses and are also interested in programming, check out Project Euler, which is an incredibly fun collection of math and programming problems to solve.
The Math Stack Exchange website is a great place to ask questions and find answers: https://math.stackexchange.com/
The Mathematics Curriculum
Overview
The curriculum of almost every undergraduate mathematics program in the United States covers the following subjects, and usually in the following order (although many students can and do study ODEs and PDEs right after calculus or linear algebra):
Four semesters of calculus
An “introduction to proofs” course
Linear algebra
Two semesters of algebra
Real Analysis
Complex Analysis
Ordinary Differential Equations
Partial Differential Equations
Electives
I’ll cover the details of each of these course topics below, and will include the best textbooks to use for independent study and any additional readings and resources you may find helpful in your journey.
1. Calculus
What It’s All About
In a nutshell, calculus is the study of change. Just like almost every undergraduate mathematics student, you will probably spend a good deal of your mathematics education studying calculus: most undergraduate math majors take two full years of calculus courses (not including precalculus or any other prerequisites) — a four-course series with names like “Calculus 1,” “Calculus 2",” etc. — and then study calculus again when, later, they study analysis. You may find that you can work through this first course much more quickly than that, but don’t be discouraged if it takes you one or more years to get through the textbook. You may also find calculus difficult, especially if you’ve never studied it before; part of the difficulty sometimes comes from the foreignness of the material (it can take a while for your brain to get used to it), and the rest of the difficulty stems from the fact that it can be a challenging subject to learn! Take your time, do as many problems as you can, and keep going. (Note: if you do end up finding calculus too difficult, make sure you go back to high school algebra and precalculus and cover everything you might have missed or not understood very well.)
Readings
Calculus: Early Transcendentals, 8th Edition by James Stewart (essential). This is an excellent textbook and it is a classic for a reason. Everything you need to know about undergraduate calculus is in this book. Everything. It contains excellent problems, good examples, and straightforward explanations. There is also a Student Solutions Manual that you may find very useful. The earlier editions of the book are just as good as this one — just make sure you are able to find the corresponding solutions manual. (Note: I also frequently recommend Thomas’ Calculus, which is also excellent, but I have lately found myself preferring Stewart; both books are fantastic and they each have their nuances and quirks — if you find you don’t like one, give the other a try.)
Calculus Made Easy by Silvanus P. Thompson and Martin Gardner (supplement). This is such a wonderful, thoughtful book. I recommend reading it alongside Stewart when you are just getting started — it may help you make sense of what you’re learning.
Additional Material
When I was a student at Penn, the most useful mathematics courses I took were ones that I snuck into (some of them I couldn’t enroll in because I didn’t have the necessary prerequisites, others because they were too full and I couldn’t register). My favorite secret classes were the calculus courses taught by Robert Ghrist. He had a way of explaining concepts in calculus that made them just click for me. He also lectured with this really infectious energy that made me strongly reconsider my devotion to physics — I’d leave his lectures wanting to devote my whole life to mathematics. Luckily, you don’t have to sneak into his classes to listen to his lectures, since he now offers them for free online through Coursera: (1) Calculus: Single Variable Part 1 - Functions, (2) Calculus: Single Variable Part 2 - Differentiation, (3) Calculus: Single Variable Part 3 - Integration, (4) Calculus: Single Variable Part 4 - Applications. I highly recommend his lectures — they pair well with Stewart’s Calculus and they will stick in your head for many, many years.
2. Introduction to Proofs
What It’s All About
Much (if not all) of advanced mathematics (i.e., things beyond high school math and introductory calculus) is not about calculating things or solving problems but about proving things. Proving is very different than calculating or solving, and there is an entire toolkit you’ll need before moving on to more advanced courses. Here, you’ll become familiar with mathematical reasoning, will learn how to read and write proofs, and will start learning to think like a mathematician.
Readings
How to Prove It: A Structured Approach by Daniel J. Velleman (essential). This book is a classic for a reason. Read it carefully, and keep it in a handy, readily-accessible place so that you can use it as a reference down the line. Supplement with How to Solve It by G. Polya and Introduction to Mathematical Thinking by Keith Devlin.
