Is Mathematics invented or discovered?
This idea comes from paper “Propositions as Types” by Philip Wadler.
Philosophers might argue as to whether mathematical systems are ‘discovered’ or ‘devised’, but the same system arising in two different contexts argues that here the correct word is ‘discovered’.
— Philip Wadler
Wadler amazed by the fact of similarities between different branches of science, like logic and type systems (Curry-Howard correspondence) and topology (Homotopy Type Theory and Univalent Foundations) and objects of a Cartesian closed category (Curry-Howard-Lambek Correspondence) etc.
So we can think of it as laws of nature being rediscovered again and again in different context (Wadler’s idea) or it is our brain which recognizes same patterns again and again in different context (my addition).
There is other interesting paper called “Physics, Topology, Logic and Computation: A Rosetta Stone” which adds quantum mechanics (QM) to “equation”. And here things getting even more interesting. Are there basic set laws of nature at level of quantum mechanic, which reappears in different science areas? Can we explain everything using QM?
Side note: there are some fields of logic and type calculus which does not have correspondence yet, generally believed that correspondence yet to be found.
Quantum Mechanics vs General Relativity
One more obstacle in the direction of unifying everything with QM is the fact that relativistic physics and QM considered to be incompatible. But there is attempt to fix this - see Space Emerging from Quantum Mechanics. There are also other works which links classical physics and QM, for example classical chaos theory. See also “There are no particles, there are only fields”.
Interesting questions arise can we describe basic laws of QM and derive all other laws of nature from them using purely mechanical way of reasoning as in Turing Machines? According to Godel if this system of rules is consistent as number theory we can not derive all truth. If we can construct simulation program based on this laws can we predict future, can we deduce what happened before? Are there physical limits that will prevent for such program (machine) to work, like uncertainty principle? Does halting problem corresponds to some limitations aroused by physics. Side note: Ultimate physical limits to computation.
Can we actually discover basic rules?
Other question if universal basic laws of QM reveal actual nature of universe or is it just the limit of our perception? Like example in “Real patterns”, where epiphenomena of movement appears in Conway’s game of life even though it is never meant to be that way, there are simple rules which describe Conway’s universe and it has nothing to do with movement. Even more disturbing is the fact that you can construct Turing Machine e.g. and represent a lot of things that TM can do in Conway’s game of life or self-replicating structure which reminds idea of DNA.