# Towards Scientific Metaphysics, Volume 2

Benedykt Bornstein’s Geometrical Logic and Modern Philosophy

## Excerpt

## Table Of Content

- 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 absolutu^{2} [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 Kanta^{6} [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}

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}

* * *

←13 | 14→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 | 15→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.

4 Krzysztof Śleziński, Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism [Benedykt Bornstein’s Philosophy, Selection and Elaboration of Unpublished Writings] (Katowice: Uniwersytet Śląski – Wydawnictwo Scriptum, 2011); Krzysztof Śleziński, Benedykta Bornsteina niepublikowane pisma z teorii poznania, logiki i metafizyki. Wybór i opracowanie oraz wprowadzenie i komentarze [Benedykt Bornstein’s Unpublished Writings about Theory of Knowledge, Logic and Metaphysics. Selection and Elaboration as well as Introduction and Comments] (Katowice: Uniwersytet Śląski – Wydawnictwo Scriptum, 2014).

5 Benedykt Bornstein’s biography can be found in: Wiktor Wąsik, “Benedykt Bornstein (1880–1948),” Przegląd Filozoficzny [Philosophical Review], Vol. 44, No. 4 (1948), pp. 444–451; Wiktor Wąsik, Benedykt Bornstein (1880–1948). Wspomnienie [Benedykt Bornstein (1880–1948). Memory], can be found in the Jagiellonian Library, ref. n. 9640 III; Bronisław Bieniek, “Benedykt Bornstein – filozof mało znany [Benedykt Bornstein.Philosopher Little Known],” Studia Warmińskie, Vol. 31, 1994, pp. 233–250; Bronisław Bieniek, Topologiczno-ontologiczne poglądy Benedykta Bornsteina [Topological and Ontological Views of Benedykt Bornstein] (Olsztyn: Wyd. Uniwersytetu Warmińsko-Mazurskiego, 2005), pp. 21–27, as well as: Śleziński, Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism, pp. 11–16.

6 Benedykt Bornstein, Preformowana harmonia transcendentalna jako podstawa teorji poznania Kanta, Kanta [The Pre-established Transcendental Harmony as the Foundation of Kant’s Theory],” Przegląd Filozoficzny [Philosophical Review - PR], Vol. 10, No. 3 (1907), pp. 261–303.

7 Apart from his scientific and educational work during the interwar period, in 1920 he took part in the Polish-Soviet War as a volunteer soldier of the 201st Polish Cavalry (Jan Bełcikowski, Opowieści szwoleżerów 1-go pułku J. Piłsudskiego zebrał i ułożył J. Bełcikowski (Radom: 1921), pp. 30–31).

8 Benedykt Bornstein, “Fragment autobiografii (9 September 1947) [Fragment of the Autobiography],” Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism, pp. 12–13. The typescript can be found in the Jagiellonian Library, ref. n. 9040 III.

9 Bornstein presented the following papers: at the I Polish Philosophical Meeting in Lvov in 1923 – Myślenie a widzenie [Thinking vs. Seeing], at the II Polish Philosophical Meeting in Warsaw in 1927 – Logiczna (topologiczna) struktura rozmnażania organicznego [Logical (topological) structure of organic reproduction], at the III Polish Philosophical Meeting in Cracow in 1936 – Ontologiczne znaczenie logiki geometrycznej [Ontological Meaning of Geometrical Logic], after which there was a discussion between: J.J. Stępniewski, G. Gawecki, Z. Zawirski, N. Łubnicki, J. Braun.

10 Kazimierz Ajdukiewicz, “Benedykt Bornstein. Prolegomena filozoficzne do geometry [Benedykt Bornstein. Philosophical Prolegomena to Geometry],” Ruch Filozoficzny [Philosophical Movement], Vol. 3, 1913, pp. 193–195.

11 Józefa Kodisowa, “Benedykt Bornstein. Zasadniczy problemat teorji poznania Kanta [Benedykt Bornstein. Fundamental Problem of Kant’s Epistemology],” Przegląd Filozoficzny [Philosophical Review], Vol. 13, 1910, pp. 343–347.

12 Bad Hersch, “Kant Immanuel. Prolegomena do wszelkiej przyszłej metafizyki, która będzie mogła wystąpić jako nauka [Kant Immanuel. Prolegomena To Any Future Metaphysics That Will Be Able to Present Itself as a Science],” Ruch Filozoficzny [Philosophical Movement], Vol. 6, 1921–1922, pp. 88–90.

13 Sanisław Leśniewski, “Teorja mnogości ‘na podstawach filozoficznych’ Benedykta Bornsteina [Set Theory on the ‘Philosophical Foundations’ of Benedykt Bornstein],” Przegląd Filozoficzny, Vol. 17, 1914, pp. 488–507.

14 Zygmunt Zawirski, “Benedykt Bornstein. Elementy filozofii jako nauki ścisłej [Benedykt Bornstein. Elements of Philosophy as an Exact Science],” Ruch Filozoficzny, Vol. 6, 1921–1922, pp. 119–122.

15 Tadeusz Kotarbiński, “Bornstein Benedykt. Teoria absolutu – Metafizyka jako nauka ścisła [Benedykt Bornstein. Theory of the Absolute. Metaphysics as an Exact Science],” Ruch Filozoficzny, Vol. 17, 1949–1950, pp. 13–16.

16 Wiktor Wąsik, Wspomnienie o zmarłych filozofach. Benedykt Bornstein (1880–1948) [Recollection of the Dead Philosophers. Benedykt Bornstein (1880–1948)],” Przegląd Filozoficzny, Vol. 44, No. 4, (1948), pp. 444–451.

17 See: Wąsik, Wspomnienie o zmarłych filozofach. Benedykt Bornstein (1880–1948), p, 447.

18 Wąsik, Wspomnienie o zmarłych filozofach. Benedykt Bornstein (1880–1948), p. 448.

Part One: Philosophical Logic and Selected Issues in Benedykt Bornstein’s Philosophy

1 From Epistemology to the Ontology of Mathematics and Category Theory

Benedykt Bornstein’s initial interests were closely related to Immanuel Kant’s epistemology. In his dissertation Preformowana harmonia transcendentalna jako podstawa teorii poznania Kanta [Pre-established Transcendental Harmony as the Foundation of Kant’s Theory], Bornstein presented theses which evolved into the foundations of his critique of Kant’s epistemology and were developed in his further papers: Zasadniczy problemat teoryi poznania Kanta [The Fundamental Problem of Kant’s Epistemology],^{19} and Krytyka immanentna filozofii gieometrji Kanta, [Critique of Kant’s Immanent Philosophy of Geometry],^{20} in which he sought his own solutions to the arising epistemological problems. Bornstein’s thorough understanding of Kant’s philosophy originated from his Polish translations of two of his essential works: Critique of Practical Reason and Prolegomena To Any Future Metaphysics That Will Be Able to Present Itself as a Science.

1.1 Overcoming the Antithesis between Sensibility and Reason

One of the most important questions that pre-Kantian philosophy was confronted with was how to determine the compliance of a priori reasoning with physical objects. According to Bornstein, Kant assumed the existence of necessary, common and objective knowledge referring merely to experience and its potential. As Bornstein pointed out in his Preformowana harmonia transcendentalna jako podstawa teorii poznania Kanta, one cannot rely solely on empirical conditions in order to reach objective knowledge. Kant’s concept ←15 | 16→of experience means that the necessary, intellectual link between intuition and perception can be achieved in the concept of the object. Experience should be understood as a unified system that consists of sensual data united by reason, necessarily and objectively, whilst experience and its objects become possible only through the connection of sensibility and reason.

Kant proved in his transcendental reasoning that there is a relation between a priori forms of sensual knowledge, intuitions and incidental perceptions. Nevertheless, as Bornstein noted, he fails to explain the relation between categories and empirical intuition. In the concept of experience, so-called empirical intuition contains the theme of phenomenon – imagined in space and time – by which one can refer to an object. As per Kant’s theory, for objects to exist in space and time they must simply exist, regardless of the content of the objects, space and time, as these are the objects’ formal conditions recognized in the human mind. Accordingly, space and time, a Kantian necessity a priori, shall be considered as pure intellectual intuitions which determine the possibility of empirical intuitions, and therefore the conditions of experience. Nevertheless, Kant failed to explain how – despite the separation of sensibility and reason – intuitions are subject to conditions expressed in categories and how these categories, independent from intuitions, may be considered useful when it comes to constructing experience of new elements that are strange to them. As Bornstein pointed out, Kant unsuccessfully attempted to prove that there is a relation between elements which are in fact independent of each other, rather than search for another far more reasonable and harmonious relation between sensibility and reason. Bornstein observed that even though Kant was on the verge of this only possible and consistent solution in the theory of epistemology, he deliberately avoided it as too cumbersome for his own philosophy.^{21}

Kant assumed that a priori forms, categories and principles – in order to attain validity – have to impose themselves upon sensibility in the form of space and time. Thereby, space and time – idealistic categories – impose themselves on a posteriori sensual data, which is the result of the noumenon’s influence on sensibility. Kant was fully aware of the difficulties he faced in his reasoning, while simultaneously adopting his Copernican hypothesis – how logical forms can impose themselves on our sensibility and at the same time overcome a given outside element that is new and strange to them. In order to overcome ←16 | 17→these difficulties, he focused on a priori sensual forms, showing their affinity with forms of reason. Then, he chose time as the more general form of the two a priori sensual forms and recognized it as the analogon of the logical categories, the schema. Therefore, logic obtains its own representation in sensual phenomena and, at the same time, one can acknowledge a correspondence between phenomena and logical principles. As Bornstein pointed out, in the solution to the basic difficulties proposed by Kant, the supremacy of logic over sensibility disappears. Here, logical forms do not dominate over sensibility and do not impose themselves on it. On the contrary, there is a clear correspondence between a priori reason and a priori sensibility. Yet, such correspondence was not in line with Kant’s Copernican Revolution, and therefore it became marginalized in his philosophy.^{22}

Bornstein’s critical studies into transcendental categories allowed him to measure the epistemic potential of philosophy. In the first place, as he concluded, one must explain and overcome Kantian opposition between reason and the senses. Comparing the concepts of categories in the first and second edition of the Critique of Pure Reason, Bornstein eventually discussed in his work Zasadniczy problemat teorii poznania Kanta the existence of the so-called harmonia praestabilita between sensibility and reason.^{23}

Then, in his work Krytyka immanentna filozofji gieometrji Kanta,^{24} Bornstein remarked that the Kantian philosophy of geometry cannot be considered as a consistent system when it comes to the dualism of sensibility and reason. According to Bornstein, while Kant’s aim was to describe the a priori status of geometry, his concepts rather demonstrate their empirical origin. Therefore, every geometrical axiom can prove its correctness only a posteriori, while geometrical space properties depend on non-subject factors.

Bornstein’s Kantianism never faded completely. Yet, from 1913, his interests turned to speculative philosophy and studies into the philosophical fundamentals of mathematics. This new chapter in his epistemological research began with the work Kant i Bergson. Studium o zasadniczym problemacie teorii poznania ←17 | 18→[Kant and Bergson. A Study of the Fundamental Epistemic Problem],^{25} in which Bornstein showed a similarity between Kantian rationalism and Bergson’s intuitionism despite often-expressed opinions as to the differences between these two epistemic concepts. Bornstein argued that Bergson’s transition from intuition to its corresponding concept was similar to the Kantian one. They also shared the same view on the relation between images and concepts. Moreover, Bergson’s deeply emphasised irrationalism has its roots in Kantian epistemology. Both Bergson and Kant shared similar ideas about the generality of concepts and the individuality of the objects of intuition. They see the essence of concept in generality, while the essence of intuition expresses itself in individuality. They do not acknowledge the existence of an individual concept adjusted to a given object. Besides, they underline the incommensurability between knowledge based on concepts and knowledge based on intuition.

According to Bornstein, by acknowledging the existence of intuitive objects that are not subject to knowledge,^{26} Kant and Bergson introduced some irrational elements into their philosophy. Yet, in both cases, the generality of concepts is misunderstood. A general concept presents not only the factors common to many objects, but also the specific, individual ones. Thus, one can formulate a concept that corresponds to an individual object – but at the same time it is clear that such a concept cannot use all the properties attached to the intuitive object.^{27} As Bornstein remarked, one can reach knowledge via concepts which do not necessarily need to have the same nature as knowable elements.

