@mithrandir

re your last paragraph: I suspect, that math sometimes plays well with the physical reality is largely a coincidence than by design. There’s this simple part of math, where things play well with Nature, you know that one where 1+1=2. And then there’s all that other, weird part with proof by contradiction, axiom of choice, undecidability, and so on. You know, the really interesting, but quite unreal (as in not “very physical”) part. The one which does not play so well with Nature any more. Explorations of connections of the latter to physical reality tend to fall on the more philosophical spectrum of ideas.

What I was referring to is that many of us drank a bit too much Kool-aid in our lifetimes and because of congruence of the first math to Nature, we started to believe that the same holds for the latter. Mind you, there’s nothing wrong about believing that! But then we get sometimes brutally reminded by the physical reality (or a computer game in this case) that we are actually quite wrong. Some call this also the “theory vs. applications/practice” gap 😄 .

@FailForward
>I suspect, that math sometimes plays well with the physical reality is largely a coincidence than by design.
Hmm, well mathematics has always been joined at the hip with physical sciences and philosophy. People first discovered pythagorean triples because they needed to deal with triangles while building buildings, for example. Even things that people usually consider "unreal" like complex numbers have very geometric, concrete interpretations and are invaluable for solving many physics equations. The notion of computability was invented because people made computers and wanted to know what they could do. So on and so on.

The "simple part of math" can get very complicated, if not more complicated than the very abstract parts. For instance, nobody is anywhere near solving the 3n+1 problem.

I actually think that the reason math is useless for so much daily life is because if you used math to model things accurately, the model would be too complicated. E.G. using Newtonian or even Hamiltonian mechanics to model the behavior of gasses is infeasible because there are billions of particles with uncertain initial states. This doesn't invalidate classical mechanics, just makes it less useful. Most things don't even have a nice convenient statistical model like gasses do.

@mithrandir This is a topic I am deeply interested in, but unfortunately have very few solid words/sentences to speak about what is it that intrigues me so much.

Part of it is a certain feeling of “muddiness”. Reality is muddy, while math is clean - something like that.

Another way to put it is this: whenever you construct a pure theory of reality and then proceed to apply it in practice, you get bogged down by all those pesky annoying details and the whole thing falls apart like a house of cards. That’s the “muddiness” in action.

Another piece is this: in math and comp-sci we love extremes: trivial case and the worst-cases. Take computational complexity. We solve trivial cases, edge cases, extremities… Typically, these tend to be in reality quite pathological cases.In reality, often that is not what matters. What matters is the average, or “median”-complexity (average complexity exists, but “median”, I am not sure). Many problems have this nature of being NP complete, or worse in the worst-case (i.e., hopeless), yet most (if not all!) real-world instances are often well solvable with the vailable tech.

What I am trying to say is this: it’s deeply intriguing to me what happens when either a good theory turns out inapplicable (as you pointed out), or when an actually bad theory (as in “not really computable) turns out to be actually useful in reality. I have some real examples of this, but that’s for another evening with a glass of a beer 🙂 .