Towards Scientific Metaphysics, Volume 2

Benedykt Bornstein’s Geometrical Logic and Modern Philosophy

by Krzysztof Śleziński (Author)
©2019 Monographs 220 Pages


In the first half of the 20th century, many Polish philosophers focused on defining scientific concepts that could lead to the development of a philosophical synthesis of reality. Benedykt Bornstein elaborated an algebraic logic and developed novum organum philosophiae in the form of geometrical logic, a mathematical system of universal science. It can also be stated that Bornstein was unquestionably the forerunner of philosophers who started research into spatial logic in the second half of the 20th century.
The aim of this study on Bornstein’s Geometrical Logic is to draw readers’ attention not only to the algebraic and geometrical tools used in philosophical research, but also to demonstrate its importance for contemporary philosophical discussions which use mathematical tools in topology and category theory.

Table Of Contents

  • Cover
  • Title Page
  • Copyright Page
  • About the author
  • About the book
  • Citability of the eBook
  • Contents
  • Introduction
  • Part One: Philosophical Logic and Selected Issues in Benedykt Bornstein’s Philosophy
  • 1 From Epistemology to the Ontology of Mathematics and Category Theory
  • 1.1 Overcoming the Antithesis between Sensibility and Reason
  • 1.2 Epistemological Decisions in the Philosophy of Mathematics
  • 1.3 Selected Issues in the Theory of Scientific Knowledge
  • 2 From the Algebra of Logic and Projective Geometry to Categorial and Dialectical Geometrical Logic
  • 2.1 Categorial Logic of Geometry
  • 2.2 Generalisations of Categorial Logic of Geometry
  • 3 Structural and Ontological Model of whole-being
  • 3.1 Universalism of the Structures of Geometrical Logic
  • 3.2 Geometrical Ontology as a Generalization of Geometrical Logic
  • 3.3 Logical and Ontological Reality
  • Part Two: Geometrical Logic.
  • Preface
  • Introduction: The Idea of Geometrical Logic
  • CHAPTER I: Geometrization of the Axioms of Algebraic Logic
  • CHAPTER II: Geometrization of the Theorems of Algebraic Logic
  • CHAPTER III: The Logical Plane and Space, and their Elements
  • CHAPTER IV: The Elements of the Categorial Plane and Complete Dual Squares
  • CHAPTER V: Sets of Four and Six Elements of the Categorial Plane
  • CHAPTER VI: Harmonic Elements in Geometrical Logic
  • CHAPTER VII: The Dichotomical and Tetrachotomical Harmonic Division of Concepts in Geometrical Logic
  • CHAPTER VIII: Specification of Mathematical Pan-Logic
  • CHAPTER IX: Spatial Forms of Specifications of Pan-Logic
  • Appendix: The Logic of Dichotomy and the Three Pythagorean Means
  • Part Three: Comments and Remarks on Geometrical Logic
  • 1 Algebraic Symbolism
  • 2 Categorial, Geometrical Logic and Whitehead’s Universal Algebra
  • 3 Selection of Axioms for Categorial-Algebraic Logic
  • 4 Interpretation of Algebraic Logic
  • 5 Algebraism of the Categorial and Projective Plane
  • 6 Topological Space of Topologic
  • 7 Idea of Mathematical Categorial Theory in Categorial and Algebraic Logic
  • 7.1 Definition of a Category
  • 7.2 Finite Categories in Bornstein’s Geometrical Logic
  • 7.2.1 The First Possible Application of the Category Theory into the Geometrical Logic
  • 7.2.2 The Second Possible Application of the Category Theory into the Geometrical Logic
  • Conclusion
  • Summary
  • Bibliography
  • Index


The endeavours of many philosophers, namely Plato, Descartes, Spinoza, Leibniz, Hegel and Hoene-Wroński, included the development of a research instrument thanks to which philosophy could be considered as a science. However, due to insufficient progress in formal and mathematical science, their studies failed to achieve their aim. Only the works of George Boole (1854), regarding the systemic study of algebraic logic, which constituted the mature concept of qualitative arithmetic as outlined previously by Leibniz, presented a new perspective of research in the sphere of qualitative mathematics. Nevertheless, only a few philosophers noticed new possibilities in the use of algebraic logic in philosophical studies. One of the first attempts to use algebra in philosophical analyses was the universal algebra developed by Alfred North Whitheade. However, these studies were not widely known or commented on by other philosophers, both Polish and foreign.