It should be underlined that when it comes to the source of knowledge, Bornstein held the view that there has to be a union between reason, the source of concepts, and the senses constituting intuition – the source of intuitive elements. Therefore, presenting the correspondence, harmony, and the structural similarity between these epistemic powers, he criticised two opposite philosophies: rationalism and sensibility. He paid much attention to the epistemic method, which should direct knowledge towards real truth. Stating that a given concept corresponds to objects, implies that there is a natural harmony between them originating from the same source.

←18 | 19→Writing about the adequacy of objects’ conceptual knowledge, Bornstein remarked that there is a structural similarity between human thought and reality. His views were different to the understandings of Kant and Bergson, to whom conceptual knowledge of reality is problematic. Since the nature of the concept should be the same as the nature of the object, Bergson considers conceptual knowledge of reality as impossible, while Kant introduces general concepts which do not give individual knowledge about the object.

Bornstein’s epistemic ideas referred critically to Kantian phenomenalism and so-called Copernicanism. As he remarked, Kant strived to explain how a priori structures and logical principles are present in reality. He built his reasoning on the assumption that categories and logical principles impose themselves upon the sensual world and arrange it in an objective way. Thus, the real world seen by human beings constitutes a world formed by the above-mentioned logical principles. These would not exist in the real world, which exists regardless of the subject, if they were not imposed upon it. The Copernican revolution in thinking about objects means that objects rely on the epistemic method. The real world, as far as it is knowable, is a world of phenomena, a world of objects as we present them.

1.2 Epistemological Decisions in the Philosophy of Mathematics

Bornstein’s epistemological studies can be found in many of his works: Poznanie rzeczywistości [Knowing Reality],^{28} Elementy rzeczywistości jako nauki ścisłej [Elements of Philosophy as an Exact Science],^{29} O najważniejszych zagadnieniach teorii poznania naukowego, [On Fundamental Questions of Scientific Knowledge],^{30} and O sądach analitycznych pochodzenia syntetycznego [On Analytic Judgments of Synthetic Origin].^{31} In Poznanie Rzeczywistości, he ←19 | 20→studied the relation between the outcome of human knowledge and reality in the context of questions on the relation between the results of our knowledge and experienced reality, and the relation between our types of knowledge and reality. The first problem was discussed as part of the philosophy of mathematics and included the question: How can pure mathematical and biological concepts refer to reality? The second problem, meanwhile, is a continuation of the discussion on intuitive and conceptual knowledge.

In his unpublished work, O sądach analitycznych pochodzenia syntetycznego (1948), Bornstein entered into the debate on the notion of analyticity and the dichotomy between analytic and synthetic truths. In this work, not only did he use Kantian terminology, but also creatively developed Kant’s concept of analytic judgments. He noticed that Kant’s view on the origin of a given proposition and its nature affected further, post-Kantian studies on analyticity.

Among Bornstein’s epistemological conclusions, the ones related to the philosophy of mathematics deserve particular attention. His studies on the philosophical foundations of mathematics led him deeper into the aspects of ontology. His theses such as Prolegomena filozoficzne do gieometryi [Philosophical Prolegomena to Geometry]^{32} and Problemat istnienia linji geometrycznych [The Question of Geometrical Lines’ Existence],^{33} although concerned with mathematical objects, methods and the nature of the existence of mathematical objects such as the point, the straight line, the plane, space, the number and the algebraic equation, are clearly aimed at metaphilosophical and ontological questions.^{34}

In Prolegomena filozoficzne do gieometryi, Bornstein described an extended line of perfectly fixed direction. He proved that the straight line exists not only as an object of human representation, but also as an object of ideal geometrical space. In Problemat istnienia linji geometrycznych,^{35} he discussed the existence of the geometrical line as a one-dimensional object which is coherent, extended, ideally exact and deprived of any sensual qualities, and which corresponds to a continuous analytic function. As Bornstein proved, among the three possibilities ←20 | 21→that geometrical lines exist along with continuous functions corresponding to them, there is only one that can be considered as true. It cannot be accepted that there is no continuous function that does not have its corresponding geometrical line, nor that a geometrical line corresponds to every continuous function. The only correct answer is that functions exist which have corresponding geometrical lines, while some do not. Such a solution leads to the conclusion that there is no contradiction between the purely conceptual domain and the intuitive one.

Bornstein proved that the infinite multiplicity of points existing between two points placed on a geometrical line is not an actual multitude but a potential one. He also demonstrated that there is a difference between the nature of the whole and the nature of its elements. Although the above-mentioned multiplicity of points is infinite, one can assign a corresponding real number to the individual points on a geometrical line. Thereby, the multiplicity of real numbers existing between two points on a geometrical line is strictly delimited as one can determine whether any real number belongs to the given multiplicity. The same can be applied to any geometrical point – one can determine whether it belongs to this multiplicity, or not. The content of the multiplicities – filled with either real numbers or points – is strictly defined, static and constant. It constitutes the amount in statu essendi, which is actual, as opposed to the multiplicities in statu fiendi, that are to exist to their fullest extent, therefore existing only potentially.

Yet, although the infinite number (multiplicity) of points existing between two points on the geometrical line actually exist, this number cannot be interpreted as one consisting of actually existing points.^{36} Moreover, it should be stressed that a given segment not only contains an infinite number of points, but also uses this completely, in line with the rule that every extension is always available for its cross section – geometrical points. The number remains equal to its segment, therefore there is an infinite number of points contained in the actually existing segment. The actual existence of the line segment refers to the static element that constitutes a given infinite number (multiplicity). Analysis conducted reveals that the identity in the statement ‘the straight line segment consists of an infinite multiplicity of points’ does not indicate any contradiction as the segment actually exists, yet it is not the multiplicity of actually existing points. Such an obvious solution, as Bornstein pointed out, makes studies on infinity clear and cleans them of possible contradictions, misunderstandings and controversies.^{37}

In experience, no geometrical object nor space is a given. Yet the reality available in experience is spatial and can be similarly extended to geometrical objects. Since geometrical objects are not a given in experience and therefore cannot be imagined, one must, according to Kazimierz Twardowski’s conclusions, create them, i.e. build concepts corresponding to objects. Bearing in mind objective and real knowledge, Bornstein made a step further when it comes to the construction of geometrical concepts, and sought object references for concepts in reality. Therefore, he created geometrical concepts which have their own object references in reality.

In Prolegomena filozoficzne do geometryi, Bornstein distinguished between the representation of physical space and the concept of geometrical space. On this account, he created the concept of geometrical space and then specified the concepts of the remaining geometrical objects. The content of the geometrical space concept consists of, on the one hand, the content of colourful and limited physical space’s foundational representation, and, on the other hand, the content of four imaginary judgments.^{38} Two of the judgments are of rational origin, meaning that they do not recognize colourfulness and the limitedness’ representation of geometrical space, while the remaining two, constituting a guarantee of experience, assign such attributes as three dimensions and the continuity of the object of the foundational representation to the geometrical space’s object. Thus, geometrical space is considered to be three-dimensional, continuous, limitless and pure. Due to the postulate of purity, it is possible to distinguish between geometrical objects’ attributes and ones assigned to physical objects. Moreover, this postulate applies merely to geometrical space and has no impact on experimental space. As for the limitless postulate, it remains valid in physical space as well and it does not lead to any conclusion contradictory to experience. The attributes of objects assigned to limited space can be derived from ones related to limitless space.

Since Bornstein aimed to prove that there is a link between geometrical space and reality, he assumed there exists an object of the foundational representation and this is physically perceived. As long as such assumption is valid, the existence of common attributes assigned both to the object of geometrical space’s concept and that of foundational representation can be proved.^{39} These attributes are not related to the subject representation’s content. In this respect, the common ←22 | 23→attribute is the representation of three-dimensional space, which does not constitute reconstructive, nor generative representation, but a perceptual one.

According to Twardowski, reconstructive and generative representations have their roots in perceptual representations: “while reconstructive representation relies on simple realization, generative representation relies on transforming and combining perceptual representations.”^{40} When defining an object existing in reality, one cannot seek generative nor reconstructive representations, as their content is recognized by the human mind as an immanent object. Despite the fact that such object is represented, it does not exist in the light of Twardowski’s distinction between content and an object of representation.^{41} Thus, speaking about the representation of three-dimensional space, Bornstein concludes that this representation is of perceptual nature.^{42} If it was a generative representation, it would mean that the concept of geometrical space is a subjective one, unrelated to the objective world, and deprived of any real meaning since one could assume such representation is four-dimensional. On the other hand, if it was a reconstructive representation, it would not have any objective value. For example, taking the space of the firmament as the basis of foundational representation, one can obtain the concept of two-dimensional geometrical space as a result. Here, although foundational representation has its corresponding object, such object, according to Bornstein, does not exist in reality, but constitutes merely the represented content. Two-dimensional objects do not exist in reality on their own and their representations do not have any corresponding nor simultaneously existing object. According to Bornstein, cognitive perceptions are not perceptual. They are not available to experience sensu stricto as they do not reach objects existing in reality. Instead, they remain in the domain of the subject. Bornstein observed that it has been proven in experience that geometrical space is a three-dimensional experience. Although it should be stressed that the above-mentioned conclusions are based on common sense and on a naive, un-scientific concept of geometrical space, nevertheless, it remains valid due ←23 | 24→to its attempt to establish geometry in experience in terms of its spatiality, not multidimensionality.

1.3 Selected Issues in the Theory of Scientific Knowledge

Studies in epistemology led Bornstein not only to ontology, but also to metaphilosophy. His views on this topic can be found in his work Elementy filozofii jako nauki ścisłej. Here, apart from conclusions on conceptual and intuitive knowledge and the types of an object’s existence, he strove to determine the relation between science and philosophy, arguing that the latter is an exact science (which is difficult to defend nowadays).

The culmination of Bornstein’s studies in epistemology and the philosophy of science is included in his work O najważniejszych zagadnieniach teorii poznania naukowego. Here, Bornstein raised other questions regarding scientific knowledge and creating scientific theory. He took an unusual perspective (in the light of the philosophy of science) and sought to overcome the limits of Kantian epistemology.

Every field of scientific knowledge has its own fundamental concepts, i.e. geometry has such terms as the point, the straight line, and the plane, while logic – genus, differentia (differentia specifica), and species. Bornstein called such concepts “regional categories of individual sciences.”^{43} Since there are some relations between regional categories, they can be considered as implementations of more general and universal categories.^{44} According to Bornstein, this indicates that the sciences are united in a common categorial relation. Such belief was also shared by philosophers who opted for a so-called unity of the world. As for Bornstein, his concept of a world united through categories was presented in Architektonika świata [Architectonics of the World] where he built his own categorial system ←24 | 25→of algebraic and geometrical metaphysics (more remarks on this system can be found later in this study). He discovered the categorial unity of respective domains of being. Thus, universal categorial structures become relevant not only in epistemology and the philosophy of science, but also in ontology. Above all, however, these categories laid the foundations of category theory which Bornstein considered as the fundamental philosophical science.^{45} Here, in category theory, Bornstein overcame the limits of Kantian epistemology.^{46}

“Therefore, a priori forms of our mind are forms not only of our mind, but also of real objects. They have their correspondence in reality. Categories as well as science, which is built upon them, have objectivity that is real, not merely ideal or phenomenal. Thus, category theory becomes both theory of real objects and ontology of the real world. Science and philosophy are authorities of truth in the proper sense.”^{47}

Here, Bornstein recalled his own theory of correspondence and harmony between reason (its forms and categories) and the objective structure of reality. He noticed that a priori forms of the human mind are at the same time forms of real objects. Thus, science based on category theory is objective and real, contrary to the subjective or phenomenal science proposed by Kant. In other words, Bornstein made scientific knowledge objective – yet this conclusion is nowadays contested, if not rejected.

Both of Bornstein’s solutions – on Kantian dualism between sensibility and reason, and between the world of thought and the world of space (which was presented in Bornstein’s qualitative-categorial geometrical logic)^{48} – led him to discover universal, ontological structures and formulate the idea of scientific metaphysics.

