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 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...
Abstract
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.
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.
AbstractIn 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.
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.
‘Historical views of infinity’ focuses on historical attitudes to infinity in philosophy, religion, and mathematics, including Zeno’s famous paradoxes. Infinity is not a thing, but a concept, related to the default workings of the human mind. Zeno’s paradoxes appear to be about physical reality, but they mainly address how we think about space, time, and motion. A central (but possibly dated) contribution was Aristotle’s distinction between actual and potential infinity. Theologians, from Origen to Aquinas, sharpened the debate, and philosophers such as Immanuel Kant took up the challenge. Mathematicians made radical advances, often against resistance from philosophers.
Dialectical Infinity and the Third Mathematical Crisis —On the Fundamental Error of Actual Infinity