In the first half of the 20th century, many Polish philosophers focused on defining scientific concepts that could lead in the future to the development of a philosophical synthesis of reality. In contrast, Benedykt Bornstein elaborated an algebraic logic and developed novum organum philosophiae in the form of geometrical logic, topologic (λόγος – τόπος),1 later making a successful attempt to develop a mathematical system referring to mathematical and logical assumption, mathematical and universal science – the much sought-after mathesis universalis – referring to the universe in Teoria absolutu2 [Theory of the Absolute].

In his work Geometrical Logic. The Structures of Thought and Space, which is published in this monograph, Bornstein developed a precise, mathematical tool that he then used in his philosophical research. What is more, this work was printed in September 1939 in Warsaw. During defensive action against German troops, the printer’s warehouses burnt down, along with Geometrical Logic. No ←7 | 8→copy was saved. All that was left was the typescript, which his wife, Jadwiga, handed to the Jagiellonian Library with the reference number 9637 III after his death. This work deserves to be re-published, due to the spatialization of logic and the question regarding formal ontology. It is clear proof of the important achievements of Polish philosophers in the first half of the 20th century.

The aim of this study on Geometrical Logic is to draw readers’ attention not only to the algebraic and geometrical tool used in philosophical research and suggested by Bornstein, but also to demonstrate its importance for contemporary philosophical discussions that use mathematical tools in topology or category theory. Moreover, in the second half of the 20th century, philosophers started research into spatial logic,3 therefore one can assume that Benedykt Bornstein was their unquestionable precursor. From the perspective of contemporary algebraic topology, his works can be considered as pioneering.

Studying Geometrical Logic, I realised that one cannot separate the mathematical tool from the individual stages of Bornstein’s scientific research, thus the first part of this elaboration contains a description of his philosophical and metaphilosophical research, which corresponds with the specification of the original concept of topology. In this part, I want to draw readers’ attention to Bornstein’s development of algebraic logic, as well as to certain spheres of research in epistemology, logic, metaphysics and ontology. The second part of this monograph contains the text of the typescript Geometrical Logic. The Structures of Thought and Space. In Geometrical Logic. The Structures of Thought and Space, numbering consistent with the original typescript has been preserved (the numbering is presented in square brackets). In the third part, I insert comments and critical remarks regarding the tool used in the philosophical research presented in the second part. These remarks are devoted to several ←8 | 9→fundamental questions: (1) to what extent Bornstein’s topology is congruent with the results of studies in the sphere of algebraic logic at the turn of 19th and 20th centuries, (2) if one translates the notation of the formulae of categorial and algebraic logic to the functional notation of category theory or the symbolism of topology, is the effectiveness and validity of this tool in philosophical research preserved? In this part, I omit many remarks regarding the symbolic notation of algebraic logic, as well as ambiguity of the formulae and the lack of precision in his studies, as these were elaborated in two of my previous studies on Bornstein’s unpublished works.4

Analysing the results of Bornstein’s scientific research, one can deduce that he was a programmatic philosopher who marked his presence in the endeavours started in antiquity to qualitatively describe reality with the use of mathematical and logical tools. Confirmation of his statements are the results of his research, which are described in the first part of this monograph, as well as the scientific biography, included underneath, for better understanding of the stages and the sphere of his scientific interests.

Benedykt Bornstein was born on 31st January 1880, in Warsaw. In secondary school, he became interested in philosophy. His neokantian teacher, Henryk Goldberg, who later worked as an editor of “Biblioteka Filozoficzna” [Philosophical Library], familiarised his students with Immanuel Kant’s philosophy in conversations and discussions. It turned out that Bornstein’s interests regarding Kantian philosophy were present during his scientific work.

Having graduated from high school, in 1900 he started philosophical and mathematical studies at the Mathematics and Physics Faculty of the (then Russian) University in Warsaw. In 1905, while participating in The Association of Progressive Youth, he was one of the organisers of the so-called January mass meeting, because of which he was expelled from university along with many other students.5 In order to continue his studies, he travelled abroad: firstly ←9 | 10→to Berlin, and then for a short period of time to Lvov, where, in 1907, under the direction of Kazimierz Twardowski, he wrote a dissertation Preformowana harmonia transcendentalna jako podstawa teorji poznania Kanta6 [Pre-established Transcendental Harmony as the Foundation of Kant’s Theory], after which he obtained his PhD. After his studies, he returned to the Kingdom of Poland, where he lived in the countryside for a few years, dedicating himself to mathematical and philosophical studies, and then permanently moved to Warsaw, where he stayed until the German destruction of the city in 1944.