According to Bornstein, epistemology should be studied within the context of other philosophical disciplines: logic, metaphysics and psychology. Epistemology as a science is broader than logic as it is not limited by conceptual elements. ←25 | 26→Bornstein observed that knowledge is related to the nature of being and therefore epistemology – just like ontology or metaphysics – is the study of the nature of being. Knowledge itself constitutes an ideal being. However, the relation between epistemology and metaphysics is built upon the recognition that there are the same corresponding structures in different spheres of being. This relation can be also seen when studying the foundations of being, and determining whether this foundation is based on matter, spirit, or both. Such questions can be applied to epistemology when asking about the origins of true knowledge.^{49}

It has to be stressed that Bornstein’s interests did not focus on the way knowledge is formulated. Instead, he sought to establish the content of knowledge, and determine how given knowledge is related to other knowledge in terms of objects. Since his area of philosophy is not the psychology of knowledge, he strove for justification of knowledge in the basic source experience of “object based” reality.

According to Bornstein, the results of philosophical studies have to be constantly analysed in order to make corrections or to bring to light mistakes made. In the history of philosophy, there are two main schools of thought – idealism and realism – and both of them are the subject of many of Bornstein’s critical studies. In idealism, there is an assumption that all objects are merely the representations of a knowing subject.^{50} In realism, objects are not only representations but also something that exists regardless of the human mind.

Bornstein proves the paradox of epistemological idealism where a given object depends on the subject and yet is merely its representation. Due to the fact that there are two types of representation: representation-perception or concept-thought, epistemological idealism divides itself into the subjective and the rational. Among the two elements of knowledge: the content and the object, both philosophies acknowledge only the content, according to the phrase: esse est percipi. Contrary to these, monistic realism limits knowledge’s content merely to its object, thus this realism can be expressed in the phrase percipi est esse.

←26 | 27→As for realism, it can be divided into monistic and dualistic. Between extreme monistic realism and extreme monistic idealism, according to Bornstein, are all philosophies which are dualistic and at the same time realistic, as well as two-sided monistic realism.^{51} In his critical studies on dualistic realism, he proposed his own realistic philosophy close to the two-sided monism of theory and the spirit in Spinoza’s works.^{52}

According to Bornstein, full correspondence and harmony between logic and forms of intuition is not just a hypothesis. It is a fact included in geometrical logic. The system of logical categories is presented in spatial form. Thus, it is space, and not time, which can be considered as a qualitative condition both for the ideal and real domains.

Bornstein assumed that there exists an extrasensory reality. Therefore, he criticized philosophers who limit themselves to an ideal world and thus cannot recognise real, valid truth hidden in a priori knowledge. As he wrote, “a priori geometrical principles are real and remain valid in the real world if the world contains them in the implicit form – that is if there is full correspondence and harmony between the a priori form of intuition and the form of trans-subjective and metaphysical reality which exists regardless of us.”^{53} Therefore, the Copernican revolution which can be observed in Kant’s philosophy and which relied on the prevalence of thought over intuition, had to be overcome by a concept proving the correspondence and equivalence of thought and intuition. Moreover, epistemic phenomenology collapsed along with the Copernican revolution. It turned out that logical and geometrical principles are not only principles of the human mind, but also principles of the reality that exists regardless of this mind.

2 From the Algebra of Logic and Projective Geometry to Categorial and Dialectical Geometrical Logic

In this subchapter, I will omit a detailed description of the construction of Bornstein’s philosophical research tools. However, I would like to underline the most important questions that go together with the elaboration of this new, ←27 | 28→formal research tool, as well as the results of its implementation in deeper knowledge of the structural unity of the world of thought and of the world of spatial objects. The tool is insightfully described in Bornstein’s work Geometrical Logic. Nevertheless, it seems impossible to tackle this scientific undertaking without taking into consideration his previous endeavours, both the successes and the failures. Thus, one can assume that only questions which allow us to demonstrate the validity of such a detailed elaboration of the tool for the qualitative, not quantitative analysis of reality are important. Therefore, only those questions allow us to use the tool in ontological and metaphilosophical studies, as well as in studies of given philosophical systems which strive to describe reality in a fundamental way, ignored by Bornstein in his work.

The development of a new formal tool in his epistemological studies of the intellectual, sensual, “logical and intuitional elements, strengthened Bornstein’s conviction that there is a harmony that exists between them. The results of these studies were presented in his work Architektonika zmysłowości i rozsądku [Architectonics of Sensibility and Reason],^{54} in which he proved that the structure of our logical thought is fully reflected in the structures of the intuitive sphere, in the domain of the visual space. Having published the work Geometrja logiki kategorialnej i jej znaczenie dla filozofji^{55} in Przegląd Filozoficzny [Philosophical Review], he demonstrated the transition from the object sphere into its objective equivalents: into the logical and geometrical world. As a result, he prepared the foundations for the concept of geometrical logic and logical geometry, a topologic, two-sided science concerned with the logical world’s structures and its corresponding structures in the spatial world. He proved that logic is not only a science of the principles of thought, but exceeds far beyond it. In the three volumes of the Architektonika świata, he analysed structures of the logical world, presenting their universality and metaphysical attributes. Bornstein discovered them not only in the real and ideal spheres, but also in the philosophy of Plato, Baruch Spinoza, Georg Wilhelm Friedrich Hegel, Józef Maria Hoene-Wroński, and Paul Natorp.

He started his studies on topologic with the thought that mathematics deals not only with numbers and measurements. There are many facts which prove ←28 | 29→that there have been studies on the qualitative scope of mathematics, among which are algebra and qualitative geometry.^{56}

The algebra of quality is generally known as the algebra of logic, algebra of concepts and judgments, algebra of meanings, or algebraic logic. It can be found in the works of Plato in his outline of qualitative arithmetic. Yet its first mature concept was developed by Gottfried Wilhelm Leibniz, and in 1854, George Boole gave it the form of a system.^{57} Thus, one can derive a number of non-numerical theorems from the system of algebraic logic, e.g. a+a=a. This means that when one adds content to the same content, the results will be equal to this content.

Endeavours to determine qualitative geometry started in ancient times. This kind of geometry can be found in works on perspective written by Euclid and Vitruvius, and in the works of Geminus or Apollonius of Perga.^{58} Geometrical studies by Gérard Desargues, Blaise Pascal, Charles Brianchon, and the nineteenth century works of Lazar Carnot, Jean-Victor Poncelet and Joseph Gergonne then led to the construction of projective geometry that analyses the properties of geometrical shapes and figures with regard to the position of points, the directions of straight lines, the positions of planes etc. where one cannot use such categories as size, length, distance or measure.^{59}

2.1 Categorial Logic of Geometry

The infinite number of points and straight lines that Bornstein limited to a few kinds exist in a plane as well as in a projective plane. The relations between the determined categories of the qualitative elements turned out to be the specific fundamental principles that were sought in the times of Plato in studies on ideal shapes.^{60} Categorial geometry of quality was brought nearer to philosophy.

Algebra of logic is also of a multitudinal nature, therefore Bornstein sought such assumptions that would give it a categorial form. Among the many sets of ←29 | 30→logical algebra’s axioms, Bornstein chose the first system of axioms determined by Edward Huntington in 1904.^{61} This system assumes the existence of at least two different elements: a and b, both of which constitute this system. In two-element algebraic logic, three operations were adopted: logical addition (+), logical multiplication^{62} (x), and negation (‘). As a result of the negation of a, one receives the element a’ (non-a). Logical addition consists of two or more elements that are merged as a sum. Consequently, merging the concept of “human being” (a) and “good” (b) one receives the concept of “good human being” (a+b). The logical addition suggested by Bornstein is an operation that resembles addition in quantitative algebra, whereas the nature of logical multiplication is contrary to that of quantitative mathematics. This is because the aim of logical multiplication is to designate the greatest amount of common content between two or more concepts. For instance, the biggest common element (ab) of the concepts “animal” (a) and “plant” (b) is the concept “organism.”^{63} One should notice that in Geometrical Logic, Bornstein does not interpret any of the symbols used ontologically. The formal approach allows Bornstein to avoid a discussion on its legitimacy. Instead, it provides a chance to modernise – or to adjust to the modern concepts in ontology – the language of predicates proposed by Bornstein. In the predicates, one can notice a convergence with modern category theory and topology. My critical study on Geometrical Logic, presented in later chapters, is based on this way of thinking.

As for the Huntington axioms for two different elements and for their sum and product, one can point to axioms of commutative and distributive properties, which in name correspond with the axioms of qualitative algebra. In addition, one should include two more axioms – regarding, directly or indirectly, the elements 0 and 1. Let us assume that the element 0 exists, whereby for any element a: a + 0 = a. Then, let us assume that the element 1 exists, whereby for any element a: a1=a. These are therefore axioms which contain addition and multiplication modules, although element 0 has the lowest logical content as it does not change the content of the element which it is added to. As per Bornstein’s earlier works, element 0 refers to the content of the concept of “something” or “object in general”. As for element 1, it represents the fullest content as the ←30 | 31→“whole” or “everything”. The logical 1 represents the upper bound of the world of concepts, while 0 refers to the lowest limit.^{64} Axioms with the elements 0 and 1 can be also shown as a+a’=1 and aa’= 0.

The system of fundamental principles of logical algebra is a system of axioms related to content logic, not the logic of the scope or class. The axioms can be divided into two types: either they can be separated into two axioms, or they are expressed in the duality of logical addition and multiplication. The duality represents the logical harmony between all axioms in logical algebra. Thus, one can easily switch between different formulas by changing the addition symbol to the multiplication symbol and vice versa, and by changing 1 to 0 and vice versa.

Apart from the three operations, algebraic logic identifies two relations: containing (<) and equivalence (=). However, no algebraic logic theorem is based on the containing relation. It can, however, be included through the use of logical operations based on equivalence.^{65} In this case, a<b=(b=a+b) means that if a is contained inside b, then a added to b (where b already contains a) does not change the content of b. In other words, if a<b then b=a+b – and vice versa. This means that every concept with a lower content is contained inside a concept of greater content. The essence of every equivalence, namely the mutual containing of elements, may be presented in the form of the definition: (a=b)=(a<b)+(b<a). Thus, the containing relations between the elements of logical content should be presented not synthetically as a<b, but analytically as ab<a or a<a+b.

Based on Huntington’s set of axioms, one can derive a number of theorems, e.g.:

– the formulas of the determining and dividing dichotomy:

a=(a+b)(a+b’), a=ab+ab’

– the rules of tautology

a+a=a, aa=a,

– De Morgan’s laws:

(a+b)’=a’b’, (ab)’=a’+b’,

←31 | 32→– extension of 1 and 0 against a and b

1= ab+ab’+a’b+a’b’, 0=(a+b)(a+b’)(a’+b)(a’+b’).^{66}

Bornstein noticed that both categorial logic of content and categorial-qualitative projective geometry are equal. This conclusion arose from the concepts of spatial schematization of logic, and the logical and categorial interpretation of the projective plane, collected in Bornstein’s 1926 work Geometrja logiki kategorialnej i jej znaczenie dla filozofii^{67} [Geometry of Categorial Logic and its Importance for Philosophy]. Referring to Huntington’s axioms, Bornstein obtained two-element logical algebra (a, b). In his concept, projective geometry was also reduced to two elements: the point and the straight line. Both fields adhered to the duality principle discovered in algebra by Charles Sanders Peirce (1867) and also independently by Ernst Schröder (1877), as well as in projective geometry discovered by Jean-Victor Poncelet (1822) and Joseph Gergonne (1826). According to this theory, the two logical operations – addition and multiplication – strictly correspond to two operations in projective geometry – cutting and joining. Recognising the structural and categorial identity of qualitatively different spheres, Bornstein developed the categorial logic of geometry – so-called topologic. Here, logic and geometry of quality proved their compatibility. According to the Bornstein’s theory, logical elements are equivalent to geometrical elements – and vice versa. Moreover, senses are positions and positions are senses (λόγος – τόποτ).^{68}

2.2 Generalisations of Categorial Logic of Geometry

Huntington’s system of axioms leads to a set of corresponding principles and theorems of two-element logic. The third axiom of the division of addition with regard to multiplication, and multiplication with regard to addition: a+bc=(a+b)(a+c) and a(b+c)=ab+ac, introduces the third element c, making the fundamental principles of logical algebra a three-element system. Bornstein developed the system in 1927 in his work Geometrja logiki kategorialnej i jej znaczenie dla filozofii [Geometry of Categorial Logic and its Importance for Philosophy].^{69} ←32 | 33→Here, spatial, three-element logical algebra became three-dimensional, categorial and geometrical logic.^{70}

Developing his philosophical tools, Bornstein worked on a study into geometrical logic where all possible relations between categories could be included. His work in this field can be found in Zarys teorii logiki dialektycznej [An Outline of the Theory of Dialectical Logic],^{71} the third volume of Architektonika świata [Architectonics of the World],^{72} and Geometrical Logic,^{73} where he presented topologic as a categorial, dialectical and geometrical logic.

