The best notation is no notation; whenever possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that you are explaining the subject to a friend on a long walk in the woods, with no paper available; fall back on symbolism only when it is really necessary.
I’d have to agree. I see this as one of the strongest aspects of, for example, Terence Tao’s expository style. His latest post on relativization is a perfect example; Tao is a master at recognizing when technical details would obscure his exposition and when they are necessary. A related point:
To explain something you must know not only what to put in, but also what to leave out; you must know when to tell the whole truth and when to get the right idea across by telling a little white fib.
This seems to be a much more common practice in science exposition; expositors of physics are never telling you all the details of, for example, the mathematics of quantum mechanics. Nearly everything is a convenient analogy. Unfortunately, I think this triggers a backlash on the part of mathematicians who are concerned with communicating “the whole truth.” Again, Terence Tao does this very well.
I have taught courses whose entire content was problems solved by students (and then presented to the class). The number of theorems that the students in such a course were exposed to was approximately half the number that they could have been exposed to in a series of lectures. In a problem course, however, exposure means the acquiring of an intelligent questioning attitude and of some techniques for plugging the leaks that proofs are likely to spring; in a lecture course, exposure sometimes means not much more than learning the name of a theorem, being intimidated by its complicated proof, and worrying about whether it would appear on the examination.
This is an interesting point. I think trying to understand undergraduate mathematics without experience with problem solving can be problematic, although I can’t speak from personal experience. It seems to me that you don’t really understand or care about a theorem until you’ve used it to solve at least one problem you were aware of before you learned the theorem. For example, I feel uncomfortable with most of the material I learned in my functional analysis class last semester because I have yet to solve an independently interesting problem using, for example, the Hahn-Banach theorem. Probably the only idea I really retained from that class is the use of well-behaved dense subspaces, since density arguments have a wide range of applicability.
Mathematics – this may surprise or shock you some – is never deductive in its creation. The mathematician at work makes vague guesses, visualizes broad generalizations, and jumps to unwarranted conclusions. He arranges and rearranges his ideas, and he becomes convinced of their truth long before he can write down a logical proof.
This statement appears as part of a longer passage explaining mathematics to non-mathematicians, and I think this aspect of mathematics deserves to be emphasized in general. To be convinced that this is the case one only has to read the comments at the n-category cafe, but I imagine most people’s concept of what mathematicians do is much duller.