The Spirit of Mathematics

This is going to be one of those posts that interests only me.

I was having a conversation at lunch today with a fellow graduate student about how to deal with this tension we have as mathematicians.

The tension is between the idea that our mathematical ideas are inevitable, and the possibility that they are all inventions, mere reflections of our own way of thinking.

I've blogged about this kind of thing before. There are all kinds of weird thought experiments that you could do if to make you think that hey, maybe there's nothing special about our approach. A jellyfish in a pure continuum would never use numbers to do math, right?

But hold on, not so fast. There's still a strong argument to be made that if you never forced any sort of discrete patterns on your perception of reality, you'd simply never have any mathematics.

I was thinking, okay, what's an example of when we need to measure continuous things, rather than discrete things?

Here's an example to make you think twice about how numbers work. Let's say I throw three marbles into an empty bucket, and you throw four into the same bucket. Now how many marbles are there in the bucket? Well, three plus four is seven, of course. Sure enough, if we empty the bucket and count the marbles, there will be seven. We could run this experiment over and over and we'd always get seven. Three and four just are seven.

But now let's do the same thing with, say, shoveling sand. I'll shovel in three loads of sand, and you shovel in four loads of sand. Now let's have someone else empty the bucket with a shovel. How many loads will it take? Seven? Perhaps. But I could see how it might only take six, or maybe it'll take eight. If we repeat this experiment, we might get different answers.

Sand is more continuous than marbles. Marbles are discrete--you count them, 1, 2, 3, 4, 5, and so on. With sand you just sort of guess--oh, that's about a shovel full. So would math have evolved the same way if we had to first deal with continuous things, rather than discrete?

Let's say a mathematician was hired in ancient Egypt to measure the rise and fall of the Nile throughout the year. The water level is a continuous thing, though. How would he describe the change over time? Maybe he'd draw a curve, instead of a numeral. Wouldn't that be perfectly legitimate?
There's some art here. The mathematician wants to make something to reproduce the flow of what he's seeing, to somehow relate to it in a natural way. In a way, he's being more poet than scientist. Most mathematicians want to be poets on some level.

But there is a practical question to be addressed. Chances are he's supposed to be tracking the water levels so that the people can know when it will flood. That's a yes/no kind of question. We have to draw a line somewhere.

Once you begin to think in this binary manner, you enter the world of the discrete. You begin to think using Aristotle's "law of the excluded middle": you can't be both A and not A. The river is either flooding or it isn't.

Moreover, it's not hard to see how this idea of distinction can progress beyond just two possible states. Maybe instead of merely drawing a curve to represent the water levels, the mathematician will decide it's useful to draw lines in banks of the river, so he can say at different times of year which lines the water level has surpassed. This provides a good estimate--a discrete way to measure a continuous thing.

And the truth is, that's all it takes to get to the integers. Once you have a way of drawing some line of distinction between yes and no, why, then you have 1 and 0. And if you have 1 and 0, then how many choices do you have? You have 2, of course. And if you have 0, 1, and 2, then how many numbers have you invented? Why the answer is 3... Could we keep going with this? Why, I think we could!

Oh, I understand perfectly well that not every culture would think to come up with the integers this way, but that's not the point. The point is how easy it really is to go from the bare concept of distinction into the vast realm of mathematics that has been built up for millennia.

It seems like the Platonists always have a point, you know? Mathematics really does feel, in a mysterious way, like it's merely discovered. It's somehow emanating all around us in the universe, and somehow we're eventually able to tap into its secrets.

The development of surprising things like non-Euclidean geometries doesn't seem to me to make a huge dent in this claim. Yes, in some sense we are merely "inventing" systems of axioms that we then proceed to work with. But the powerful and unexpected thing about mathematics is how such a tiny, bare concept can lead to such an extraordinarily large body of knowledge.

For instance, if my little thought experiment is of any value, then it seems like the bare idea of distinction gives birth to arithmetic, which gives birth to number theory. It's kind of unreal if you know anything about all the powerful theorems that exist in this area.

Yet even here, there is this desire to be poetic. It is not enough simply to be able to manipulate the world around us using the power of arithmetic. Mathematicians desire elegance and symmetry. Solving particular computations are not as interesting as proving results about all the numbers that have spun out of this strangely simple thought experiment.

I suppose there must always be this mysterious blend of two forces: the scientific and the artistic. The scientific force is the force of distinction; it categorizes and quantifies. The artistic force is the force of expression; it responds intuitively.

If we had no ability to draw distinction, how could we really know anything? We might be free to respond to the world, but we would never understand it. We would never even think of ourselves as ourselves--we would have no line between ourselves and the world, or between different things in the world.

And yet if we have no ability to respond intuitively, how could we really know anything? We might be able to categorize and quantify, but we would never care to. How can we learn about the world unless we are first swept away by its beauty? Occasionally mysticism must trump rationalism.

When I do mathematics, I actually do experience a bit of both. The scientific force is easy for everyone else in the world to see, but for any real mathematician, the artistic is there to see, as well. And if I could describe what I mean, well, then it wouldn't be what I mean. But you can always take a look for yourself.

It all comes back to Trinity. Three in One, One in Three. In Threeness, God has quantity, distinctness, the scientific force. In Oneness, God Is Who He Is, unquantifiable, beautiful, to which we but respond, overwhelmed, because we cannot describe.

Yeah.