Terence Tao

If I talk about x, is x a real number or a natural number or a function or something?

If you write things informally,

It's often in some context.

You say, you know, clearly X is equal to, let X be the sum of Y and Z, and Y and Z were already real numbers, so X should also be a real number.

So Lean can do a lot of that, but every so often it says, wait a minute, can you tell me more about what this object is, what type of object it is?

You have to think more at a philosophical level, not just sort of computations that you're doing, but sort of what each object actually is in some sense.

It's using much more traditional, good old-fashioned AI.

You can represent all these things as trees, and there's always algorithm to match one tree to another tree.

So it's actually doable to figure out if something is...

a real number or a natural number.

Yeah, every object sort of comes with a history of where it came from and you can kind of trace it.

Oh, I see.

Yeah, so it's designed for reliability.

So, modern AIs are not used in, it's a disjoint technology.

People are beginning to use AIs on top of Lean.

So, when a mathematician tries to program a proof in Lean, often there's a step, okay, now I want to use the fundamental calculus, say, okay, to do the next step.

So, the Lean developers have built this massive project called MethLib.

a collection of tens of thousands of useful facts about mathematical objects.

And somewhere in there is the fundamental theorem of calculus, but you need to find it.

So a lot of the bottleneck now is actually lemma search.