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Philosophy of Mathematics

In this issue

Journal NotaBene Issue 57 (2022) Editor: Silviya Kristeva

Topic of the issue: Philosophy of Mathematics

How Mathematics is Applicable to the World?

The aim of the paper is to reach an answer to its title question. In this connection two approaches are being considered for the answer that is looked for – a historical and a theoretical one.
Keywords: mathematics, philosophy, mathematical structures, application of theories.

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The Idea of Creativity in Whitehead’s Writings of Mathematics

The question that will interested me in the present paper is, first, when the idea of creativity appears in Whitehead’s writings on mathematics and, second, how this idea works in mathematics itself and in philosophy of mathematics? So, I shall follow chronologically the appearance of the idea of creativity in Whitehead’s writings on mathematics and the evolution of his intellectual development with regard to the function and role of creativity in mathematics. The key role of creativity in Whitehead’s metaphysics is emphasized in his last publication Immortality (1941). The connection of creativity with mathematics is emphasized by Whitehead in his last publication on mathematics Mathematics and The Good (1941). The conclusion of my paper is that Whitehead’s views concerning the idea of creativity in mathematics was formed in his last mature period of intellectual development after 1920ties and he has supported firmly these views till the end of his life.
Keywords: philosophy of mathematics, creativity, A.N. Whitehead

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Contemporary Mathematical Theories and Methods with Worldview Significance

The article examines recently developed mathematical theories and apparatus in terms of their worldview significance, related to discovering opportunities for new, deeper and more complete knowledge of phenomena and processes in nature and society, for their more adequate philosophical understanding, for opening new cultural horizons. The theory of fuzzy sets and the chaos theory with the associated fractal geometry are presented in detail. The first of them transforms the bivalent perception of the world and the associated dichotomization, providing methods for more accurate and adequate reflection of the phenomena around us (especially in the social sciences), which are characterized by diversity, complexity and nuance. Chaos theory, in turn, transforms our notions of determinism, predictability, reproducibility, self-organization, etc., while providing us with an effective apparatus for understanding the meaning of various phenomena and processes, for their more accurate prediction and embodiment of our knowledge in innovative technological solutions.
Keywords: modern mathematical theories, worldview, fuzzy set theory, chaos theory, fractal geometry

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Numerical Identity, Algorithmic Operationality and Self-Referential Paradoxes

The problem of numerical identity is closely related to identity of objects and self-identity, as formalized in the logical Law of identity. Here I propose a novel approach to analyzing numerical identity within the general framework of the concepts of operatio and operandum, which, as I argue, are to be found at the foundations of mathematics, logic and natural language. The approach introduces the notion of algorithmic history of sentences, both formal and in natural language, where the emergence of a well formed sentence develops on strict stages, reflected by the steps of standard algorithms. This algorithmic approach allows for two novel results: first, it permits to rigorously discriminate computationally accessible structure for sentences in natural and formal languages, which makes them rigorously available for computational operations and programs development. Second, it presents a fresh look at classical self-reference paradoxes as the Liar’s paradox and Russell’s paradox, where, as I argue, the paradox is only apparent since it cannot form on a single step of the algorithm of any sentence S, since any of S’s interpretations and self-references develops, necessarily, on a different step of the algorithm.
Keywords: identity, numerical identity, algorithm, metaphysics, operandum.

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The Reasonable (In)Effectiveness of Mathematics in Behavioral Sciences

The question concerning the applicability of mathematics is one of the focal points of interest in the philosophy of mathematics. One of the areas where the mathematical models are considered as descriptively inadequate is the science of human behavior. The so-called rational choice theory for example, is usually considered as a purely normative theory which poses insurmountable difficulties when applied to real world situations. A collection of counterexamples that aim to show this were introduced by Amartya Sen. He uncovers the difficulties that arise from the classical idea that rational behavior reduces to maximization of an acyclic ordering. The choices that people make and which we treat as rational in the intuitive sense of this word are contextually dependent and are affected by the presence of alternatives, which bear information about the nature of the objects of choice and the choice situation itself. The paper offers a formal reconstruction of the functioning of these alternatives – as “flags” that signal the transition from one ordering to another, that mark the conceptualization of alternatives as chances or as risks. This gives rise to an interesting mathematical structure, related to two of the key concepts of order theory (ideal and filter). This structure can be presented axiomatically by means of simple axioms. This shows, in accord with the position of David Hilbert, that there are no areas where mathematics is inapplicable, just fields of research where it is still not appropriately applied.
Keywords: applicability of mathematics, behavioral sciences, choice operators, rational choice, revealed preferences.

