Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

Some protestant Mathematicians had a strong preoccupation with the Day of Judgement. Stifel, Faulhaber, Napier and Newton made calculations in order to determine the date of the end of the world. Craig gave mathematical rules for a decline in the reliability of Christian tradition; in order to prevent a reliability of nearly zero, the Day of Judgement must come before. Furthermore, some conflicts between theology and mathematics are discussed. The Council of Konstanz condemned Wyclif's theory of the continuum. It seems (...) that the relation between part and whole was a crucial point between the Aristotelians and the “moderns”. (shrink)

Until recently, historians of mathematics usually agreed in refusing to consider the numerous geometrical publications of Thomas Hobbes as a contribution to the development of mathematics in the seventeenth century. From time to time, one could find statements that although Hobbes did not find new theorems he undoubtedly had profound insights into the logical foundations of mathematics, but these occasional remarks did not encourage historians to go deeper into Hobbes's mathematical thought. In the end, the general conclusion was that Hobbes's (...) preoccupation with squaring the circle, doubling the cube (starting when the philosopher was more than forty years of age), and challenging Euclid's definitions were better ignored, at least in the history of science. In particular, his controversy with the Savilian professors Seth Ward and John Wallis was seen as a ‘deplorable affair’, liable only to damage the reputation of the protagonists. (shrink)