Conceptions of Infinity and Set in Lorenzen’s Operationist System

In the late 1940s and early 1950s, Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as a precursor of the better-known dialogical logic (Notable exceptions are the works of Schroeder-Heister 2008; Coquand and Neuwirth 2017; Kahle and Oitavem 2020.), and one might assume that the same philosophical motivations were present in both works. However, we want to show that this is not everywhere the case. In particular, we claim that Lorenzen’s well-known rejection of the actual infinite, as stated in Lorenzen (1957), was not a major motivation for operative logic and mathematics. Rather, we argue that a shift happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operationism is concerned with a critique of the Cantorian notion of set and with related questions about the notions of countability and uncountability; it is only later that his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity.

Historical and Cultural Heritage of Russian Cities As an Element of the Cultural Universe

. The article reveals the idea of the cultural universe as the universality and potential infinity of the existence of a person and society, defines its components and approaches to understanding. The historical and cultural heritage of cities is a part of the cultural universe; it performs important functions of preserving the historical memory of the territories. The author examines the features of the historical and cultural heritage of Russian cities, analyses the problems and prospects of its research, preservation and use. The spiritual and material culture of cities, formed in their history, contributes to the ideological, patriotic, and moral education. The article substantiates the importance of the historical and cultural heritage of Russian cities in the development of the cultural universe, in the process of socio-cultural transformation of society, in the formation of the worldview of the individual. It is concluded that it is necessary to study the historical and cultural heritage of cities as a specific phenomenon of the urbosphere.

The Immanence of Truths and the Absolutely Infinite in Spinoza, Cantor, and Badiou

The following article compares the notion of the absolute in the work of Georg Cantor and in Alain Badiou’s third volume of Being and Event: The Immanence of Truths and proposes an interpretation of mathematical concepts used in the book. By describing the absolute as a universe or a place in line with the mathematical theory of large cardinals, Badiou avoided some of the paradoxes related to Cantor’s notion of the “absolutely infinite” or the set of all that is thinkable in mathematics W: namely the idea that W would be a potential infinity. The article provides an elucidation of the putative criticism of the statement “mathematics is ontology” which Badiou presented at the conference Thinking the Infinite in Prague. It emphasizes the role that philosophical decision plays in the construction of Badiou’s system of mathematical ontology and portrays the relationship between philosophy and mathematics on the basis of an inductive not deductive reasoning.

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.

The Concept of Reality and the Possibility of the Novel

This chapter describes Hans Blumenberg's theorization of the novel as a genuinely modern genre in “The Concept of Reality and the Possibility of the Novel” (1964). Art claims as its subject matter the formal proof of reality and not the material content that presents itself with this proof. Without doubt, the nonpossible would represent the fulfillment of this claim — namely, the infinite context, which alone could be counted as the normal equivalent to the open-endedness of physical experience. This is the starting point from which modern literature — and the aesthetics appropriate to it — proceeded toward the novel as the most comprehensively “realistic” genre, representing a context which, though finite in itself, presumes and indicates infinity. The potential infinity of the novel represents its ideality, arising out of the concept of reality, as well as the aesthetic irritation inevitable in view of the fact that its task of representing an infinite context can be fulfilled only by aesthetically binding principles of form.

Why heaven doesn't make earth absolutely meaningless, just relatively

In this article, I seek charitably to develop an argument suggested by Thaddeus Metz. This is an argument against the view that it is consistent to hold that, while our lives may have some meaningfulness even if there is no heavenly afterlife awaiting us, if there is such an afterlife, they are even more meaningful, because heaven adds a potential infinity of meaningfulness. Having developed this argument on Metz's behalf, I criticize it. I conclude that – while throwing out a number of interesting ideas and possibilities along the way – no argument along Metz's lines can finally reach the conclusion aimed for.

Introducing the Main Characters

This first chapter introduces the central concepts and distinctions that Leibniz uses in articulating his view of infinity. In other words, the author introduces the main players in this book. These include: Leibniz’s rejection of infinite number; his distinction between infinite being and infinite number; degrees of infinity; the distinction between actual and potential infinity; indivisibility; his syncategorematic approach to infinite terms; his distinction between infinite number and infinite series; the law of the series; and the distinction between primitive force and derivative force. The chapter’s aim is to present at the outset some of the terminology and concepts used in the book in order to present Leibniz’s approach to infinity—that is, to clarify the major resources needed in order to present his complex views. At the same time, this serves as a sketch of (what the author takes to be) Leibniz’s approach to infinity.

PHILOSOPHICAL AND RELIGIOUS VIEWS OF FLORENSKY AND THE IDEA OF ACTUAL INFINITY

The article briefly observes the history of the idea of the actual infinity in European culture until the beginning of the 20th century. Special attention is paid to the role of Cantor set theory in reviving interest in the idea of actual infinity in Western Europe and Russia. The influence of the Cantor’s philosophy of religion on the Western European theology of the late 19th century - early 20th century is given. The influence of Cantor’s ideas on the formation of Florensky’s views is described. A detailed analysis of the application of the idea of actual infinity in the book “The Pillar and the Statement of Truth” is given. Florensky describes the understanding of the connection of Kant’s antinomical of reason and the idea of a potential infinity. The potential infinity is considered by Florensky as a source of imperfection and sinfulness. Special attention is paid to the understanding of truth as actual infinity. The introduction of the actual infinity allows Florensky to remove the one-sidedness of the law of identity and the law of sufficient basis in the Supreme unity...