There's this little pet project of mine, the Functional Universe [0]. It explores a direction suggested by Dirac: treating mathematical structure and transformation not merely as a language for describing physical reality, but as potentially constitutive of it. FU models physical reality in terms of functional state evolution, with an emphasis on composition, aggregation, and transitions. In that sense, I think there's an interesting point of contact with Dirac's emphasis on transformations: Dirac points toward transformations as fundamental mathematical structures from which physics might be derived; FU asks what happens if physical reality itself is formulated in terms of transitions and their composition.
I think Michael Levin is also doing work around platonic math being a thing that exists moreso than an actual concept. His work is related to biologies discovery of algorithms, the complexity levels of simple algorithms, and n-order effects that life is finding way to exploit for computational purposes.
While not being deeply familiar with the work but I think the short of it is that life has found algorithms at many different orders that we've not discovered them yet, and by 'running' these different algorithms it turns many P problems into NP problems.
The interesting thing about it is you can make falsifiable tests around this and do actual science around it.
Curious, in this model, is there any discussion on whether the mathematical structures underlying space-time are continuous or discrete? Usually the former is assumed, but it seems your "functional" approach may be able to accommodate the latter possibility.
universe is continuous / cant be proven otherwise but any measurement or simulation of it is discrete by nature same reason pi is computed infinitely and why calculus exists and why math exists at all to try and explain continuity of nature even if all of it is perceived discretely.
The question is what exactly is continuous. The quantum fields themselves? And we have no evidence the the quantum fields are discrete quantum states themselves, but the particles they produce appear otherwise.
It's always possible that continuous maths are a side effect of this field. Of course, the causality could be revered.
if you think of universe as minecraft lattice (which is arguably best way to think of it and explain all of physics currently in existence in intuitive level) then yea, it can be like eg a spin lattice discrete at some plank length from which field perturbations as particles emerge..look up spin lattice from https://arxiv.org/abs/hep-th/0507118 so basically bringing back idea of quantum ether cuz explaining empty space with waves without some underlying structure doesn't really make sense anyway since all other waves have mediums in nature.
It is an interesting discussion you bring up but will take much more than a comment to talk about. There is a divide between mathematics and physics regarding numbers. Real numbers have supposedly infinite precision, and arbitrary size, which are not true in our physical universe. Indeed infinities have given physicists quite some headaches. It's a bit of a dark art as to where you can and cannot just use infinity willy nilly and many have expressed distaste to it. Also at some point you have to talk about computability, like are these infinite precision and magnitudes even real if nothing in the universe even gets close, what is omega to the omega, statements dream up by the utterly deranged?
I remember there being a talk or some article about this but can't find it anymore, but it is an interesting thing.
Only infinite magnitude causes problems in physics, infinite precision doesn't. Real numbers have infinite number of digits only in digital model of numbers, so their infinite size is a modelling artifact. e=1 in base e - one digit.
[0] https://voxleone.github.io/FunctionalUniverse/