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How to Learn Math and Physics

John Baez

October 14, 2024

Introduction

"How to learn math and physics" — the title is deliberately provocative. Everyone has to learn their own way. I don't know how you should learn math and physics. But presumably you came here looking for advice, so I'll give you some.

My advice is aimed at people who are interested in fundamental theoretical physics and the math that goes along with that. (By "fundamental" physics I mean the search for the basic laws concerning matter and the forces of nature.) If you want to do experiments instead of theory, or other of kinds physics like condensed matter physics and astrophysics, or math that has nothing to do with physics, my advice will be of limited use. You should still learn the basics I mention here, but after that you'll have to look elsewhere for suggestions.

Learning math and physics takes a whole lifetime. Luckily, it's a lot of fun... if you have a reasonably patient attitude. A lot of people read pop books about quantum mechanics, black holes, or Gödel's theorem, and immediately want to study those subjects. Without the necessary background, they soon become frustrated — or worse, flaky.

It can be even more dangerous if you want to plunge into grand unified theories, or superstrings, or M-theory. Nobody knows if these theories are true! And it's hard to evaluate their claims until you know what people do know.

So, especially when it comes to physics, I urge you to start with slightly less glamorous stuff that we know to be true — at least as a useful approximation, that is — and then, with a solid background, gradually work your way up to the frontiers of knowledge. Even if you give up at some point, you'll have learned something worthwhile.

This webpage doesn't have lots of links to websites. Websites just don't have the sort of in-depth material you need to learn technical subjects like advanced math and physics — at least, not yet. To learn this stuff, you need to read lots of books. I will list some of my favorites below, and also some you can get free online.

But, you can't learn math and physics just by reading books! You have to do lots of calculations yourself — or experiments, if you want to do experimental physics. Textbooks are full of homework problems, and it's good to do these. It's also important to make up your own research topics and work on those.

If you can afford it, there's really nothing better than taking courses in math and physics. The advantage of courses is that you get to hear lectures, meet students and professors, and do some things you otherwise wouldn't — like work your butt off.

It's also crucial to ask people questions and explain things to people — both of these are great ways to learn stuff. Nothing beats sitting in a cafe with a friend, notebooks open, and working together on a regular basis. Two minds are more than twice as good as one!

But if you can't find a friend in your town, there are different ways to talk to people online. In all cases, it's good to spend some time quietly getting to know the local customs before plunging in and talking. For example, trying to start a rambling discussion on a question-and-answer website is no good. If you've got physics questions, try Physics Stack Exchange. For research-level questions, try Physics Overflow. For questions about math, try Math Stack Exchange, or for research-level questions, Math Overflow.

For more free-wheeling discussions of math and physics, try Physics Forums.

There are also lots of interesting blogs and free math books online.

Finally, it's crucial to admit you're wrong when you screw up. We all make tons of mistakes when we're learning stuff. If you don't admit this, you will gradually turn into a crackpot who clutches on to a stupid theory even when everyone else in the world can see that it's wrong. It's a tragic fate, because you can't even see it's happening. Even bigshot professors at good universities can become crackpots once they stop admitting their mistakes.

To avoid looking like a fool, it's really good to get into the habit of making it clear whether you know something for sure, or are just guessing. It's not so bad to be wrong if you said right from the start that you weren't sure. But if you act confident and turn out to be wrong, you look dumb.

In short: stay humble, keep studying, and you'll keep making progress. Don't give up — the fun is in the process.

How to Learn Physics

There are 5 cornerstone topics that every physicist should learn: and in roughly that order. Once you know these, you have the background to learn the two best theories we have: and And once you know these, you'll be ready to study current attempts to unify quantum field theory and general relativity.

If this seems like a lot of work... well, it is! It's a lot of fun, too, but it's bound to be tiring at times. So, it's also good to read some histories of physics. They're a nice change of pace, they're inspiring, and they can show you the "big picture" that sometimes gets hidden behind the thicket of equations. These are some of my favorite histories:

Next, here are some good books to learn "the real stuff". These aren't "easy" books, but they're my favorites.

First, some very good general textbooks:

Then, books that specialize on the 5 cornerstone topics I listed above:

Classical mechanics:

Statistical mechanics:

Electromagnetism:

Special relativity:

Quantum mechanics:

These should be supplemented by the general textbooks above, which cover all these topics. In particular, Feynman's Lectures on Physics are incredibly valuable.

After you know this stuff well, you're ready for general relativity (which gets applied to cosmology) and quantum field theory (which gets applied to particle physics).