In 1915, he started work in the Philosophy Department at the Faculty of Humanities in the Association for Scientific Courses, which later transformed into the Free Polish University. From 1928, he worked as the head of the Department of Systematic Philosophy in the affiliated Free Polish University in Łódź.7 After the German invasion of Warsaw in 1939, the university was closed, and Bornstein was arrested. From 4th October to 13th January 1940 he was imprisoned for 101 days in the so-called “Serbia prison” in Pawiak prison.

After his release from prison, he taught logic and epistemology at underground meetings of the Free Polish University – from 21st February to the first day of the Warsaw Uprising on 1st August 1944. He gave lectures three to seven times a week for two hours, usually at the school on Plac Wilsona [Wilson Square] in Żoliborz. The school was run by Stanisław Trojanowski, who subsequently worked in the Łódź School District. The lectures were held in places located in the immediate vicinity of German offices, facilities or military premises.8

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After the uprising, in October 1944, Bornstein travelled to Częstochowa, where he also took part in underground education, in so-called Academic Courses, which were organised in the same way as the underground universities in Warsaw. The courses lasted until the middle of January 1945, when Częstochowa was liberated from the occupying power.

In February 1945, prof. Teodor Vieweger, the last head of the Free Polish University, organised the national university in Łódź. Bornstein cooperated with him and inaugurated lectures in March. When the University of Łódź was created, he became the head of the department of Logic and Epistemology, which was transformed later into the Department of Ontology, where he gave lectures until the end of the academic year 1947/48. He died suddenly on 11th February 1948 after surgery, leaving many manuscripts and outlines of work for the following years.

For his whole life he worked alone, although he did not isolate himself from philosophical society: he participated in Polish Philosophical Meetings where he presented his papers,9 published his articles in “Przegląd Filozoficzny” [Philosophical Review], “Wiedza i Życie” [Knowledge and Life] and “Przegląd Klasyczny” [Classical Review]. Many of his papers were published in Sprawozdania Łódzkiego Towarzystwa Filozoficznego [Reports of the Philosophical Society of Łódź]. He also published abroad, for example in “Imago” and “Ruch Filozofický”.

Bornstein’s philosophical views were not very popular. However, this does not mean that his research went unnoticed. Very frequently, philosophers expressed their opinions in reviews of his works, which were published in “Ruch Filozoficzny” [Philosophical Movement] and “Przegląd Filozoficzny”. Among philosophers interested in Bornstein’s work, one can distinguish: Kazimierz Ajdukiewicz,10 Józefa ←11 | 12→Kodisowa,11 Bad Hersch,12 Stanisław Leśniewski,13 Zygmunt Zawirski,14 Tadeusz Kotarbiński,15 and Wiktor Wąsik.16 Only at the end of his scientific work did many foreign philosophers notice his endeavours. One professor from Sorbonne University, Etienne Sougian, highlighted the similarity between his own research and Bornstein’s studies. What is more, he also pointed out the correspondence between their results, even though they used diverse methodology. Moreover, professor Uuno Saarnio from Helsinki was interested in Bornstein’s Architektoniką świata [Architectonics of the World], especially the problem of the geometry of logical algebra.17

It is worth emphasising that Bornstein became interested in Kantian philosophy very early, in his secondary school, thanks to Henryk Goldberg, who was his neokantian teacher and also an editor who published in “Biblioteka Filozoficzna” [Philosophical Library]. His initial interests were visible during his whole scientific work.

After his studies, Bornstein started his research with a critique of Kant’s epistemology and philosophy of mathematics, realising what the biggest problem of epistemology and the understanding the structure of reality was. He dedicated a huge part of his work to solving the problem of dependency between the world ←12 | 13→of thought and the spatial world, and therefore he can be classified as a programmatic philosopher, who creatively developed a specified programme of research, in contrast to those philosophers who are systematic and who organize their knowledge at each given stage of research.

