Mathesis Universalis (Latin universal science, from the Greek μάθησις – knowledge, science and Latin Universalis – general) is a philosophical term, especially popular in the 16th and 17th centuries and denoting the concept of a hypothetical universal science built on the type of some calculus. He goes back to the idea of the universal organ of knowledge (Greek ὄργανον – tool, instrument), which is still coming from Aristotle, and also to the occult knowledge, which is peculiar to the cultures of the Ancient East, as a set of effective methods. In the medieval Arabic mathematics, in the spirit of this tradition, the standardization of methods for solving equations begins, a prototype of modern algebra is created (the words “algebra” and “algorithm” of Arab origin, from the corresponding Arabic “al-Djebre” – rule of signs in solving equations, “al-Khwarizmi” – the name of the Arab scientist of the 9th century).

In the 13th century, the Franciscan missionary Raymond Lully tried to construct some universal algorithm for automating the process of logical reasoning (Lullius’s “Great Art” was printed only in 1480). The entire 16th century passes under the sign of a persistent search for convenient algebraic symbols that would allow the creation of some calculus for solving various problems (K.Rudolf, M.Stifel, R.Bombelli, P.Ramus, S.Stevin, F.Viet). In the late 16th century, in the Sorbonne, J. Bruno propagandizes his version of lulism. The 17th century can be called the “blossoming age” of the idea of mathesis universalis. At the same time, the idea of an anthropoid machine, the “golem,” is very popular.

Then R.Dekart creates his own method of analytic geometry, which allows us to reduce the solution of geometric problems (of a certain class) to the solution of algebraic equations. Descartes wants to reduce physics to geometry, and the latter – to algebra, which thereby appears to be the embodiment of the desired mathesis universalis. The Descartes method in geometry becomes extremely popular, although critical voices are also heard. Thus, I. Newton, who brilliantly possessed the method of analytic geometry, believed nevertheless that the reduction of geometry to “calculations” ignores the nature of geometric science. Even further in the direction of realizing the idea of mathesis universalis, GV Leibniz advanced. He, systematically sketching the version of this science, called it “universal characteristic”.

Discovered by Leibniz (independently of Newton), the differential and integral calculus should have been only part of it, namely the part devoted to the problems of infinity. He also makes sketches of a “geometric characteristic” – a kind of algebraic-topological method for studying curves and surfaces. But the main direction was the task of formalizing logic, reducing it to such an algebra of calculus. With the help of the latter, Leibniz hoped to effectively build a system of natural science and solve “the main moral and metaphysical problems.” However, it was only in the 19th century that a real creation of mathematical logic began, which became a powerful tool for studying the foundations of mathematics. And it was the means of mathematical logic that succeeded in proving some theorems of “undecidability”, showing the impossibility of realizing the Leibniz project mathesis universalis in its entirety.

At the same time, from the 30s of the 20th century, a theoretical study of “machine thinking” (the theory of algorithms) begins, which leads to the construction of the first electronic computers in the 1950s, and to the powerful development of computer technology in the 1970s. The progress of the latter, the widespread introduction of computers into the sphere of management of complex technological processes, their application not only in physical and mathematical sciences, but also in humanities – linguistics, economics, psychology – make it possible to solve the problem to a certain extent in human management, information processing, exploratory and partly research activities.

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