Sunday, 26 January 2014

soft question - Why do so many textbooks have so much technical detail and so little enlightenment?

I also suffer from this problem -- I used to learn best from books, but in grad school, I'm having real trouble finding any book I can learn from in some subjects. There are a few reasons for this sad state of affairs that come to my mind. I'll list them first and expand on them below.



  1. Providing real enlightenment well is very, very hard, and requires a very intimate relationship with a subject.


  2. Different mathematicians need vastly different motivations for the same subject.


  3. Mathematics needs to age before it can be presented well.


  4. Good writing is not valued enough in the mathematical community.


The first of these is true to such a strong degree that it surprises me. Even for well-established subjects, like undergraduate mathematics, where there are a million mathematicians who know the subject very well, I find that all the really good books are written by the true titans of the field -- like Milnor, Serre, Kolmogorov, etc. They understand the underlying structure and logical order of the subject so well that it can be presented in a way that it basically motivates itself -- basically, they can explain math the way they discovered it, and it's beautiful. Every next theorem you read is obviously important, and if it isn't then the proof motivates it. The higher-level the subject, the fewer the number of people who are so intimate with it that they can do this.
It's interesting how all the best books I know don't have explicit paragraphs providing the motivation - they don't need them. (Of course there are exceptions -- some amazing mathematicians are terrible writers, and there are people with exceptional writing ability, but the point stands).



Regarding the second point, different people want completely different things for motivation. The questions that pop into our heads when we read the theorems, the way we like to think, the kind of ideas we accept as interesting, important, etc., is different for all of us. For this reason, when people try to explicitly describe the motivation behind the subject they almost always fail to satisfy the majority of readers. Here, I'm thinking of books like Hatcher, Gullemin & Polluck, Spivak, etc., where some people find that they finally found the book that explains all the motivation perfectly, and others are surprised at the many paragraphs of text that dilute the math and make finding the results/proofs they want harder and reading slower.
At the same time, the amount of effort each of these authors must have spent on organization of their book seems absolutely immense. For this reason, unless there are 50 books written on a subject, the chances that you will find a book that seems well-motivated for you are low.



The third reason is simple: it takes time for a new subject to stop being ugly, for people to iron out all the kinks, and to figure out some accepted good way to present it.



Finally, it seems to me that good writing, especially expository writing, is not particularly valued in the community, and is valued less now than it was before. Inventing new results seems to be the most respectable thing to do for a mathematician, teaching is second-best, and writing has the third place. People like Hatcher & co. seem to be rare, and I don't know of many modern titans of mathematics who write any books at all, especially on a level more elementary than their current research.




So what do we do? I think what algori said in his answer is the only way to go.

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