No numbers, no formulas

A few months back, Mark Chu-Carroll complained (and rightfully so) about some idiot on NPR who said that, since sudoku puzzles don't have to use numbers but can use any symbols, they're not mathematical puzzles at all.

It's a common misconception, I think: If something doesn't have lots of numbers or formulas in it, it's not math. In reality, math isn't about either of those things. In fact, we could in principle rewrite the entire body of mathematical knowledge so that it wouldn't contain one single formula. It would be extremely difficult and utterly pointless, of course, but nonetheless possible; we use formulas only because they're convenient, not because they're necessary. Scroll through Gottlob Frege's Foundations of Arithmetic; it's mostly just plain text, not a whole lot of formulas there. And yet it's definitely mathematics, and important mathematics at that; it contains the first logically correct definition of the concept of number.

Here's another example of this misconception at work. Back when I was in college, a friend of mine who was a psychology undergrad asked me to participate in an experiment he was conducting for his thesis (why he called something that had no control group an "experiment" I'll never know, but let's leave that for now). He gave me a sheet of paper with integers 1 through 10 randomly scattered about the page, interspersed with (also randomly scattered) letters A through J. He asked me to connect the integers in ascending order as quickly as I could, and timed me. Then he gave me another sheet just like the first one, and asked me to connect the letters in alphabetical order; this whole thing was then repeated a few times. When, after it was finished, I asked him what he was trying to get at he said he was interested in relative speeds with which brain hemispheres process information, and that this would allow him to do it because the task of connecting numbers was managed by the right hemisphere whereas the task of connecting letters was managed by the left one. I asked him how he knew that. He said: "Because the right hemisphere is responsible for mathematical reasoning so it must control connecting numbers, and the left hemisphere is responsible for verbal skills so it must control connecting letters." In other words, to him, the first task was mathematical (because it was about numbers), and the second one was verbal (because it was about letters), even though, quite clearly, conceptually it was the same damn task!

So now that we know that math is not about numbers or formulas, can we say what it is about? It's about a certain type of reasoning, I guess. What type of reasoning? Who knows, really. In this respect math is like porn: very hard to define, but very easy to recognize when you see an example.