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Felix Klein’s Erlangen Program – Methodological Genesis and Perspectives. Part I

Felix Klein’s Erlangen program is conceived and presented by some researchers in the field of history and philosophy of mathematics as a very important factor, with status, I would say the “Copernican turn” in geometry. The present study seeks the metaphysical and metamathematic position, i. e. the philosophical-mathematical topology of the German’s Mathematician’s Program in the logical scheme of the whole geometry. Emphasis is placed on the works of Evarist Galoa and their methodological influence in the operationalization of the theoretical-group approach by Klein, in the field of geometry. All this is achieved and summarized through a conceptual analysis of Klein’s methodological guidelines related to the study of the diversity of any number of dimensions and abstraction from the geometric figure, which he developed as necessary for the emergence of more complex geometries with other types of invariants. Thus, the Erlangen program implicitly contains, but also explicitly states its guiding idea – the unity of concept and reality.
Keywords: Felix Klein, Erlangen program, history of mathematics, philosophy of mathematics.

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Felix Klein’s Erlangen Program – Methodological Genesis and Perspectives. Part II

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Number Sets and the Absolute Number Continuum. Part I

Dedekind raises the question of continuity in the construction of number-sets, and Cantor implements this concept, deducing infinite numbers as a new kind of number. How these significant inventions relate to general number theory will be the subject of the present study. Cantor defined numbers and number-sets in their order of infinite succession in power, thus constituting the “absolutely infinite totality of numbers”, or the Absolute Continuum. Thus, however, the construction of the number itself and the series of numbers must interact with the absolute continuum, which provides an extension and even completion of number theory. In this possibility, a transition and connection between number-sets is found. Dedekind demonstrated this connection through the correspondence of the set of points on the straight line and the set of real numbers. Dedekind’s conception of the continuous domain of the real numbers requires a certain strengthening through the deep geometry of the straight line.
Keywords: Cantor, Dedekind, theory of numbers, infinite numbers, deep geometry.

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Number Sets and the Absolute Number Continuum. Part II

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Reasons Connecting Jacopo de Barbari’s “Portrait of Luca Pacioli” with Albrecht Durer’s “Melancholia”. Part I

The text constructs the visual history linking Jacopo de Barbari’s attributed “Portrait of Luca Pacioli and His Pupil”, 1495, starting from the circumstances: 1/ Jacopo de Barbari and Albrecht Durer met twice, sharing a passionate interest in proportion; 2/ Albrecht Durer made the font antique in the work “On the Divine Proportion”, 1498 by Luca Pacioli; (it has been suggested that the pupil depicted in the “Portrait” may be Albrecht Durer); 3/ the common point of contact between the artists Jacopo de Barbari and Albrecht Durer is the application of mathematics in art. And if “Portrait of Luca Pacioli and His Pupil”, 1495, only indirectly outlines Durer’s invisible figure as a geometer, then “Melancholia I”, 1514, presents him as an innovative algebraist, confirming his position as a Northern Renaissance polymath artist of the rank of Leonardo da Vinci. Also, the text suggests the alchemical-occult prehistory of melancholy.
Keywords: Albrecht Durer, Jacopo de Barbari, Luca Pacioli, Vitruvius: perspective, anatomy, proportion, humanistic antiquity, magic square, melancholy.

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Reasons Connecting Jacopo de Barbari’s “Portrait of Luca Pacioli” with Albrecht Durer’s “Melancholia”. Part II

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Podium

Philosophy of Bulgarian History: Petar Mutafchiev. Part I

The principal notions and values of the Petar Mutafchiev’s philosophy of Bulgarian history are described and commented: Bulgarian, native, nation, people, paganism, cultural-political influence of byzantinism, past and future, political leaders, intelligence, etc. A special attention is paid to P. Mutafchiev’s interpretation of the role of those phenomena in the processes, trends and dramatic events in the Bulgarian history: to his ctiticism to the leaders and the intelligence; to his interpretation of people as a victim and sufferer. The evaluation of his ideas from the point of view of past and contemporary authors is emphasized.
Key words: Bulgarian, native, nation, people, cultural-political influence of byzntinism, political leaders.

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Philosophy of Bulgarian History: Petar Mutafchiev. Part II

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Minima moralia

Guardians of Faith and Memory

The article aims to refer attention to the preservation of material cultural heritage as part of the small projects of memory embedding. The connection with the past and the heritage built with faith and hardship must be preserved for future generations. The preservation of these material values, their restoration and popularization is the necessary connection for the predetermination of cultural development, based on memory and with vision to the future.
Keywords: philosophy, faith, memory, connection with the past, material cultural value

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Announcement

Philosophical Reflection on the “Null” as “Emptiness” or Unfolding the “Nothing” into “Something” out of Numbers

Why does the Pythagorean teaching not say anything about the zero? Why, provided it has been clarified that the number three for instance symbolizes the completeness, nothing has been said about the beginning “before the initial ONE”? Why numbers in the Pythagorean teaching explain the existence however nothing explains the non-existence as a factual undoubtedness – the non-existence of things is undoubted as much as their existence. Could the Pythagorean teaching have been “scared” of the “non-existing” things and their wisdom, which was not practically realized as a numeric expression and image?
Keywords: Pythagorean teaching, monades, universe, Åastern philosophy.

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Poiesis

Proportions and Similarities. Photos by Teodora Todorowa

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