# Learning math on your own

## A guide, heartfelt and opinionated

I’ve spent a few years trying to learn math, and I thought I would write down some advice for others trying to do the same, in case it might be useful.

*Disclaimer: I’m not a **Fields medalist, nor even a research mathematician. I wish Fields medalists *would* write guides like this, but they haven’t, and it took me a lot of time and effort to learn all this, which I don’t think you should have to spend. Perhaps I’ll be embarrassed in a few years by this guide, but then, perhaps it’ll help someone.*

*Why learn math?**A bird’s-eye view of math**Advice**Books, etc.*

# Why learn math?

There are two different reasons why you might want to learn math:

- you might be interested in some field that uses math (science, engineering, programming, etc.) and want to use it
- you might have been enchanted by some surprising or interesting piece of math and want to learn more

These are both valid reasons; don’t let anyone convince you that one of them is better than the other. If you start with one of usefulness or beauty, you may well find that after studying for a while the other one captures you too.

Let me give you some encouragement. Mathematics is (as all mathematicians know) one of the greatest creations of humanity. It’s one of the very few in which continuous improvement happens in an unambiguous sense. It has a monopoly on certain truth, and indeed, after learning math you will see that ‘logical’ arguments can only take place within mathematics. Also, the culture of mathematics is, relatively, very charming, with a noble history and strong sense of fair-mindedness.

# A bird’s-eye view of math

Math at school tends to be about calculating answers, and so people get the impression that this is what all math is about. It’s not; math is really about understanding interesting things like numbers, shapes, motions, randomness, and processes. Math at university tends to be about proving facts about these topics, although again there’s more to math than this.

*A thumbnail sketch of the difference between calculating and proving: when you calculate you tend to answer questions like, *what is 2 ⨉ 3?* When you prove you tend to answer questions like,* is 6 prime?

The school curriculum is designed to build up to calculus. You learn about numbers and shapes, then about using symbolic variables and the idea of functions, and finally about the most important functions (exp, log, sin, cos). This makes you ready for calculus (which is indeed quite interesting).

The university curriculum is usually broader and deeper, and contains many different subjects which are related in interesting ways. There is a traditional division of these subjects into the fields of *geometry*, *algebra*, and *analysis* — very roughly, the mathematics of space, time, and motion, respectively.

# Advice

First of all, believe in yourself. Just because you didn’t do well at school, or somebody told you you’ll never understand it, or you’ve always struggled, doesn’t mean you can’t do it. Of course, it will take time and effort.

Question: How do professional mathematicians learn new things?

Answer: Slowly and with difficulty, just like amateur mathematicians.

If you are learning on your own, the main way to learn is books. The last section of this essay is all about specific books, so, some general advice:

- Don’t feel you have to read math books from cover to cover. I’ve only read one math book this way! It’s normal to skip around, omit things you don’t understand or aren’t interested in, and focus on what captivates you.
- Don’t buy or check out loads of books; instead, pick a few that seem good to you, buy them, and immerse yourself in them.
- Follow your own goals. You don’t need to ‘learn everything’, or even everything in a given book.
- Buy books used. Some of the main mathematical publishers (Springer, Cambridge) now print ‘new’ copies of books that you might order from e.g. Amazon using ‘digital printing’, which has a nasty grainy look to it, feels cheap, and is hard to read. The first printings will have been done nicely, and you can get those by ordering used copies.

Of course, you cannot learn from books alone. If you can, attend lectures, go to math circles/groups/seminars, socialise with other mathematicians. In any case, invent your own problems and try to solve them, and do as much as you can to interact with mathematical topics physically i.e. using models, toys, games, programs, drawings, etc.

*A note on mathematical notation and terminology:*

An important part of learning math is getting to grips with the notation and the terminology. Unfortunately mathematical notation and terminology is not exactly perfect, but it does have some lovely aspects and it’s slowly getting better. A brief primer: mathematicians like to treat any concept, no matter how abstract, as an ‘object’, and label it with a single letter (*H*, say); often they run out of normal letters and start using Greek, Hebrew, or German letters. Often this ends up with certain objects receiving conventional single-letter names e.g. the famous numbers π or *e*. Many things in mathematics are also named after their discoverers, which becomes frustrating after a while (‘the Euler characteristic of the Calabi-Yau threefold’). In order to organise their books, mathematicians call the more important facts ‘theorems’, and the less important ones ‘lemmas’.

# Books, etc.

So, what should you read?

The very best introduction to math is *1089 and All That** *by Acheson. This is an amazing little book that gives you a glimpse of practically every feature of math and is fantastically written and very entertaining.

