How to read a book in mathematics?

How is it that you read a mathematics book?
Do you keep a notebook of definitions? What about theorems?
Do you do all the exercises? Focus on or ignore the proofs?

I have been reading Munkres, Artin, Halmos, etc. but I get a bit lost usually around the middle. Also, about how fast should you be reading it? Any advice is wanted, I just reached the upper division level.

Answer

This method has worked well for me (but what works well for one person won’t necessarily work well for everyone). I take it in several passes:

Read 0: Don’t read the book, read the Wikipedia article or ask a friend what the subject is about. Learn about the big questions asked in the subject, and the basics of the theorems that answer them. Often the most important ideas are those that can be stated concisely, so you should be able to remember them once you are engaging the book.

Read 1: Let your eyes jump from definition to lemma to theorem without reading the proofs in between unless something grabs your attention or bothers you. If the book has exercises, see if you can do the first one of each chapter or section as you go.

Read 2: Read the book but this time read the proofs. But don’t worry if you don’t get all the details. If some logical jump doesn’t make complete sense, feel free to ignore it at your discretion as long as you understand the overall flow of reasoning.

Read 3: Read through the lens of a skeptic. Work through all of the proofs with a fine toothed comb, and ask yourself every question you think of. You should never have to ask yourself “why” you are proving what you are proving at this point, but you have a chance to get the details down.

This approach is well suited to many math textbooks, which seem to be written to read well to people who already understand the subject. Most of the “classic” textbooks are labeled as such because they are comprehensive or well organized, not because they present challenging abstract ideas well to the uninitiated.

(Steps 1-3 are based on a three step heuristic method for writing proofs: convince yourself, convince a friend, convince a skeptic)

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