I’ve been thinking about how one could get continuous physics from a discrete process. Suppose you had a differential equation, and you wanted to make a discrete approximation to it. Furthermore, suppose you had a discrete algorithms for simulating this differential equation that takes in a parameter, say, dt which controls the resolution of the simulation. As dt tends toward zero, the dynamics of the simulated diff eq will tend towards the dynamics of the real diff eq.
Now suppose, we have a a turing machine that implements this algorithm as a subroutine. More precisely, the turing machine runs a simulations of diff equation at a resolutions of 1 then 1⁄2, then 1⁄3 and so on and so forth.
Finally, suppose their we’re a conscious observer in this simulation, at what resolution would they expect their physics to be simulated? Depending on one’s notion of anthropics, one could argue that at any resolution, there is a finite amount of observers in lower resolution simulations, but an infinite amount in higher resolution simulations. Consequently, the observer should expect to live in a universe with continuous physics.
I’ve been thinking about how one could get continuous physics from a discrete process. Suppose you had a differential equation, and you wanted to make a discrete approximation to it. Furthermore, suppose you had a discrete algorithms for simulating this differential equation that takes in a parameter, say, dt which controls the resolution of the simulation. As dt tends toward zero, the dynamics of the simulated diff eq will tend towards the dynamics of the real diff eq.
Now suppose, we have a a turing machine that implements this algorithm as a subroutine. More precisely, the turing machine runs a simulations of diff equation at a resolutions of 1 then 1⁄2, then 1⁄3 and so on and so forth.
Finally, suppose their we’re a conscious observer in this simulation, at what resolution would they expect their physics to be simulated? Depending on one’s notion of anthropics, one could argue that at any resolution, there is a finite amount of observers in lower resolution simulations, but an infinite amount in higher resolution simulations. Consequently, the observer should expect to live in a universe with continuous physics.
Related: https://en.wikipedia.org/wiki/Super-recursive_algorithm#Schmidhuber’s_generalized_Turing_machines