Advice for Learning Math

Introduction

The best way to learn math is to pursue a degree. Obviously, not everyone has the opportunity to do that. The next best thing is a solid path for self-study, which tends to be quite book-heavy. Of course, if for some reason burning the midnight oil reading math textbooks isn’t your idea of a fun time, then I will jingle some shiny keys in the pop-math section at the end of the article that you will hopefully enjoy. Joking aside, there’s a lot of fantastic (and accurate) pop math out there for a general audience: this article has plenty of links to 3blue1brown (my favorite video if his is the hardest problem on the hardest test), who I highly recommend.

The Essentials

Discrete math and an introduction to proofs is a much better gateway to “real” math than the standard calculus route. It requires only a halfway decent grasp of high school algebra (which you can brush up on from Khan Academy).

The ideal discrete math course should provide a strong foundation in all the essentials:

• Boolean algebra
• Set theory
• Modular arithmetic
• Equivalence relations
• Group theory
• Graphs

as well as the elementary proof techniques like induction, contradiction, and so on. You absolutely don’t need to master all of these topics, but covering the basics and getting a rough feel for each of these different objects will be a huge benefit for studying any other area of math.

To get started, I recommend Discrete Mathematics: An Open Introduction by Levin or Book of Proof by Hammack. If you want to try something interactive, the natural number game is an introduction to proofs using Microsoft’s proof assistant, Lean. Lean is nice because it helps to break down proofs step by step in a super organized way.

Linear Algebra

After you have a solid foundation in discrete math, I heavily recommend studying linear algebra, which is an essential backbone in virtually every area of math. Linear Algebra Done Right by Sheldon Axler lives up to its name. Most university linear algebra courses are taught with engineers in mind, so the intuitive explanations take a back seat to more applied topics. Axler’s book takes a more “high brow” view of the subject, and gives a far more intuitive approach from first principles. Any linear algebra class should be supplemented with 3blue1brown’s linear algebra series, which helps to visualize difficult concepts like linear transformations (as actual transformations!) and span. Linear algebra has a steep learning curve, but a lot of the challenge is just peeling back the rigorous, abstract definitions and finding the deeply intuitive concepts underneath.

For the more applied parts of linear algebra, I would recommend Gilbert Strang’s Linear Algebra as a reference text.

You’ll then be equipped to understand his videos on machine learning.

Abstract Algebra

The final essential piece of essential math I recommend is abstract algebra (or just “algebra” to mathematicians), which essentially abstracts over many patterns that appear all over mathematics. For example, we can add, subtract, and multiply matrices, just like we can add, subtract, or multiply integers; that’s because matrices and integers are both examples of structures called rings. I often think of abstract algebra as the backbone of math.

There are three major objects you’ll encounter in a standard abstract algebra class:

• Group theory, which is the most “basic” (and therefore most common) structure. Group theory is the mathematical study of symmetry, which provides the foundation for virtually all of modern mathematics and has major applications in physics, chemistry, cryptography, and many other areas. Here’s an excellent video to motivate the subject.
• Ring theory, which generalize the notion of “numbers” to anything that has operations that behave like addition, subtraction, and multiplication. If you study number theory, you may see that a lot of the proofs from number theory can be generalized to rings. They can also be used to generalize vector spaces to a broader class of object called modules.
• Fields, which we often think of as a special kind of ring that has addition, subtraction, multiplication, and division. Fields also have a deep relationship with vector spaces. (In fact, fields are vector spaces, just maybe with only one dimension.)

I’d recommend reading the through Algebra: Chapter 0 by Aluffi. Many people enjoy the book’s tone, and it teaches Algebra in a much more “modern” way than some of the standard texts (like Dummit \& Foote, which is the standard text in most universities). Aluffi introduces abstract algebra from the perspective of category theory, which helps organize abstract algebra much more neatly. You can supplement your reading with with Socratica’s abstract algebra playlist.

Graph Theory

Graph theory is a branch of math that studies graphs (also known more accurately as “networks” – they are totally different from the graphs you study in highschool algebra.). In graph theory, a graph is essentially a flow chart: there are “nodes” or “vertices” which are connected by lines called “edges”.

Graph theory is applicable to everyday life and essential to computer science. Maybe you want to schedule tasks efficiently, find the most efficient route from one place to another, or solve puzzles like “who is the liar”. In fact, graph theory was invented to solve an 18th century riddle asking if it was possible to find a path through the city of Königsberg that crosses each bridge exactly once.

Trudeau’s Graph Theory book is one of the standard texts, and reads quite nicely. Deistel’s Graph Theory is a faster, more comprehensive introduction with less focus on applications.

