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I think this can be meaningfully asked. Intentional states

**Ideally I'm asking this about: mathematical nominalists, constructivists, intuitionists, and realists (and ideally I'd ask this about logical objects but that was getting too complicated for me).

My instinct would be that nominalists could be either way (e.g. fictionalism yes if an intention could be formed about any desired object defined vs. Chomsky "no angels" aka no we are too finite), constructivists and intuitionists would say there are possibly at least as many intentional states because of how they do proofs differently(?)(including no law of excluded middle), and realism is harder to answer about but would be the most likely to say no.

Would they agree or disagree?

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  • How many mathematical objects are there? Infinite infinities, and just saying "infinite infinities" doesn't come anywhere close to capturing the magnitude of infinity we're talking about. See the cardinal numbers. The number of possible brain states may also be infinite, but it's a far, far smaller infinity, probably not larger than the power set of the real numbers.
    – causative
    Sep 3, 2022 at 2:20
  • As an aside, formal mathematical formulas are strings of symbols from a finite alphabet, and there are a countable infinity of those. But there is an uncountable infinity of real numbers, which is a lot more than a countable infinity. It follows from this, that the vast majority of real numbers cannot be named by any formula. If you select a real number at random between 0 and 1, the probability that there exists a formula that names just the number you selected is exactly 0. In this sense, real numbers have "outgrown" the formulas that represent them.
    – causative
    Sep 3, 2022 at 3:56
  • I am guessing the idea is that realists believe in "intention transcendent truths" while others should not. I am afraid this gets terminological before it gets interesting. Whose "intentional states" are we talking about? Human (in the fullness of time and "in principle"), some idealized "transcendental subject's" a la Kant and Husserl, God's? Since few philosophers of math would want to limit math too much they'd probably all allow for the negative answer for a sufficiently restricted range of intention holders (that, nonetheless, includes all past and future humans).
    – Conifold
    Sep 3, 2022 at 4:34
  • I'd assume that intentional states correspond to brain states. An "intention" is something that happens in the brain, after all. So all we need is a rough upper bound on the number of possible brain states (whether or not these brain states ever actually happened), and this would be an upper bound on the number of intentional states as well. And unless physics is very weird, this upper bound is far smaller than many infinities used in mathematics. If a brain state is representable by a real-valued function in N dimensions, then the cardinality of brain states is the power set of R.
    – causative
    Sep 3, 2022 at 6:03
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    Per Brentano's intentionality inexistence mark of abstract math objects indeed there must be at least one possible intentional state for each postulated or defined math objects since mental state aboutness relatum is nothing but the grounding variables committed by its practitioners. Even better if you combine structuralism with psychologism, you don't even need to commit to any disputed Meinongian free encoding ontology unless a few base generic objects such as defined in category theory, you can directly ground every structural (arrow) relation as its practitioner's intentional relata...
    – Double Knot
    Sep 6, 2022 at 1:48

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Insofar as the intuitions in intuitionism are intentional states, then yes, I'd expect a closer to one-to-one correspondence between the numbers of possible intuitions and of possible intentional states. I should also agree when you say that realists do not end up with that correspondence, but I will also go into a little detail about why those realists end up where they do.

Not, however, then, that this is an absolutely universal explanation, but: in model theory, there are some juxtapositions where you can have a model, as a set, be only countably infinite, and yet it "logically" covers uncountably many objects. More broadly, the Löwenheim/Skolem number of a logic, say, might be so large that you have an uncountable base, yet upwards-wise you can still push the model-theoretic functions to "cover" more objects than are enumerable modulo the relevant base. Insofar as models are semantic structures that are about proper classes of objects, it seems possible to have a mismatch between the number of intentional (semantic-aboutness) states in a system, and the number of objects proposed by the system.

A similar moment occurs in finitely-axiomatized proper-class theorizing, except here we can have a relatively finite model-theoretic set yet "cover" (or, more accurately, "refer to") an absolutely infinite one. Hypothetically, you can use one class axiom (an axiom for a specific class) to comprehend class-many objects. (For some reason, having one axiom scheme with arbitrarily many instances is not entirely the same thing; but it is comparable thereto.) So perhaps you might think of this as having a single intentional state comprehending plural objects; and eventually, we'll find ourselves in the dark caverns of the unrestricted quantifiers, wondering if our flashlights' reach is as far as possible, or if we can't quantify over "absolutely everything," esp. not by a single symbol! (So then not just bare unrestricted quantification, but internally plural quantification, are opened up as questions, i.e. do we refer by a single term to {the category of all sets} or directly, but then plurally, just to {all sets}?)

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    Thanks. I’m going to keep rereading this because I think I get your points but want to be more sure. Realism I think does and is intended to, most suggest real yet unknown mathematical truths. And I want to stretch the limits of what things we can think do exist yet can’t “ make contact with”, doing so in the form of what we can actually form intentionalities about. I’m trying to better understand why math (especially under realism) is seemingly far and away so beyond intentional grasp comparatively. But maybe it isn’t so unique in that respect. Taking progress where I can manage to articulate
    – J Kusin
    Sep 3, 2022 at 15:09

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