Although Bornstein’s scientific work concerns only one main question, it can be divided into three stages, which permeate and which constitute evidence of maturation on the path to reaching his own philosophical concept. The first stage starts with publication of his dissertation: Preformowana harmonia transcendentalna jako podstawa teorji poznania Knata (1907), in which he undertakes above all epistemic questions. Simultaneously, Bornstein translates Kantian works and comments on them, developing his thought in a creative and critical way. Epistemological research directed Bornstein’s attention towards the relation between the world of thought and the world of spatial objects.

The second stage of his work includes research on the philosophy of mathematics and logic, the foundations of which were the aspects of logic and qualitative mathematics (algebraic logic and projective geometry), not quantitative mathematics. It turned out that both of these spheres can be considered as one and can be presented as geometrical logic in qualitative and categorial form. One can attribute the world of senses and concepts to spatial elements. Such representation resulted in clarity of the elements of the world of thought, whereas spatial elements were introduced to the sphere of philosophical research.

Having construed a mathematical and logical tool of philosophical research in the form of categorial geometrical logic, Bornstein proceeded to ontological research. He developed a general theory of objects, the so-called theory of the absolute. He discovered the presence of the structures of categorial geometrical logic in the formal and real spheres: the world of sound, the periodic table, genetics and the world of numbers. Also, he discovered the structures of categorial geometrical logic in the philosophical systems of Plato, Descartes, Spinoza, Hegel and Hoene-Wroński. The tool construed by him was continually improved and generalised until the development of categorial dialectical logic.

Bornstein was generally respected thanks to his forbearance and kindness. Wiktor Wąsik, who knew him personally, considered him a reliable academic and a man of principle.18

* * *

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I would like to thank Andrzej Obrębski, the Head of the Extraordinary Collections Department in the Jagiellonian Library, for sharing the typescript of Geometrical Logic. The Structures of Thought and Space, and for granting permission for this work to be published.

←14 |

1 Benedykt Bornstein, “Co to jest kategorialna geometria algebraiczno-logiczna? [What is the Categorial Algebraic Logic Geometry?],” in: Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism [Benedict Bornstein’s Philosophy, Selection and Elaboration of Unpublished Writings], ed. Krzysztof Śleziński (Katowice: Uniwersytet Śląski – Wydawnictwo Scriptum, 2011), pp. 74–75.

2 Benedykt Bornstein, Teoria absolutu. Metafizyka jako nauka ścisła [Theory of the Absolute. Metaphysics as an Exact Science], Łódź: Łódzkie Towarzystwo Naukowe, 1948.

3 Amongst the works dedicated to spatial logic, geometric space or studies on mutual dependence of the sphere of logic and geometric space, one can distinguish: Marco Aiello, Ian Pratt-Hartmann, “What is Spatial Logic?,” in: Handbook of Spatial Logic, eds. Marco Aiello, Ian Pratt-Hartmann, Johan Benthem (Springer 2007), pp. 1–11; Davide Lewis, Counterfactuals (Oxford: Blackwell, 1973); Mormann, Thomas. “Accessibolity, Kinds and Laws: A Structural Explicatio,” Philosophy of Science, Vol. 61, No. 3 (1994), pp. 389–406; Philippe Balbiani, “The Modal Multilogic of Geometry,” Journal of Applied Non-Classical Logics, Vol. 8, 1998, pp. 259–281; Yde Venema, Points, Lines and Diamonds: A Two-Sorted Modal Logic for Projective Planes,” Journal of Logic and Computation, Vol. 9, No. 5 (1999), pp. 601–621; Thomas Mormann, “On the Mereological Structure of Complex States of Affairs,” Synthese, Vol. 187, 2012, pp. 403–418.


ISBN (Hardcover)
Publication date
2019 (April)
philosophy of science logic and ontology Polish philosophy spatial logic category theory
Berlin, Bern, Bruxelles, New York, Oxford, Warszawa, Wien, 2019. 217 pp., 30 fig. b/w, 4 tables

Biographical notes

Krzysztof Śleziński (Author)

Krzysztof Śleziński is a professor of philosophy at the University of Silesia in Katowice, Poland. His research and teaching interests focus on philosophy of nature, philosophy of science, ontology, philosophy of education and the history of philosophy in Poland.


Title: Towards Scientific Metaphysics, Volume 2
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