What Is Math? (Baby Don’t Hurt Me, No More)

Today’s XKCD

Now, I generally avoid philosophy at all costs. All costs. For many reasons. Any good thought has likely been thought and better expressed before. It’s hard to separate philosophy from the pretensiousness of many who like to discuss it. Many reasons. But I am a mathematician, and it’s hard not to think about what it is I’m actually doing.

So, the comic expresses the traditional stacking of sciences. Biological, Chemical, Physical, Mathematical. And somewhere, burning in some deep, dark pit of hell, Computational Linguistics. It’s an easy stack to make. As in the quantum physics course I just took, it’s often difficult to tell which came first, the physics of quantum mechanics, or the math of it. Indeed, it often appears as though quantization in physics, is a result of a necessity of the math. Relativity, I have argued, is a logical necessity – from the math. Sometimes, it’s even more difficult to separate the math from ‘reality’. So it begs the question, I think so at least, what is math?

To which I would like to make a proposal.

I don’t think math belongs in the stack at all. Math, as a subject, represents something far more encompassing. Yes, math describes the physical world to a phenomenal degree. But then again too, it can also describe the chemical world, and the biological world, and even the computational linguist’s world. At least, I hope so. Math doesn’t belong above all the other sciences, but rather along side, because it’s function is less as a science itself, but much more as a language for expressing thought. Indeed, one of the more astounding features of mankind is the extent to which mathematics so universally expresses thought. If I’d drawn the comic, the mathematician would sort of be floating on a cloud above the other scientists, waving down at them cheerily.

But there is much to argue about that. I mean, Mathematics as a whole seems far more than just a language. It seems to be a subject unto itself, from logic to knot theory and back again. But I do wonder if this is any different than any other language’s ability to express thoughts with regard to itself. Books could be written in analysis of the sentence “Colorless green ideas sleep furiously.”. A simple dictionary contains a vast array of information and knowledge about a given language, without ever really describe what words are. Is this necessarily any different than turning the language of Mathematics in on itself to study and explore what it says about … itself? Awkwardly constructed, perhaps, but I think the thought is valid.

The flipside of this is, let’s assume theat Mathematics is nothing more than a language, especially in the way it interacts with the other sciences. If you agree then, that Mathematics does seem like more, perhaps it is because, as with the other sciences, we are merely describing something with it, something that literally defies description except through the language of Mathematics. A sort of phantasmal sixth scientist, standing right behind the physicist, representing … who knows what.

Of course, there is a sixth scientist standing behind the physicst – the Mathematician. And maybe that suffices.