Another incredible book is ** Indra’s Pearls** by Mumford, Series, & Wright (now sadly rather expensive to buy, so try your local library). Their idea was to explain a piece of completely grown-up, cutting-edge mathematics but assuming you didn’t touch mathematics after school. There’s no book like it — beautifully written, beautifully illustrated, beautifully made.

If you’re struggling with the school curriculum, try ** Measurement** by Lockhart. He essentially covers all of school mathematics (including calculus) after arithmetic, having thought long and hard about how to explain it after quitting his job as a research mathematician at a university and teaching in a school himself for 10+ years. (I find some of his philosophical opinions overbearing, though.)

Once you get interested in the university curriculum, let ** Mathematics: Its Content, Methods, and Meaning** (ed. Aleksandrov, Kolmogorov, & Lavrentev) be your guide. This is a deeply touching book, a long survey of all of mathematics written by 20 of the best Russian mathematicians of the mid-20th century. It’s comprehensive — basically it covers an entire undergraduate mathematics degree — yet, lucid — for many topics, it has the clearest explanations I’ve seen.

Some other ideas. There’s a famous series of books, all written by mathematicians associated with the University of Göttingen in Germany in the early 20th century, which are some of the most influential pedagogical books in mathematics (though now considered a little old-fashioned). Give any of them a try; they are all rather noble. In increasing order of difficulty:

by Rademacher & Toeplitz*The Enjoyment of Mathematics*by Courant & Robbins*What is Mathematics?*by Hilbert & Cohn-Vossen*Geometry and the Imagination*by Pólya*How to Solve It*by Klein*The Development of Mathematics in the 19th Century*

Three other books I’d like to point out at the beginner’s level are ** Crocheting Adventures with Hyperbolic Planes** by Taimina, if you crochet,

**by Acheson, if you like physics, and**

*From Calculus to Chaos***by Cromwell, if you find this sort of thing appealing:**

*Polyhedra*Now onto the bulk of mathematics books. Most modern mathematics books use notation taken from fields called ‘set theory’ and ‘mathematical logic’. It’s very simple, but when you first encounter it you might think you are learning ‘real mathematics’. Actually, this notation is more like a language — a way of expressing mathematics — and not the actual content of mathematics itself. I think a particularly good place to learn it is the first chapter of ** Galois’ Dream** by Kuga (an intruiging book anyway), but you can pick it up anywhere.

Mathematics textbooks tend to be terrible. Mathematicians are always complaining about how dry and boring they are, how hard to read, how lacking in examples and pictures and motivations and applications, and so on. The key thing to understand is that textbooks on a given topic bear a similar to relation to that topic as do manuals for electronic gadgets to the gadget: they give a dry, correct description of how it works, but to really understand it you’ll have to play with it yourself and explore its possibilities.

I won’t recommend any particular textbooks; instead I’ll point you towards four other lists of books, which can help you find a good book on any particular topic you might be interested in:

- The Chicago undergraduate mathematics bibliography
- Viktor Blåsjö’s reviews on Amazon
- Roy Smith’s reviews on Amazon
- Owl’s list of graduate textbooks

Instead, I’ll tell you a bit about different authors’ styles, and recommend some books that are special in one way or another.

One interesting feature of 20th century mathematical writing is the contrast between the Russian and French styles. The French tended towards formalism & completeness, while the Russian tended towards intuition & casualness. This contrast can be nicely seen in two series of books that they published: the French ** Elements of Mathematics** by Bourbaki (a fictional author) are exhaustive, logical, from-the-ground-up expositions of advanced mathematics, while the Russian

*Encyclopaedia of Mathematical Sciences**contains discursive, wide-ranging, pleasant overviews of many fields of mathematics.*

As for particular authors, some to seek out are: Miles Reid, humorous, insightful, & well-illustrated; Jean-Pierre Serre, purveyor of elegant jewels; Vladimir I. Arnold, original, opinionated, and intuitive; Igor R. Shafarevich, a very humane author; Pierre Samuel, charming and enthusiastic; John H. Conway, delightfully different.

A few more particular recommendations. To learn a bit more about the culture, philosophy, history, and meaning of mathematics, Rota’s ** Indiscrete Thoughts**. For a fascinating panorama of what mathematics looked like a century ago, Klein’s

**. A dated but dazzling & immensely cultured tour, Weyl’s**

*Lectures on Mathematics***. Finally, if you become interested in the cutting edge, browse the enormous**

*Symmetry***.**

*Princeton Companion to Mathematics*Online, read all the essays by William Thurston, a beautiful soul, linked here. You can find thousands of insights at mathoverflow, and ask any questions you may divise at the Math StackExchange. And lastly, for more book reviews, see the MAA, Zentralblatt, and Math Reviews.

If you found this useful, I’d be grateful to know — please comment!