Sunday, June 17, 2007
Math textbooks should teach the student the material, and how to think mathematically, as soon as possible and with the least amount of work as possible. If we view learning math as learning classical piano, we'd want a tutorial that clearly walked us through the basics and showed us the guts of a piece of music rather than force us to figure most of it out on our own. It would be arrogant for the tutorial to consist of: "You want to learn classical piano? All right then, listen to this virtuosic recital and repeat it." *
In particular, I'm griping about Rudin's Principles of Mathematical Analysis, apparently "the Bible" of introductory analysis textbooks. I stopped reading after I finished the second chapter, which is a primer on topology -- containing exactly one diagram. I guess all those figures I had to draw for high school geometry class were just to help the stupids grasp what must actually be a verbal enterprise. What a joke. I don't know if this is due to a puerile belief that grown-up books contain no pictures, a puritanically Platonic hatred of the sense of sight as opposed to reason, or simply a misguided belief that students should expend their mental energy in visualizing what the author is explaining. The author should just show the student what the basic picture is, and direct their mental energy toward hard work for advanced problems.
With no guide, the student is left to conjure up his own mneumonic devices, which is bad since he may not be very imaginative but may still want to grok the material; and even the more imaginative students would be better served by using their imagination for complicated problems. It's like having first-graders make up their own alphabet for a creative writing class -- just teach them the damn alphabet, and focus their energy on the creative composition part.
Several complaints at the Amazon entry for PMA mention its lack of motivation, and this is very true: Rudin gives the impression that the ideas sprang, Athena-like, fully formed from the head of their discoverer. The other principal complaint is that the proofs are too terse, requiring the reader to fill in lots of gaps -- this isn't so bad, although again it does waste the reader's limited time and resources. But the former flaw is even worse because it conveys no sense of discovery: some guy had a hunch, used his visual intuition (or something similar) in order to conceive the idea, which he then fleshed out in more detail, assuming it was on the right track. To quote from George Simmons' introductory calculus text, presentations like those of Rudin "produce belief without insight, and are therefore fundamentally unsatisfying. It is important to know that a mathematical theorem is true, but it is often more important to understand why it is true" (p.852 with original italics; he is discussing a weakness of using mathematical induction in explaining a theorem).
For instance, almost every introduction to induction that I've seen uses the example of the sum of the first n natural numbers, 1 + 2 + 3 + ... + n. The formula is given -- it is n(n + 1) / 2 -- and the author walks through the proof himself or asks the reader to do so. In either case, the formula materializes ex nihilo, as though this configuration of symbols just appeared to someone in a dream. In fact, it it easy to provide a visually intuitive reason that this formula is correct; a well known example is shown below:
We start with 1 block at the bottom, then 2 in the row above, each successive row containing 1 block more. Adding up the blocks from the n rows tells us what the sum of the first n natural numbers is. Now, if we make a copy of the array and flip it, it interlocks nicely with the original array to form a rectangle whose width is n and whose length is (n + 1). Since the sum only comes from the blocks in the original array, it must be half the area of this rectangle -- that is, n(n + 1) / 2. Anyone who has passed algebra and geometry can see that this is true, so the fact that this insight -- or indeed any insight into the formula -- is left out of most introductions to inductive proofs just shows how committed many writers are to producing "belief without insight." This is a serious error because, in the words of the great Henri Poincare, "It is by logic that we prove, but by intuition that we discover."
As an example of a book that strikes more of a balance between the two, Pugh's Real Mathematical Analysis works well (I found out about the book by reading through some disgruntled reviews of PMA at Amazon). Another good example is Artin's Algebra, also considered a Bible of its field, but whose author isn't primarily concerned with showing off his virtuosity. Ironically, Artin's book on abstract algebra has more diagrams and pictures than Rudin's book on real analysis! I'm sorry, but I just can't get over that. It's even more bizarre that Rudin was awarded a Steele Prize for mathematical exposition, since he's (in)famous for barely expounding at all on the ideas in his books. In fairness, I haven't read "Big Rudin," and I hear it is better.
To sum up, textbooks are meant to instruct, not necessarily to awe. Euclid's Elements is one of the most impressive feats of human reasoning, but without pictures or sense of purpose it stinks as a textbook on geometry. A confused review (PDF) of the matter apologizes for textbooks written a la Euclid since they are more terse, elegant, and cleansed of temporal impurities like images, "behind the scenes" motivation, and applications. At least the writer of the Elements was also the creator of many of the ideas presented therein, similar to a Bach fugue if it were performed by Bach himself, who was a keyboard virtuoso as well as a first-rate composer. But in the context of training new students, marveling at the beautiful should take a back seat to cultivating their ability to create original ideas, which is obviously not to say it should be jettisoned altogether.
* On a related note, reading through the reviews at Amazon and elsewhere, it seems that the mathematics community currently suffers from a sickness similar to that of the classical music community, whereby a non-trivial portion of the group focuses on and argues over the merits of the interpreters of great ideas, more so than on the great ideas and their creators themselves. Perhaps this is just a sociological way to signal in-group status: any boob may like this or that piece by immortal composers like Bach and Beethoven, but only the initiated have a preference for Rubinstein or Horowitz. Grow up and focus on what's important.