If proofs exist in a social context, how have they changed over time?

It all starts with Aristotle. He said that there needs to be some sort of deductive system — that you can only prove new things by basing them on things you already know and are certain of, going back to certain “primitive statements,” or axioms.

So then the question is: What are those basic things that you know to be true? For a very long time, people just said, well, a line is a line, a circle is a circle; there are a few things that are simple and obvious, and those should be the assumptions we start from.

That perspective has lasted forever. It’s still around today to a large extent. But the Euclidean axiomatic system that developed — “a line is a line” — had its problems. There were these paradoxes discovered by Bertrand Russell based on the notion of a set. Moreover, one could play word games with the mathematical language, creating problematic statements like “this statement is false” (if it’s true, then it’s false; if it’s false, then it’s true) that indicated there were problems with the axiomatic system.

So Russell and Alfred Whitehead tried to create a new system of doing math that could avoid all these problems. But it was ludicrously complicated, and it was hard to believe that these were the right primitives to start from. Nobody was comfortable with it. Something like proving 2 + 2 = 4 took a vast amount of space from the starting point. What’s the point of such a system?

Then David Hilbert came along and had this amazing idea: that maybe we shouldn’t be telling anyone what’s the right thing to start with at all. Instead, anything that works — a starting point that’s simple, coherent and consistent — is worth exploring. You can’t deduce two things from your axioms that contradict each other, and you should be able to describe most of mathematics in terms of the selected axioms. But you shouldn’t a priori say what they are.

This, too, seems to fit into our earlier discussion of objective truth in math. So at the turn of the 20th century, mathematicians were realizing that there could be a plurality of axiomatic systems — that one given set of axioms shouldn’t be taken as a universal or self-evident truth?

Right. And I should say, Hilbert didn’t start off doing this for abstract reasons. He was very interested in different notions of geometry: non-Euclidean geometry. It was very controversial. People at the time were like, if you give me this definition of a line that goes around the corners of a box, why on earth should I listen to you? And Hilbert said that if he could make it coherent and consistent, you should listen, because this may be another geometry that we need to understand. And this change in viewpoint — that you can allow any axiomatic system — didn’t just apply to geometry; it applied to all of mathematics.

But of course, some things are more useful than others. So most of us work with the same 10 axioms, a system called ZFC.

Which leads to the question of what can and can’t be deduced from it. There are statements, like the continuum hypothesis, which cannot be proved using ZFC. There must be an 11th axiom. And you can resolve it either way, because you can choose your axiomatic system. It’s pretty cool. We continue with this sort of plurality. It’s not clear what’s right, what’s wrong. According to Kurt Gödel, we still need to make choices based on taste, and we hopefully have good taste. We should do things that make sense. And we do.

Speaking of Gödel, he plays a pretty big role here, too.

To discuss mathematics, you need a language, and a set of rules to follow in that language. In the 1930s, Gödel proved that no matter how you select your language, there are always statements in that language that are true but that can’t be proved from your starting axioms. It’s actually more complicated than that, but still, you have this philosophical dilemma immediately: What is a true statement if you can’t justify it? It’s crazy.

So there’s a big mess. We are limited in what we can do.

Professional mathematicians largely ignore this. We focus on what’s doable. As Peter Sarnak likes to say, “We’re working people.” We get on and try to prove what we can.