Monday, July 02, 2007

How many ways can one interpret "understanding analysis"?

One of the things I wanted to do this summer was flesh out some of my math background a bit since I'm interested in logic but didn't do enough math as an undergrad. For example, I've learned a fair amount of discrete math, but comparatively little continuous math. To fill in the holes in my education, I thought I'd work through a few of the yellow UTM books. I'm nearing the end of Understanding Analysis by Abbott. It is an introduction to single variable real analysis, and I must say I rather like it. There are chapters on infinite series, topology of the real line, functional limits and continuity, derivatives, integrals, series of functions, and some odds and ends on metric spaces, Fourier series, and the construction of the reals. The exercises in each section have a range of difficulty and the proofs in the chapters are fairly explicit. The second to last section of each chapter tends to contain some of the more interesting results, leaving large chunks of their proofs to the reader. One thing I like is that each chapter starts and ends with a little bit of historic and philosophical background, for example the relative timeline of development of the notions of continuity and derivative. Not a lot, but they suffice to provide motivation for why mathematicians were interested in developing these specific notions. My biggest gripe with the book is that there are a few (too many really) small typos in various proofs. Most of them are easy to spot, but a couple left me confused for a spell. I think the book labors too long on the introduction of quantifiers, but maybe it is needed when taking this book as a freshman. In any case, I've enjoyed the book and think I've gotten a decent amount out of it. One of Abbott's aims in writing the book is to bolster mathematical intuition (in the non-loaded sense), and, generalizing from a single case, he seems to do alright. I'd recommend it to any philosopher who wants to get a bit of a feel for analysis.


N. N. said...


What level of mathematical ability is required to read Abbott's book? How much math did you have as an undergraduate?

Shawn said...

Abbott's book really doesn't require much mathematical ability or training at all. It is as gentle an introduction to this subject as I'd expect to find. As an undergrad I didn't take much math. Only a couple of classes really, although I did take a fair number of logic classes.