How much math is knowable? [video]

The fact that Turing machines are abstract and infinite is a simplification. Turing machines are an attempt to create the simplest possible model of "computation", without compromising on what "computation" means. It's a way to prove statements about the nature of computation, that's all. A model of computation that is finite, concrete, and/or physical is less useful because it contains unnecessary details that are secondary to the nature of computation. If you're not trying to prove fundamental things about computation, you probably never need to care about Turing machines.

Most cases where infinity is invoked are done to simplify things. Forget whether infinity exists in the real world or not: ask only whether it's useful. After all, does the number 17 really exist? All models are wrong; some models are useful.

> What precisely is meant by simulating the laws of physics?

IIRC That is in reference to the work around the Church–Turing–Deutsch principle which just classic physics.

Basically Turing machines only have access to the computable reals, which is a zero measure set, thus cannot simulate, but only approximate Newtonian physics. The claim is that quantum computers have access to all the reals, and thus could possibly simulate those real valued functions.

IMHO, "simulating the laws of physics" is an example of a common way of avoiding a very common and challenging issue of getting people to accept that we do not have access to almost all of the reals from algorithms, books, pens, papers, computers, etc...

> Why not pick 10^180 or something as the highest number/states then?

Because then we have to find all sorts of limits to all of our proofs, and qualify everything, and waste time proving that our results are under this limit? If we prove something about an algorithm parameterized by some N, and we rely on a Turing Machine in our proof, then we'd also have to figure out which N hits our limit of 10^180 states on the tape, so we can qualify the limits of our proof. This could be very difficult, since the translation to a Turing machine model can be arbitrarily complex. Why bother? It doesn't help anyone.

> To understand a finite computation, we (some amount of scientists and mathematicians) must understand at least ZFC?

Turing machines are for proving things about the nature of computation itself. You seem to be imagining that you need to understand Turing machines before you'll be allowed to make statements about some specific computation. That's simply not true. If Turing machines help you prove something, use them; if not, ignore them. Putting an arbitrary limit on them only makes them worse at proving things.

I'm not really sure why you're so focused on ZFC. On one hand, the vast majority of math proofs (and therefore CS proofs that are based on them) assume ZFC, usually without even bothering to mention it. On the other hand, the axiom of choice seems completely irrelevant for Turing machines. Although the tape is infinite, it's almost always implicitly assumed that the state is finite at every step (the rest of the tape is 0). You'd have to initialize the tape with some infinite pattern to get around this, and that's definitely out of the ordinary. It's probably better to think of the state as "arbitrarily large" rather than infinite. I suspect you could get away with assuming Peano arithmetic for most CS proofs anyone cares about.

But also, yes, some number of scientists and mathematicians are expected to understand the basic axioms of math if they're going to prove mathematical theorems.

Computations are finite, but that limit keeps going up. If we picked a number that seemed like "way more than enough" in 1960, it might be less than our phones can do now. If we picked a number from cosmology, what happens if that science changes? "Heat death" is the current consensus of our distant future, but even that is challenged. Until we know what dark energy is, we can't be sure about "heat death". So if cosmology has some breakthrough that changes our understanding, why make it so we have to go revisit all our CS proofs? Using infinity decouples the proofs from irrelevant externalities.

> the indispensability of modern math is still hotly talked about in philosophy

Meh, philosophers have to talk about something. I'll worry about what philosophers say when I see a single useful insight from the entire field. We've been waiting thousands of years, but I'm sure a useful thought is right around the corner.

And the concept of infinity is hardly modern.