Does Math Evolve?

Since the beginning of human language and civilisation we have gradually developed a formal language called mathematics. All human natural languages are intimatly coupled with our sophisticated imagination. They only have a limited self-contained logics. Mathematics has gradually been simplified and formalized. It is a language that has been optimized to expressed relations in the simplest possible ways. A driving force in the evolution of this language has been the simplification in the expression of the relations existing mathematical structures.

There are many physical realities which have been discovered based exclusively on the assumption on equivalence a certain physical structure to certain mathematical structures. But the success of mathematics is mostly concentrated in the large scales and the small scales of the universes. At the biological scale and at the planetary scales the entities of interests are not equivalent to mathematical structures.

For the ancient and modern platonists, the intimate connections between some aspects of the world and the mathematical world is taken as a proof of the objective existence of the mathematical world. The inventions of mathematicians are discoveries of eternal realities. Modern cosmologists by positing the a priori existence of mathematics and of quantum theory and general relativity as initially existing prior to the big bang are all platonist dualists.

It is possible to imagine a creation scenario which would begin with the co-creation/ co-evolution of mathematics and the universe. The laws of nature would gradually evolves with the evolution and stabilization of the relations. Evolution would consist in the gradual explicit embodiement in physical structure of a subset of the regular implicit relations of the world. The platonic world being gradually constructed in the structures of the world.

It is a gradual passage from culture/process (implicit relation) to nature/structures (explicit embodied relations).

In this scenario all the layers of nature correspond to the appearance of a structure which accelerate the culture/nature transfer speed. The passage from biology to human culture correspond to the culmination of the evolution of mammalian dreaming into human consciousness which is a splitting of the awareness field between external and self-awareness (self-controled induce dream). The implicit embodied platonic wold which is the human bodied can then become explicit and able to be express in natural language and mathematics.

In my opinion, it is a very hard choice to make between perceiving mathematics as evolving along with the universe and perceiving mathematics as what you called "supranatural" or beyond spacetime.

To stand by the former would mean that we see mathematics as inherently part of the universe - that is, universe somehow has a mathematical and physical side at the same time. They are dual in the sense that as the physical universe evolves in certain ways, the mathematical universe follows suit such that there is always a correspondence between physical objects (matter, energy, etc), physical laws and mathematical objects (numbers, operators, spaces etc) along with their operations (addition, multiplication etc). This opinion itself is probably difficult to argue for or against, for lack of evidence to show that indeed the world is mathematical in nature, i.e. this matter is inherently a philosophical issue that most likely cannot be settled by means of experimentations or tests.

To stand by the latter view would mean that mathematics is beyond our universe although from ontological perspective, it is existent. To explain the interaction or even how mathematics work in view of physical universe is harder by means of this stand, since we do not know how mathematics comes to be applicable to reality at all. However, it does make it easier to swallow the idea that mathematics does not need to produce physically feasible results/statements.

The latter stance still mean we are facing the same problem as the former - the status of mathematics as being supranatural would be quite unprovable physically and hence in the face of science, it most likely cannot gain a place. Both interpretations require philosophical considerations.

Pertaining the feasibility of applying simple mathematical rules to Planck scales, let us consider a Kantian perspective: that humans do have 'categories' imposed upon our experiences like rose-tinted spectacles. For example, Immanuel Kant postulated that space and time are both structures imposed by our minds to make sense of whatever sensory input we obtain - they rearrange and organize our perceptions. Let us suppose that mathematics is in this particular case adopting similar role. What can we gain from it?

I believe what we gain is the dissolution of your question. If mathematics is a default structure imposed by our minds (this is different from claiming mathematics is an invention) - that addition and multiplication is a method to organize our perception on quantities, shapes, transformations and abstractions, then if we suppose there exists a human who can experience an instance of Planck scale observation or even thought experiment, he may in attempt to make sense of what it means to be at Planck scale try to come up with a formalism that addresses the problem. This is analogous to how physicists try to search for quantum theory of gravity: the physicists realize they cannot make sense of certain phenomena by brute-force combination of quantum physics and general relativity, and thus in that state they are unable to do anything concrete to the phenomena - just like how our hypothetical thinker faces the Planck scale dilemma. Upon meticulous thoughts over time, we hope physicists can resolve the problem, and thus we can hope the same for this hypothetical thinker.

What is the point of the thinker above? It serves as an example that a third interpretation is possible: that mathematics is a mental structure of the mind that organizes perception. If this interpretation is chosen, the question "whether simple arithmetical operation is applicable" is a little bit weird in view of this interpretation: for unless there is something for the thinker to test that view, otherwise the hypothetical thinker cannot even decide the answer to the question.

If mathematics truly evolve over time, I think we will be faced with big trouble. That means mathematical statements are all time-dependent, and regardless of how little the effects of time on those statements, now mathematicians have to find relationships between mathematics and time functions. This will, in turn, reduce mathematics to physical science - because mathematics cannot work by virtues of logic and a set of axioms anymore. Mathematics that depends physically on time will make it a physical science in our current definition of science as a field of knowledge, and this epistemological impact is, in my opinion, self-defeating because mathematics was meant to abstract things out of physical sciences and crystallize them to generalize many issues.

Supranatural mathematics poses no such problems, but it does provide another set of epistemological issues: for example, how would we account for countless instances of highly close correspondence between mathematics and reality? For example, how can we deny certain physical statements because mathematics show them to be so by means of proofs by contradiction? All these issues are not any less thorny.

I would prefer my third interpretation, though admittedly it has its own set of problems too. But I sure like the idea that mathematics may start to work only after it becomes computationally meaningful. Maybe someone would like to critique my ideas too?

I think maths just describes the structure of transcendentally ideal space and time in which all phenomena appear.

https://www.academia.edu/7347240/Our_Cognitive_Framework_as_Quantum_Computer_Leibnizs_Theory_of_Monads_under_Kants_Epistemology_and_Hegelian_Dialectic