Concreteness September 5, 2010

Posted by Sarah in Uncategorized.
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A friend sent me a quote from Don Zagier:

Even now I am not a modern mathematician and very abstract ideas are unnatural for me. I have of course learned to work with them but I haven’t really internalized it and remain a concrete mathematician. I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, imagine you have a series of numbers such that if you add 1 to any number you will get the product of its let and right neighbors. Then this series will repeat itself at every fifth step! For instance, if you start with 3, 4 , then the sequence continues: 3, 4, 5/3, 2/3, 1, 3, 4, 5/3, etc. The difference between a mathematician and a nonmathematician is not just being able to discover something like this, but to care about it and to be curious about why it’s true, what it means, and what other things in mathematics it might be connected with. In this particular case, the statement itself turns out to be connected with a myriad of deep topics in advanced mathematics: hyperbolic geometry, algebraic K-theory, the Schrodinger equation of quantum mechanics, and certain models of quantum field theory. I find this kind of connection between very elementary and very deep mathematics overwhelmingly beautiful. Some mathematicians find formulas and special cases less interesting and care only about understanding the deep underlying reasons. Of course that is the final goal, but the examples let you see things for a particular problem differently, and anyway it’s good to have different approaches and different types of mathematicians.

It does seem true that some research mathematicians like concrete examples and some like generalities. I’m not sure how that distinction develops. But as a student I find it much easier to learn beginning with examples, as concrete and elementary as possible, and it’s hard for me to believe that anybody learns well otherwise.

Which is why it frustrates me when an introductory book dealing with representation theory begins by letting G be a group acting on a G-module over a field. It’s not that the generality isn’t useful in the subject. It just seems pedagogically lousy for a first-time reader.

Start instead with groups acting on finite-dimensional vector spaces over the complex numbers — in fact start with SO(3) — and you get a clear sense of the structure and original purpose of representation theory. You can see why it’s fundamentally about symmetry. You can even use (gasp!) polyhedra as an example. It’s so satisfying when a book goes down to the basic level. Call me childish — but I think we all start out as children before we work up to greater sophistication. (The book is Shlomo Sternberg’s Group Theory and Physics and it really is wonderful.)