Terence Tao

But again, it shows it's possible.

the thing is you can get this emergent very complicated structures but only with very carefully prepared initial conditions yeah so so these these these glider guns and and gates and and software machines if you just plunk down randomly some cells and you when they get left you'll not see any of these um and that's the analogous situation of navier-stokes again you know that that with with typical initial conditions you will not have any of this weird computation going on um but

basically through engineering, you know, by specially designing things in a very special way, you can make clever constructions.

This is a recurring challenge in mathematics that I call the dichotomy between structure and randomness.

That most objects that you can generate in mathematics are random.

They look like random, like the digits of pi.

Well,

we believe is a good example but there's a very small number of things that have patterns but now you can prove something as a pattern by just constructing you know like if something has a simple pattern and you have a proof that it does something like repeat itself every so often you can do that but

And you can prove that most sequences of digits have no pattern.

So if you just pick digits randomly, there's something called low-large numbers.

It tells you you're going to get as many ones as twos in the long run.

But we have a lot fewer tools to, if I give you a specific pattern, like the digits of pi,

How can I show that this doesn't have some weird pattern to it?

Some other work that I spend a lot of time on is to prove what are called structure theorems or inverse theorems that give tests for when something is very structured.

So some functions are what's called additive.

Like if you have a function that has natural numbers, they're natural numbers.

So maybe, you know, two maps to four, three maps to six, and so forth.

Some functions are what's called additive, which means that if you add

If you add two inputs together, the output gets added as well.

For example, a multiplying by a constant.