While its a little antiquated — 013zen

I'm not sure if this is antiquated at all. To me it looks like many contemporary discussions in the constructivist and finitist context is very much in line with Kant's views on the nature of mathematics.

Great post.

Just as a photon is either a wave or particle depending on how it is measured, it seems like these difference in math philosophy may all be neither wrong nor right - it depends on how the topic is approached. — EricH

When it comes to the metaphysics of mathematics, it could very well be a matter of perspective, yes. But when it comes to the foundations of mathematics, there isn't much room for perspective I imagine.

Thanks for the links. From what I know, modern set theory's axioms are ZF(C). And from Ernst's article, ZF(C) axioms are expressed using a first-order logic/language whose only undefined term is ∈ (membership).

From a few different sources, it seems that second-order logic takes first-order logic and stacks upon it quantification of properties/functions/relations (whatever term the source chooses), which are sets of elements. Some sources seem to imply that SOL may have functions while FOL may not but others clarify that SOL's distinction is in fact that it quantifies over those relations, like ∃A(Xxy), there is a relation A that x bears with y.
I did study set theory throughout many years of school but not as deeply as a math grad would, naturally. What I gather from your link, however, is that what is called function in set theory (mapping) is a bit different from what a function is in SOL (or FOL), and a set theory function can also be represented as sets of sets, differently from FOL functions which are not but can be represented as sets in some contexts. That being, it is clear to see how set theory can be built off ZF(C) which is expressed with FOL, which (FOL) the SEP says is what is typically used for building set theory. If SOL is weaker than set theory as the SEP says and SOL applies to set theory as you say, I wonder how SOL is applied then. I am not a fan of ChatGPT, but it gave me this:

For example, certain principles or properties involving collections of sets may be more naturally expressed in SOL.

Second-order logic plays a significant role in model theory, which is the branch of mathematical logic that deals with the study of mathematical structures (such as sets) using formal languages.

After writing all this, I realised I made a mistake in my question, modal-structuralism does not reduce mathematics to a SOL but it is expressed instead in a SOL S5. So I guess I will edit the question later today.

Pretty much. So mathematical expressions are true only if there is a proof-path that shows it to be true. There are, one concludes, mathematical expressions that are neither true nor false. — Banno

That is something that I tend to agree with.

This is opposed to Platonism — Banno

to platonism* :smile:

But then the claim "it is not the case that this proof-path pre-exists our construction of it", the syntax being the proof-path, and in our case being the FOL that we see in things such as ZFC, did we really construe relations such as ∧ and →? If so, it would then bring up "how did we"?

But then again, prima facie there is nothing necessary about the idea of cats, protons, or communism. It could be that numbers are innate ideas, being then "world-independent".

Well, this sort of depends on one's view of the world. In ontic structural realism, things sort of are the mathematics that describe them. A proton or a cat can be described as a sort of mathematical entity, and so might be thought to exist in the way number do.

There is, however, a boundary issue here in that "things" tend to have fuzzy boundaries. That's why these sorts of proposals tend to take the entire universe as a single mathematical object and things as just parts. But this does not rule out the possibility that there are morphisms across various subprocesses within the universal process that describe "things," and these could exist in the ways numbers exist or sets exist, as abstract objects.

I do think there is something to this, but to boundary issue remains tricky. Any "thing" only exhibits some of its properties over any given interval, and its properties are defined by how it relates to other things or by relations between parts of the thing and itself. There is a contextuality in how the world works that makes it so that one can not describe any given thing without recourse to describing other things and how they relate. This isn't that far off the formalist mantra re mathematical entities: "a thing is what it does."

How things relate to minds is a very special sort of relationship then. When a thing is considered in thought, a great deal of a thing's relational properties become "present" at once to some observer. This is where things "most fully are what they are," because it is where more relations are pulled together into a unity.

The fullest realization of any entity possible would be in a perfect mind, to which all relations are present at once. In medieval thinkers, there is a general acknowledgement of the fact that truth exists in relation to minds, not being simply equivalent with being. The "mind of God," then is the place where this sort of "knowing" relation, all properties made present at once, exists. I've often thought that the "view from nowhere," would be better termed the "God's eye view," because what it really wants is to see things in this way. God was seen as problematic, so God gets axed, and then we get this problem where truth is the way things can be conceived of with no mind to pull any relations together.

