Actual Infinity: A Pseudo-problem or a Meta-Foundation of Western European Philosophy and Science?

The author of the article reveals the theological context of the origin of the concept of actual infinity and clarifies the problem of actual infinity. The author shows that this problem is not a paradoxical category of thinking, but a problem of the unity of two realities (eternal, unchanging and infinite, and temporary, changeable and finite), which has been misunderstood. The author raises the question of the relevance of the problem of actual infinity brought by Christianity for modern secularized science and philosophy. The author shows that the problem of the unity of the two realities was declared much earlier than Christianity. This problem was already dealt with by the ancient Eleans. They initiated the one-sided view and incorrect understanding of this problem, which opened the main path of development of the entire Western European philosophy. With the advent of Christianity, all the dangers identified by the Eleans (and above all by Zeno) and then still unclear on this path received a new sharpness and now real force. The author of the article shows that the regularity of the relation of the finite, the actually infinite, and the potentially infinite, revealed by Zeno, was the basis for changing the classical rationality to the non-classical one. In turn, the fact of the collapse of the classics has become evidence of modernity that the problem of actual infinity is not a mental paradox, but contains the real possibility of changing the finite nature. But this change is not carried out in the direction suggested by the recognition of actual infinity itself, but in another direction, opposite to it, but closely connected with it. The disclosure of the essence of this connection will be the disclosure of the problem of actual infinity.

TWO TRADITIONS OF THE ACTUAL INFINITY INTERPRETATION

The article considers two traditions in the interpretation of the actual infinity. One is associated with the name of Nicholas of Cusa, the other with the name of Rene Descartes. It is shown how Nicholas of Cusa within the framework of his idea of the coincidentia oppositorum overcomes the traditional Aristotelian norms of philosophizing, while Descartes puts the finitist ideology at the foundation of both his theology and the theory of knowledge.

FLORENSKY’S PHILOSOPHICAL IDEAS AND THE THEORY OF SETS OF CANTOR

The article considers the role of the idea of actual infinity in the works of Florensky. The introduction briefly traces the history of ideas about the actual infinity in European culture to the works of George Cantor. The reaction of European scientists and religious figures to the emergence of the “naïve” theory of Cantor sets is characterized. A detailed analysis of the connection between Florensky and George Cantor’s ideas is given. Many quotations from the 1904 work on the symbols of Infinity are given, which illustrate the influence of Cantor’s works on Florensky. The presentation of Florensky’s religious and philosophical ideas of Cantor about the actual infinity is given. Emphasized understanding Florensky transfinite numbers Cantor as symbols.

An Analysis of Zeno’s Paradox and Non-measurable Sets Based on Dialectical Infinity

The Problem of Continuity and Discreteness is the basic problem of philosophy and mathematics. For a long time, there is no clear understanding of this problem, which leads to the stagnation of the problem, because the essence of the problem is a problem of finity and infinity. The essence of the philosophical thought on which the mathematical definition of “line segment is composed of dots” is the idea of actual infinity, and geometric dot is equivalent to algebraic zero in terms of measure properties. In view of the above contradictions, this paper presents two solutions satisfying both the philosophical and mathematical circles based on the view of dialectical infinity, and the authors make a deep analysis of Zeno’s paradox and the non-measurable set based on both solutions.

Kant’s Mereological Account of Greater and Lesser Actual Infinities

Recent work on Kant’s conception of space has largely put to rest the view that Kant is hostile to actual infinity. Far from limiting our cognition to quantities that are finite or merely potentially infinite, Kant characterizes the ground of all spatial representation as an actually infinite magnitude. I advance this reevaluation a step further by arguing that Kant judges some actual infinities to be greater than others: he claims, for instance, that an infinity of miles is strictly smaller than an infinity of earth-diameters. This inequality follows from Kant’s mereological conception of magnitudes (quanta): the part is (analytically) less than the whole, and an infinity of miles is equal to only a part of an infinity of earth-diameters. This inequality does not, however, imply that Kant’s infinities have transfinite and unequal sizes (quantitates). Because Kant’s conception of size (quantitas) is based on the Eudoxian theory of proportions, infinite magnitudes (quanta) cannot be assigned exact sizes. Infinite magnitudes are immeasurable, but some are greater than others.

