Created by humans, our science is anthropomorphic, but not universal. Even in the basics of mathematical logic and axioms of set theory, it emanates from the everyday experience of people. This science meets fundamental challenges when trying to describe, for example, quantum systems,
One of the main achievements of mathematics of XX century are Godel’s incompleteness theorems which state that any formal system in mathematical logic, capable of expressing elementary arithmetic, can not be both consistent and complete. Namely, there are statements expressible in its language that are unprovable. Godel’s theorems developed an axiomatic theory of natural numbers of R. Dedekind and G. Peano. Published in 1931, they showed the failure of Hilbert's program to formalize mathematics. At present, Godel’s incompleteness theorems provide the main principle of methodology of modern science.
Indeed, contemporary theoretical physics forces us to conclude that any complicated physical system is not described by a unique theoretical model, but one needs several models, each of them has its own area of application and describes only a part or a certain aspect of a physical system. Moreover, these models at the intersection of their application areas fail to be consistent in principal.
In particular, until recently, theoreticians followed famous Dirac’s thesis: "A physical law should have mathematical beauty", written by him on the wall of D.D. Ivanenko’s office in
. However, almost none of existent realistic theories satisfy this thesis. For example, the unified Standard Model of electroweak interaction is far from to be mathematically elegant. At present, only classical field theory admits the comprehensive mathematical formulation in terms of fibre bundles. Fundamental problems remain in classical mechanics: for instance, there is no intrinsic definition of inertial reference frames. In quantum mechanics, we have different non-consistent quantization techniques, e.g., algebraic quantization (the GNS construction) and canonical quantization. Moscow State University
However, the main "headache" of contemporary theoretical physics is quantum field theory. Some its parts (algebraic quantum theory, perturbative quantum theory, quantum electrodynamics) themselves look rather satisfactory. However, an integrated mathematical formulation of quantum field theory fails to exist yet. Moreover, there are doubts whether such a formulation within the existent mathematics is possible at all.
This mathematic is based on the mathematical logic which formalizes the logic of human thinking. It results from evolution of mental processes of a human mind, and it is the logic of statements in a language of words. This logic is not universal, it is "anthropomorphic". For example, an intelligent ocean in "Solaris" of Stanislaw Lem exemplifies a different logic, not the logic of statements.
In addition to the mathematical logic, the foundation of contemporary mathematics also contains the axiomatic set theory. In the initial period of its development at the fall XIX century (e.g., by G. Cantor), set theory was based on the intuitive notion of a set. However, soon it turned out that the uncertainty of this notion led to contradictions. The most famous of them are antinomies of Russell (1902) and Cantor (1899). Unfolded around antinomies debate has stimulated the development of axiomatic set theory, although its axioms are based on intuitive ideas, too. First axioms of set theory were suggested by Zermelo in 1908. At the present, there are several axiomatic systems of set theory, which are divided into four groups. Let us mention the Zermelo - Fraenkel system and the Von Neumann – Bernays – Godel one. The latter mainly is used in mathematical physics since it is a base of theory of categories. In the framework of this axiomatics, in addition to sets, another basic concept of the class is introduced in order not to consider too "big" sets that leads to contradictions. For example, all of the sets form a class, but not a set. Classes, unlike sets, can not be elements of classes and sets. With all the variety of axiomatic systems of set theory, all of them include some basic concepts and axioms, e.g., the notions of that a set consists of elements, the subset, the complement of a subset, the empty set, and axioms of the existence of the union and intersection of sets. All of these concepts came from the everyday experience of people dealing with classical macroscopic objects. However, they are not so evident, for example, in a quantum world. In particular, a quantum system may not consist of elements, or not admit a subsystem, or a subsystem has no a complement, etc.
Thus, our mathematics based on the logic of statements and set theory fails to be adequate in order to study the inanimate nature, where there are no “statements” and "words”. Therefore, our science fails to be universal, and it is both limited in its subject and incomplete in the image.
About twenty years ago, the idea was put forward to develop a new "quantum" logic and new "quantum" mathematics. However, the problem is not in that a new system of axioms must be offered, but in the fact that such a system could lead to “rich” mathematical theory. It is not possible yet. At the same time, the existent mathematics is meaningful because it simply follows an observable reality. Figuratively speaking, it solves a problem which has a solution a posteriori, and this solution needs to be recorded only. Developing one or another "quantum" mathematics, we do not know whether the problem has a solution in principle. Unfortunately, we can not put ourselves in the place of quarks and, therefore, we do not understand something important in a quantum world.
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