Leibniz in Cantor’s Paradise: A Dialogue on the Actual Infinite

2019 ◽  
pp. 71-109
Author(s):  
Richard T. W. Arthur
Keyword(s):  
Actual Infinite

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.

Bradwardine and Buckingham on the extramundane void

2014 ◽  
Vol 17 ◽  
pp. 123-149 ◽  
Author(s):  
Edit Anna Lukács
Keyword(s):  
Fourteenth Century ◽  
Critical Edition ◽  
Philosophy Of Nature ◽  
Divine Power ◽  
Actual Infinite

In the corollaries to Book I, Chapter 5 of De causa Dei, Thomas Bradwardine assumes the existence of an actual, infinite, God-filled extramundane void. Thomas Buckingham, Bradwardine’s former student, develops in the unedited Question 23 of his Quaestiones theologicae a rejection of the void’s existence precisely in opposition to the theory of his master. His argumentation is not only remarkable in its own; it also allows us to reassess essential concepts from Bradwardine’s De causa Dei, such as divine power, causality and ubiquity. This paper first presents the Aristotelian notion of the void in rendering it in the context of the philosophy of nature at fourteenth-century Oxford; it is then dedicated to the analysis of the chapter in question from De causa Dei along with Buckingham’s answer. It is accompanied by a critical edition of Question 23 from Buckingham’s Quaestiones theologicae, »Utrum sit necesse ponere Deum esse extra mundum in situ seu vacuo imaginario infinito«.

A MODIFIED PHILOSOPHICAL ARGUMENT FOR A BEGINNING OF THE UNIVERSE

Think ◽  
2013 ◽  
Vol 13 (36) ◽  
pp. 71-83 ◽  
Author(s):  
Andrew Loke
Keyword(s):  
Dynamic Theory ◽  
Philosophical Argument ◽  
Actual Infinite ◽  
The Universe

Craig's second philosophical argument for a beginning of the universe presupposes a dynamic theory of time, a limitation which makes the argument unacceptable for those who do not hold this theory. I argue that the argument can be modified thus: If time is beginning-less, then it would be the case that a person existing and counting as long as time exists would count an actual infinite by counting one element after another successively, but the consequent is metaphysically impossible, hence the antecedent is metaphysically impossible. I defend the premises and show that this argument does not presuppose the dynamic theory.

scholarly journals De la determinación del infinito a la inaccesibilidad en los cardinales transfinitos

1994 ◽  
Vol 26 (78) ◽  
pp. 27-71
Author(s):  
Carlos Álvarez J.
Keyword(s):  
Point Of View ◽  
Inaccessible Cardinal ◽  
General View ◽  
Very Old ◽  
Cardinal Numbers ◽  
Actual Infinite ◽  
Potential Infinity

In this paper I deal with two problems in mathematical philosophy: the (very old) question about the nature of infinity, and the possible answer to this question after Cantor’s theory of transfinite numbers. Cantor was the first to consider that his transfinite numbers theory allows to speak, within mathematics, of an actual infinite and allows to leave behind the Aristotelian statement that infinity exists only as potential infinity. In the first part of this paper I discuss Cantorian theory of transfinite numbers and his particular point of view about this matter. But the development of the theory of transfinite numbers, specially the theory of transfinite cardinal numbers, has reached with the inaccessible cardinal numbers a new dilemma which makes us think that Aristotelian characterization of the infinity as potential is again a possible answer. The second part gives a general view of this development and of the theory of the inaccessible cardinal numbers in order to make clear my point of view concerning Aristotelian potential infinity.

Horizonal Hermeneutics and the Actual Infinite

1982 ◽  
Vol 8 (1) ◽  
pp. 36-97
Author(s):  
Robert S. Corrington ◽  
Keyword(s):  
Actual Infinite

scholarly journals Beginningless Past, Endless Future, and the Actual Infinite

2010 ◽  
Vol 27 (4) ◽  
pp. 439-450 ◽  
Author(s):  
Wes Morriston ◽  
Keyword(s):  
Actual Infinite

scholarly journals Aristotle’s Actual Infinities

2021 ◽  
pp. 133-186
Author(s):  
Jacob Rosen
Keyword(s):  
Actual Infinite ◽  
Central Case ◽  
The Continuum

In histories of thought about the infinite, Aristotle is constantly said to have rejected any form of actual infinite, and to have allowed quantities to be at most potentially infinite. Aristotle does reject actual infinites in spatial magnitude: nothing is infinitely big or infinitely small. But in the central case of plurality, the evidence for potentialism is much weaker. This paper argues that Aristotle had no principled objection to the idea that there are actually infinitely many things. One part of the argument concerns the distinction in Aristotle between plurality (πλῆθος‎) and number (ἀριθμός‎). Another part concerns the meaning of phrases like ‘infinite by division’, arguing that such phrases do not refer to how many times something has been divided, but rather to how small something is. The argument of this paper, if successful, affects how we should think about the metaphysics of parts in Aristotle’s theory of the continuum.

Theology and the Actual Infinite: Burley and Cantor

2011 ◽  
Vol 9 (1) ◽  
pp. 101-108 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Actual Infinite
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