Platonism though, makes more sense if things don't exist most fully arelationally, "in themselves," but rather both relationally and in-themselves, which I take it is how Plato himself understood the Forms, and this entails a sort of God-mind to which all relations are present at once. When modern Platonisms drop this for abstract entities existing "in-themselves," I think they start to lurch off the rails. The "view from nowhere," really wants to be the "God's eye view," but it has pulled the rug out from underneath it.

Psychologism is sort of the opposite extreme. It excepts the need for mind in an explanation, but then has mind floating off by itself, a sui generis entity in which abstract entities appear to be emerging ex nihilo. But this doesn't make sense either, it's the relation of abstract entities, nature, and mind from which knowledge emerges.

But then the claim "it is not the case that this proof-path pre-exists our construction of it", the syntax being the proof-path, and in our case being the FOL that we see in things such as ZFC, did we really construe relations such as ∧ and →? If so, it would then bring up "how did we"? — Lionino

I'm not following what you say here.

I'm not following what you say here. — Banno

It says the proof-path does not pre-exist our construction of it, "it" I am guessing here being the proof-path. The proof-path is the syntax we use. If the syntax we use is a FOL, it implies it (and its operators → ¬∀) is construed. That leads to the question how are they construed? I thought the connectives of FOL at least (except ¬) were taken as primitives based on the possible binary outcomes of an operation applied to two binary variables $p \otimes q$.

under construction

This would be impossible if the contradictory theory was erected on a logical foundation containing the Boolean principle Ex Contradictione Quodlibet ECQ, from a contradiction everything follows. So ECQ has to be abandoned, but fortunetely that proves possible, indeed mathematically straightforward. What remains is a rich field, of novel mathematical applications interesting in their own right, which sidestep the vexing questions of which foundational principles to adopt, by developing contradictions in areas of mathematics such as number theory or analysis which are far from foundations. — SEP's Inconsistent Mathematics

In a parallel with the above remarks on rehabilitating logicism, Meyer argued that these arithmetical theories provide the basis for a revived Hilbert Program. Hilbert’s program was the project of rigorously formalising mathematics and proving its consistency by simple finitary/inductive procedures. It was widely held to have been seriously damaged by Gödel’s Second Incompleteness Theorem, according to which the consistency of arithmetic was unprovable within arithmetic itself. But a consequence of Meyer’s construction was that within his arithmetic R# it was demonstrable by finitary means that whatever contradictions there might happen to be, they could not adversely affect any numerical calculations. Hence Hilbert’s goal of conclusively demonstrating that mathematics is trouble-free proves largely achievable as long as inconsistency-tolerant logics are used.

The arithmetical models used by Meyer and Mortensen later proved to allow inconsistent representation of the truth predicate. They also permit representation of structures beyond natural number arithmetic, such as rings and fields, including their order properties. Axiomatisations were also provided. — SEP's Inconsistent Mathematics

It should be emphasised again that these structures do not in any way challenge or repudiate existing mathematics, but rather extend our conception of what is mathematically possible. This, in turn, sharpens the issue of Mathematical Pluralism; — SEP's Inconsistent Mathematics

Various authors have different versions of mathematical pluralism, but it is something along the lines that incompatible mathematical theories can be equally true. The case for mathematical pluralism rests on the observation that there are different mathematical “universes” in which different, indeed incompatible, mathematical theorems or laws hold. Well-known examples are the incompatibility between classical mathematics and intuitionist mathematics, and the incompatibility between ZF-like universes of sets respectively with, and without, the Axiom of Choice. It seems absurd to say that ZF with Choice is true mathematics and ZF without Choice is false mathematics, if they are both legitimate examples of mathematically well-behaved theories. — SEP's Inconsistent Mathematics

Shapiro’s distinctive position has other ingredients: mathematics as the science of structure, and mathematical pluralism implying logical pluralism (on logical pluralism see also Beall and Restall 2006); but we do not take these up here. — SEP's Inconsistent Mathematics

The primacy of theories fits, too, with the natural observation that the epistemology of mathematics is deductive proof. It is only if one takes as a starting point the primacy of the mathematical object as the truth-maker of theories, that one has to worry about how their objects manage to co-exist. — SEP's Inconsistent Mathematics