In two-categorial, algebraic logic, there are basic dialectical operations (joining contradictions: a+a’=1, aa’=0) and basic dialectical relations (containing contradictions: 0<1). Thus, algebraic logic becomes dialectical. As Bornstein proves, if there are various levels of dialectics, there are various corresponding dialectical logics. Since each of the logics has its own spatial form, one can distinguish various levels of categorial and dialectical logic of geometry. The common basis for the levels constitute the specifications of the formulas of joining two opposite elements: a+a’=1 and aa’=0.^{74} In geometrical logic of the lowest dialectical intensity, the a and a’ elements do not contain themselves in each other. Thus, in the above expressions, 1 or 0 can be reduced neither to a nor a’.^{75}

In one-sided, dialectical, geometrical logic, there is a relation a<a’ (or a’<a) which is equal to the expression a’=a+a’, while a=aa’. Knowing that a+a’=1 and aa’=0, then if a<a’, we obtain a+a’=1=a’ and aa’=0=a. Further, if a’<a, then a+a’=1=a, and aa’=0=a’.^{76}

In two-sided, dialectical, geometrical logic, the equation a=a’ means that a<a’ and a’<a. Thus, a+a’=1=(a=a’) and aa’=0=a, where also a=a’.^{77} An example ←33 | 34→of two-sided dialectical logic is Hegelian logic where it is possible for dialectical elements to exist which are contained inside each other, so a<a’ but also a’<a.^{78}

It has to be stressed that Bornstein, writing on dialectics, focused particularly on logic, which he presented as a general form of panlogic. Thus, the universal forms of panlogic contain the diversity that can be observed in the world. Due to Bornstein’s spatiality of the logic of content, one can justifiably adopt the term “panlogotopic”, which, according to Bornstein’s terminology, can describe the levels of dialectics in geometrical logic.

3 Structural and Ontological Model of whole-being

Bornstein continued the endeavours of philosophers such as Plato, Aristotle, Leibniz and others to understand reality while discovering the fundamental principles of being and its structure. Moreover, he was an adherent of the classical, Aristotelian concept of metaphysics, namely ontology, as well as the principles of being. In his scientific work, Bornstein decided to consolidate metaphysics on scientific knowledge, which Kant failed to achieve. The results of this work can be found in Architektonika świata [Architectonics of the World] (1934–1936), in which he used geometrical logic, “mathematical, architectonical topology”. However, in this work, he concentrated on the structure of the world itself, not on the fundamental metaphysical question – the absolute principles, known from his Teoria absolutu [Theory of the Absolute] (1948).

In both works, he proceeded from algebraic and logical analyses to ontological ones. This became possible thanks to the fact that mathematics comprises not only quantitative research spheres, but also qualitative ones, e.g. projective geometry and the algebra of logic. As we are concerned with research on qualitative mathematics, we can use it as a tool in work on the qualitative sphere of reality, namely the field of research into philosophy. If there were no qualitative sphere in mathematics, the attempt to create a mathematical tool for philosophy would be contradictio in adiecto. However, as Bornstein suggested, considering that qualitative research connects mathematics and philosophy, one needs to try to construct a scientific metaphysics, a feat which was not achieved by earlier philosophers, i.a. Plato, Descartes, Spinoza, Leibniz, Hegel, Hoene-Wroński and Natorp.

←34 | 35→An important step in developing scientific metaphysics that made reference to the formal sciences included the results of critical epistemological analyses in Kant’s philosophy. Bornstein’s acceptance of the compatibility between reason and sensibility created a new research perspective on the parallelism between the world of thoughts, concepts, and content, and the world of spatial objects. According to Bornstein, such compatibility can be seen in the real and ideal spheres of scientific research: the world of sound,^{79} the structure of the periodic table of elements,^{80} genetic laws,^{81} as well as theorems of arithmetic regarding Pythagorean medians,^{82} and structures of syllogism.^{83} Analysing individual spheres of scientific research, Bornstein noticed the universal, architectonical structures of geometrical logic. However, he did not try to speculate on reality. What is more, when analysing the philosophies of Plato, Descartes, Spinoza, Hegel or Hoene-Wroński, he showed the first suggestions of the elements of the architectonical structures he discovered. Thus, he showed how far their research towards fundamental laws and structures of reality goes to explaining the sum of actual being. The results of individual studies on the philosophical systems demonstrated the ←35 | 36→presence of the same intuitions regarding the unity of being, although they did not grant the arguments the mathematical precision necessary to create scientific philosophy according to the mathesis universalis.

3.1 Universalism of the Structures of Geometrical Logic

Bornstein noticed the presence of structures of geometrical logic in the ideal and real spheres of scientific research, as well as in some of the philosophies, and then came to the conviction that there exists a principle of analogy in the unity of the world. He claimed that the difference between elements that belong to diverse spheres of being does not exclude their affinity. Bornstein continued Aristotle’s studies on analogy, where this principle referred to the identity of qualitative relations.^{84} Bronstein argued that elements of the individual relations equate with their role, and therefore, they constitute either categories of simple elements (so-called determining) or categories of complex elements (so-called determined). Despite the substantial difference between elements, it can be seen that the roles they perform are identical – consequently, they belong to the same category, and are therefore called isocategorial. In different spheres of research, Bornstein noticed not only the identical relations between pairs of elements taken into consideration, but also the categorial identity that connects all of the different elements. According to Bornstein, one should understand this analogy as isocategories of elements including their substantial distinctiveness, or isocategories of relations along with their substantial distinctiveness: “not only the relations between the two pairs are identical, but there also exists categorial identity, which connects these elements, and only thanks to this identity do different elements become analogical.”^{85}

Further research in the sphere of geometrical logic led to the discovery of structures that had not been known in the algebra of logic. These structures include i.a. polar structure, which was discovered first and which was known by Heraclitus; dual and neutral structure; quaternary structure; multiplying structure along with dual structure (which is additive in relation to this structure and which creates the so-called agential structure); and the five-element structure of ←36 | 37→development. The greatest number of these structures were discovered after connecting logic and geometry using the system of geometrical logic.

While reading Geometrical Logic, one may have the impression that analyses of geometrical logic are only a theoretical suggestion to elaborate on the new sphere of formal, mathematical ontology expressed in algebraic symbols. In critical remarks to this study, I will present numerous comments regarding the notation of the predicates and the whole of the structure of categorial, geometrical logic. It will turn out that this work includes intuition of the mathematical theory of categories and topologic issues. I would like to point out that Bornstein’s aim was not to create a new sphere of mathematics for theoretical purposes, but to distinguish a model of a mathematical structure out of the known spheres of qualitative mathematics, which could be a suitable tool in philosophical research on reality leading to the development of a general model of being. Such a model could not be achieved without previous interpretation of ontological symbols and structures presented and visualised in the diagrams of the topologic model, categorial geometrical logic and, simultaneously, categorial logical geometry.

If logical geometry or geometrical logic can be understood to be the universal method of qualitative research into reality, its elements, operations, relations and structures should be used in various spheres of real qualities. They should manifest themselves in individual spheres, allowing us to discover unknown principles in these domains, while, at the same time, if these laws are known, permit us to capture their categorial essence and understand their “unity” with analogous laws in other spheres. Is this really true? The answer to this question will at the same time be an assessment of the philosophical values of Bornstein’s achievements. A complete answer would most certainly exceed the framework of this critical study. I will therefore present only the most relevant elements, which will allow us to conduct a preliminary assessment of Bornstein’s philosophy.

The construction of topologic should not raise any doubts. It contains elements that are completely different, but at the same time congruent, and which represent themselves reciprocally – the sphere of spatial objects and the sphere of concepts and non-spatial thoughts. Despite the diversity of the substrates of these domains, all of their activities, relations and theorems can be expressed in an identical way, with symbols and formulae of the same algebra. If both the spheres display the same construction, only one explanation presents itself for this parallelism: they must have something fundamentally in common with each other. One may wonder whether this can be expressed only according to the Platonian model. Plato noticed the involvement of spatiality in the world of forms, but failed to explain it, claiming that this involvement is somehow ←37 | 38→mysterious,^{86} and asking whether we should accept this coincidental answer, or whether we should follow Leibniz’s concept of harmony established by God.^{87} According to Bornstein, the affinity of these spheres demonstrates itself in their qualitative nature. Space and its elements, similarly to concepts or ideas, are formed according to the same general quality, present in different spheres.

Bornstein’s research into the ideal and real spheres, i.e.: acoustics and music, the biological laws of inheritance, and the properties of the elements of the periodic table – has shown the presence of relations and operations that are subject to the laws of qualitative mathematics. In these spheres, the structures of geometrical logic exceeded the limits of the spheres of the ideal world and stepped into the field of real domains. The results of these studies allowed Bornstein to make the next step toward generalization of logical and geometrical regional categories to the level of the ideal and real spheres. This generalization, thanks to further ontological interpretations of the categories of topologic, introduced analyses into the new sphere of geometrical ontology.

3.2 Geometrical Ontology as a Generalization of Geometrical Logic

Structures of geometrical logic that are present in geometrical ontology manifest their universality, and, at the same time, their validity and real meaning for philosophy, which aims to know reality.^{88} An example of the real meaning of logical and geometrical structures is their presence in the world of colours. For instance, the principle of duality is in force in the world of colours, where the mixing of ←38 | 39→colour rays constitutes their additive combining. As a result, one receives the colour white (a+a’=1), whereas mixing colourful substances constitutes their multiplicative combining into the colour black (aa’=0). In Oettingen’s theory of acoustics and music, the principle of duality has a fundamental role. Taking into consideration the duality in this theory, one can notice the dual compatibility between the major scale and minor scale of a given tone.^{89} “The fact that the major scale is included in one and the same tone, and that tones include a minor scale of the same common tone, constitutes the basis of the contemporary dual belief in the world of tones and musical harmony.”^{90}

The principle of duality is also observed in biology. When it comes to the reproduction of organisms, there are elements present in two forms: zygote (a1) and gamete (a0).^{91} A zygote includes a gamete (similarly as in the point – a1, where exists a straight – a0), an element which is simpler and which presents a set of genetic primordia. This relation is the same as the relation of sound to tone, or colour to a ray of colour, or a point to a straight line. The duality of these elements constitutes the relation of the whole to an equivalent component, which can be generalised. One can claim that between these two elements there is the relation of a concrete element to an equivalent abstract, or even more generally – the relation of a substance to its equivalent characteristics.

The phenomenon of the reproduction of organisms can be described with the formulae of geometrical logic. In a reproductive system, two zygotes: male (a1) and female (b1) join themselves in a multiplicative way into a commonality (ab), however, their gametes (a0 and b0) connect themselves in an additive way, creating a new zygote (a+b), which is dual when it comes to the element that connects the zygotes in the commonality (ab). In Geometrical Logic [p. 13], Fig. 2 presents a dual six-part structure which is interpreted by Bornstein ontologically, and therefore called a generative and creative structure.

Bornstein discovered the individual elements, operations, relations and structures of geometrical logic, which manifest their universal character in chosen real spheres. Yet, only analyses of the qualitative structures of real spheres led Bornstein to expand their use to the whole of reality being studied by philosophy. Along with this expansion, the regional, logical categories (concepts), as well as spatial (points and straights) are generalised to the categories of general beings. Bornstein suggests calling them categories of the concrete and of the abstract,^{92} rather than concepts, points and straights. Transforming logical geometry (geometrical logic) into ontological geometry (geometrical ontology) is therefore justified.

