Euclid's If
Excitingly, I now have a new website. Hello, internet! My previous blog was great, but it was basically that: a regular blog without much room for anything else. I will be posting here occasionally, but I’ll also be using this site as a place to host a CV, link to my papers, and maybe carry out a few slightly left-field experiments.
With that in mind, I thought it would be fitting to start out by thinking (for which read worrying) for a while about what it is I actually do. I’m a mathematician, which means I prove mathematical statements. Two and two, I can demonstrate, make four. This is mostly fine. The problem arises when we ask what those statements, in and of themselves, mean. I can talk about adding them all day, but in the end actually saying what two is isn’t easy.
Euclid understood this on at least a subconscious level. The first proposition in his Elements is, in translation, “On a given straight line to construct an equilateral triangle.” He never asks or determines where we get the line, and he never considers a world in which a straight line is not given. In certain geometries, the idea of a line being “straight” doesn’t make any sense at all, and, indeed, Euclid’s constructions tend to fail in these situations. Mathematics, as practiced by Euclid and many others, is a grand tower of if-then statements. “If”, the mathematician says, “two exists, then I can add it to itself and get four.” This is fundamentally unsatisfying: we’d like to be able to draw conclusions, not just give conditionals.
We get around this with the Zermelo-Fraenkel axioms: a way of rigorously defining numbers in terms of sets. If you don’t know what a set is, think of it as a generalized box. You can have an empty box, or you can have boxes with things in them, so the set of all even numbers would be a box with every even number in it. The first Zermelo-Fraenkel axiom, in this analogy, says that two boxes with the same things in each must be the same box. There are seven more, plus the axiom of choice, which I won’t go into now. Suffice it to say that it was controversial for a while because it does strange things when sets become infinitely large. All nine axioms together form ZFC (Zermelo-Fraenkel-Choice), which is the foundation of mathematics today. From these, we can rigorously define numbers, addition, and all sorts of other lovely goodies, escaping Euclid’s if.
Sadly, it’s not quite that simple. None of us have ever seen a box holding all the even numbers, and yet, by ZFC, we can work with the set of all even numbers. We need a way of justifying the existence of abstract sets that obey the ZFC rules, or we’re just staring Euclid’s if in the face yet again. We’ve had this problem for decades. Zermelo was writing about axioms in 1930. Surely, by now, we’ve found some line of reasoning, some way of showing that mathematicians aren’t just talking crazy talk all day. What’s this great proof? Why do we know that sets exist?
Because we say so, that’s why.
This sounds like a massive cop-out. Let me assure you that it isn’t. The problem with just asserting the existence of the number two is that we don’t know what that number really, really is. We know it’s the number that comes after one, but to define it that way we’d have to define one and “comes after” and rubbish like that. ZFC, on the other hand, provides a complete recipe for making sets. Anything – anything at all – that obeys the ZFC rules can be a set. All we have to do now is assert that one exists. Mathematics becomes the study of any and all systems that obey the ZFC axioms. Crucially, we don’t need these systems to exist in front of us to study them: the ones we imagine in our minds work just as well. This is different from just asserting that two and two make four, in that we know exactly what we’re asserting. We aren’t relying on a lot of badly-defined cultural assumptions. The if is still there, but satisfying it is trivial.
There’s still a trade-off. By reducing mathematics to sets and things that come out of them (which does include numbers), we lose our intuition that mathematical methods work. There’s no link between ZFC and the real world, which means there’s no guarantee that, if I use math to calculate how to build a bridge, the bridge won’t collapse under me. Even if two and two made four, which they do, there’s no reason why two gallons of milk, added to two gallons of water, shouldn’t give five gallons of liquid. It turns out that you do indeed get four gallons, but, if you define math by starting from ZFC, that is not at all obvious. The effectiveness of mathematics in describing the physical world is a scientific law, codified by centuries of observation, not a mathematical one proven beyond all doubt.
We’ve come a long way since Euclid. Third graders today use methods that he would have killed to have, and modern mathematicians are far beyond where he ever dreamed of being. In making this advance, though, we’ve had to give up on a few things. As mathematics became more precise, it lost its inherent connection to the world. A mathematician’s intuition is no longer an advanced version of that of a builder or engineer. Personally, I think it’s been worth the loss, but engineers, builders, and even some mathematicians may disagree.