Friday, January 13, 2012

Framing mathematics

No, not framing it for a crime. I just read the article A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today by Frank Quinn (math professor). It establishes a surrounding explanation for how modern mathematics came to be structured as it is. In the process, it contrasts two ways of thinking about mathematics: the "old" style, and mathematical sciences, wherein math relates to observable facts and is intuitive (but often yields incorrect results), and the "new" style, which he calls the "core" of mathematical research, wherein math is the study of abstract rules which do not relate to reality (but whose results are provably, rigorously correct).

It is fascinating.

I had never paused to consider the evolution of my discipline. Yet Prof. Quinn highlights and summarizes my experience of grade-school math education: everything seemed much clearer and more reasonable when -- finally! -- worked as abstract symbols according to rules. This is how I learned geometry (my first proofs!), trigonometry, and calculus. I cannot imagine attempting to learn calculus through intuition. What terror! (Does this infinite series feel like it converges? What's your hunch about the derivative of f(x)?) And of course now in retrospect I think of the math I learned earlier -- multiplication, fractions, arithmetic -- in the more advanced terms I learned later.

The article was summarized for me by this: "the old dysfunction was invisible, whereas the new opacity is obvious." Yes, math is opaque; I've studied for years and this is the first thing I'd admit. And my topics are squarely in the "new/core" section: I've done research in precise definitions, logical proofs, completeness. Carefully justifying each step is a technique that I use in my dreams. Math for me has always been its own arena of knowledge, one of three (the humanities, sciences, and math), each with its own methods. Even though we use the science word "discovery," a mathematical discovery is nothing of the sort. And as a grad student, I am amazed at what other grad students do as "research."

This post's theme word is anemometer, "an instrument for measuring the speed of wind." This magical anemometer predicts the trends in sociology research!

1 comment:

David Kordalewski said...

I recently stumbled across a somewhat related paper. It discusses the fallibility of mathematicians and difficulties inherent in full formality. The main argument of the paper is that formal verification of software is not going to work. Good reading, even though I can't agree with it entirely.
Social Processes and Proofs of Theorems and Programs