Natural Cybernetics of Time, or about the Half of any Whole

Norbert Wiener’s idea of “cybernetics” is linked to temporality as in a physical as in a philosophical sense. “Time orders” can be the slogan of that natural cybernetics of time: time orders by itself in its “screen” in virtue of being a well-ordering valid until the present moment and dividing any totality into two parts: the well-ordered of the past and the yet unordered of the future therefore sharing the common boundary of the present between them when the ordering is taking place by choices. Thus, the quantity of information defined by units of choices, whether bits or qubits, describes that process of ordering happening in the present moment. The totality (which can be considered also as a particular or “regional” totality) turns out to be divided into two parts: the internality of the past and the externality of the future by the course of time, but identifiable to each other in virtue of scientific transcendentalism (e.g. mathematical, physical, and historical transcendentalism). A properly mathematical approach to the “totality and time” is introduced by the abstract concept of “evolutionary tree” (i.e. regardless of the specific nature of that to which refers: such as biological evolution, Feynman trajectories, social and historical development, etc.), Then, the other half of the future can be represented as a deformed mirror image of the evolutionary tree taken place already in the past: therefore the past and future part are seen to be unifiable as a mirrorly doubled evolutionary tree and thus representable as generalized Feynman trajectories. The formalism of the separable complex Hilbert space (respectively, the qubit Hilbert space) applied and further elaborated in quantum mechanics in order to uniform temporal and reversible, discrete and continuous processes is relevant. Then, the past and future parts of evolutionary tree would constitute a wave function (or even only a single qubit once the concept of actual infinity be involved to real processes). Each of both parts of it, i.e. either the future evolutionary tree or its deformed mirror image, would represented a “half of the whole”. The two halves can be considered as the two disjunctive states of any bit as two fundamentally inseparable (in virtue of quantum correlation) “halves” of any qubit. A few important corollaries exemplify that natural cybernetics of time.

L’infini entre deux bouts. Dualités, univers algébriques, esquisses, diagrammes

The article affixes a resolutely structuralist view to Alain Badiou’s proposals on the infinite, around the theory of sets. Structuralism is not what is often criticized, to administer mathematical theories, imitating rather more or less philosophical problems. It is rather an attitude in mathematical thinking proper, consisting in solving mathematical problems by structuring data, despite the questions as to foundation. It is the mathematical theory of categories that supports this attitude, thus focusing on the functioning of mathematical work. From this perspective, the thought of infinity will be grasped as that of mathematical work itself, which consists in the deployment of dualities, where it begins the question of the discrete and the continuous, Zeno’s paradoxes. It is, in our opinion, in the interval of each duality ― “between two ends”, as our title states ― that infinity is at work. This is confronted with the idea that mathematics produces theories of infinity, infinitesimal calculus or set theory, which is also true. But these theories only give us a grasp of the question of infinity if we put ourselves into them, if we practice them; then it is indeed mathematical activity itself that represents infinity, which presents it to thought. We show that tools such as algebraic universes, sketches, and diagrams, allow, on the one hand, to dispense with the “calculations” together with cardinals and ordinals, and on the other hand, to describe at leisure the structures and their manipulations thereof, the indefinite work of pasting or glueing data, work that constitutes an object the actual infinity of which the theory of structures is a calculation. Through these technical details it is therefore proposed that Badiou envisages ontology by returning to the phenomenology of his “logic of worlds”, by shifting the question of Being towards the worlds where truths are produced, and hence where the subsequent question of infinity arises.

The boundless riches of God

This paper explores the concept of infinity in mathematics and its relation to theological considerations. It begins by seeking to answer the question of whether mathematical enquiry into the character of infinity may cast some light on the infinite character of God. Drawing on the work of Euclid, Cantor, and Gödel in particular, it considers concepts of potential and actual infinity and how mathematical discoveries have implications for (i) the relation of the finite and infinite (which has theological implications for the incarnation); (ii) the relation of theory and reality; (iii) the future scope of discovery and invention; and (iv) further reflection on the givenness of revelation.

Dialectical Infinity and the Third Mathematical Crisis —On the Fundamental Error of Actual Infinity

This paper discusses the problem of finity and infinity based on the philosophical perspectives of opposing idealism and receiving dialectical materialism. Based on Hegel’s dialectical infinity view, this paper makes a comprehensive criticism of the thought of actual infinity. After Hegel’s dialectical infinite thought scientifically explained the limit concept in calculus, the Second Mathematical Crisis caused by the contradiction of infinitesimal quantity was solved thoroughly. However, the mathematics world has not learned the experience and lessons in history, has always adhered to the idealist thought and methodology of actual infinity, this thought finally brought the third crisis to mathematics. At the end of this paper, based on the infinite view of dialectical materialism, the author analyzes the Principle of Comprehension and the Maximum Ordinal Paradox, and points out that the essence of the Principle of Comprehension is a kind of actual infinity thought. Only by limiting the Principle of Comprehension to a potential infinity can we solve the Third Mathematical Crisis completely.

Pavel Florensky: At the Boundary of Immanence and Transcendence

This chapter discusses four themes in the religious philosophy of Pavel Florensky (1882–1937): Georg Cantor’s mathematics, truth, philosophy of language, and the visual arts. Apart from Church doctrines, the key ideas that emerge in his work are ‘antinomy’, ‘discontinuity’, ‘actual infinity’ and ‘realism’. Deeply rooted in the Christian-Platonic tradition, Florensky is critical of rationalism, empiricism, Kantianism, and positivism. He anticipates postmodern insights in the sense that his worldview allows for synchronic difference as well as time and diachronic change. But unlike postmodern thought, which tends to interpret synchronic difference and the flux of time in terms of relativistic perspectivism and historicism, Florensky provides difference and change with a realist underpinning. And despite his emphasis on antinomicity and discontinuity in his conception of truth, he affirms the grandeur of reason and rejects irrationalism and fideism.