Shortly after finishing his book on geometry, he abandoned the metaphysical idealism that was to have provided the framework for this grand synthesis. A much greater influence on his thought at this time, however, was a group of German mathematicians that included Karl Weierstrass, Georg Cantor, and Richard Dedekind, whose work was aimed at providing mathematics with a set of logically rigorous foundations.
Russell’s abandonment of idealism is customarily attributed to the influence of his friend and fellow Apostle G. For Russell, their success in this endeavour was of enormous philosophical as well as mathematical significance; indeed, he described it as “the greatest triumph of which our age has to boast.” After becoming acquainted with this body of work, Russell abandoned all vestiges of his earlier idealism and adopted the view, which he was to hold for the rest of his life, that analysis rather than synthesis was the surest method of philosophy and that therefore all the grand system building of previous philosophers was misconceived.
This is rather like defining the village barber as “the man who shaves all those who do not shave themselves” and then asking whether the barber shaves himself or not.
At first this paradox seemed trivial, but the more Russell reflected upon it, the deeper the problem seemed, and eventually he was persuaded that there was something fundamentally wrong with the notion of class as he had understood it in Frege saw the depth of the problem immediately.
When Russell wrote to him to tell him of the paradox, Frege replied, “arithmetic totters.” The foundation upon which Frege and Russell had hoped to build mathematics had, it seemed, collapsed.
Whereas Frege sank into a deep depression, Russell set about repairing the damage by attempting to construct a theory of logic immune to the paradox.
Like a malignant cancerous growth, however, the contradiction reappeared in different guises whenever Russell thought that he had eliminated it.
Eventually, Russell’s attempts to overcome the paradox resulted in a complete transformation of his scheme of logic, as he added one refinement after another to the basic theory.
The tragedy of Russell’s intellectual life is that the deeper he thought about logic, the more his exalted conception of its significance came under threat.
He himself described his philosophical development after Russell’s Paradox—at the very heart of the system of logic upon which he had hoped to build the whole of mathematics.