alt Hacker NewsI find this topic of precision and of the computation that happens in the universe to be pretty interesting, but also very hard to pin down.
For example, let's take two arbitrary real numbers, call them x and y. They're almost certainly irrational uncomputable numbers, of course. Now, what is their sum? It's x + y, of course - I can tell you that without doing an infinity of computational work. Now, is this a useful representation? It very much depends on the use you have for it. For two unrelated uncomputable numbers, their sum is almost certainly uncomputable anyway, so there's not going to be any meaningful representation of it either. However, we can still derive useful properties about it, and use it in physical simulations as we need. For example, if both x and y are positive, I can tell you that x+y is greater than either of them, so I can also tell you that the electromagnetic force between two particles that are at distance x+y will be weaker than the force between two particles at distances of just x or just y.
So, what would it mean for the universe to "compute the exact value" of these two, or of their sum? What can physics do, assuming that some aspects are indeed continuous, that a simulation running on a TM actually can't?
But you have a problem when you want to compare two numbers (to see if the particles overlap). Or if you want to divide one number by another.
In theory, in classical physics if you start a simulation on a Turing machine, then all the initial states must be computable. And you can carry on using symbolic representation of numbers, even with transcendental functions (as long as you compute them to a precision necessary to carry out the next step of the simulation).
But with quantum mechanics, we might have true randomness, so precise classic simulation can become impossible after the first interaction.