3. Linear Algebra
What It’s All About
In this class, you’ll learn how to solve systems of linear equations. You’ll cover real and complex vector spaces, eigenvalues and eigenvectors, determinants, linear transformations, applications of linear algebra, and more. I found linear algebra to be incredibly fun to learn, and I hope you will too!
The Best Textbooks to Use
Introduction to Linear Algebra, Fifth Edition by Gilbert Strang (essential). This is a wonderful textbook, and I have found it especially accessible for independent study. Gilbert Strang posted the solutions to the textbook’s problems on his website, and he regularly updates the site with new materials.
4. Algebra
What It’s All About
In abstract (or “modern”) algebra, you’ll learn about algebraic structures like groups, fields, rings, and more. You’ll study basic group theory, ring theory, field theory, Galois theory, algebraic geometry, and more (you’ll also learn more about vector spaces). Algebra is not for the faint of heart, and you’ll need to be comfortable with proofs and be prepared to take it slow. Most mathematics undergraduate programs will have a two-semester (one full year) abstract algebra program, and mathematics graduate programs will then spend another entire year studying the same things over again, so be prepared to spend at least that much time and energy. The good news is that it is very much worth the time and effort — don’t get discouraged!
The Best Textbooks to Use
Abstract Algebra, 3rd Edition, by David S. Dummit and Richard M. Foote. This absolutely enormous book may seem overwhelming at first, but once you jump in, you’ll find that it’s huge because it’s incredibly detailed, filled with examples and exercises, and relatively straightforward to follow once you get used to it. It’s worth reading and studying carefully, and don’t be afraid to take your time — remember that many full-time math students will study this for at least an entire academic year as part of a two-semester (sometimes even three-semester) advanced undergraduate or beginning graduate course in abstract algebra. You don’t need to read the entire thing unless you’re so inclined — I would recommend working your way up to Chapter 9 if you can.
Additional Material
Benedict Gross, who is a professor of mathematics at Harvard, taught a truly wonderful course on abstract algebra at the Harvard Extension School and made his lecture videos, notes, and problem sets available online for free to the public. The school has since taken the course down, but you can still access it using the Wayback Machine (there are also some lectures on YouTube). His lectures and course materials are a nice supplement to the textbook by Dummit and Foote.
5. Real Analysis
What It’s All About
Mathematical analysis is divided into two groups: real analysis and complex analysis, which are the study of the real numbers and real functions and the complex numbers and complex functions, respectively. To jump into real analysis, you’ll need the four semesters (two years) of foundational calculus courses as well as the familiarity with proofs that you’ll get from abstract algebra (however, I think you could start studying real analysis after working through the first chapter of Dummit and Foote, and study the two topics in parallel if you really wanted to).
The Best Textbooks to Use
Understanding Analysis, Second Edition, by Stephen Abbott (essential). This book is quite good. Not only does Abbott explain the concepts, he explains the history, the importance, the meaning behind everything, every step along the way. Study this alongside Rudin, and supplement with Spivak.
Principles of Mathematical Analysis, by Walter Rudin (essential). This book is also known as “baby Rudin” to contrast it with his more advanced (graduate-level) textbook, Real and Complex Analysis. Study this alongside Abbott, supplement with Spivak.
If you haven’t gotten your fill of real analysis, I highly recommend Michael Spivak’s Calculus and the accompanying Answer Book. Spivak has a way of explaining things that I have found very useful. It’s easy to get stuck and confused when trying to learn math (especially when you’re learning it on your own!), and one trick I’ve discovered that helps me keep going is picking up another book and seeing how that author explains a particular concept that confused me.
6. Complex Analysis
What It’s All About
Complex analysis is the study of complex numbers and their functions. I enjoyed studying real analysis as much as one can, but — but!! — I absolutely loved complex analysis. Don’t jump into complex analysis until you’ve spent time studying real analysis and its prerequisites.
The Best Textbooks to Use
Complex Analysis, Third Edition, by Joseph Bak and Donald J. Newman (essential). This is a very good book but you may find it too bare-bones to study on its own. Read alongside Zill and Shanahan and supplement with Needham if necessary.
