Created by humans, the existent mathematics is anthropomorphic, but not universal. Even in the basics of mathematical logic and set theory, it emanates from the everyday experience of people dealing with classical macroscopic objects. This mathematics meets fundamental challenges when trying to describe, for example, quantum systems.
Indeed, it is the mathematics of sets. Based on this mathematics, theoretical physics treats any physical system as a set. This treatment, a posteriori, seems to be adequate in the case of macroscopic classical systems.
However, what is about quantum systems? Is any quantum system a set? Does it consist of elements? For instance, is a photon a set? The concepts of an element, a subset, the complement of a subset, an empty set, the union and intersection of sets etc are not so evident in a quantum world. In particular, if a quantum system (e.g., a hydrogen atom) is made up by two quantum systems A and B, it does not contains neither A nor B as a subsystem. This is the well known entanglement problem in quantum theory.
Reference:
G.Sardanashvily's site: Frontier problems
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