General relativity — to get intuition for the subject before tackling the details:

General relativity — for when you get serious:

General relativity — for when you get really serious:

Cosmology:

Quantum field theory — to get intuition for the subject before tackling the details:

Quantum field theory — for when you get serious:

Quantum field theory — two classic older texts that cover a lot of material not found in Peskin and Schroeder's streamlined modern presentation:

Quantum field theory — for when you get really serious:

Quantum field theory — so even mathematicians can understand it:

Particle physics:

While studying general relativity and quantum field theory, you should take a break now and then and dip into this book: it's a wonderful guided tour of the world of math and physics:

And then, some books on more advanced topics...

The interpretation of quantum mechanics:

This is a reasonable treatment of an important but incredibly controversial topic. Warning: there's no way to understand the interpretation of quantum mechanics without also being able to solve quantum mechanics problems — to understand the theory, you need to be able to use it (and vice versa). If you don't heed this advice, you'll fall prey to all sorts of nonsense that's floating around out there.

The mathematical foundations of quantum physics:

Loop quantum gravity and spin foams:

String theory:

How to Learn Math

Math is a much more diverse subject than physics, in a way: there are lots of branches you can learn without needing to know other branches first... though you only deeply understand a subject after you see how it relates to all the others!

After basic schooling, the customary track through math starts with a bit of:

and

To do physics you need calculus, multivariable calculus and linear algebra:

To do physics you also need to know about ordinary differential equations, which rely on calculus: Most of the fundamental laws of physics are partial differential equations; these rely on multivariable calculus: Multivariable calculus is a prerequisite for complex analysis, which is like 'calculus using complex numbers': Calculus is a prerequisite for real analysis, which you can think of as 'heavy-duty calculus': To do topology, which is the study of general spaces, it's good to already know the concept of 'continuous function', which you may have seen in calculus. You can study set theory and logic without any of the previously listed subjects: Linear algebra is a good warmup for abstract algebra, where you leave the real and complex numbers behind and consider more general number systems: not necessarily in exactly this order. Proofs become very important at this stage. You need to know a little set theory and logic to really understand what a proof is, but you don't even need calculus to get started on number theory: From then on, the study of math branches out into a dizzying variety of more advanced topics! It's hard to get the "big picture" of mathematics until you've gone fairly far into it; indeed, the more I learn, the more I laugh at my previous pathetically naive ideas of what math is "all about". But if you want a glimpse, try these books: I haven't decided on my favorite books on all the basic math topics, but here are a few. In this list I'm trying to pick the clearest books I know, not the deepest ones — you'll want to dig deeper later:

Finite mathematics (combinatorics):

Probability theory:

Calculus:

Multivariable calculus:

Linear algebra:

This is a great linear algebra book if you want to understand the subject thoroughly:

These books are probably easier, and they're free online:

Ordinary differential equations — some free online books:

Partial differential equations — some free online books:

Set theory and logic:

Complex analysis:

Real analysis:

Topology:

Abstract algebra:

I didn't like abstract algebra as an undergrad. Now I love it! Textbooks that seem pleasant now seemed dry as dust back then. So, I'm not confident that I could recommend an all-around textbook on algebra that my earlier self would have enjoyed. But, I would have liked these:

Number theory:

These are elementary textbooks; for more advanced ones read on further.

More Advanced Math

I'll start with some books on mathematical physics, because that's been one of my favorite subjects for a long time. Out of laziness, I'll assume you're already somewhat comfortable with the topics listed above — yes, I know that requires about 4 years of full-time work! —l and I'll pick up from there. Here's a good place to start:

It's also good to get ahold of these books and keep referring to them as needed:

Here's a free online reference book that's 787 pages long:

Here are my favorite books on various special topics:

Group theory in physics:

Lie groups, Lie algebras and their representations — in rough order of increasing sophistication:

Geometry and topology for physicists — in rough order of increasing sophistication:

Geometry and topology, straight up:

Algebraic topology:

Geometrical aspects of classical mechanics:

Analysis and its applications to quantum physics:

And moving on to pure mathematics...

Knot theory:

Homological algebra:

Ring theory:

Combinatorics:

Algebraic geometry:

I found Hartshorne's famous book quite off-putting the first ten times I tried to read it. I think it's better to start by getting to know some 'classical' algebraic geometry so you see why the subject is interesting and why it's called 'geometry' before moving on to delightful modern abstractions like schemes. So, start with this introduction:

Then try these:

Number theory:

Category theory:

I have always imagined that Paradise will be a kind of library. - Jorge Luis Borges

baez@math.removethis.ucr.andthis.edu
© 2019 John Baez

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