Terence Tao

People played with notions of curved space.

Not because they thought that the actual world was not Euclidean, but... Wait, wait, back up for a sec.

So anyhow, so now... Yeah, so Euclidean geometry has all these amazing theorems, like the sum of the angles of triangles, always 180 degrees.

It's a classic theorem.

But they were very hard to prove, very complicated to prove.

And mathematicians have tried for a long time to see if...

if there was any simpler way to prove these theorems, like if you take away some of the axioms of geometry, could you still prove these theorems are at right angles and whatever?

And then by doing so, they discovered these non-Euclidean geometries, like where you're on a sphere or some hyperboloid instead of that space.

And now the sum of angles of a triangle is not 180 degrees.

The area of a circle is not pi r squared.

And these are very weird geometries.

Yeah.

Well, fiction and art is like that, too.

You explore worlds that aren't real, and there's value to that.

So, yeah, so these were kind of geometries that weren't real in some sense until Einstein, when he was developing general relativity.

realized he needed a theory of curved space, and he asked a mathematician friend, I think Marcel Grossman, hey, do you know of any mathematics that deals with curved space?

He says, oh yeah, there's this non-Euclidean geometry stuff, there's this British bright guy, Riemann, who developed this wonderful Riemannian geometry, and he took a look, and it was almost exactly what he needed, almost word for word, the language he needed to express the theory of relativity.

Yeah, so my theory is that both pure math and science are motivated by compressing the world around them.

So they have all this data.

In the case of pure mathematicians, it's mathematical data.