Terence Tao

Within mathematics itself, there's also a theory and experimental component.

It's just that until very recently, theory has dominated almost completely.

99% of mathematics is theoretical mathematics, and there's a very tiny amount of experimental mathematics.

I mean, people do do it, you know, like if they want to study prime numbers or whatever, they can just generate large data sets.

So once we had the computers, we began to do it a little bit.

Although even before, well, like Gauss, for example, he discovered, he conjectured the most basic theorem in number theory, which is called the prime number theorem.

which predicts how many primes, up to a million, up to a trillion.

It's not an obvious question.

And basically what he did was that he computed, I mean, mostly by himself, but also hired human computers, people whose professional job it was to do arithmetic, to compute the first 100,000 primes or something, and made tables and made a prediction.

That was an early example of experimental mathematics.

Theoretical mathematics was just much more successful because doing complicated mathematical computations was just not feasible until very recently.

Even though we have powerful computers, only some mathematical things can be explored numerically.

There's something called the combinatorial explosion.

If you want to study, for example, Zermatt's theorem, you want to study all possible subsets of numbers 1 to 1,000.

There's only 1,000 numbers.

How bad could it be?

It turns out the number of different subsets of 1 to 1,000 is 2 to the power of 1,000.

which is way bigger than any computer can currently enumerate.

In fact, any computer ever will ever enumerate.

There are certain math problems that very quickly become just intractable to attack by direct brute force computation.