Читать книгу The ideological foundations of technological singularity - Boris Shulitski - Страница 11

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