I was going to post something similar and then saw yours, and it was better. We can add that math is language, and sometimes language describes things that are real, and sometimes not.

So when an application is found to a previously pure theory area (e.g. number theory and RSA) does math becone real?

Is realness not an instrinsic property but just a judgement on how useful something is?

Is being real a real property of something or just a construct?

Math is in language. In a similar way that we can have fiction and non-fiction, with different uses of language, you can have 'pure' and 'applied' uses of math. Using 'real' as the goal kind of misses the point. It's not imaginative, if that's the antithesis of real in this whole conceptual discussion. But, like fiction to non-fiction, it's still a public phenomenon that "touches" the world in language as a shared, communal, element in a form of life. Math is remarkably useful as a representation of the world and for us solving problems, but you can also give directions to a location to someone with just hand signals and barking like a dog. In a way, math is not special. But, the degree of accuracy that we've developed in math, however, is what makes it remarkable.

> Using 'real' as the goal kind of misses the point

It misses the point of the question "is math real"?

Tbf, i am actually sympathetic to the position that "realness" is not really relavent or a well defined term when it comes to what is basically a descriptive language of patterns and relations. But "is it real" is the question we started the thread on.

The study of math in language is called semiotic, and thanks to a bunch of french poseurs it’s become a joke word among the genuinely intellectually curious. Nevertheless, attempts have been made to tie it all together, most notably by the brilliant and mostly forgotten Charles S. Peirce.

Whence the pragmatic maxim:

  Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.

Pragmatically speaking, math is no more or less real than that.

Okay, here is precisely where the Simulation Hypothesis becomes useful and interesting. I know, I know. But bear with me for a minute...

The universe, as we know it, is simulatable. In other words, it can apparently be reduced to mathematics in precisely those three senses I outlined above. (If it helps, you can imagine electrons and other subatomic particles as bits of information in a coordinate space, which are constrained to operate in accordance with rules that govern logical relations between things.)

What mathematics is real? Anything that would relate to that universe or any of its constituent parts, in _any_ meaningful sense. We don't need to have found a use for it, and surely there's a great deal that is still undiscovered.

What mathematics is unreal? Whatever is, a priori, absolutely unnecessary to the existence of such a universe -- or, even worse, would break a universe described in terms of logic were it somehow made manifest.

Given that logics (of all sorts) are subsets of mathematics, I'm not sure one can convincingly hoist (any) logic above all other mathematics to make arguments.

> When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal.

Are you considering the set of natural numbers to be “unreal”? Or, if you consider the set of natural numbers to be real, do you not consider power sets to be a logical relation?

To be fair it's not too strange to consider both the natural numbers and the power sets to be to greater or lesser extents abstractions to allow us to reason about all the things we can logically put in one of them.

After all both of them are mostly full of things we will never need, encounter, or even be able to define.

I kind of think the core of the question of if math is real, is if abstractions are "real".

I'm not a finitist, but it's a highly defensible philosophical position.

As for this:

> Or, if you consider the set of natural numbers to be real, do you not consider power sets to be a logical relation?

Skolem solved that quite neatly already: Every "uncountable" set has a countable model. Thus the power set is, in fact, no larger than the set of natural numbers, because both can be fully described in a countable manner. What I'm describing is necessarily an abstraction of an abstraction, though, so I'd consider it "unreal" by definition.

I know what “a model of a theory” (such as a model of a theory of sets, even one which asserts that some expression defines an uncountable set) means.

I’m not quite sure what a “model of a set” means. I suppose maybe you mean like, the set in some model, which is described by the given description of a set in the theory?

> Thus the power set is, in fact, no larger than the set of natural numbers, because both can be fully described in a countable manner.

I don’t think this follows.

There are countable models of set theory. And for these models, there is (in the meta-theory) a bijection between the set representing the set of real numbers, and the set (in the meta theory) of natural numbers.

I don’t think this establishes that “in fact” there is a bijection between the real numbers and the natural numbers.