Intermediate Topics

Number Theory

At this point, there are several avenues open to you. If you liked group theory and some of the early topics from Beachy and Blair, I recommend pursuing number theory, which is the enormous field studying patterns in the integers. A Friendly Introduction to Number Theory by Joseph H. Silverman is a wide introduction and an excellent place to start; if you want a more advanced introduction to the subject (or a second course), A Classical Introduction to Modern Number Theory by Ireland and Rosen is a good choice.

My undergraduate capstone project on elliptic curves drew heavily from Rational Points on Elliptic Curves by Joseph H. Silverman and John T. Tate. Elliptic curves are fascinating objects that have a central place in modern mathematics, and were used to settle Fermat’s Last Theorem, which was math’s biggest unanswered problem for over 350 years until it was solved with elliptic curves. The Birch and Swinnerton-Dyer Conjecture is an open problem about elliptic curves; as a Millennium prize problem, solving it would net a $1,000,000 prize.

Theoretical Computer Science

With a crash course in proofs, set theory, and graph theory, you can jump into Introduction to the Theory of Computation by Michael Sipser, which is an excellent read. I recommend that you do exercise 28 in chapter 5, which covers Rice’s Theorem, a very useful theorem in theoretical computer science. (I’m honestly surprised Sipser put such little emphasis on it!)

I also recommend reading Scott Aaronson’s excellent blog Shtetl-Optimized, particularly Five Worlds of AI, the 8000th Busy Beaver number eludes ZF set theory, and Logicians on safari.

Calculus

Calculus is the star of every ambitious highschooler’s senior year, and is the reason that I became a mathematician. It is the study of change, specifically “instantaneous change”. It has applications virtually everywhere: business, engineering, physics, computer science (particularly machine learning), biology, statistics, and everywhere else. You’re best served by picking up a copy of Calculus: Early Transcendentals by Stewart and supporting it 3blue1brown’s calculus series; if you’re a bit ambitious and want to see some of the nitty-gritty details that often get swept under the rug, consider adding on Calculus by Spivak. When you get to multivariable calculus, add on Khan Academy’s lecture series. (You might recognize the narrator.)

Real Analysis

Calculus is a treat, but eventually the time comes for every mathematician to discover out how the sausage is made. You may have noticed that the calculus textbooks were very light on proofs. (This is because most calculus students are physicists or engineers; the most important part of being a mathematician is learning to hate those “people”.) Unfortunately, the subject can’t meaningfully advance until you build up a real, rigorous foundation for the calculus on the real numbers: this field is called real analysis, and my two preferred books on the subject are Analysis I by Terence Tao and Understanding Analysis by Stephen Abbott. After you’re done with those and want to study calculus in a more general setting, the standard text is Calculus on Manifolds by Spivak.

Topology

After the beauty of calculus has been thoroughly shredded by real analysis, you can either learning some machinery to abstract over and simplify the heinously ugly proofs of important theorems (such as the intermediate value theorem). Abbott’s book points towards the subject of topology, which is the abstract study of space and continuity, which has applications all over physics and advanced math. Introduction to Topological Manifolds by John Lee is a nice read; Topology by Munkres is the standard text in the field; it’s a bit terse but worth referring to as a secondary text.

Complex Analysis

If you find yourself pining for the days before real analysis where you could be forgiven for believing fairy tales like “every continuous function is differentiable” and “every differentiable function is infinitely differentiable and converges to its Taylor series”, consider studying complex analysis, where those statements are actually true. Complex analysis the extension of calculus to functions of a complex variable, and is often considered the most beautiful field of math. Complex analysis has applications in physics, engineering, and elsewhere. You can get a visual understanding from Visual Complex Analysis by Needham; for a more rigorous standard introduction, I’ve heard good things about Complex Analysis by Stein and Shakarchi.

Differential Equations

A natural next step after learning calculus is solving differential equations; I recommend learning about separable ordinary differential equations, the Laplace transform, the eigenvalue method, and the Fourier transform. The theory of differential equations is actually something I know very little about, so I can’t make any solid recommendations there.

Probability

All knowledge degenerates into probability.
— David Hume

Now that you’ve learned calculus, it’s a good time to jump into probability and statistics, which are used literally everywhere. My textbook was An Introduction to Mathematical Statistics and its Applications by Larsen and Marx, which only uses calculus.

The probability book with the real meat is Probability Theory with Examples by Durrett, which teaches probability using measure theory, which is a much more powerful foundation and really gets to the core of probability. Durrett gives a crash course on measure theory and hits all the major topics in probability theory, but is a bit terse.

The Hard Stuff

Galois Theory

The crowning theorem of an introductory abstract algebra sequence is Abel-Ruffini and Galois theory. Galois theory, like complex analysis, is renowned for its beauty and power. The motivating question for Galois theory is determining why quadratic, cubic, and quartic polynomial all have explicit formulas for their roots, but the general quintic polynomial doesn’t. The answer – the Abel-Ruffini theorem – just kind of tumbles out at the end of the course, and by then you won’t even care: it pales in comparison to the fundamental theorem of Galois theory and the Galois correspondence. This is one of the final topics in Aluffi, so just keep reading and you’ll get a solid overview.

