Good point. So linguistic physicalism is too strong. But then what does a metaphysical physicalist mean when he says everything is "physical" ? So in the number case... we can say that ZFC require the existence of infinity of objects that can't be described with a formula. Suppose instead of a metaphysical physicalist, I'm a metaphysical numberist... everything imo is a number... so I say that the smell of garlic is some non-computable number. We don't know which, but it's a number nevertheless. Does that view make sense? (lol... I meant in jest, but just found out Pythagoras believed this!)

I don't know if the statement "Most real numbers are not describable by mathematics" makes sense. It's certainly not possible to write down every single real number using a finite number of symbols, but that seems to me like a different statement than saying you can't describe it. Mathematics can describe plenty of things that are impossible to write down.

@Cubic the point is that no matter how you turn it, there are countably infinite many such descriptions, and strictly more real numbers. The aforementioned diagonalization argument by Cantor is that you line up your descriptions (whatever they are) one by one, and boom, there's a real number that didn't get a description. It's a description-agnostic argument.

This is known as the "math tea argument" and is wrong or at least poorly phrased. There are models of ZFC where every real is definable in the language of ZFC. math.stackexchange.com/a/3954453/913783

@S.C. The first sentence of the post you linked is, "I would describe the situation as follows: it's not that the math tea argument is wrong, so much as that it is easy to misapply it." The post goes on to say how hard it is to specify and prove the math tea argument within ZFC. There are countable models of ZFC (Lowenheim-Skolem), so it's not so surprising that every real might be definable in one of those models, but these models, being countable, are nothing like our intuitive conception of the reals.

@S.C. The basic problem is that the math tea argument is something we can say from a meta perspective, on the outside of ZFC looking in. It's hard to translate it into a theorem inside ZFC (though the post you linked does describe some partly successful attempts to do this), but that doesn't invalidate what we can say from an outside perspective.

Don’t formulae like the laws of motion produce infinitely-many real-valued positions depending on their arguments…?

@JosephWeissman yes, the crux of the argument isn't that you can't produce infinitely many reals; it's trivial to write a function that can take the value of any real (e.g. $f(x)=x$ trivially meets this criteria). The idea though is that there must an infinite number of concrete $x$ that do not have a finite representation, no matter what that representation looks like (this is a property of the representation, not of the numbers themselves). The question though is why this would matter. Something not being representable doesn't mean it's impossible to talk about it.

"The mathematical formulas are countable because you could list them all one by one, in order of increasing length." This argument seems underdeveloped. Why wouldn't Cantor's diagonalization argument apply the same way here as it does to the real numbers?

@Cubic Something not being representable does mean it's impossible to talk about it in particular. Virtually all real numbers can not be specified as an individual number. It's possible to talk about the first person who ever came within a hundred meters of your current position, but unless you're in Antarctica or space, merely as an abstract, not as a person.

@KarlKnechtel A mathematical formula is of finite length, unlike a real number, so Cantor's diagonalization argument doesn't apply.

@KarlKnechtel like I just said, there's no such thing as an unrepresentable real in general; Representability is not a property of any one number, it's a property of the chosen representation. We can just switch to a different one where needed, nothing is stopping us from doing that. The only thing that's not possible is to have a single system in which all reals have a finite representation (for practical considerations of course it doesn't matter because for any real number you can find an arbitrarily close rational number, which can all be finitely represented)

You jest @AmeetSharma but Kronecker, much closer to our time than Pythagoras, averred that π doesn't exist. And these arguments against Cantor have a sound pedigree

@Cubic How do you describe your "chosen representation"? That description is going to be a string of characters R. You can enumerate over strings R just as well as you enumerate over formulas F within a particular R; the strings R are countable. This means the pairs (R, F) are also countable, so even appealing to more "chosen representations" still leaves you with only a countable number of ways to specify a real number.

"Well, suppose that the universe fundamentally works based on continuous real numbers, such as the position of a particle or the magnitude of a wave function at a particular point." Isn't this a bad assumption? More specifically, doesn't quantum mechanics in general and the solution to the ultraviolet catastrophe in particular show this to be a poor model?

Yes @Chuu. Discrete spacetime is an active question

@Chuu: No. Anyone who knows just a tiny bit of QM would not ever claim that the UV catastrophe shows anything about whether the magnitude of a wave function is real or not.

@Chuu Wave function in QM is a kind of generalization of classical probabilities. Even with classical probabilities it is possible to have a finite discrete system with outcome probabilities being irrational numbers.

@Chuu You say "Isn't this a bad assumption?" I think that this misses the point of the answer. I am sure that they will correct me if I am wrong, but I believe that causative's point is that linguistic physicalism is less plausible than metaphysical physicalism. One way to justify this claim is to demonstrate a "physical" system which does not have a "linguistically physical" description. The real numbers, in the "usual" setting, provide such an example. The idea that space-time might be continuous (hence requiring real numbers to describe) is kind of icing.