About Math
Question: What is math?
Math
Really, try to come up with an answer to that question. It's not easy. In fact, it might be easier to start by answering the question What is math not?
A few things math isn't:
- Numbers - Math is not just the mundane tallying of endless sums and differences of random numbers. It is not memorizing times tables. It is not seemingly arbitrary rules about when to carry that digit in the 1's place.
- Boring - This might be hard to believe for some, but I swear math really is not boring. There's a deep beauty in math, an awesome interweaving of the most simple and the most complex ideas, and a joy and satisfaction that comes from studying it and being able to see the world in a different way.
- Science - Math is not the same as science, although it is used all the time in science. Why not? Because science, in the modern sense, is emperical. That is, it involves gathering data from experiments in the real world. Whereas math can pretty much entirely be done with pencil and paper.
So what is it then? It's part art, part logic, part curiosity, part pictures, part symbols, part concrete, part abstract, and of course, part frustration. It's different things to different people and you won't find one exact definition for it. My rough attempt at defining mathematics would be something like the rigorous study of quantity and patterns. By "rigorous" I mean that math is built around and upon strict logical rules and proofs. By "quantity" I mean anything that can be measured (money, volume, numbers themselves, ...). And by "pattern" I mean any thing that seems to behave according to some rule.
Math as a Language
It is important to note that math really is its own language. It has symbols and characters that look just as foreign to most people as Chinese or Arabic would. But in many respects, math could be considered a 'stronger' language because it can be so information-dense. Take the following equation for example:
$$ S=-\sum _i {P(x_i)log(P(x_i))} $$
It's fine if you don't know what that equation is. Or even if you don't get the weird way it's written or the symbols used. Just notice that it has some sense of structure and deliberateness to it. There's an i below that weird sideways "M" thing and also hanging off the bottom of each x. The P(xi) appears twice. The expression log(P(xi)) seems to be two nested expressions: log([something]) and P([something]).
This is the equation for entropy in Information Theory. In just 20 characters it expresses so much. On the surface level, the "first translation" if you will, this equation says "Entropy equals the negative of the sum of the probabilities times the log of the probabilities." There are two things to note: (1) that's a lot more than 20 characters, and (2) this "translation" has it's own, deeper conceptual meaning that can't be gained just by reading the symbols. You have to put mental effort into figuring out your own, internal meaning of what that is. And in this case, it has something to do with the uncertainty in messages and signals.
It's also important to note that, just like any other language, if you don't practice math you won't be good at it. Just like you wouldn't be surprised that you're bad at Spanish if you never practice Spanish.
Here is a (hopefully) much more familiar example of math-speak, the surface area of a cube with side-length \(s\):
$$ A=6s^{2} $$
Here the "first translation" is something like Surface area equals 6 times side-length squared. Why does this make sense? What's the 'deeper' connection between the symbols and the reality? It's this: A square with length L has area L2. And the surface of a cube is made up of six squares of the same length and area. So the total surface area is 6 x (area of one square) which is what's written above.
Math as an Art
It might sound odd to talk about something so technical, so precise, or so calculated as an art. But I can think of at least two keys ways that mathematics and "the arts" are very similar.
First, math can be every bit as beautiful as any visual artwork out there. But beyond the fact that "pretty images exist in math" (which is true) is the fact that math itself is beautiful. Just as a famous piece of music or a famous sculpture can be beautiful in different ways, math can too. In music, one might appreciate the wonderful sounds played and enjoy the experience of listening. In sculpting, one could appreciate the fine-tuned skill of the sculptor and his delibrate, expert choices to do things a certain way. In math, the things you come to appreciate are the logical arguments and the elegance of it all. And in both math and art creativity is highly valued and necessary. And nowhere moreso than math is the saying "simple is beautiful" true.
The second way math can be considered an art is that it too is human. Most people would consider "art" to be a thing only humans take part in. Whether this is actually true or not could be intensely debated. But one key argument that it is true would probably include something like "humans are the only creatures with enough intellect to 'do art' ". And if it is true that 'to do art' requires intelligence (a big if) then surely math, the product of intellect, reasoning, and collective knowledge gained over centuries, must be considered an art. As the famous mathematician Burtrand Russell once put it:
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty [...] The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry."
