Antecedents

Abstract

The main goal of this introductory chapter is to describe some of the mathematical and philosophical ideas that will allow the reader to understand the significance of Russell’s contributions. Contrary to the general concensus, mathematicians were well aware, in the last few decades of the nineteenth century, of important issues on the principles of mathematics. They were especially aware of problems concerning the foundations of the concept of number. In this chapter, my analysis will not be historical, but will provide a logical foundation for material to be presented later. Considering that the set-theoretic ‘paradoxes’ were discussed as such in Russell’s The Principles of Mathematics (published in May 1903) for the first time and that some of Russell’s ideas on the concept of number rely on Cantor’s concept of transfinite numbers, then it is natural to focus the discussion on Cantor’s contributions to the theory of transfinite numbers. In particular, the discussion will be centered upon those ideas presented by Cantor in his articles of 1895 and 1897, which played a fundamental role in Russell’s final presentation of his concept of number. I will try to very briefly narrate the reactions of some mathematicians and philosophers to Cantor’s ideas.

The time will come when these things which are now hidden from you will be brought into the light.

Georg Cantor