DiBeos

Have you ever opened a math book and found yourself banging your head for hours just to get through 2 or 3 pages? Usually, that’s how it goes: Lemma. Theorem. Proof. Definition. Proposition. Remark. Corollary. Lemma. Theorem. Proof. Definition. Proposition. Remark. Corollary. And so on… Next chapter. Repeat. No intuition. No context. No concrete examples. Or at least, no examples that are beginner-friendly.

Here’s my problem with most advanced math books: they’re way too abstract from the start.

Don’t get me wrong, abstraction is beautiful, and it is definitely necessary if a student wants to deeply master any subject. Actually, I’d go even further and say that abstraction is exactly what makes math elegant and powerful. But if a book is meant to teach, then it should help the reader build up to that abstraction, not start there and expect them to catch up on their own.

You’ll often hear people say that a book is great because it’s ‘rigorous’ or ‘complete’. But ‘rigorous’ doesn’t mean ‘good at teaching the subject’. Just because someone’s a brilliant mathematician doesn’t mean they know how to teach.

Let’s take as an example the book “Algebraic Topology” by Allen Hatcher. 

Now, let me be absolutely clear: Hatcher is a brilliant mathematician. The book is rigorous, the subject is beautifully exposed, and it covers a vast amount of material with very complete precision. And it’s available for free, which is extremely generous, and it’s often considered THE reference for anyone who’s serious about algebraic topology.

Personally, I really like it, but as a reference, not as a first learning resource. It’s the kind of book I love having on my iPad when I need to remember a definition, look up a theorem, or check the precise formulation of a result.

But as a book to learn from for the very first time? As a beginner in the subject? I mean, it’s a disaster. It is abstract from the get go, and because of that it’s extremely hard to follow. Definitions flow into lemmas, which flow into theorems and proofs, and everything is so tightly packed that someone who is just starting to learn about algebraic topology would take hours just to kind of get some of the ideas. 

The exercises are great, but they assume a level of intuition and context that hasn’t yet been built up in the text itself. The concepts are introduced formally, but the why (the motivation) is just ignored. 

This isn’t just my opinion, by the way. Many students online say the same thing. For instance, check out this Reddit thread, where the original poster wrote:

Anyone else find Hatcher's Algebraic Topology unreadable?
by inmath

“Ironically the exposition is supposed to make it easier to understand, but 559 pages, each of them being a literal wall of text doesn’t make it an easy read by any means… It’s difficult to distinguish the key results from his ramblings, trivial deductions from important ones…”

Others commented:

“The exposition could be much more economical and clearer.”

And

“It is the most beautiful unreadable book ever self-published.”

So it’s not that the book is bad, but just like most advanced math books, it assumes way too much from the reader. 

This is exactly what I mean when I say that rigor alone doesn’t teach. Great math books need to start with intuition, context, and simple examples, and from there build up to formalism and proofs. Hatcher is an extreme example of course, but it does illustrate how many books invert that order, and that’s why so many students struggle with it.

And it makes sense, because being a good mathematician, or researcher, doesn’t necessarily mean being a good teacher. Of course, the opposite is also not true. But when you’re looking for a resource to study any subject (especially in advanced math) you should not look for the best mathematician or the best teacher in the world. What you need is the sort of material that both teaches well and has a solid grasp of the subject, so it can transmit the right amount of information accurately and clearly.

These are the steps you should look for in any material:

1. Intuition: it needs to start as simply as it can. You need to build intuition first.

2. Concrete Examples: it must give you the motivation and concreteness for why and how to use what you are about to learn, even if it’s only in the context of pure math.

3. Rigor: Once you’ve built solid intuition and worked through concrete examples, that’s when rigor becomes not just important, but essential. It’s what makes math math. It’s the ability to justify every step, to prove results and to speak a universal/logical language. If you want to really own a topic (like, really master it), you need to study rigor.

4. Practice (exercises): I honestly do not understand why a book would give you exercises without solutions. It makes absolutely no sense to me. I would go even further and say that it would be preferable for a book to contain not only exercises, but also their detailed solutions. That’s the way students learn: you need to try on your own until you run out of ideas on how to solve the exercise, then look at the detailed resolution in the book, and then try it again some time later on your own, without looking at the resolution. That’s the best way to practice. 

It is a tricky task, but if you find books and resources like that, you will learn advanced math REALLY well and REALLY fast.

Because we noticed this problem, we’re trying to fix it through our Youtube channel and this website. That’s why every single week we publish a video and a PDF file, completely free, that contains a lot of extra useful information. We try to follow this structure: 1. Intuition; 2. Concrete Examples; 3. Rigor; 4. Practice.

Let us know how we can improve our videos and PDFs to help. They are not perfect, but we are committed to gradually improve their quality for the next 10-20 years. So, please leave a comment or email us.