There are two distinct but interrelated questions concerning Aristotle’s account of infinity that have been the subject of recurring debate. The first of these, what I call here the interpretative question, asks for a charitable and internally coherent interpretation of the limited pieces of text where Aristotle outlines his view of the ‘potential’ (and not ‘actual’) infinite. The second, what I call here the philosophical question, asks whether there is a way to make Aristotle’s notion of the potential infinite coherent (...) and rigorous with modern tools that can stand as a rival to the widely-accepted view of the infinite as characterized in a mathematical theory of sets. In this paper, I argue that the theoretical roles that Aristotle intends his account of the potential infinite to fulfill lead to a deep and irresoluble tension that can help explain the persistence of debates on both of these questions. I do so by turning to the places where Aristotle attempts to argue for or against the existence of particular infinite processes to show that he slides between different underlying notions of when changes are possible. Making these underlying notions clear can help us better understand the role of Aristotle’s account in the history of philosophy, the possible pitfalls for a contemporary theory of the potential infinite, and what each of these debates might learn from each other. (shrink)

Aristotle is said to have held that any kind of actual infinity is impossible. I argue that he was a finitist (or "potentialist") about _magnitude_, but not about _plurality_. He did not deny that there are, or can be, infinitely many things in actuality. If this is right, then it has implications for Aristotle's views about the metaphysics of parts and points.

This paper proposes a formalization of the class of sentences quantified by most, which is also interpreted as proportion of or majority of depending on the domain of discourse. We consider sentences of the form “Most A are B”, where A and B are plural nouns and the interpretations of A and B are infinite subsets of \. There are two widely used semantics for Most A are B: \ > C \) and \ > \dfrac{C}{2} \), where C denotes (...) the cardinality of a given finite set X. Although is more descriptive than, it also produces a considerable amount of insensitivity for certain sets. Since the quantifier most has a solid cardinal behaviour under the interpretation majority and has a slightly more statistical behaviour under the interpretation proportional of, we consider an alternative approach in deciding quantity-related statements regarding infinite sets. For this we introduce a new semantics using natural density for sentences in which interpretations of their nouns are infinite subsets of \, along with a list of the axiomatization of the concept of natural density. In other words, we take the standard definition of the semantics of most but define it as applying to finite approximations of infinite sets computed to the limit. (shrink)

In my introduction to Kontinuität und Mechanismus, I expressed surprise at the lack of work which was being done at the time on the young Leibniz in spite of the fact that conditions for investigating the period up to 1676 are almost ideal—certainly in Leibnizian terms. Most of the letters and papers from this period of immediate philosophical significance have now been published in the Akademie-Ausgabe so that there is here an incomparably better starting point for detailed studies than in (...) the case of his later work. (shrink)

It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has (...) good reason to resist the traditional arguments offered in favour of the existence of the infinite and that, while there is a lacuna in his own ‘logical’ arguments against actual infinities, his arguments against the existence of infinite magnitude and number are valid and more well grounded than commonly supposed. (shrink)

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other hand, Aristotle says (...) that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

This volume announces a new era in the philosophy of God. Many of its contributions work to create stronger links between the philosophy of God, on the one hand, and mathematics or metamathematics, on the other hand. It is about not only the possibilities of applying mathematics or metamathematics to questions about God, but also the reverse question: Does the philosophy of God have anything to offer mathematics or metamathematics? The remaining contributions tackle stereotypes in the philosophy of religion. The (...) volume includes 35 contributions. It is divided into nine parts: 1. Who Created the Concept of God; 2. Omniscience, Omnipotence, Timelessness and Spacelessness of God; 3. God and Perfect Goodness, Perfect Beauty, Perfect Freedom; 4. God, Fundamentality and Creation of All Else; 5. Simplicity and Ineffability of God; 6. God, Necessity and Abstract Objects; 7. God, Infinity, and Pascal's Wager; 8. God and (Meta-)Mathematics; and 9. God and Mind. (shrink)

Infinite regress arguments are a powerful tool in Aristotle, but this style of argument has received relatively little attention. Improving our understanding of infinite regress arguments has become pressing since recent scholars have pointed out that it is not clear whether Aristotle’s infinite regress arguments are, in general, effective or indeed what the logical structure of these arguments is. One obvious approach would be to hold that Aristotle takes infinite regress arguments to be per impossibile arguments, which derive an infinite (...) sequence. Due to his finitism, Aristotle then rejects such a sequence as impossible. This paper argues that this obvious approach does not work, even for its most amenable cases. The paper argues instead that infinite regress arguments involve domain-specific infinities, and so there is not a general finitism which underpins infinite regress arguments in Aristotle, but rather domain-specific reasons that there cannot be an infinite number of entities in each domain in which Aristotle invokes an infinite regress argument. (shrink)

Avicenna believed in mathematical finitism. He argued that magnitudes and sets of ordered numbers and numbered things cannot be actually infinite. In this paper, I discuss his arguments against the actuality of mathematical infinity. A careful analysis of the subtleties of his main argument, i. e., The Mapping Argument, shows that, by employing the notion of correspondence as a tool for comparing the sizes of mathematical infinities, he arrived at a very deep and insightful understanding of the notion of mathematical (...) infinity, one that is much more modern than we might expect. I argue, moreover, that Avicenna’s mathematical finitism is interwoven with his literalist ontology of mathematics, according to which mathematical objects are properties of existing physical objects. (shrink)

In this paper, I shall be arguing for what I hope is a modern version of a very traditional view, which is that God can explain two very basic phenomena: the first is the existence of the universe as we know it: the second is the particular way in which the universe is organised. I shall also, though briefly, try to counter the view that the totally unwelcome features of our universe make it impossible to reconcile the universe as it (...) is with anything like traditional theistic belief. This project, however, is quite a daunting one. So I would wish to make it clear right at the start that, while I would claim that my views are reasonable, and indeed more reasonable than belief in the denial of these views would be, I still do not hold that it is unreasonable for someone to reject each of the conclusions for which I shall argue. For plainly anyone, whether myself or any opponent, can be both reasonable and mistaken. (shrink)