Mathematics has had its share of historical shocks, beginning with the discovery by Hippasus the Pythagorean that the integers could not possibly be the elements of all things. Likewise with Kurt Gödel’s Incompleteness Theorems, which presented a serious (even fatal) obstacle to David Hilbert’s formalism, and Bertrand Russell’s own discovery of the paradox inherent in his intuitively simple set theory. More recently, Paul Benacerraf presented a problem for the foundations of arithmetic in “What Numbers Could Not Be” and “Mathematical Truth.” (...) Drawing out difficulties he found in mathematical realism, Benacerraf ended up proposing his own structuralist view of mathematics, according to which numbers could not be considered objects at all, but mere placeholders, wholly defined in terms of the structures within which they were found. While we do not intend to work through the intricacies of structuralism or other types of mathematical nonrealism, or even of any of the many other competing theories in contemporary philosophy of mathematics, our question concerns problems one finds in an earlier, and perhaps more naïve, view of mathematics which, though a realism of its own, begins with seemingly more problematic assumptions than these. Classically expressed, the question is: do numbers exist in the world, whether as substances or as attributes of things? These are among the things Benacerraf holds numbers could not be; still, we hope to dispel some of the confusion which might bring one prematurely to reject this account. Most precisely, we wish to address the problem whether there can be a science of arithmetic, as that term is understood by Aristotle and Thomas Aquinas. (shrink)

In this book, philosopher Jean W. Rioux extends accounts of the Aristotelian philosophy of mathematics to what Thomas Aquinas was able to import from Aristotle’s notions of pure and applied mathematics, accompanied by his own original contributions to them. Rioux sets these accounts side-by-side modern and contemporary ones, comparing their strengths and weaknesses.

Georg Cantor argued that pure mathematics would be better-designated “free mathematics” since mathematical inquiry need not justify its discoveries through some extra-mental standard. Even so, he spent much of his later life addressing ancient and scholastic objections to his own transfinite number theory. Some philosophers have argued that Cantor need not have bothered. Thomas Aquinas at least, and perhaps Aristotle, would have consistently embraced developments in number theory, including the transfinite numbers. The author of this paper asks whether the restriction (...) of arithmetic to the natural numbers that is apparently assumed by Aristotle and Aquinas is necessary in the light of their stated principles. The author concludes that, while some texts from Aristotle and Thomas suggest that such discoveries as zero, rational, and real numbers, and even Cantor’s own transfinite numbers, are legitimate objects of scientific knowledge, a careful analysis shows that they are incompatible with the ultimate arithmetical principle, the unit. (shrink)