Arithmetic and the Reality of Numbers in the Late Latin Middle Ages

The late Middle Ages (ca. 1270-1400) in the Latin West witnessed an extraordinary rise of interest in the metaphysical status of numbers. This paper is a case study of one of the most popular arguments in favour of realism about numbers: the view according to which numbers are extramental entities distinct from the things that they number. Part one is a reconstruction of the realist argument, which is based on the commonly accepted division of sciences into real sciences and rational sciences. It is an equally commonly accepted claim that arithmetic is one of the real sciences. On the realist interpretation, for a science to be real, its object must be real. Thus, since the object of arithmetic is number, numbers must have extramental reality. Part two is an analysis of several most interesting anti-realist rebuttals of the above argument.