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)

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)