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.

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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.

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