Philosophy 290-2

Description: Contemporary set theory provides a simple criterion for measuring sizes of infinite collections: Two sets A and B have the same size if and only if A and B can be put into one-to-one correspondence. This criterion, along with a criterion for ordering unequal sizes of infinity, allowed Cantor to develop a theory of transfinite numbers. But the road that led to Cantor was a complex and fascinating one. Historically, several conflicting intuitions concerning the proper way to extend counting from the finite to the infinite were put forward. For instance, many mathematicians and philosophers during the middle ages (and later) defended intuitions (based on frequency or on part-whole relations) that were in conflict with the criterion of one-to-one correspondence. In addition, attempts to develop an arithmetic of infinite numbers were carried out using intuitions and principles that were quite distant from the system of Cantorian cardinalities. Fascinatingly, some of these early intuitions and ideas have recently found a coherent mathematical implementation. In this seminar we will explore the genealogy of the issues related to the problem of assigning sizes to infinite collections and the attempts to develop systems of infinite numbers. We will start from Greek times and then progressively reach the thirteenth century. We will read texts by, among others, Aristotle, Euclid, Proclus, Philoponus, Al-Kindi, Ibn Qurra, Avicenna, Robert Grosseteste, and William of Auvergne. In the second half of the seminar we will go over a book by Prof. Mancosu titled “The wilderness of infinity. Robert Grosseteste, William of Auvergne and mathematical infinity in the thirteenth century” (forthcoming for OUP). While the focus of the seminar is mostly historical and philosophical, we will also give some attention to the contemporary mathematical theories that implement the non-Cantorian intuitions mentioned above.