Why are higher/graduate level books discussing deep topics/results written if the author clearly doesn't want the reader to understand any of it?

Question

Have you noticed how almost anyone writing a text in the above mentioned category starts their work with examples and proofs fleshing out the minute details (stuff that you can do without the author doing it for you), but as you progress further, the details are lost and all you're left with are wordy, hand-wavy reasonings and appeals to the reader to fill in the pot-holes in the exercises? I understand that the reader has to walk the walk also, but it feels like no body is actually interested in teaching the stuff and most of the stuff is just left to the reader as an exercise. I've noticed this happens more frequently in the area of algebraic geometry than anywhere else. Look at the papers some of these people are writing too. The easy stuff is explained ad nauseam but when the going gets tough, you're presented with paragraphs of 'explanations' instead of just writing the full proof clearly in the language of mathematics like you're supposed to be. Anyway, would anyone else like to comment - in agreement or not - and share their thoughts on this? Or is it just me that feels this way?

Answer 1

A graduate maths book is not a novel.

What I mean by this is that it's not something you pick up and just read your way through, nor it is something where the difficulty level doesn't change as you're reading it (though there may, certainly, be experimental novels where the difficulty level changes as you progress). This is also where the graduate text differs from an undergraduate text where the difficulty level stays fairly constant throughout the text.

Why? Because graduate text books are written to serve different purposes. An undergraduate text serves as an introduction to a subject, and while it might go into more depth on some topics, those are still considered introductory topics. All the details are provided because they are an introduction and part of the aim is to show the reader how to think about these things for themselves. Like when reading an annotated chess game, the author is providing key insights into what's gone on and highlighting points of interest to help the reader. In a graduate text the reader is annotating as they go.

A graduate text can be a reference text: its job is to provide lots of results that are commonly known, or expected to be known, by researchers in the area so they can reference them. This can be for citing them in papers, and it can be for looking the result up and checking what the hypotheses are and how/if they can be weakened. These books are definitely not novels and are not intended for reading straight through.

A graduate text can also be a teaching text, in which case the expectation is that the reader will work out the details for themselves. Results are given with directions for thought and key insights provided, but the reader now has to really think about things. A chapter isn't bedtime reading, it's three weeks of work. But the payoff is significant; you get a deeper understanding of the subject and how everything fits together. And, crucially, you can now go back and read that chapter easily because all the things needed to understand are now in your mind as well as on the page.

I said these books aren't like novels. If they were, they'd be linear and most aren't; they have a core and then they move out to areas that the authors are most interested in, missing out whole swathes of possible material and leaving it to be covered by another book or other papers/teachers/internet articles. If a graduate text were a crime novel then the victim's name would be revealed somewhere in the middle and then there'd be four chapters on why the room the murder was committed in was especially interesting. The best way to view these books, really, is as a very large exercise in understanding a subject properly.

Finally, no-one wants the book to be not understood. The author has a certain amount of pride at stake, in that they want their book to be referenced and talked about. The published wants to make (some) money from this book. The readers want something valuable in return for their money.

Answer 2

I think that it's a part of good training to try to take the fundamentals and explain them in overly amount of detail. As students comfort with the subject increases, they should be asked to fill in more and more details on their own. Now it's only natural to get stuck or bogged down on some detail, but that's what professors and friends are for - to discuss the subject. Even if you get stuck and can't figure it out, but you practice communicating what you're stuck on and why you're stuck you'll form better relationships with the people around you and learn as you go.

Plus it's nigh impossible to include absolutely all of the details in an argument. Try to prove the crossing number of the trefoil is three. I bet in full rigor it's at least 10-15 pages. This is absurd, it's best to communicate the idea rather than full rigor ~ but if you're asked to supply the rigorous argument one should be able to.