In ontological geometry, just as in logical geometry, Bornstein distinguishes: simple concrete elements (simple substances), which correspond with points (a and b), concrete complex elements (complex substances), which correspond with points a+b, and simple abstract elements (properties), which correspond with straights, for instance ab. The highlighted elements constitute ontological categories. Striving to explain the difference between the co-ordinates a and b, between the straights a and b, and the features a and b, Bornstein referred to Aristotelian philosophy. The features of the co-ordinates a and b that constitute the element a+b in logic are called the differentia specifica and genus proximum, which together show the species. Bornstein noticed their logical categories differentia specifica and genus, the manifests of general, ontological categories that have an active, generative role and a passive role, which in Aristotelian philosophy correspond to the form (in the first case), and to matter (in the second). Matter and form that are abstract create a concrete substantial relation. As per the law of the general duality of geometry and logic for the abstract figure of matter and form, we have their concrete forms (points a and b), which indicate in a multiplicative way their commonality relation ab. As a result, there is a further, ontological interpretation of the structure of the triangular plane of categorial ←40 | 41→topologic. Not having distinguished the concrete and abstract elements, one can reduce this structure to four elements: a, b, ab, and a+b, which represent the four Aristotelian causes,^{93} due to the fact that it is a creative and generative structure. The formal cause a, the material cause b, and the efficient structure ab, are proper causes, whereas the result, which is concrete, constitutes the final cause a+b, the proper result. It should be stressed that categorial geometrical ontology becomes an ontological model of topologic.^{94}

Since the categorial plane contains the other points, straights and co-ordinate system not yet interpreted ontologically, Bornstein exceeds Aristotelian metaphysics with his interpretations. If it turns out that by ontologically interpreting the other elements of topologic one receives a deeper understanding of Aristotle’s thought, one could consider it as proof of the relevance of Bornstein’s tool for studying reality in philosophy.

In the diagrams of the categorial, logical and geometrical plane, apart from the straight a, there is also straight a’, vertical axis 0bb and dual elements, concrete elements that are in the horizontal axis 0aa. In this way we can obtain a system of all the possible categories of abstract and concrete forms.^{95} For the straights b and b’ and the horizontal axis 0aa’ as well as elements that are dual and that are placed on the vertical axis 0bb’, we obtain a system of all the possible categories of abstract and concrete matter. The vertical and horizontal system of geometrical ontology manifests a system of matter and forms, while a system of slanting co-ordinates – a system of causes and effects of the system of all the oblique lines and all the points that are placed on them. What is more, all the points on the categorial plane are the vertex of four straight lines, and every straight line is the base of the four points that are placed on it.

Referring to the concept of harmonic conjugacy of four elements, one may notice that there lacks a fourth straight line, even though there are three abstract forms in the straight a, a’ and vertical axis 0bb’. It turns out that the lines a and a’, which unite ←41 | 42→in the vertex, the point 1a+a’ (that is placed at infinity), indicate a fourth straight line at the infinity 1(a+a’)(b+b’). In the ontological interpretation, the point 1a+a’ can be understood as the whole of the abstract forms. The axis 0aa’, dual in relation to the point 1a+a’, is the base of the harmonic four of the points a and a’, and the beginning of the system of co-ordinates 0aa’+bb’ and points at the infinity 1b+b’. The axis 0aa’, which constitutes the common base of the four concrete forms, can be ontologically understood as the commonality of all the concrete forms. These two structures, dual in relation to each other, are strictly connected together, and therefore one can consider this as a 10-element structure that presents the system of all forms.^{96} Similar reasoning as well as ontological interpretation can be conducted for all abstract and concrete forms. Also, in the case of a given 10-element structure that presents the system of all of the substances, we can distinguish the commonality of all of the concrete substances in the form of an axis 0bb’ and all of the abstract substances, presented in the point 1b+b’.

Bornstein especially highlights the border elements of ontological geometry. In the two-dimensional categorial diagram of panlogic, one may notice three different zeros (two co-ordinates and the origin of the system of co-ordinates: 0aa’, 0bb’, 0aa’+bb’) and three units (two points at infinity and a straight line at infinity: 1a+a’, 1b+b’, 1(a+a’)(b+b’)). One can consider the zeros as elements of the prefinity of the categorial plane, which corresponds to the nature of the logical zero as a concept which contains the minimum concept in itself. However, the units considered as elements out of the finity, which are placed at the infinity of the plane, correspond to the concepts that contain in themselves the maximum content. Both zeros and units are infinite elements, and between them is a whole world of finite elements. Infinite elements of a dialectical nature correspond with the content differently from any value or number. They do not represent any multiplicity, as in other categorial and general concepts. Zeros and units are a whole and the maximal recognition of the nature of logic, geometry, and the real world. Bornstein, taking into consideration their characteristics of infinity and unity, calls them principles with absolute meaning, limit, and universal, differentiating them from the finite categories.^{97} Zero, which is contained in every element, expresses its common base universality, the principle that accepts everything. The unit as a whole that contains everything constitutes a specific kind of universum, of whole-being.

Ontological geometry also explains the emergence of finite elements from infinite elements. For instance, the absolute element 1a+a’=a+a’ constitutes the ←42 | 43→whole of the ontological elements a and a’, and is an additive connection of polar elements, which are generated by development, the dialectical division 1a+a’. In geometrical interpretation, finite elements in the form of two straight lines, are projections from a point placed at infinity. The absolute element 0aa’=aa’ constitutes the commonality of the two polar elements connected in a multiplicative way. Bornstein included a full study on the ontology of absolute elements and on the genesis of finite elements out of infinite ones in his work Teoria absolutu. Metafizyka jako nauka ścisła [Theory of the Absolute. Metaphysics as an Exact Science].

3.3 Logical and Ontological Reality

The logical and ontological reality discovered by Bornstein is not a methodological construction nor an ontological phenomenon, but is being independent from our knowledge. The logical, geometrical and ontological forms show that they immanently belong to the spheres of real qualities. According to Bornstein, the ontological a priori of the categorial geometrical logic is valid for real domains, as it corresponds with real a priori of these spheres. “Both of these constructions, ontological and real, correspond with each other, because both of them contain the common, ontic source of being.”^{98} According to Bornstein, the fundamental mistake in Kant’s phenomenalism and methodological idealism consists of assuming that knowledge is independent from being, and as a result, knowledge is governed by principles of its own method which are relevant only to it, and which have nothing in common with being. The world of the phenomenal being, the objective world, is created as a result of laws and methods that are relevant only to knowledge. By definition, the independence of knowledge from being leads to further consequences: one can, like Kant, assume the existence of the real world, unknown and unrecognisable for us with its own, objective construction, or, like neo-kantianism claims, real being gains objective character only when it becomes known being. In the second consequence, knowledge is treated as generative, divine knowledge, because out of the as yet non-existent, objective world, knowledge created beings formed objectively.^{99}

The consequences that stem from the assumption of the independence of knowledge from being are not able to connect the a priori knowledge with reality and give them unambiguous understanding. One acceptable assumption which avoids the paradoxicality of the relation of knowledge to reality is, according to Bornstein, treating knowledge and reality according to the same ontological forms, which are adapted to the proper nature of knowledge and being.^{100}

Thanks to geometrical logic, unity and the isomorphic construction of reality are shown. According to Bornstein: “one needs to avoid the temptation to understand this unity in an exaggerated way, while also noticing the second aspect of the world that exists in it, which is diversity. The world is created economically, but not extravagantly, not poorly. I do not mean here the unknown richness in the essence of the world, but rather the formal side of the world, its principles and structures. Reality cannot be closed in a framework of only one undifferentiated form, which will not accept an architecture too poor and too unvarying, in which the relations between elements will not take into account diverse possibilities used comprehensively by reality.”^{101} Bornstein, considering the universality of geometrical logic, thought about its most general form, yet undifferentiated, but embracing all of the possibilities. This figure is panlogic, which is derived from mathematical logic and which contains the dialectical logical elements 0 and 1 (aa’=0, a+a’=1).

In his numerous works, Bornstein demonstrates an endeavour, quite common for other philosophers, to understand reality through demonstrating such principles and axioms, out of which one could understand their complete polymorphism and diversity. Especially important for him is the philosophy of Baruch Spinoza. Here, he saw that his thought corresponds with Spinoza’s ideas. According to Spinoza, attributes of God, namely extension and thought, express identical order, which is present in all of the other attributes. Understanding the order, which can be found in the extended sphere, as well as knowledge and the structuring of its objective principles, could lead to considerable progress in metaphysical knowledge. One would know not only the order which exists in the spatial sphere, but in general its universal structure that can be seen in the construction, order and organisation of the world. One can also use the attribute of thought, which contains the same universal order of the world, as “the order and connection of ideas is the same as the order and connection of things.”^{102} The ←44 | 45→symptom of extension and the idea of this symptom are one and the same thing, but expressed in two ways.

According to Spinoza, following a range of specific, individual, changeable things is impossible for our mind, therefore one should seek such generality and category that can embrace all of the individual things that exist in time and explain how multiplicity emerges out of this generality. “Thus we can see that it is before all things necessary for us to deduce all our ideas from physical things – that is, from real entities, proceeding, as far as may be, according to a series of causes, from one real entity to another real entity, never passing to universals and abstractions, either for the purpose of deducing some real entity from them, or deducing them from some real entity. Either of these processes interrupts the true progress of understanding.”^{103} Following this method, Spinoza discovered a range of things permanent and eternal as an infinite symptom of the second type. The range of permanent and eternal things connects one substance with the multiplicity of its symptoms, represents a range of levels that lead from the most general things to detailed things, and indicates the level of development and differentiation of the substance and its attributes. For the attribute of extension, the symptoms of infinity of the first type are motion and quiescence. The infinite symptom of the second type represents the figure of the whole world. This symptom exceeds nature in its multiplicative form and constitutes its source as the a priori law that enables and embodies it. As per Bornstein’s interpretation “the infinite symptom of the second type manifests itself in a different way in terms of logic and category than the infinite symbol of the first type. The latter consists of (…) a general, commonality nature, whilst the first has undeniably the characteristics of the whole.”^{104} Having read Bornstein’s categorial, logical and geometrical interpretation of both of the infinite symptoms, it turns out that they manifest themselves geometrically as a disc, which has its centre everywhere, whereas its perimeter is nowhere.^{105}

Since the mathematical topologic assumes ontological, logical and geometrical form, and within its elements there are border, unit, and whole elements that have universal meaning, it also has philosophical meaning. Thanks to this, philosophy acquires the accurateness and certainty of mathematics. In the history of philosophy, many have tried to make philosophy more scientific. However, the lack of proper development in logic and mathematics makes it impossible to achieve these endeavours. According to Bornstein, only Boole’s algebraic logic system and the development of research into projective geometry could be used to make the first successful attempt to construct philosophy as a mathesis universalis. To construct such a philosophy, Bornstein used a tool created by himself, which he presented in Geometrical Logic.

←46 | 47→19 Benedykt Bornstein, Zasadniczy problemat teoryi poznania Kanta [Fundamental Problem of Kant’s Epistemology], Warszawa: Skład Główny w Księgarni G. Centnerszwera i S-ki, 1910.

20 Benedykt Bornstein, “Krytyka immanentna filozofii gieometrji Kanta [Critique of Kant’s Immanent Philosophy of Geometry],” Przegląd Filozoficzny [Philosophical Review - PR], Vol. 14, No. 3 (1911), pp. 317–328.

21 Benedykt Bornstein, “Preformowana harmonia transcendentalna jako podstawa teorji poznania Kanta [The Pre-established Transcendental Harmony as the Foundation of Kant’s Theory],” Przegląd Filozoficzny [PR], Vol. 10, No 3 (1907), pp. 295–302.

22 Benedykt Bornstein, Teoria absolutu. Metafizyka jako nauka ścisła [Theory of the Absolute. Metaphysics as an Exact Science] (Łódź: Łódzkie Towarzystwo Naukowe, 1948), pp. 51–53.

23 A critical analysis of the antithetic question of Kantian epistemology – see: Krzysztof Śleziński, “Benedykta Bornsteina recepcja filozofii Immanuela Kanta [Benedykt Bornstein’s Reception of Immanuel Kant’s Philosophy],” Folia Philosophica, Vol. 38, 2017, pp. 75–84.

24 Bornstein, Krytyka immanentna filozofji gieometrji Kanta, pp. 317–328.

25 Benedykt Bornstein, “Kant i Bergson. Studium o zasadniczym problemacie teorii poznania [Kant and Bergson. A Study of Fundamental Epistemic Problem],” Przegląd Filozoficzny [PR], Vol. 16, 1913, pp. 129–199.

