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
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