scholarly journals Actual and Potential Infinity

Noûs ◽  
2017 ◽  
Vol 53 (1) ◽  
pp. 160-191 ◽  
Author(s):  
Øystein Linnebo ◽  
Stewart Shapiro
Keyword(s):  
Potential Infinity

scholarly journals Conceptions of Infinity and Set in Lorenzen’s Operationist System

2021 ◽  
pp. 23-46
Author(s):  
Carolin Antos
Keyword(s):  
Constructive Mathematics ◽  
Dialogical Logic ◽  
Actual Infinite ◽  
Potential Infinity ◽  
And Mathematics

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.

Introducing the Main Characters

2019 ◽  
pp. 14-34
Author(s):  
Ohad Nachtomy
Keyword(s):  
Infinite Series ◽  
Infinite Number ◽  
Potential Infinity ◽  
The Law

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.

3. Historical views of infinity

2017 ◽  
pp. 32-53
Author(s):  
Ian Stewart
Keyword(s):  
Immanuel Kant ◽  
Physical Reality ◽  
Human Mind ◽  
Space Time ◽  
Time And Motion ◽  
Potential Infinity ◽  
Zeno's Paradoxes ◽  
And Mathematics

‘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.

Pretense and Presupposition

Aboutness ◽  
2014 ◽  
Author(s):  
Stephen Yablo
Keyword(s):  
Twentieth Century ◽  
Philosophy Of Language ◽  
Semantic Content ◽  
First Century ◽  
What Is Said ◽  
Potential Infinity ◽  
Good Guide ◽  
Twenty First Century

A great puzzle of twentieth-century philosophy of language was, how are finite beings able to understand a potential infinity of sentences? The answer is supposed to be that understanding is recursive: infinitely many sentences can be constructed out of finitely many words combined according to finitely many rules; we understand a sentence by understanding the words in it and knowing the relevant rules. A great puzzle of twenty-first-century philosophy of language is shaping up to be this: how do we reconcile the solution to the previous puzzle with what sentences actually strike us as saying? It's a puzzle because S's compositionally determined meaning is not always a very good guide to what S intuitively says, or to its contribution to what is said by sentences in which S is embedded. This chapter focuses on the more radical case where a sentence says something its meaning positively disallows, such as the case where a sentence's real content is not a possible semantic content.

The Concept of Reality and the Possibility of the Novel

2020 ◽  
pp. 499-524
Author(s):  
Hans Blumenberg
Keyword(s):  
Subject Matter ◽  
Formal Proof ◽  
Modern Literature ◽  
The Novel ◽  
Material Content ◽  
Starting Point ◽  
Potential Infinity ◽  
The Aesthetic

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.

4. The flipside of infinity

2017 ◽  
pp. 54-69
Author(s):  
Ian Stewart
Keyword(s):  
Modern Concept ◽  
Logical Framework ◽  
Standard Analysis ◽  
Potential Infinity ◽  
Non Standard Analysis

‘The flipside of infinity’ examines a logical counterpart of the infinite: infinitesimals. These are quantities that are infinitely small, instead of infinitely large. Historically, such quantities formed the basis of calculus, one of the most useful branches of mathematics ever invented. However, they caused considerable head-scratching, starting an argument that took about two centuries to resolve. This was achieved using a version of Aristotle’s potential infinity—namely, potential infinitesimality. Exhaustion is also explained, along with the modern concept of a limit, which abolished infinitesimals. Then it considers how infinitesimals were reinstated and outlines non-standard analysis, which provides a logical framework for infinitesimals.

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

2020 ◽  
pp. 1-20
Author(s):  
T. J. MAWSON
Keyword(s):  
Potential Infinity ◽  
The Way

Abstract 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.

Paradox and Potential Infinity

2011 ◽  
Vol 42 (1) ◽  
pp. 195-219 ◽  
Author(s):  
Charles McCarty
Keyword(s):  
Potential Infinity

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

2020 ◽  
Vol 3 (2) ◽  
pp. p73
Author(s):  
Zhang Hong
Keyword(s):  
Dialectical Materialism ◽  
Limit Concept ◽  
Actual Infinity ◽  
The Third ◽  
Potential Infinity ◽  
Fundamental Error

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.

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