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Hello,

I have a bit of a general question about math books. I get the feeling that in a lot of math books, the organization for the theorems and lemmas are not explained well (ex. Topics in Algebra Herstein, Linear Algebra Hoffman).

So I'm curious what you all think about the advantages/disadvantages of this. Because on the one hand, it forces one to create the connections for themselves. But on the other hand, it makes the presentation less clear and interesting.

In the classes that I have taken, I find that I have to constantly bother the Professors to motivate what is going on instead of just going along and providing the stripped down proofs.

Furthermore, I am wondering what you all do to rectify this problem. Last semester, I had a great opportunity to speak with a professor about the reading that I did each week but it makes it much harder to study things independently when there is nobody to give perspective on why we are doing what we are doing. Any thoughts would be appreciated!

--Alex

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3 Answers

I think you are doing exactly the right thing in demanding that your professor motivate the theorems and lemmas, and to show the ideas for the proof and not just the proof itself. Too often, mathematical writing is stripped of the motivation, leaving bare the logical structure as if it arose out of nothing.

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I think the advantages and disadvantages depend on the purpose of the book and on the specific subject area. Some books are reference books, packed with results, and it's okay to be light on the motivation there. However, if you want or expect people to follow your proofs line-by-line, a little motivation goes a long way. Personally, when I'm reading (or listening to) a proof, I cringe whenever I see a magic number or function pulled out of thin air that "just happens" to work. Tell me what you were looking for and how you found it. You don't have to go into excruciating detail, but a little context makes it much clearer, and it helps make sure the reader knows how to find these "magic" functions in his or her own research.

I think almost any field of math would benefit from a conversational introductory book. Why are certain theorems important? Which ones are routine but tedious, and which ones rely on very novel ideas? What are some good heuristics in the field, and when do they break down? Which theorems generalize nicely, and for which is it not known? Reading such a book would be like discussing the field with a knowledgeable colleague, and I think it would encourage mathematicians to learn more math outside their field.

Come to think of it, the Princeton Companion achieved this somewhat. Now let's expand each article to a whole book!

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The single best effort I've seen is David Williams' "Probability with martingales" and "Weighing the odds". The theorems, lemmas, etc, are given movie-style ratings to draw attention to which are `big' and which are curiosities, and which fall between.

...makes it easy to skim, and enjoyable to study.

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