26 Bornstein, “Kant i Bergson. Studium o zasadniczym problemacie teorii poznania,” p. 141.

27 Bornstein, “Kant i Bergson. Studium o zasadniczym problemacie teorii poznania,” p. 152.

28 Benedykt Bornstein, “Poznanie rzeczywistości [Knowing Reality],” Przegląd Filozoficzny [PR], Vol. 17, 1914, pp. 48–62.

29 Benedykt Bornstein, Elementy filozofii jako nauki ścisłej [Elements of Philosophy as an Exact Science], Warszawa: Skład Główny w Księgarni E. Wendego i S-ka, 1916.

30 Benedykt Bornstein, “O najważniejszych zagadnieniach teorii poznania naukowego [On Fundamental Questions of Scientific Knowledge],” in: Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism [Benedict Bornstein’s Philosophy, Selection and Elaboration of Unpublished Writings], ed. Krzysztof Śleziński (Katowice: Wydawnictwo Uniwersytet Śląski – Wydawnictwo Scriptum, 2011), pp. 51–61. The typescript can be found in the Jagiellonian Library, ref. n. 9027 III (1942).

31 Benedykt Bornstein, “O sądach analitycznych pochodzenia syntetycznego [On Analytic Judgments of Synthetic Origin]” in: Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism, pp. 97–101. The typescript can be found in the Jagiellonian Library, ref. n. 9035 III (1948).

32 Benedykt Bornstein, Prolegomena filozoficzne do gieometryi [Philosophical Prolegomena to Geometry], Warszawa: Wyd. E. Wende i S-ka, 1912.

33 Benedykt Bornstein, „Problemat istnienia linji geometrycznych [The Question of Geometrical Lines’ Existence],” Przegląd Filozoficzny [PR], Vol. 16, No. 1 (1913), pp. 64–73.

34 Bornstein, Prolegomena filozoficzne do gieometryi, pp. 68–73.

35 Bornstein, Problemat istnienia linji gieometrycznych, pp. 64–73.

36 Bornstein, Elementy filozofii jako nauki ścisłej, p. 37.

37 Bornstein, Elementy filozofii jako nauki ścisłej, p. 39.

38 Bornstein, Problemat istnienia linji gieometrycznych, pp. 2–14.

39 Bornstein, Problemat istnienia linji gieometrycznych, p. 13.

40 Kazimierz Twardowski, “Wyobrażenia i pojęcia [Imageries and Concepts],” in: Wybrane pisma filozoficzne [Selected Philosophical Writings] (Warszawa: PWN, 1965), p. 128. (Original work published in 1898. English translation: “Imageries and Concepts,” Axiomathes, Vol. I, 1995, pp. 79–104).

41 Kazimierz Twardowski, “O treści i przedmiocie przedstawień [On the Content and Object of Presentations],” trans. Izydora Dąmbska, in: Wybrane pisma filozoficzne, pp. 24–28.

42 Bornstein, Prolegomena filozoficzne do geometryi, pp. 6–7.

43 Benedykt Bornstein, “O najważniejszych zagadnieniach teorii poznania naukowego [About the most Important Issues of the Theory of Scientific Knowledge],” in: Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism [Benedykt Bornstein’s Philosophy, Selection and Elaboration of Unpublished Writings], pp. 56–57.

44 For instance, in various ontological domains one can notice the specification of the universal category of stability, as in the law of conservation of energy, or heredity, Bornstein suggested that in sociology and political science this category existed as conservative principles. See for instance: Benedykt Bornstein, Pierwsze prawo Mendla w świetle logiki geometrycznej [Mendel’s First Law in the Light of Geometrical Logic] (unpublished work written in 1938, Biblioteka Jagiellońska ref. n. 9023 III). Other numerous examples that can be found in the three volumes of Architektonika świata [Architectonics of the World].

45 When it comes to ontology and the theory of the object, Bornstein discussed the ontology questions in his Teoria absolutu [Theory of The Absolute].

46 Kant discussed category studies in his Critique of Pure Reason. He argued that the categories of our reason are the a priori concepts that organise our sensual experience. Objectifying them at the same time, they move from the domain of subjective sensation to the sphere of scientific knowledge.

47 Bornstein, O najważniejszych zagadnieniach teorii poznania naukowego, p. 59.

48 Bornstein prepared a common structure for both logic and geometry in 1922. See: Benedykt Bornstein, “Zarys architektoniki i geometrji świata logicznego [Outline of Architectonics and Geometry of the Logical World],” Przegląd Filozoficzny [PR], Vol. 25, No. 4 (1922), pp. 475–490.

49 Benedykt Bornstein, „Teoria poznania [Epistemology],” in: Benedykta Bornsteina niepublikowane pisma z teorii poznani, logiki i metafizyki [Benedykt Bornstein’s Unpublished Writings about Theory of Knowledge, Logic and Metaphysics. Selection and Elaboration as well as Introduction and Comments], ed. Krzysztof Śleziński (Katowice: Wydawnictwo Uniwersytetu Śląskiego – Wydawnictwo Scriptum, 2014), pp. 46–49. The typescript can be found in the Jagiellonian Library, ref. n. 739684 III (1945–1946).

50 Epistemological idealism is not connected with metaphysical idealism, which is objective and realistic.

51 Bornstein, Teoria poznania, p. 96.

52 Benedykt Bornstein, “Podstawowe pojęcia metafizyki Spinozy w świetle logiki geometrycznej [Basic Concepts of Spinoza Metaphysics in the Light of Geometrical Logic],” in: Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism, pp. 137–156. The typescript can be found in the Jagiellonian Library, ref. n. 9024 III (1939).

53 Bornstein, Teoria absolutu, p. 54.

54 Benedykt Bornstein, Architektonika zmysłowości i rozsądku [Architectonics of Sensibility and Reason], Warszawa: Wyd. F. Hoesicka, 1927.

55 Benedykt Bornstein, “Geometrja logiki kategorialnej i jej znaczenie dla filozofji [Geometry of Categorial Logic and Its Importance for Philosophy],” Przegląd Filozoficzny [PR], Vol. 29, 1926, pp. 173–194 and Vol. 30, 1927, pp. 69–85.

56 See: Bornstein, Teoria absolutu. Metafizyka jako nauka ścisła, pp. 9–10.

57 Georg Boole gave geometrical logic the form of a system in 1854. See: George Boole, Collected Logical Works, part I and II (Chicago, London: P. Jourdain, 1916).

58 See: Lucio Russo, Zapomniana rewolucja. Grecka myśl naukowa a nauka nowożytna [The Forgotten Revolution: How Science Was Born in 300 BC and Why it Had to Be Reborn], trans. Ireneusz Kania (Kraków: Wyd. Universitas, 2005), pp. 75–82.

59 See: Richard Courant, Herbert Robbins, Co to jest matematyka?[What is Mathematics?], trans. E. Vielrose (Warszawa: PWN, 1962), pp. 217–295.

60 Benedykt Bornstein, “Początki logiki geometrycznej w filozofii Platona [Beginnings of Geometrical Logic in Plato’s Philosophy],” Przegląd Klasyczny [Classical Review], Vol. 4, 1938, pp. 8–9.

61 In this publication: Benedykt Bornstein, Geometrical Logic. The Structures of Thought and Space, [p. 7], and: Edward Huntington, “Sets of Independent Postulates for the Algebra of Logic,” Transaction of the American Mathematical Society, Vol. 5, No. 3 (1904), pp. 292–296.

62 Bornstein presents multiplication as: axb or a.b or ab; the most common is the last one.

63 See: Bornstein, Teoria absolutu, p. 20.

64 One can present it as: a < 1 and 0 < a. See: Bornstein, Teoria absolutu, pp. 21–22.

65 Bornstein, Geometrical Logic, [pp. 9–11].

66 Bornstein, Geometrical Logic, [pp. 18–21].

67 Bornstein, Geometrja logiki kategorialnej i jej znaczenie dla filozofii, pp. 173–194.

68 Benedykt Bornstein, “Co to jest kategorialna geometria algebraiczno-logiczna? [What is the Categorial Algebraic-Logical Geometry?],” in: Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism, pp. 63–77.

69 Bornstein, Geometrja logiki kategorialnej i jej znaczenie dla filozofii, pp. 69–85. Bornstein also connected geometry with three-element algebraic logic, obtaining three-category geometrical logic, along with its laws and principles, see Bornstein, Geometrical Logic, [pp. 24–29].

70 Bornstein, Geometrical Logic, [pp. 24–28].

71 Benedykt Bornstein, “Zarys teorii logiki dialektycznej [An Outline of the Theory of Dialectical Logic],” in: Benedykta Bornsteina niepublikowane pisma z teorii poznania, logiki i metafizyki, pp. 127–184; the typescript of the unpublished work can be found in the Jagiellonian Library, ref. n. 9031 III.

72 Benedykt Bornstein, Architektonika świata. Logiczno-geometryczna architektonika uniwersalna [Architectonics of the World. Universal Logical-Geometrical Architectonics] (Warszawa: Skład Główny Gebethner i Wolff, 1936), pp. 51–172.

73 Bornstein, Geometrical Logic, [pp. 76–91].

74 Bornstein, Zarys teorii logiki dialektycznej, p. 133.

75 Bornstein, Zarys teorii logiki dialektycznej, pp. 141–154; see also: Bornstein, Geometrical Logic, [pp. 74–79].

76 Bornstein, Zarys teorii logiki dialektycznej, pp. 163–168.

77 Bornstein, Zarys teorii logiki dialektycznej, pp. 179–183.

78 Śleziński, Benedykta Bornsteina niepublikowane pisma z teorii poznani, logiki i metafizyki. Wybór i opracowanie oraz wprowadzenie i komentarze, pp. 135–139.

79 Benedykt Bornstein, “Logika tonów harmonicznych [The Logic of Harmonic Tones],” in: Architektonika świata. Logika geometryczno-architektoniczna [Architectonics of the World. Geometrical-Architectonical Logic] (Warszawa: Skład Główny Gebethner & Wolff, 1935), pp. 161–182.

80 Benedykt Bornstein, “Teoria akustyczna układu periodycznego pierwiastków chemicznych [Acoustic Theory of the Periodic System of Chemical Elements],” in: Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism, pp. 103–135. The typescript of the unpublished work can be found in the Jagiellonian Library in Cracow, ref. n. 9022 III.

81 Benedykt Bornstein, Pierwsze prawo Mendla w świetle logiki geometrycznej [Mendel’s First Law in the Light of Geometrical Logic]. The typescript of the unpublished work (1938) can be found in the Jagiellonian Library, RN 9023 III. Bornstein, Benedykt. “Jakościowo-matematyczny aspekt podstawowych praw mendlizm [Quality-Mathematical Aspect for the Principal Rules of Mendel’s Theories].” Sprawozdania z Czynności i Posiedzeń Łódzkiego Towarzystwa Naukowego za I półr. 19 47 roku [Reports of the Philosophical Society of Łódź], Vol. 2, No. 1/3 (1947), pp. 75–81.

82 Bornstein, Geometrical Logic, [pp. 91–98].

83 Benedykt Bornstein, “Sylogizm a przyczynowość czyli o związku dziedziny logicznej z dziedziną realną [Syllogism and Causality, or the Relation between Logic and Real],” Przegląd Filozoficzny [PR], Vol. 33, 1930, pp. 106–130; Bornstein, Architektonika świata. Logiczno-geometryczna architektonika uniwersalna, pp. 146–150; Bornstein, Geometrical Logic, [pp. 91–98].

84 Aristotle, Metafizyka [Methaphysics]. V 6, 1017b.

85 Benedykt Bornstein, “Wstęp do metafizyki jako nauki ścisłej [Introduction to Metaphysics as an Exact Science],” in: Benedykta Bornsteina niepublikowane pisma z teorii poznani, logiki i metafizyki, p. 204.

86 “In the same way that which is to receive perpetually and through its whole extent the resemblances of all eternal beings ought to be devoid of any particular form. Wherefore, the mother and receptacle of all created and visible and in any way sensible things, is not to be termed earth, or air, or fire, or water, or any of their compounds or any of the elements from which these are derived, but is an invisible and formless being which receives all things and in some mysterious way partakes of the intelligible, and is most incomprehensible. In saying this we shall not be far wrong;” Plato, Timaeus, translated by B. Jowett, in Plato, The Dialogues, p. 1871: [access 08.08.2018]: http://webs.ucm.es/info/diciex/gente/agf/plato/The_Dialogues_of_Plato_v0.1.pdf.