Complex Analysis: A First Course with Applications by Dennis G. Zill and Patrick D. Shanahan (essential). Just like Zill’s other (amazing) textbooks, this book is geared toward undergraduate physics and engineering students rather than math majors, which makes it ideal for independent study. (However, if you find yourself able to follow Bak & Newman with no issues, then feel free to skip this one.)
Visual Complex Analysis, by Tristan Needham (supplement). This book is pure magic. I found it especially useful when studying physics and trying to learn the required and necessary math on the side.
Additional Material
Wesleyan has a free Coursera class on complex analysis that you may find a helpful supplement to the textbooks. (As of the date of publication of this math guide (03/06/2022), there was a course starting on 02/26/22.)
7. Ordinary Differential Equations
What It’s All About
In this class, you’ll learn about ordinary differential equations — what they are, how to solve them, and how they are used to model the physical world. You can study ODEs in this part of the sequence (as the second-to-last course in the curriculum) or you can jump in right after you’ve studied calculus or calculus + linear algebra — it’s totally up to you. (I should note that if you are also studying physics and engineering on the side, I do recommend studying ODEs sooner rather than later because of how important they are for physics and engineering, but that’s not necessary if you’re only studying math.)
The Best Textbooks to Use
Ordinary Differential Equations by Morris Tenenbaum and Harry Pollard (essential). This delightful book has pretty much everything you need. If you feel like you want more problems/exercises to solve, you can supplement with Blanchard et. al.
Differential Equations by Paul Blanchard, Robert L. Devaney, and Glen R. Hall (supplement). If you want more exercises and explanations, I suggest checking this one out. It’s a pretty solid book, and it has a Student Solutions Manual.
Additional Material
There’s a fantastic series of video lectures from Arthur Mattuck’s ODE course on MIT OCW. They go well with Tenenbaum and Pollard, and they even go beyond it in ways that are really fun.
8. Partial Differential Equations
What It’s All About
You’ve made it this far, and now you get to study PDEs, which are just completely magical and incredible and model the most important things in the world around us. Here, you’ll study what PDEs are and learn all about Fourier Series and harmonic functions and Green’s Identities and Green’s Functions, and so, so much more.
The Best Textbooks to Use
Partial Differential Equations: An Introduction by Walter A. Strauss (essential). You can get the Student Solutions Manual, too. When you’ve finished this book, it’s worth diving into Tolstov.
Fourier Series by Georgi P. Tolstov (essential). This is probably my favorite math book of all time. It’s just incredible. I envy you for getting to read it for the first time!
9. Electives
What They’re All About
Now that you understand all of the fundamentals of undergraduate mathematics, you have a solid foundation and can study more advanced and specialized topics. There is so, so much to be discovered and so much joy to be found. Good luck :)
Some Recommendations
Any and every topic imaginable: Springer publishes a few amazing mathematics series that you should be familiar with: Undergraduate Texts in Mathematics (UTM), Springer Undergraduate Mathematics Series (SUMS), Graduate Texts in Mathematics (GTM), and Texts in Applied Mathematics (TAM). There is a volume for any and every topic imaginable, and I have loved every book I’ve studied. You can pick and choose based on your interests. I recommend staying with books from the UTM and SUMS series until you’ve finished courses 1-8 of this curriculum, and then you can start studying books from the GTM and TAM series.
Discrete Mathematics: Discrete Mathematics with Applications by Susanna S. Epp.
History of Mathematics: A History of Mathematics by Carl B. Boyer and Uta C. Merzbach.
Number Theory: Read A Friendly Introduction to Number Theory by Joseph H. Silverman alongside An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright.
Philosophy of Mathematics: Thinking about Mathematics: The Philosophy of Mathematics by Stewart Shapiro, Philosophies of Mathematics by Alexander George and Daniel J. Velleman, and Philosophy of Mathematics: Selected Readings. Supplement with On Formally Undecidable Propositions of Principia Mathematica and Related Systems by Kurt Gödel and Gödel's Proof by Ernest Nagel and James Newman.
Topology: Experiments in Topology by Stephen Barr and Topology by James Munkres.
“What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men.” - G.H. Hardy
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