The work of Ivan Punchev entitled “Introduction to the system of dialectical logic is composed by four books dedicated to one of the most fundamental and most difficult problems in philosophy. This is building of integral system of rational dialectical logic. Currently such system exists (as a mystical type) only in Hegel, and in rational form this logic was applied only in Marx’s “Capital”. Despite the efforts of many philosophers for more than a century, solution to this problem has not been achieved ​​entirely. The problem is further complicated with the advent of the mathematical logic as a historic new stage in the development of formal logic. This situation is related also with the famous “third crisis” in the foundations of mathematics started by the introduction of the mathematical concept of infinity in Set Theory and the consequently raised antinomies. At the same time, the mathematical logic created a new standard for contemporary systematization of any scientific knowledge and thus put a requirement for mathematical modeling of the classic dialectical logic. This issue is central to the four volumes. In its context are studied problems of the history the idea of a mathematical dialectical logic and dialectical mathematics. The scale and depth of this problematic situation require exploration of all main ideas in philosophy, logic and mathematics. This determines the nature of this study as integrative, interdisciplinary and complex.

Part I traces the idea of a single dialectic mathematical reasoning in the history of philosophy, logic and mathematics from Antiquity and Middle Ages to the Modern times. Main part of the content is devoted to unexplored until now “philosophical program” of Nicolas of Cusa to build a new, non-Aristotelian, dialectical logic in which the traditional categories of logic are merged with the mathematical structures. Cusa was first to voice the idea to build mathematical dialectical logic and dialectic mathematics. Thanks to this, Nicolas of Cusa became widely recognized forerunner as of the science revolution done by Copernicus, Galileo and Newton and the whole mathematics of the New Time, particularly integral and differential calculus and analytical and projective geometry. In Part I, these ideas of Cusa are associated with the modern mathematical logic and Set Theory.

Part II is a systematic reconstruction of Hegel’s dialectical logic. It discusses the potential for its mathematical modeling, particularly using Cantor’s Set Theory and presentation of dialectical categories as concepts with actual volume and content. Particular attention is paid to the remarkable idea of Hegel that dialectical interpretation of the inferences transforms them into logical system objects, so called endless “threefold inferences.” In connection with this, in Part III devoted to the reconstruction of the dialectical-logical structure of “Capital”, this important Hegelian logic discovery of the “threefold inference” is revealed in its rational form as the main structure of Marx’s epochal book. So far, that fact was not discovered by none of the scholars of rational dialectical logic.

Part IV discusses the current situation in the universe of mathematical logic and mathematics. The hypothesis that dialectical-logical method, if applied to the Set Theory as a base of modern mathematics, can help overcoming the antinomies in its foundations is systematically examined. The antinomies are summarized and new dialectical notion for the category “set” is developed. The author proposes a new concept, namely “absolute or perfect class”, which treats the set as its element. It resembles algebra of sets, which are fitted with arithmetic signs plus and minus. Because of this, to them are applicable arithmetic groups of symmetry and asymmetry. Meanwhile, the antinomy “sets” are presented as single units of these groups – 0 (zero) and 1 (unit), and thus, are resolved rationally. In general, the idea of mathematical and dialectical logic and dialectical-logical mathematics is developed in the context of a “cosmic philosophy”, which reflects the modern revolution in quantum cosmology and the vision for endless multitudes of universes with different nomological structures populated with intelligent life.

Further to this, Ivan Punchev intended to offer its own “system of mathematical and dialectical logic” developed by mathematical modeling of the traditional dialectical logic. Unfortunately, he left us before this to be accomplished.* This remains as a main goal of the Stream Logos of Human Future Project.

 

* This text was prepared by Ivan Punchev before his death.

Ivan Punchev, PhD

Ivan Punchev, PhD

(1942 - 2009)

For more than 40 years, Ivan Punchev worked in Philosophy Institute at Bulgarian Academy of Sciences. He has a unique contribution to the theory of dialectical-logic systems. With his profound knowledge in Hegel’s philosophy and the history and theory of mathematics, he is perhaps the best mathematician among philosophers and best philosopher among mathematicians. He is continuer of the concept of Cosmic Philosophy with considerable innovative ideas.