Category Theory

After seeing the isomorphism theorems proven multiple times and seeing the fundamental theorem of Galois theory, I recommend reading Category Theory In Context by Emily Riehl to get “a bird’s eye view of math”.

Algebraic Geometry

If you’ve gotten this far and still want more, look at the most recent version of The Rising Sea by Ravi Vakil and start sending out PhD applications.

Pop Math

• For high-quality and accessible videos about math, I recommend 3blue1brown (especially the hardest problem on the hardest test) and some of Veritasium’s videos. Numberphile has some good videos too, but their -1/12 video is a real stinker.

• The natural number game is quite fun and a way to do short, guided, and rigorous mathematical proofs online using Lean, a popular proof assistant.

• The only Millenium Problem to be solved, The Poincaré Conjecture, was proven by Grigori Perelman in 2003. Perelman famously declined the $1,000,000 prize and quit mathematics shortly after. The story of the proof and the drama on the periphery was famously captured in Manifold Destiny, a controversial New Yorker article.

• In a similar vein, I recommend this phenomenal article about Alexander Grothendieck — the luminary who revolutionized the now-booming field of algebraic geometry and perhaps the greatest mathematician of the 20th century — and his abrupt disappearance to eat dandelion soup in the Pyrenees.

• If you like books but not textbooks, I loved Fermat’s Enigma by Simon Singh, about a 350 year-old math problem and Andrew Wiles’s proof in 1994. The book is famous for its quotation of Andrew Wiles’s moment of triumph:

  “Suddenly I had this incredible revelation. … It was so indescribably beautiful; it was so simple and so elegant. I couldn’t understand how I’d missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I’d keep coming back to my desk looking to see if it was still there. It was still there. I couldn’t contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much.”

• Knot theory is a branch of topology that studies knots, which has applications to string theory and the study of DNA.

• One day, a PhD student named George Dantzig showed up late to a lecture and saw that his teacher had written two problems on the board; Dantzig thought they were homework problems, and began working on them after lecture. He noticed that they were “a little harder than usual”, but managed to solve both after a few days and handed in his solutions. Six weeks later, his professor told him that those “homework” problems had both been major unsolved problems in statistics – until Dantzig solved them! Dantzig’s story was used as inspiration in the film Good Will Hunting.

• You can’t comb all the hair on a coconut without making a cowlick
• Any Rubik’s cube can be solved in at most 20 moves
• The Rubik’s cube group needs only 2 generators
• You can’t design a regular expression that will parse only valid HTML, Python, or regular expressions
• Subgroups of finitely generated groups may not be finitely generated
• See also: many other theorems that disappointed mathematicians

Fun Calculus Facts

• There are functions that are continuous everywhere but differentiable nowhere. In some sense, most continuous functions are like this
• Every continuous function on a closed interval is integrable. (This means that there are functions which are only differentiable a finite number of times)
• We can at least approximate any continuous function with polynomials
• There are some functions that are differentiable on some interval but not integrable on that interval, like \(f(x) = 1/x\) on \((0, 1)\).
• Even functions that are infinitely differentiable might not converge to their Taylor series
• However, complex-differentiable functions are much nicer: if a complex function is differentiable on a disk, then it is infinitely differentiable and converges to its Taylor series. This follows from the fact that we can integrate a complex-differentiable function to get its derivative
• In stark contrast to the single-variable case, partial derivatives can exist at points where the function is not continuous

• If an infinite series doesn’t converge absolutely, you can rearrange the terms to get the series to diverge or converge to any value you want

• The greatest mathematician that never lived
• Mathematical “urban legends”
• Too simple to be simple
• Widely accepted mathematical results that were later shown to be wrong
• Harvard’s Math 55

Philosophy of Math

Math is fairly mysterious, even to experts. Why is mathematics so good at solving problems across such a wide variety of domains? In particular, why is it such a useful tool for the natural sciences like physics, biology, chemistry, and so on? One famous essay raises this question:

There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.” Naturally, we are inclined to smile about the simplicity of the classmate’s approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense.

These are the opening paragraphs to The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner, and is expanded on in a follow up The Unreasonable Effectiveness of Mathematics by R. W. Hamming. (The guy who made Hamming codes.)

The philosophy of math stretches all the way back to Plato, and there are a handful of books an articles worth reading to understand the history and debates of the philosophy of math:

• Here’s the Stanford Encyclopedia of Philosophy entry on The Philosophy of Math
• Philosophy of Mathematics: Selected Readings, edited by Paul Benacerraf and Hilary Putnam

The Numberphile Video

• If you want an explanation of that abominable -1/12 Numberphile video, see here for an overview and here for an explanation of the Riemann zeta function and analytic continuation.