Intuitionism

Mathematics is a human construction, but it certainly has a real substance. What does "mechanically provable" even mean, if there is no absolute truth? Do you believe in the definition of a proof or not? Do you believe that whenever the assumptions of your theorem are true, and you have a proof of a conclusion, that then the conclusion is also true? If you do believe that, that's your absolute truth, then. If you don't believe that, a proof is meaningless, isn't it?

That doesn't mean the external physical world doesn't exist, the information we use to construct that model must come from somewhere, and we can deduce that the source is a persistent and consistent one.

The philosopher Husserl said: “The tree plain and simple, the thing of nature, is as different as it can be from this perceived tree as such, which as perceptual meaning belongs to the perception, and that inseparably.”

He came up with the idea of the noema which is our experience of something, and noesis which is our conscious act of perception. For me, that's our act of interpretation of our sensory perceptions. Sometimes this all goes wrong and we construct a flawed model that does not correspond perfectly to actual external reality, such as when we are deceived by optical illusions, stage magic or just hallucinate. Fortunately we can test and correct our perceptions through action in the physical world.

I'm an out-and-out physicalist but I think he is quite correct, we must distinguish between our internal perception of things and how things actually are. Fortunately science is extremely powerful in this regard. It has allowed us to decouple our model of the world from the limitations of our perceptual system, and come up with rigorous models of reality such as Relativity and Quantum Mechanics that are not tied to direct interpretation by our perceptive systems.

I have a similar but somewhat different take on this. The universe comes first, it behaves - for what ever reasons - in the way it does. Than we humans show up and invent logic, a set of rules that is useful to reason about our universe. It either rains or it doesn't makes sense in our universe. But the universe could potentially have been different, for example something like the many-worlds interpretation but where the inhabitants of that universe can experience all the branches. Their logic might say it rains and it doesn't.

Other ideas like objects, properties, space, time, causation or countability might also be influenced by the way our universe works and how we perceive it and they might be far from universal or useful across different possible universes. On top of that we than construct mathematics and explore what all the things are that we can build from those ideas and our laws of logic. It should probably not be especially surprising that we can on the one hand construct things that are not realized in our universe and on the other hand find structures that are useful for describing and understanding our universe.

Bold claim :)

Even while maintaining a willful agnosticism about Platonic realism, it seems clear that the business of doing mathematics - intuitionistic or otherwise - depends on the ability to state and follow unambiguous rules, or else how to establish a proof within some axiomatic system or other?

But if mathematicians have this ability, even as an asymptotic ideal, and are able to make judgments as to when rules are not being followed correctly, mustn't this in turn depend on some pre-existing regularity (i.e. order) in the universe? Ineluctable physical law seems a natural candidate for this order.