Intuitionism

> observation enforces a certain kind of consistency in the laws of physics.

That sounds like a typical chicken-and-egg problem.

> how can it be that this ordering is so consistent?

One thing that I didn't explicitly state is that the universe only needs to seem ordered. You are making an assumption that isn't necessarily founded. 99.9% of the time we aren't checking on the ordered state of the universe. It is the exceedingly rare case that we measure with sufficient granularity to expect to see quantum effect, for example.

> I would expect a lot more miracles to happen

I totally understand this. You might expect a table upon which your keyboard rests to change color as we pass through a chaotic portion of some multi-verse. But that assumes order can't exist at the time-scales of universes.

Imagine it like a picture. The set of all possible pictures at 640x480 is massive but finite. The vast majority of images in that space are like white noise. But you've probably seen millions of images at that resolution that are totally coherent. You don't expect the middle block of some specific image to randomly be a different color. Same with a movie. The set of all sequences of 640x480 images at 30 fps with 1 minute duration is finite. It is full of total chaos. But when you watch a movie you don't expect chaos to ensue at any moment.

If you consider our entire universe like a single image from the set of all images or a movie from the set of all movies, it isn't surprising that it is entirely ordered from beginning to end. It just feels weird because the scale in both time and space is so large in the case of the universe. But given the scale of the entire set, such a large subset having internal consistency isn't totally unreasonable.

Something to consider in the context of "repetition", is that it requires abstraction, and possibly memory. As noted before, I do not see any kind of repetition (identical things, counting) in nature. I think abstraction and memory are both emergent properties from human brains (or machines, brains in other mammals, octopuses, etc.) My pet theory also initially discards "things", because that again requires abstraction.

For reference, my views are somewhat related to "emergentism", "connectionism", and "realism", but I haven't found a school of philosophy that I feel comfortable with.

> how could you even study this?

This is indeed the biggest challenge. I am currently studying this from a conceptual art perspective, because philosophy and science do not seem adequately equipped for this kind of problem.

I get the appeal, it's a minimalism thing. The thinking mind makes up patterns and checks them against each other, there is no ultimate reality, blah blah blah. Besides, a mathematical ground truth would be a kind of transcendence, in that it's prior to human thought, which makes it uncomfortably close to ideas of God.

Still, I've come around basically to mathematical platonism. The structure is out there, we just happen to be smart enough to tease some of it out.

I have two arguments for this. The first is the very existence of long-standing problems, and their eventual resolution either way. For centuries we were able to wonder whether Fermat's last "theorem" (which was really a conjecture at the time) was actually true, and eventually Wiles came around with an extremely complicated proof and it was settled. And we do believe that math/logic is consistent enough that someone else couldn't just have followed a different train of thought and come up with a proof of the opposite. How does a strict intuitionist account for this kind of situation?

The second, and possibly deeper argument, has to do with structural equivalences. I've been out of the field for decades, but I know that a standard trick in academic math is to develop structural equivalences between disparate fields. You want to prove something in an area of math, but it's hard, so you prove that the whole structure of that subfield has a one-to-one correspondence with the structure of another subfield, and then prove the corresponding theorem in the other subfield, which happens to be easier (see: analytic number theory). Again, this sounds like exploring an existing territory, not like arbitrarily building thought-bridges here and there. The bridges are where they are, and if you try to build one where reality didn't put it, your proof will get nowhere.

An even stronger form of this is that, in advanced mathematics, all kinds of notions of universality appear all the time. One of the most famous is probably computability theory. Just using a few basic symbols (say, integers, first order logic and some additive operations), you get theories of varying power. But as soon as you hit a certain level of richness, bang, all of a sudden, you've hit computability. Your theory is rich enough to embed a Turing machine, and therefore is exactly as rich and expressive as any other computable - even if one is based on numbers and multiplication, and the other on graphs or some such other weird thing.

Universality shows up in lots of places. I'm too far out of the field to remember them, but it starts with the very integer numbers - there are plenty of ways to formalize their initial construction, but the eventual result is exactly the same.

At this point my general thinking is that the bulk of the structure is pre-given. I have no special conjecture to make about how that comes to be - it's all a logical structure, prior to matter or thought, so unlike physics, it's not like there could be another universe out there with different basic mathematics.

Again, definitional issues are critical here.

I can make up all sorts of math that isn't really "discovered", it's more "invented". Most of it won't correspond usefully to the physical universe. See e.g.: https://plato.stanford.edu/entries/formalism-mathematics/ :

> "mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."

I take the brevity and lack of commitment to a position in your reply as an indication that you're not really willing or able to defend your position. That's fine, it's a complex topic, as the many articles on the SEP linked above attest. But if you want to claim "the world is made of math", then the onus is on you to define what you mean by that. To me, it looks a little incoherent.

All information that exists is physical, encoded in a physical substrate. Beads in an abacus, holes in a punched card, distributions of charge in a computer memory. Hypothetical information that is not encoded physically cannot be causal. A book or computer program that have not been written can have no effects. Only information that exists in a physical encoding can be causal, by virtue of it’s physicality.