Rather, in any model of any of the usual set theories, there will be no bijection between the set of real numbers and the set of natural numbers. (Of course there will not be, because these theories entail that there is no such bijection.)

If I take a non-standard model of arithmetic, and for some non-standard natural number n, and consider the uniform distribution of (non-standard) natural numbers less than n, then for every standard natural number, the probability of getting that natural number from that distribution, will be equal. Are we therefore to conclude that there is a uniform probability distribution over all the standard natural numbers? By no means!

Skolem's own proofs are a little bit opaque, so I've constructed a new algorithmic proof of the downward Löwenheim-Skolem Theorem which is easier to comprehend. I have yet to publish it, but it'll probably turn up somewhere else towards the end of the year:

Preliminaries: (1) Let T be a first-order theory with a countable or uncountable infinite model M. (2) Let L be the language of T. (3) T is assumed to be a set of sentences (closed formulas) in L. Algorithm Steps:

Initialize Countable Set S: Start with S = Const(L), the set of all constant symbols in L.

Extend Language: For each n-ary relation R in L and each n-tuple (c1, ..., cn) of elements in S, if M⊨R(c1, ..., cn), then add a new constant symbol c to S and extend L to L' by adding c.

Iterate for All Formulas: For each formula ϕ(x1, ..., xn) in L and each n-tuple (c1, ..., cn) of elements in S: If M⊨∃xϕ(c1, ..., cn), then pick some a in M such that M⊨ϕ(a, c1, ..., cn). Add a new constant symbol c to S to represent a and extend L to L' by adding c.

Closure: Repeat Steps 2 and 3 until S no longer changes. Since T and L are countable, this process will eventually result in a countable set S. Construct Countable Model N: Take the substructure of M generated by S as N. By construction, N is countable.

Elementary Submodel Check: By construction, N is an elementary submodel of M. This is because for any formula ϕ and any n-tuple (c1, ..., cn) from S, M⊨ϕ(c1, ..., cn) if and only if N⊨ϕ(c1, ..., cn).

Conclusion: N is a countable elementary submodel of M, and therefore T has a countable model. End.

You're right about the bijection, but, in the countable model, the set that "represents" the real numbers is countable. (From the perspective of the meta-theory.) However, within the model itself, this set still satisfies all the axioms that make it "look" uncountable. For example, there's no bijection between this set and the set representing the natural numbers within the model, even though such a bijection exists in the meta-theory.

This is why Skolem's solution can be most succinctly described: "Every set has a countable model if one steps outside the set and constructs a countable model from its elements."

In Kleene's “Introduction to Metamathematics,” he describes the situation as follows: "Either we must maintain that the concepts of an arbitrary subset of a given set, and of a non-enumerable set, are a priori concepts which elude characterization by any finite or enumerably infinite system of elementary axioms; or else (if we stick to what can be explicitly characterized by elementary axioms, as we may well wish to in consequence of the set-theoretic paradoxes ) we must accept the set-theoretic concepts, in particular that of non-enumerability, as being relative, so that a set which is non-enumerable in a given axiomatization n may become enumerable in another, and no absolute non-enumerability exists."

> When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal.

I find that to be quite a bold statement. The question as to whether "mathematics is real" feels firmly rooted in philosophy. First, I would question what is meant by real and unreal. Second, I would question if the answer matters whether the existence of the real and the unreal are not yet discovered, or even imagined.

Your describing physics more than math in my opinion. But it's hard to say things like set theory isn't real when a lot of calculus works nicely and describes the world very accurately. That toolbox is heavily built on the "unreal" things in math your describing to be not intrinsically relevant. The weird parts of math often speak to some deeper thing your discounting. Hopefully I'm not reading into your comment to much

And yet irrational numbers are real.

I think the tie between Mathematics and the real world is more of a fortunate coincidence than a necessary tie or even causality. Human will still study math (as we know it now) if it real world doesn't exist or it somehow becomes not describable by math, because there's the innate desire for logic and solving puzzles.