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

(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.

FYI: I have posted a link to this blog-entry in my blog — https://feddialectics-miguel.blogspot.com/2018/03/mathematical-dialectics-and-dialectical.html .

Regards,

Miguel

Thank you Miguel, Couple of years ago I found your web site dialectics.org, and it was one of the sources of my own reflections on dialectics. I am trying to carry on further what Dr. Punchev achieved. In this context currently I am doing my own PhD thesis called “Ontology of Infinity: Mathematics, Physics and Metaphysics”. I started from a hint from Dr.Punchev that the algebra of dialectics could be developed using the algebra of complex numbers. Further to this I discovered isomorphism between the basic mathematical abstracts and the dialectical notions. I see that in many aspects we (your project and what we have tried to do with Dr. Punchev) share the same insights regarding the dialectic as a cosmic principle. Probably the main difference (also between me and Dr.Punchev) is that currently I hold that the dialectics is already formalized in the whole mathematics itself, actually the mathematics could be called а symbolic dialectic. In this way one would not need to invent and teach new special dialectic symbolic language: this would be enough to negate the mystification of the mathematics though a critique of its Aristotelian shell, which I believe is a historic product, not an essence of mathematics. Here I cannot go into details, but I will be very glad if we keep in touch to discus all these quite interesting for me topics. You efforts so far have been enormous, and I think that we through a dialog we can help each other.

Yes, we too believe that our continuing dialogue with you will prove fruitful for both projects.

With your concurrence, we would like to add your web site URL — http://humanfuture.info/en/home/ — to the list of links on the Links Page of the http://www.dialectics.org web site:

http://www.dialectics.org/dialectics/Links.html .

Your view of modern mathematics as a whole as embodying a “symbolic dialectics”, though with a mystical “philosophy” [actually an “ideology” in Marx’s sense] about what this mathematics really is, reminds us of Marx’s comment to the effect that “reason always exists, but not always in a rational form”. We might adapt this quote, for your application, to something like: “dialectical reason has always existed in mathematics, but not always in an explicitly dialectical rational form”.

Nevertheless, for our part, we find that, just as Marx’s immanent critique of the mystification-permeated science of classical political economy led to a new, improved science of Marxian ‘psychohistorical socio-politico-economics’, as set forth in the volumes of Marx’s Das Kapital, so too an immanent critique of the mystification-enfettered ‘ideo-science’ of modern mathematics tends to yield new, improved mathematics.

To begin with, this new, improved mathematics includes, in our experiences, through the immanent critique of the standard Peano/Dedekind “Natural” numbers system of arithmetic, a Lakatosian “counter-example” axioms-system of arithmetic, which is one of the “predicted” non-standard models of the “Natural” numbers arithmetic, whose algebra is also a ‘contra-Boolean algebra’, modeling a categorial-dialectical logic: an algebra in which a strong contrary to Boole’s “fundamental law of thought” [asserting an “inescapability” of LINEAR algebraic logic] is a theorem that arises deductively from the axioms of this “non-standard” arithmetic.

Miguel, I sent you email with one paper on the mathematics and dialectics. Something like starting point in our dialogue.

I fully agree with you regarding “dialectical reason has always existed in mathematics, but not always in an explicitly dialectical rational form”. That’s why I am trying to do in my PhD thesis. I am doing this as a historic system analysis of economic development of modern society as a source of the mystification of the truth in mathematics and sciences in generally. My point is that the industrial society, its political form, the national state, and the spirit of this national state, the bureaucracy, require widespread adoption of so called by me “concrete dialectics” focused on study only and exclusively of the finity. As its opposite term in the philosophy (continental philosophy, existentialism, phenomenology, etc.) the focus is on the abstract continuum of the mind as actual infinity (the reflexive negation, negation of negation). So, I call this “abstract dialectics”. The thirty contemporary form of dialectics is the “symbolic dialectic”, the mathematics, that however is still existing in its alienated form defined by the non-contradictory truth, coming from the social influence of the “concrete dialectics”. Something like this.