David Peat

34 From Certainty to Uncertaintylists all catalogs that do not refer to themselves.” He’s almost finishedhis work when the thought strikes him, “Do I list the catalog I’ve justcreated within its own pages or not?”“If I leave out that entry then my catalog is incomplete,” he reasons,“for it has one missing entry, the title of the Grand Catalog itself.”And so he begins to add the reference to the Grand Catalog. But as hedoes so, he realizes that he is being inconsistent because this catalog isonly supposed to contain entries for catalogs that don’t refer to themselves,and here he is adding a reference to the catalog within the catalogitself.The librarian is in a double bind. If he wants to be consistent, thenhis catalog is incomplete. If he completes it, then it is at the expense ofbeing inconsistent. What applies to catalogs, Russell argues, also appliesto the definition of the class of “number.” In one stroke Russellhad demolished Frege’s work and exposed something very fishy at thefoundations of mathematics.Paradoxes like this one made it even more important to establishmathematics on a firm basis in which every step is logical and everyargument is transparent. As it turned out, Russell himself was oneof those philosopher–mathematicians determined to undertake thisprogram.Principia MathematicaRussell’s interest in these questions began in that auspicious year of1900 at the First International Congress of Philosophy held in Paris.On August 3 Russell heard the philosopher and mathematicianGiuseppe Peano address the meeting. He was so impressed with Peano’sclarity of mind that it marked the turning point in his intellectual career.He believed that Peano’s abilities arose out of a mind that hadbeen disciplined by the study of mathematical logic. This clarity wasthe key that Russell had been seeking for many years; he returned hometo England and began to study Peano’s work.As he did so he recalled his school days when, while learning geometry,he had puzzled about its logical foundations. Now, with his