Ways and words about infinity have frequently hidden a continuing paradox inspired by Zeno. The basic puzzle is the tortoise's – Mr T's – Extension Challenge, the challenge being how any extension, be it in time or space or both, moving or still, can yet be of an endless number of extensions. We identify a similarity with Mr T's Deduction Challenge, reported by Lewis Carroll, to the claim that a conclusion can be validly reached in finite steps. Rejecting common solutions (...) to both challenges, we use a Wittgensteinian approach for dissolution, noting also that other paradoxes, such as the Surprise Examination and Yablo's, trap us inconsistently into similar endlessnesses. In so doing, we encounter a hopping flea, generate a proportionality paradox and consider the knight's move in chess. (shrink)

In this paper we attempt to clear out the ground concerning the Aristotelian notion of density. Aristotle himself appears to confuse mathematical density with that of mathematical continuity. In order to enlighten the situation we discuss the Aristotelian notions of infinity and continuity. At the beginning, we deal with Aristotle’s views on the infinite with respect to addition as well as to division. In the sequel, we focus our attention to points and discuss their status with respect to the actuality–potentiality (...) distinction. Then we focus our attention to the nature of continuity, which Aristotle tends to consider as leading to infinite divisibility and potential density of the extended. (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)

On Zeno Beach there are infinitely many grains of sand, each half the size of the last. Supposing Aristotle denied the possibility of Zeno Beach, did he have a good argument for the denial? Three arguments, each of ancient origin, are examined: the beach would be infinitely large; the beach would be impossible to walk across; the beach would contain a part equal to the whole, whereas parts must be lesser. It is attempted to show that none of these arguments (...) was Aristotle’s. Indeed, perhaps Aristotle’s finitism applied to magnitude only, not plurality. (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)

In Physics VI.9 Aristotle addresses Zeno's four paradoxes of motion and amongst them the arrow paradox. In his brief remarks on the paradox, Aristotle suggests what he takes to be a solution to the paradox.In two famous papers, both called 'A note on Zeno's arrow', Gregory Vlastos and Jonathan Lear each suggest an interpretation of Aristotle's proposed solution to the arrow paradox. In this paper, I argue that these two interpretations are unsatisfactory, and suggest an alternative interpretation. In particular, I (...) claim that what seems on the face of it to be Aristotle's solution to the paradox raises two puzzles to which my interpretation, as opposed to Lear's and Vlastos's, provides an adequate response. (shrink)

I reconstruct the original versions of Zeno's Racetrack and Achilles paradoxes, along with Aristotle's responses thereto. Along the way I consider some of the consequences for modern analyses of the paradoxes. It turns out that the Racetrack and the Achilles were oral two-party question-and-answer dialectical paradoxes. One consequence is that the arguments needed to be comprehensible to the average person, and did not employ theses or concepts familiar only to philosophical specialists. I rely on this fact in reconstructing the original (...) dialectical versions of the paradoxes. I show that both paradoxes rely for their success on forcing the dialectical answerer to reflect on his own potentially unending experience of imagination, and show that this renders the most popular contemporary critique of each paradox unworkable in an ancient dialectical context. In responding to the Racetrack, Aristotle seeks to replace the answerer's first-person experience with something more objective, a visible diagram. He then argues that the Racetrack involves an equivocation, an equivocation resulting from the fact that one and the same visible diagram can be interpreted in two ways. He frames his charge of equivocation in standard dialectical fashion, but his use of a diagram is his own innovation. While first employing his response to the Racetrack to construct a response to the somewhat different Achilles paradox, Aristotle later proceeds to offer a revised critique of the Racetrack itself. His revised critique is heavily influenced by his reaction to a variant pre-Aristotelian version of the paradox, a version involving counting. This leads him to reflect on the possibility of mentally experiencing an infinite collection. He also reflects on his earlier diagrammatic methodology, and on the conditions that individuate points, especially halfway-points. He concludes that points which are individuated in diagrams need not represent points that are individuated in reality. His revised critique of the Racetrack hinges on the distinction between actual points and potential points, and I show that this critique maintains the dialectical form of its predecessor, while constituting a reflection on the potentially misleading nature of diagrams. (shrink)

This work analyzes two perspectives, Atomism and Infinite Divisibility, in the light of modern mathematical knowledge and recent developments in computer graphics. A developmental perspective is taken which relates ideas leading to atomism and infinite divisibility. A detailed analysis of and a new resolution for Zeno's paradoxes are presented. Aristotle's arguments are analyzed. The arguments of some other philosophers are also presented and discussed. All arguments purporting to prove one position over the other are shown to be faulty, mostly by (...) question begging. Included is a sketch of the consistency of infinite divisibility and a development of the atomic perspective modeled on computer graphics screen displays. The Pythagorean theorem is shown to depend upon the assumption of infinite divisibility. The work concludes that Atomism and infinite divisibility are independantly consistent, though mutually incompatible, not unlike the wave/particle distinction in physics. (shrink)