Plato's neglected problem and the property view of numbers : back to Euclid with some differences

It is seldom noted in the philosophy of mathematics that Plato divides his world of forms or ideas into two different realms. In the higher realm, each pure number exists only as a unique entity, but in the lower realm it exists as a multiplicity. Such a bifurcation of numbers is not usually made in modern philosophy of mathematics, but this book argues that it needs to be made. Otherwise, the philosophical analysis of numbers cannot explain how, for instance, an addition such as 3 + 3 = 6 can refer to two distinct numbers 3. 

One way to account for this bifurcation is by means of the usually neglected property view of numbers. This book defends a version of this view, according to which the natural numbers are properties of collections, and the real numbers are properties of proportion-relations. It is also argued that the property view of numbers harmonizes well with the findings of modern empirical research on numerical cognition. 

Central to the book is Euclid´s notion of ratio, which is put forward in his Elements – here it is called proportion-relation . It is claimed that we need to revive Euclid´s distinction between numbers and magnitudes, as a distinction between two number lines, one for the original natural numbers and one for the real numbers. Strictly speaking, the original natural numbers cannot be regarded as a subclass of the real numbers. The line for the real numbers contains, however, numbers that are counterparts of the original natural numbers.

Ingvar Johansson is emeritus professor of theoretical philosophy at Umeå University, Sweden. Most of his writings are centered around questions in the philosophy of science and ontology.