Aristotle’s Actual Infinities

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.

The Philosophy of the Cosmonomic Idea and the Philosophical Foundations of Mathematics

Since the discovery of the paradoxes of Zeno, the problem of infinity was dominated by the meaning of endlessness—a view also adhered to by Herman Dooyeweerd. Since Aristotle, philosophers and mathematicians distinguished between the potential infinite and the actual infinite. The main aim of this article is to highlight the strengths and limitations of Dooyeweerd’s philosophy for an understanding of the foundations of mathematics, including Dooyeweerd’s quasi-substantial view of the natural numbers and his view of the other types of numbers as functions of natural numbers. Dooyeweerd’s rejection of the actual infinite is turned upside down by the exploring of an alternative perspective on the interrelations between number and space in support of the idea of infinite totalities, or infinite wholes. No other trend has succeeded in justifying the mathematical use of the actual infinite on the basis of an analysis of the intermodal coherence between number and space.

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.

Endless and Infinite

It is often said that time must have a beginning because otherwise the series of past events would have the paradoxical features of an actual infinite. In the present paper, we show that, even given a dynamic theory of time, the cardinality of an endless series of events, each of which will occur, is the same as that of a beginningless series of events, each of which has occurred. Both are denumerably infinite. So if (as we believe) an endless series of events is possible, then the possibility of a beginningless series of past events should not be rejected merely on the ground that it would be an actual infinite. What would be required to rebut our argument is a symmetry breaker – something that motivates treating the past relevantly differently to the future. We consider several attempts to provide a symmetry breaker and show that none of them is successful.

On beginningless past, endless future, God, and singing angels: An assessment of the Morriston-Craig dialogue

SummaryWhether the past or future can be infinite is an interesting question for theologians working on the relationship between God and Time as well as Eschatology. In a recent exchange, Wes Morriston concluded that if William Lane Craig’s familiar line of argument against the possibility of a beginningless series of events worked, it would work just as well against the possibility of an endless series of predetermined events. He argued that neither Craig’s claim that an endless series of events is potential infinite nor the claim that future events don’t exist is successful in blocking this conclusion. I argue that a proponent of the Kalam Argument does not have to follow Craig’s denial of an actual infinite number of propositions, and I show how Morriston’s conclusion can be blocked. In particular, I argue that an asymmetric treatment of past and future is justified on a dynamic theory of time, while the distinction between abstract and concrete infinities is helpful for responding to Morriston’s counter-argument based on the number of angelic praises yet-to-be-said.

Bradwardine and Buckingham on the extramundane void

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«.