The ideological foundations of technological singularity. Boris Shulitski

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СКАЧАТЬ objects” (9, 251). One can agree with this group of French mathematicians. However, where do these structures come from and how do they relate to the world of reality?

      If these are abstractions of some sides of the real world, then the position of Bourbaki is quite consistent with the point of view of F. Engels. N. Bourbaki themselves wrote "…the main problem lies in the interaction of the experimental world and the mathematical one. The fact that there is a close connection between material phenomena and mathematical structures is what seems to be completely unexpectedly confirmed by recent discoveries of modern physics, but the profound reasons for this are completely unknown to us, and perhaps we will never know them” (9, 258). This is a pessimistic conclusion, and, according to academician B. V. Gnedenko, it means only that N. Bourbaki only superficially touched the most important question: what is the object of the mathematical study (17). They did not attempt to reveal the process of the basic concepts and basic tasks of mathematics formation in the historical aspect.

      Such questions cannot arise in connection with the definition of F. Engels, since it already contains the statement that mathematical concepts are only abstractions derived from certain relations and forms of the real world, they are taken from the real world and therefore are naturally associated with it. In essence, this explains the amazing applicability of the results of mathematics to the phenomena of the world around us, explains the success of the process that we are now witnessing and which is called the “mathematization” of knowledge. A number of examples is known when abstractly created mathematical theories were far ahead of the discovery of the corresponding real physical processes in the field of natural science. “The amazing, incomprehensible efficiency of mathematics in natural science, the fact that its modern models often describe quite well the complex processes of material reality, evidence of the fact that that mathematics reflects not only the quantitative, but also to some extent qualitative aspect of the objective reality phenomena, and that was noticed yet by Kant and Hegel” (20, 16).

      1.7 Hypothesis of “associative analogy”

      If we analyze the state of modern mathematics as a field of science, as a language of science in a historical aspect, and reveal the process of the basic concepts formation, it becomes obvious that modern mathematics has a logical internal structure, elements of which are, in turn, the same mathematical structures, amazing applicability of which is so surprising (“the principle of hierarchy of structures” by N. Bourbaki).

      But if mathematical concepts are abstractions of relations and forms of the real world, are taken from the real world and are naturally associated with it, then the question arises – whether the internal structure of modern mathematics, formed in the process of historical abstraction of forms and relations of the real world, can reflect the underlying fundamental structure of the real the world? Isn’t the internal structure of mathematics a model of the real world? If this is so, then there is a unique opportunity to look at objective reality through the prism of the internal structure of modern mathematics. So, what is the basis of modern mathematics?

      In accordance with the research of the N. Bourbaki school, the set theory is the foundation of modern mathematical knowledge. “It is possible to derive almost all modern mathematics,” Bourbaki write, “from a single source, the theory of sets” (43, 26). The theory of sets, as it is well known, is based on two concepts – the concept of “set” and the concept of “relation”. “Set” is a collection of elements. The element of the set is the main structural unit in the simulation of objective reality by the means of mathematics. The concept of “relationship” reflects the presence of connections between elements of a set. The combination of the elements of a set and connections, relations between them form a specific mathematical structure (43). Thus, the concepts of “set” and “relation” can be considered as the foundation of the logical structure of mathematics.

      Consider some “set of elements”. The relation (the law of composition) between the proper elements of this set is defined as internal (unary, binary, ternary – depending on the number of elements). The simplest mathematical structure – the groupoid 

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