A fundamental question is whether continuous variables are really things that can concretely exist in our universe, or whether they’re just mathematical abstractions. (Or in the words of Kronecker: “God made the integers; all else is the work of man”.)

My guess is that ultimately the universe is discrete. But even so, it can be useful to use continuous variables (e.g. for the pressure of a fluid, even though ultimately the fluid is made of molecules). But in practice, continuous variables tend to be hard to work with except in simple settings. It’s great when one can get an exact algebraic answer to something (using Mathematica!). But when things get complicated, that’s impossible… and one ends up having to use “numerical approximations”, which in turn involve discretization. And the worst part is that after that discretization it’s very hard to tell if the answers one gets are “correct” in terms of the idealized continuous variables.

In the book I actually do talk a certain amount about continuous systems: e.g. https://www.wolframscience.com/nksonline/chapter-4 But my main conclusion is that the basic phenomena are the same as in discrete systems—just more difficult to identify.

I even have my favorite PDE, which shows rule-30-like behavior—but which ultimately kills all known numerical PDE solving schemes (it’s been a great test for NDSolve in Mathematica for years).