The universal characteristic of G. W. Leibniz and the prospective developments in the foundations of mathematics

G. W. Leibniz was not the first who expressed the idea of creating a universal perfect language, but he proposed a very different approach in his project. The purpose of Leibniz’s project was to represent all human knowledge in the form of a kind of “universal algebra.” Leibniz intended to implement the translation into this universal language with the help of a universal characteristic, to which some special computing operations could be applied. By “characters” Leibniz meant symbols used to denote both numbers and geometric quantities. Since logic is similar to algebra, according to Leibniz, it must be built as a “universal mathematics.” As part of the development of this idea, he undertook a number of attempts to arithmeticize and algebraize syllogistics. Despite the setbacks that befell Leibniz in this work, two points must be specially emphasized. First, logic was subjected to arithmetization (logical relations were represented in the form of arithmetic or algebraic ones). Secondly, it should be noted that the resulting systems were intensional (the relationship between concepts, as well as the structure of propositions acquired an intensional interpretation). Leibniz discovered some kind of correspondence between the statements of logic and the statements of geometry in the course of his research. Further development of the ideas of Leibniz was carried out by two somewhat opposite ways. The first way is to build up logic as a calculus free of all content, as a study of a pure logical form. This path is associated primarily with the algebra of logic. The second way can be traced through the ideas of Neo-Kantianism, Grassmann, J. Peano and G. Frege, through the emergence and development of type theory and, finally, in modern approaches to the foundations of mathematics, and primarily in the homo-topy type theory. In modern variations of the type theory, the distinction between logical form and non-logical content ceases to be clear. In the homotopy type theory, the logical form has a geometric nature. In connection with the Leibniz’s program, it should be noted that indeed the homotopy type theory has the required characteristics: it is intensional, it has a computational interpretation and directly describes calculations, it does not require the use of an external deductive system, its “logical” and “geometric” interpretations are equivalent, it does not assumes the reduction of the whole variety of mathematical entities to the entities of some special kind, and it is structuralist in its nature.

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