I’m currently reading Bert Mendelson’s Introduction to Topology. My goal is a modest one: I want to be able to say—in my own words, and as a mathematician would—what topology is actually about.
This was inspired in large part by a sentence from another book. In Unknown Quantity: A Real and Imaginary History of Algebra, John Derbyshire writes:
The adjective “algebraic,” when used by mathematicians, can usually be translated as “concerned with polynomials.”
When I first read Derbyshire’s statement, I was taken aback by its profundity. Most people would say algebra is about manipulating symbols or balancing equations. Derbyshire’s is a mathematician’s statement—rich in perspective. I’d like to be able to say something similar about topology.
Anyway, progress has been slow. Painstakingly slow. I hit a point in Mendelson’s book where it became clear that my proof writing wasn’t up to snuff. I made all kinds of errors. Most often I’d make what software engineers call type errors: I’d assume things about mathematical objects that weren’t true, and it would shipwreck my proofs. I’d botch notation, or invoke theorems I hadn’t yet earned the right to use. Unsurprisingly, most of what I got wrong were things a computer would have caught for me.
I first came to proof writing through Daniel Velleman’s How to Prove It, which is a fantastic—and reasonably gentle—introduction to the subject. Velleman teaches students to move by telling them how. Reading Mendelson has felt more like getting hit in the face until the idea sticks.
Fortunately, the path to improvement is tried and true: practice and fast feedback.
While I originally set out to learn topology, what I’ve really been learning is how to write proofs. Topology has just been the medium for it, which is probably why progress has felt slow.
However, I believe this is healthy. I think of proof writing as the engine that moves me through the ocean of mathematics. I’ve also come to believe that as long as you’re moving in the right direction, the path doesn’t matter so much. The information you need will find you.
I’ll write more about topology once I can actually say something about it.