I have looked at Tao’s book on Measure Theory, and they are perhaps the best math books I have ever seen. Besides the extremely clear and motivated presentation, the main feature of the book is that there is no big list of exercises at the end of each chapter; the exercises are dispersed throughout the text, and they are actually critical in developing the theory.
Question: What are some other math books written in this style, or other authors who write in this way? I am open to any fields of math, since I will use this question in the future as a reference.
That was the question; the following is just why I think Tao’s style is so great.
- When you come to an exercise, you know that you are ready for it. There is no doubt in the back of your mind that “maybe I haven’t read enough of the chapter to solve this exercise”
- Similarly, there is no bad feeling of “maybe I wasn’t supposed to use this more advanced theorem for this exercise, maybe I was supposed to do it from the basic definitions but I can’t”. It makes everything feel “fair game”
- It makes it difficult to be a passive reader
- It makes you become invested in the development of the theory, as if you are living back in 1900 and trying to develop this stuff for the first time
I think you can achieve a similar effect with almost any other book, if you try to prove every theorem by yourself before you read the proof and stuff like that, but at least for me there are some severe psychological barriers that prevent me from doing that. For example, if I try to prove a theorem without reading the proof, I always have the doubt that “this proof may be too hard, it would not be expected of the reader to come up with this proof”. In Tao’s book, the proofs are conciously left to you, so you know that you can do it, which is a big encouragement.
Vakil’s notes on Algebraic Geometry http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf are written in the same style. A nice contrast to the terse, traditional nature of the standard reference, Hartshorne https://www.springer.com/gp/book/9780387902449