What is Math Really About?

I often find that people are confused about what math is or what its about. This is understandable- I only came to understand this late in my college education, and I was very surprised by what I learned. Notice that I am not a mathematician- I'm a computer scientist. What I want to share is my experience and knowledge having learned this myself over the last several years. I'm going to talk about what math is, what it does, and how its used. I want to express that math is not one big unified system that explains the universe, its not all about numbers, that subjects like calculus are not "higher math", and math is not even one thing.

So, what do we do in math? The basic situation is that we observe something like a relation between people (a family tree for example), a physical phenomenon (gravity, electricity, and lots of other things), or something as simple as a quantity of some object. Having found something we want to understand, we construct some mathematical object that describes the situation. Instead of describing a single family tree, we would describe a general structure of all family trees. To do this, we have a large number of objects to combine from the usual practice of mathematics. Often we need many objects collected together to describe our situation that together give the information we need to describe the object we are interested in. The information we need give the "theory" of the thing ("the theory of family trees"), and the particular objects the models (your particular family tree).

This begs the question- where do all those objects we use come from? They in turn encode some intuition about the world- relations encode relations between things, numbers encode quantity and order, lattices encode complex orderings, etc. In other words, we can combine these objects to describe a new situation, but how do we arrive at the "first" objects- how do we found our system? The answer is that there are many ways to do this, and they are *not* all the same. Found math in different ways, and you get different results. The reason that this is not as much a problem as it might seem is that for the most part we work in some "ambient" system of logic with some notion of sets. While some of the more abstract and obscure (but still important) parts of math may change depending on the system, there is a large part of it that is generally (but not universally) agreed on. The usual system to found math on is called ZFC, and the usual logic to use is classical logic, but just as a writer doesn't wonder about how transistors work when they use a computer, a mathematician doesn't necessarily worry about how math is founded when doing their work.

While though there are many different foundations for math (category theory, set theory, logic, and more). there is some similarity to them. This brings us to the heart of math- writing down symbols that follow rules. Thats really it. We write down symbols, rewrite them according to rules, and interpret them as having a meaning. I said that when we see something interesting, we try to encode it as a mathematical object. Therefore, we should be able to encode the notion of encoding notions as an object, which we call a theory. Logic is the study of these systems of symbols and their rules, so logic is math. Notice that there are many systems of logic, which vary hugely in what they can describe. They can be very strong, and define all of what we think of as math, or very weak and be able to describe nothing, or anything in between. These systems usually will have some meaning to us as humans- they give a systematic way to write down and manipulate symbols, and therefore to write down and manipulate the concepts that the symbols are intended to stand for. Ideally, if two people agreed on some logic- the meanings of the symbols and what it means to combine the symbols in different ways- they would agree on anything that could be defined using that logic.

This brings me to the end of the first part in this series of posts. I still want to talk about models in general, and the mechanical process of writing and rewriting symbols, but the main point is here: math is logic, logic is writing symbols (often on chalkboards). The meaning comes from the person doing the writing, and what they believe in. You can refound math any way you want, you can construct all sorts of exotic objects, you can go into the highest abstractions, but if there is no reason or justification for your work then it is not interesting. This is why there is no "problem" with the fact that there is no number 1/0, or that are many sizes of infinity, or with the various things commonly called paradoxes. Even if you could come up with a construction that does not have these "problems" (it would be more accurate to call them properties), it is not necessary very interesting or useful. If there is some application, or even if it just help people understand something that is already well known, then it can become a part of the huge, distributed effort that we call math, even if it is a new mathematics based on different axioms (constructive math for example). The activity of math is more a way of thinking then a single unified system, and if it describes the universe as we experience it, then its because it was designed to.