Feels like a misuse of intuition.
Ordinarily the argument for intuition is that you're able to take in so much data subconsciously and the gross outcome of all that data becomes apparent to your consciousness. It's useful in processing data your conscious mind did not or could not process because consciousness is in such high demand.
But for this, there's a lack of evidence for your subconscious to witness. What's more, while all it would take for a mathematical model to be born from nature is observation of a consistent pattern, it's not so easy to explain how nature would spring forth from maths, as maths is not sufficient for existence. I can go through all kinds of hypotheticals mathematically but those hypotheticals don't breathe life into the maths. So you're assuming some third factor that actualises the content of a mathematical model, one for which there is no evidence (or, at least, you're begging the question).
I'm inclined to agree with Tann on the maths/cognition distinction. The brain is very modular and complex, and while I can't tell you exactly which part of the brain is used for which processes, it's highly unlikely they're all the same. For instance, all things spatial can be reduced to maths, but you need spatial awareness at all times, even while doing math. The motor cortex is used to guide the pencil you use to complete your maths exam, and each movement is reducible to a mathematical model, but you don't need to stop thinking about maths in order to write down the maths you're thinking about.
All of the abstract higher level processing is in the cerebral hemisphere, including maths. But a lot of stuff that can be mathematically modeled is subcortical, and is stuff that many non-human animals are capable of (emotion, memory, hormonal nonsense etc.).
So I guess the question becomes "is all modeling of reality necessarily maths". Is the perceptual and predictive model that the dog uses to catch the thrown ball actually maths? To which my answer would be that it's interesting but probably no maybe? As far as I'm aware maths has a symbolic element and is characterised by an absolute level of precision (or the option to have it, yes people round numbers). Instead I'd say that these other brain structures enact processes parallel to maths without actually being maths. It's causal. The information enters the brain and is computed just like raw material enters a factory and comes out as ice-creams. While there is a model that explains how the factory/brain came to be effective at doing what it does, the thing that it does requires no additional maths. The blades are set to cut pieces at a particular length, ingredients are added based on the time a hose is open rather than weight, the wrapper is applied perfectly without anyone having thought about how this specific wrapper should fit the specific icecream. Individual components of the process are so far removed from the outcome that you might not recognise their function if viewed in isolation.
Guh sorry for ramble. Just woke up.
Super interesting topic, even if I disagree with your conclusions.
