Univalence wrote: ↑Wed May 22, 2019 1:58 amThis is where you are wrong.

Any system for which we have a state-space model is an example of Mathematics saying something about the universe. It predicts how the system is going to behave.

I'll let you join the dots between state-space and finite state machine.

The moment you mention "time" in a Mathematical model it becomes physics. And computer science.

We'll have to agree to disagree about that. Here's my argument.

Consider Euclidean geometry. For two thousand years everyone believed it described the universe.

In the 1840's Riemann and others showed how to make perfect mathematical sense of non-Euclidean geometry. Then in 1915 Einstein published his theory of general relativity, showing that Riemann's math better modeled the world we live in. It was then understood that math did not tell us the truth about the world. The best a mathematical theory could be was consistent and interesting.

Now you say the world is a computer. I disagree but for purposes of this post I'll stipulate to your view. And yet there can be theories of math that disagree with that view. You can have classical set theory with its many noncomputable numbers and functions. You can say it doesn't model the world, and that's fine. But you can't banish it from existence. Set theory is interesting in its own right. Set theory is simply the study of mathematical structures that satisfy the axioms of set theory, just as group theory is the study of all the models of the axioms for groups. No connection with reality is implied or expected. If it's consistent (and we set theory we can't even be sure!) and interesting, it's good pure math. If the physicists find a use for it, it's good applied math. It need not be "true" in any meaningful sense.

People have understood this since 1840. That's really when mathematicians started paying attention to foundations in the first place, once Riemann showed that math doesn't tell you what's true.

So whether the universe is or isn't a computer is one thing. And HOTT is another. You can't possibly claim that any given mathematical theory is the absolute truth of the world. Math doesn't do that.