87 „I find the true principles of things in the Unities of Substance, which this system introduces, and in their harmony pre-established by the primitive Substance.”, see: Gottfried Wilhelm Leibniz, New Essays Concerning Human Understanding, trans. Alfred Langley, p. 66 [access 08.08.2018]: https://archive.org/stream/newessaysconcern00leibuoft/newessaysconcern00leibuoft_djvu.txt

88 Bornstein, “Wstęp do metafizyki jako nauki ścisłej, pp. 193–200.

89 Krzysztof Śleziński, Benedykta Bornsteina koncepcja naukowej metafizyki i jej znaczenie dla badań współczesnych [Benedykt Bornstein’s Concept of Scientific Metaphysics and its Meaning for Contemporary Investigations] (Kraków, Katowice: Wydawnictwo Scriptum, 2009), pp. 183–185.

90 Bornstein, “Logika tonów harmonicznych,” in: Architektonika świata. Logika geometryczno-architektoniczna, pp. 103–104.

91 The notation of the elements a1 and a0, highlights the difference that exists on the diagram of categorical, geometrical logic between the point a (a1), and line a (a0) - the straight a0 that is in the point a1 (however, when there is a straight (ab), it is determined by two points: a1 and b1). Initially, Bornstein used symbols that differentiate point and straight, but they turned out to be unwieldy, therefore he made the notation on the diagrams simpler and assigned the symbol a to the homonymous straight as well as to the point. The point is the element that, according to Bornstein’s terminology, is more determined than a straight, therefore the point has the role of the precise element, and the straight – the abstract element. The symbols that correspond to the elements on the categorial plane of geometrical logic cannot introduce the ambiguity of their understanding, thus one should present them in a language that will deprive them of their ambiguity. The solution to this problem can be found in the part which includes commentary on the work Geometrical Logic.

92 Bornstein, Teoria absolutu, p. 47.

93 Bornstein, Teoria absolutu, p. 48.

94 In philosophy we may find such proceedings – an example of deductive science in the study of axioms and the abstract can be the semantic model suggested by Kazimierz Ajdukiewicz, presented in Logika pragmatyczna: Kazimierz Ajdukiewicz, Logika pragmatyczna [Pragmatic Logic] (Warszawa: Państwowe Wydawnictwo Naukowe, 1975), pp. 188–191. The ordering of objects which denote the primal terms contained in axioms and realizing axioms of a given deductive theory, Ajdukiewicz called the model of this theory. Therefore, one can admit, that any semantic, and in this case, ontological interpretation, will designate a given model of Bornstein’s topologic.

95 Bornstein, Geometrical Logic, [p. 14, Fig. 3].

96 Bornstein, Teoria absolutu, p. 50.

97 Bornstein, Teoria absolutu, pp. 63–65.

98 Bornstein, Teoria absolutu, p. 55.

99 For instance, Wilhelm Windelband claimed that “the a priori knowledge of the world in itself is possible only for its creator. The claim to know a priori noumens would be identical to the claim to create them [translation mine]”. Andrzej Noras, Historia neokantyzmu [History of the neo-Kantianism] (Katowice: Wydawnictwo Uniwersytetu Śląskiego, 2012), p. 468.

100 Bornstein, Teoria absolutu, p. 57.

101 Bornstein, Teoria absolutu, p. 57 [translation mine].

102 Benedict Spinoza, Ethic. On the Improvement of Understanding, trans. R.H.M. Elwes, p. 6 [access: 08.08.2018]:http://www.naturalthinker.net/trl/texts/Spinoza,Benedictde/Spinoza,%20Benedict%20de%20-%20The%20Ethics%202.%20Of%20the%20Nature%20and%20Origin%20of%20the%20Mind.pdf.

103 Spinoza, Ethics [access: 08.08.2018]:https://en.wikisource.org/wiki/On_the_Improvement_of_the_Understanding/Part_2#94

104 Baruch Spinoza, “Podstawowe pojęcia metafizyki Spinozy w świetle logiki geometrycznej [Basic Concepts of Spinoza’s Metaphysics in the Light of Geometrical Logic],” in: Filozofia Benedykta Bornsteina oraz wybór i opracowanie niepublikowanych pism, p. 147 [translation mine].

105 The symbol of a disc or a ball is very often present in philosophy. Throughout the centuries, this symbol could be found for instance in the philosophy of Plato, Dionysius the Areopagite, Nicholas of Cusa, in the philosophy of the medieval mystics and renaissance philosophers: Ficino and Giordano Bruno, and is very important in Pascal and Leibniz’s philosophy of metaphysics. See: Dietrich Mahnke, Unendliche Sphäre und Allmittelpunkt. Beiträge zur Genealogie der Mathematischen Mystik (Halle: Max Niemeyer, 1937), pp. 43–61.

The Structures of Thought and Space

Algebraic-geometrical logic or, simply geometrical logic likewise known as topologic, appeared as a spatial representation of algebraic logic (the first, preliminary outline having been published in the „Przegląd Filozoficzny” („Philosophical Review”, Warsaw 1926 and 1927). The representation was effected by the application of Descartes`s co-ordinates to logic and by making use of the correspondences between duality in logic and in (projective) geometry. The spatialization of logic enabled us to give it a marked structural and architectonic character – one which brought out the „order” which reigns in this domain. This geometrical logic is of great philosophical, and particularly of ontological, significance; the whole philosophical aspect has, however, been disregarded in the present work. Only the foundations of geometrical logic as such have been dealt with, and the architectonics ruling its elements have been brought out (inter alia, the concept of harmonic sets of four and that of the neutral mean, etc. have been introduced).

Logic (pan-logic) can in its specifications acquire a character which is distinctly dynamic (dialectic) and parallel with physical geometry (non-Euclidean). In the Appendix, the correspondence between the logical sum, logical product, the logical neutral mean and the harmonic, arithmetical and geometrical means has been introduced, thus leading to the discovery of a number of hitherto unknown elementary arithmetical theorems which bind together these three Pythagorean means.

Introduction: The Idea of Geometrical Logic

Geometrical logic, the most general principles of which we shall develop below, is not isolated in the domain of science in general, and in particular in that of mathematics. For it is in mathematics above all that the age-old thought, which has with irresistible, instinctive force imposed itself on the spirit of humanity, the thought that the worlds of spatial and non-spatial elements are connected in some profound manner, has met with unqualified success.

It was in this direction that Greek arithmetic, as developed in the school of Pythagoras, first went; we see here how fruitful were the scientific results of the conception of giving spatial models of so completely non-spatial entities such as ←47 | 48→are numbers. Pythagorean arithmetic was in its entirety a geometrical arithmetic, the philosophic foundation of which was based on the postulate connecting the world of space with that of numbers, and stating that a point in space is none other than a unit (a component of all numbers) possessing a position in space. Accordingly, the Pythagoreans represented numbers as separate points, being their component units, and, according to the shapes produced by these points they distinguished triangular, square, and right-angled numbers. As a result of the spatial interpretation of whole numbers they found that by adding to unity the succeeding odd number, 3, a square could be obtained and that adding similarly to the figure and number so obtained the following odd numbers (5, 7, etc.), the resulting figures and numbers were also quadratic (9, 16, etc.). In this way we find at the inception of Greek arithmetic that extremely interesting theorem of the theory of numbers which says that the sum of series of consecutive odd numbers, beginning with unity will always represent a quadratic number [1 + 3 + 5 + … + (2n – 1) = n2]. This example shows how important a role the method of spatial representation of numbers played in Greek arithmetic.

This interconnection of the worlds of numbers and of space attains its culmination in the analytical geometry of Descartes, in which the multiplicity of numerical pairs (x, y) is represented as a plane geometrical figure, and of triple numbers (x, y, z) by a three-dimensional figure. The harmony of these two worlds – the analytical and spatial – is [p. 3] complete; although one these domains consists of completely non-spatial elements, such as are numbers, and the other of elements existing in space, yet the laws and relations prevailing in these worlds are identical.

It now becomes apparent that the concept of geometrical logic is not so strange as it might at first sight have appeared. On the contrary, it is perfectly natural, and its realization is a necessary complementation of such branches of mathematics as geometrical arithmetic, geometrical algebra, and analytical geometry, for in all these cases we have to deal with the same fundamental matter. This is the correspondence of the domains of spatial and non-spatial worlds, and of the coincidence of these two realms, with this difference only that the non-spatial domain is in one case represented by numbers, and in the other by concepts. Actually numbers and concepts are of the same nature, and have themselves nothing in common with extension; they are par excellence of a non-intuitive, non-figurative, purely conceptional nature. If then numbers are, in spite of their non-spatial nature in such astounding, yet undoubted harmony with spatial objects, and if they can be represented in their forms, it is but a short step to the presumption that this will apply to concepts in general, and that, in the same way as geometrical arithmetic and algebra, and analytical geometry already exist, so ←48 | 49→also must exist geometrical logic. Moreover, since we already recognise the ordinary quantitative geometrical algebra, what can be more natural than to suppose that for qualitative algebra, which is identical with exact logic, there would be a corresponding certain qualitative geometry – the geometry of logic.

This conception of the affinity of the domains of thought and of space existed, although only in a rudimentary form, in the mind of the founder of logic, Plato, who first discovered the world of concepts, of logic. This world is, according to the profound convictions of its discoverer, a model of order, a cosmos, a system, in which each concept is in some exactly defined relation to any other concept, and occupies an exactly defined „position” in this system. If this is so, however, the concept of this world of logic should present an analogy to the spatial world; the elements of this logical world should be distributed and correlated in some „logical space”. Thus thought Plato, when he spoke of „the space of thought” (τόπος νοητός), in which ideas were to exist.

The tendency to geometrize the logical world is undoubted in Plato; the detailed realization of this tendency w [p. 4] as not possible at the time, since the world of ideas discovered by him itself too little explored – and it was not until two thousand years later that the genius of Leibnitz found means of furthering the exact, mathematical knowledge of this world. Leibnitz similarly attached great importance to the visualization of the elements and principles of logic by their spatial representation, by „driving lines” (per linearum ductum), and there can be no doubt that if he had succeeded in creating a coherent system of algebraic logic he would at the same time have given us a system of the geometry of logic.

We can also find in the usual, traditional non-mathematical logic elements testifying to our instinctive, unconscious conviction as to the affinity of the worlds of thought and of space. We have here in mind the employment of diagrams for the expression of logical relationship. In general, even logicians, not to speak of those learning logic, do not realize the exceptional importance of the circumstance that the relations between non-spatial concepts can be illustrated by the relations between spatial elements, such as, for example, circles. The use of diagrams is usually regarded as an ingenuous didactic device, facilitating orientation in the relations prevailing in the realm of thought, and making it more easy for us to understand these relations than would be possible with direct consideration of the problems in question. Those holding such views would assert that the use of such diagrams is in no way strange, since the human mind actually does possess a tendency towards the spatial representation of thoughts or of spiritual entities in general. It might be replied that, in the given case, it is immaterial that such spatial representation is helpful in the teaching of logic, ←49 | 50→and that it is not the fact that we quite naturally tend to apply this means of representation, under the urge of some natural intellectual instinct, but rather we should note that the use of such diagrams withstands objective tests, and that ordinary spatial diagrams are, if even to only a limited extent, able accurately to represent thought relations. It is in this that consists the objective aspect of the above-discussed spatial correspondence of logical elements; this is a fact of the highest scientific and philosophical importance, although at first sight it appears to be so insignificant, and apparently of purely didactical-psychologic value. Yet these spatial schemes of classical logic are incontrovertible evidence of the coincidence of these two so dissimilar domains, and allow of the hope that, with the moment when logic attains the dignity of an exact science, this as yet fragmentary, disconnected, and limited representation of the world of logic in space will also be enabled to attain a fundamentally higher level. We may already, [p. 5] by the appropriate selection of spatial elements (that is, above all, straight lines, and not things of a higher type, such as are circles), a priori hope to attain an adequate and systematic representation in space of the elements, operations, and relations of the world of logic, and this would be equivalent to the spatialization of exact logic, in other words with the creation of the fundamentals of exact geometrical logic.

