I’ve got a bunch of stuff queued up to be posted over the next couple of days. It’s

been the sort of week where I’ve gotten lots of interesting links from

readers, but I haven’t had time to finish anything!

I thought I’d start off with something short but positive. A reader sent

me a link to a post on Reddit, with the following question:

Throughout elementary and high school, I got awful marks in math. I always

assumed I was just stupid in that way, which is perfectly possible. I also

hated my teacher, so that didn’t help. A friend of mine got his PhD in math

from Harvard before he was 25 (he is in his 40’s now) I was surprised the

other week when I learned he isn’t particularly good at basic arithmetic etc.

He said that’s not really what math is about. So my question is really for

math fans/pros. What is math, really? I hear people throwing around phrases

like “elegant” and “artistic” regarding math. I don’t understand how this can

be. To me, math is add, subtract, etc. It is purely functional. Is there

something you can compare it to so that I can understand?

This hits on one of my personal pet peeves. Math really is a beautiful

thing, but the way that math is *taught* turns it into something

mechanistic, difficult, and boring. The person who posted this question

is a typical example of a victim of lousy math education.

So what is math? It’s really a great question, and not particularly

an easy one to answer.

You’ll get lots of different answers depending on just who you

ask. It’s a big enough thing that you can describe it in a lot of

different ways, depending on your perspective. I’m going to give

my own, and you can pipe in with your own in the comments.

To me, math is the study of how to create, manipulate, and understand

abstract structures. I’ll pick that apart a bit more to make it more

comprehensible, but to me, abstract structures are the heart of it. Math

*can* work with numbers: the various different sets of numbers are

examples of *one* of the kinds of abstract structures that we can work

with. But math is so much more than *just* numbers. It’s numbers, and

sets, and categories, and topologies, and graphs, and much, much more.

What math does is give us a set of tools for describing virtually

*anything* with structure to it. It does it through a process

of *abstraction*. Abstraction is a way of taking something

complicated, focusing in on one or two aspects of it, and eliminating

everything else, so that we can really understand what those one

or two things *really* mean.

For example, look at topology. Topology is basically a way of

understanding shapes. But it does it in a completely abstract way. It throws

away everything except the concept of *closeness*. You have a

collection of points, and you’ve got a concept of things that are

*close* to one another, defined in terms of *neighborhoods*. By

playing with different notions of what things are close to each other, you can

create any shape you can imagine, and some that you probably can’t. But you

don’t really need numbers at all: you can just create and play with shapes in

topology – as long as you’ve got the set of points, and you’ve got set

relations, you can figure out what it really means for something to be a

torus. You can see what’s really strange about a moebius strip. You can

take the moebius strip, and add a dimension to it, and see exactly how you

produce a klein bottle.

For another example, look at category theory. It’s a way of understanding

function. What’s a function? At it’s core it’s a *mapping* from one

thing to another. But what does that really mean? What can you *do*

with that basic idea? What can you *make* with it? The answer is:

virtually anything you can imagine.

But math is more even than just those abstract things. Why does music

sound good to us? Because it’s got an underlying structure. That structure

can be described mathematically. Personally, I’m a huge Bach fan. I believe

that he was the greatest composer of music that ever lived. His music

is magnificently beautiful, and incredibly moving. But to really understand

it, to really grasp all of what he was doing in his music, you need to understand

that it’s structure on structure on structure on structure. That structure

is mathematical. If you’re really understanding the structure of Bachs music –

if you sit down and analyze it, *you’re doing math*

When you look at a cubist painting, you’re looking at a strange kind of

projection of something. The artist has taken the subject of the painting

apart, viewed it from different perspectives, different points of views,

different ways of understanding it or seeing it, and assembled them together

into a single image. When you look at a cubist painting, and try to understand

what the artist was seeing, how they were seeing it, and how the pieces

of the final image really fit together – *you’re doing math*.

When a scientist tries to analyze something about the world, to understand

how it works, and describe it in a way that tells us something important about

how things behave – they’re doing math. They’re *abstracting* the

world to come up with a precise, formal, descriptive way of stating what

they’ve learned.

When you look at a road map, and figure out how to get from one place to

another – you’re doing math. The map is an *abstract* representation of

the world that allows you to do certain useful things with it. (And frankly,

this is one example that I’ve never been able to understand. I can’t read

maps. Quite literally a bit o’ brain damage – some scar tissue in the left

frontal lobe of my brain.)

When a jazz musician improvises, part of what they’re doing is

math. For an improvisation to make sense, for it to sound good, and fit

with what’s going on around it, there are a set of constraints on it:

on pitches, pitch progressions, rhythm, chords. Those are all abstract

properties of the music, which are mathematical!

Math is unavoidable. It’s a deeply fundamental thing. Without math,

there would be no science, no music, no art. Math is part of all of those

things. If it’s got structure, then there’s an aspect of it that’s

mathematical.