But if we attempt to find in the ordinary, traditional logic elements of affinity of the domain of concepts with that of space, we must also discuss the language, the terminology of logic, in which our intuition of this affinity has found its unconscious expression. And so the „terms” of judgment betray their spatial origin, since they designate the limits of judgment, and in this way suggest the idea of a judgment as a segment joining two limiting points: the terms of judgment, subject and predicate. We see the same in the terminology of syllogisms, in which we speak of the „middle term”, i.e., of the term situated between two extreme ones, as if we were speaking not of concepts, but of points on a line. Expressions such as that one idea „is contained” in another, or the „crossing” of ideas – these are all redolent of the spatial factor. Further, we speak of the „definition” of concepts, i.e., of the assignment to them of limits, of their situation in space, as it were. If it be objected that the transference of terms from the domain of space to that of logic is perfectly natural, and due to the circumstance that our knowledge of spatial objects is of more remote origin and is far more complete that is that of abstract ideas, and that it for this reason that we characterize a concept by its situation, we must of course agree to this. Yet this in no way detracts from the significant fact that these spatial terms very successfully and exactly characterize the correlation of logical elements, and that they give rise to spatial representations which correctly depict logical structures.

←50 | 51→We thus see that indications for the geometrization of the domain of logic abound on all sides. In the domain of ordinary logic the mere fact of its terminology, so deeply imbued with spatial elements, should lead us to search not only for the subjective sources of this so significant fact, but also for its objective basis, to the existence of which testify the geometrical diagrams of this traditional logic. The possibility and necessity of the spatialization of logic are suggested by still other evidence, viz., the relationship between the objects of logic and of mathematical analysis. Arithmetic, algebra, and analysis in general apply, similarly to logic, to a world of non-spatial elements, and yet [p. 6] geometrical arithmetic, geometrical algebra, and geometrical analysis (known as analytical geometry) exist – why therefore should geometrical logic not be possible? This is the more probable as the more fact of the existence of logic in the form of qualitative algebra (algebra of logic) requires the creation of a geometrical counterpart of this qualitative algebra, such as we already possess, to a certain extent, for ordinary algebra. And such an analogue would be termed geometrical logic.

CHAPTER I: Geometrization of the Axioms of Algebraic Logic

Before we undertake an examination of the features proper of geometrical logic (also called by us topologic), we must first deal more closely with the algebraic logic which we are to geometrize. Algebraic logic (the algebra of logic) is a system based on a number of axioms which permit us to deduce further theorems. We shall base ourselves in the following representation of the algebra of logic on the first system of axioms, given by E. Huntington in his Sets of Independent Postulates for the Algebra of Logic.^{106}

Huntington takes into consideration a multiplicity of elements which contain at least two elements differing from each other; moreover assuming that the elements a and b belong to this system, the elements a+b and ab (the logical sum and product of a and b) will likewise belong to it. (Huntington formulates the above properties of the system of elements by means of two postulates.)

The remaining postulates which govern this system of elements can be expressed as follows:

1a. There is an element 0, such that a+0=a for every element a.

1b. There is an element 1, such that a.1=a for every element a.

2a. a+b=b+a

2b. ab=ba

3a. a+bc=(a+b)(a+c)

3b. a(b+c)=ab+ac

4. There is an element a’ such that for every element a:

4a. a+a’=1, and

4b. aa’=0

Postulate No. 1. The elements 0 and 1 are two limitary elements of the system of algebraic logic as regards which we shall later ascertain that 0 is the smallest logical comprehension and 1 the largest (cf. p. 32, theorem 12). Postulate No. 1 states the existence of these elements as the moduli of addition and multiplication, i.e. such elements (as 0 and 1 in arithmetic) which do not change the elements to which [p. 8] they are joined by the symbol „+” (modulus 0) or by the symbol „x” (modulus 1).

The essential nature of logical moduli can be better understood if we bear in mind that 0 is the smallest and 1 the largest logical comprehension. The smallest, poorest, least determinate, logical comprehension is the concept: „object” („being” = „something”). If so, to say a given element is a, is clearly equivalent to stating that it is a+0, since a+0 signifies „object a”. The formula a.1=a, in line with the significance of logical multiplication expresses that when we seek the greatest part common for the logical comprehension of a and 1, we find it equals a. As 1 is the largest comprehension, the richest, the one which involves in itself all the other comprehensions, the comprehension a is naturally the greatest part common to a and 1.

Postulate No. 2. The equations a+b=b+a and ab=ba express what is called the law of commutation in logical addition and multiplication, whereby the logical sum and product do not depend on the order of the operations: we receive the same results when we add the concept b to the concept a as we add the concept a to concept b; similarly in the case of multiplication.

Postulate No.3. Formulae Nos. 3a and 3b express what is known as the law of distribution. Equation 3a, a(b+c)=ab+ac, is known in ordinary algebra. In the logic of algebra it has the following significance: the greatest common element of the concept a and of the concept b+c consists of two parts: the common element of the concept a and of the concept b, as also the common element of the concept a and of the concept c (and vice versa). This common element of a and b+c has therefore here been divided into two parts, hence the name given to the law of distribution (in this case the distribution of logical multiplication). In logical addition, equation 3a, i.e., a+ bc=(a+b)(a+c) applies. This means: in order to add ←52 | 53→to a the logical product b.c, it is necessary to add b to a, then c and to multiply these sums by each other (and vice versa).

Postulate No.4. The equations 4a and 4b express the negative element a’ (i.e., the one received from the element a by the operation of negation) by means of the corresponding positive element a and the elements 0 and 1. In accordance with these formulae, the element a’ is such an element which supplements the element a to 1 (a+a’=1), and which has the minimum community with the element a (a.a´=0).

A few more explanations must now be added to the above preliminary information on the system of axioms of the algebra of logic. Attention [p. 9] is first drawn to the fact that the elements a, b, c are in our understanding primarily not classes of objects falling under any given concepts, but they are the concepts a, b, c themselves –mental comprehensions. Thus, the concept „man” means here not a „class of man” but „the totality of the features common to all mankind”. In short, the above system of axioms is above all one of the postulates of the logic of comprehension and not of the logic of extension (class).

Let us now pass to a certain characteristic feature of this system of postulates. All the postulates appear in it in two forms; they split into two, and express themselves in twofold manner: the first equation refers to addition, the second to multiplication. We therefore have to deal with what is known as the duality of logical addition and multiplication. This duality is indicative of the unusual harmony reigning in the logical world and is expressed by this correspondence of the formulae for logical addition and multiplication. The law of duality affects, as we have seen, all the foregoing postulates of the algebra of logic. It permits us, in the case of expressions of the type of our postulates, to pass from a given formula to one dual with it simply by changing „+” for a „x”, and the 1 for a 0, and conversely, as in the case of equations 1–4.

It will have been noticed that the symbol „<“ does not figure among the above postulates although it signifies the relation of inclusion (implication) which is so important in logic. This relation (the inclusion of a in b), however, can be reduced to the relation „=“ and to the logical operations included in the above postulates, so that

a<b=(b=a+b) (Ia)

This definition of the relation of inclusion becomes more obvious when we take it into account that if comprehension a is contained in that of b, then addition to b of comprehension a, already contained in the former, does not change comprehension b; thus a+b=b, if a<b, and conversely: if by addition of comprehension ←53 | 54→a to comprehension b the latter undergoes no change, this means that the added comprehension a is already contained in comprehension b.

The equivalence established above, as for that matter all logical equivalence, signifies the mutual inclusion (implication) of the members of such equivalence; thus when (a<b) it follows that (b=a+b) and vice versa. This essential feature of all equivalence, the fact that it consists in the mutual implication of its members, is expressed by the following definition:

(a=b)=(a<b)+(b<a) (II)

[p. 10] In connexion with the relation < now under examination, attention is drawn to the fact that the law of duality applies to formulae expressing the equivalence between elements connected by the operation „+” and „x”, as also to formulae expressing inequalities (<) between such elements. Thus, for instance, the formula ab<a corresponds to a<a+b and expresses the indubitable truth that the greatest common part of the elements a and b is contained in the element a. Hence, in order to pass from the formula for inequality to the dual one, it is not only necessary to change the „+” for a „x” and vice versa (or the 0 for 1, and vice versa) but also to transpose the members of the relation of implication. By so doing, we pass, for example, from the formula 0<a to the corresponding dual one: a<1.

The postulates of algebra of logic and the questions most closely connected with it should now be clear and we can pass to the most important matter in this work, viz., to the establishment of a system of logico-geometrical co-ordinates which will make it possible to geometrize the algebra of logic.

The fact that logical relations, primarily the relations of the logic of extension, permit of spatial schematization, is known to all who have studied logic and applied Euler’s circles. And it is in the possibilities of these Eulerian diagrams (or of similar schemes) that lies the essential point of the problem to be attacked. Instead of using the circular diagrams which have gained such popularity since Euler began to use them (they were known, of course, before), we shall, however, consider the simpler straight-line diagrams which Leibnitz preferred to apply. Thus, if we desire to show that the two classes a and b have a common part ab, we usually take two segments (a and b) of the same straight line in such wise that they partially overlap. The part common to both represents class (ab), common to both classes, their logical product.

Such a schematization as the above (in which a and b, as also ab are represented by straight lines) has the fundamental fault that it does not express the fact that ab is a derivative in relation to a and b, and therefore belongs as it were, to another generation, to another dimension. In the above scheme, ab is ←54 | 55→represented by means of a straight line in the same way as the elements a and b which go to form it. Should we desire to depict in a spatial scheme this indubitable heterogeneity of dimension of the species on the on hand, and of the genus and [p. 11] specific difference on the other, the classes a and b would have to be depicted in the form of two intersecting straight lines a and b: the point of intersection would then symbolize the class ab, contained both in a and b: then this class product would be of a dimension other than that of the classes which formed it.

In the present work, however, we are primarily concerned with the logic of comprehension and not of extensions.^{107} The query arises: Can two intersecting straight lines (e.g., at right angles) likewise symbolize an operation referring to logical comprehensions? The answer is, of course, in the positive; but the point of intersection of the straight lines will now form the comprehensional point of view, not signify here a formation common to the classes a and b, but a new comprehension, formed from the cumulation of the comprehensions a and b, corresponding to these classes. It will be the point a+b.^{108} This point in not now contained in a and b (as in the case of multiplication of classes); the concept (comprehension), for instance, of „an equilateral right angle” is neither contained in the concept of „a right angle” nor in that of „equilateral”, but, conversely, the constituting concepts of „a right angle” (a) and „equilateral” (b) are contained in the derivative concept of „an equilateral right angle” (a+b). In other words a<a+b, b<a+b. The concepts a and b, taken (separately) as logical comprehensions are only a possibility of the sum a+b are something less determinate, less definite, than the sum a+b. In the present schematization of a logical concept, the point a+b will represent a notion fundamentally richer, a notion which is more differentiated and specified (species) than the straight lines a and b (genus and specific difference); the point a+b will not here be an element of the straight line a or b, but their determination, that which the straight lines a and b constitute – in which they are contained. It is in such wise that the spatial relation of the straight lines a and b to the point a+b should be understood: these lines pass through it, and ←55 | 56→they exist in it since it is their synthetic formation – and in this sense they are included within it (e.g., the lines of the pencil are included in the point-vertex of the pencil) [p. 12].

It now remains to elucidate how we are to schematize the comprehension ab, comprehension which is more general and less determinate than the comprehensions a and b. On the basis of the above we can understand that to this process of abstraction corresponds in the spatial field the passage from more specified elements (points) to fundamentally less specified ones (straight lines); in other words, if a and b are represented by points the scheme of the logical product ab will be the straight line joining these points. This straight line will represent what the points have in common – qualitatively the nearest element of which they are the specifications. Just as the comprehension ab, as less determinate, is implied both in comprehension a and comprehension b, so straight line ab, as a less determinate product, will be implied in the points a and b which are more determinate that it. Hence, both in the scheme of multiplication and in that of addition of comprehensions, the straight line is implied in the points. Let the straight lines a and b intersect at right angles and the points a and b on these lines be joined by means of a straight line; the scheme thus yielded will represent spatially both the multiplication and the addition of concepts-comprehensions (see Fig. 1).

Fig. 1:

We have, in this case, a spatial representation of the relations:

## Details

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- philosophy of science logic and ontology Polish philosophy spatial logic category theory
- Published
- Berlin, Bern, Bruxelles, New York, Oxford, Warszawa, Wien, 2019. 217 pp., 30 fig. b/w, 4 tables