Russell graduated as a ‘wrangler,’ as first-class mathematics degrees are known at Cambridge, but if this makes his success sound effortless, that is misleading. Russell’s finals so exhausted him (as had happened with Einstein) that afterward he sold all his mathematical books and turned with relief to philosophy.45 He said later he saw philosophy as a sort of no-man’s-land between science and theology. In Cambridge he developed wide interests (one reason he found his finals tiring was because he left his revision so late, doing other things). Politics was one of those interests, the socialism of Karl Marx in particular. That interest, plus a visit to Germany, led to his first book, German Social Democracy. This was followed by a book on his ‘ancestor’ Leibniz, after which he returned to his degree subject and began to write The Principles of Mathematics.
Russell’s aim in Principles was to advance the view, relatively unfashionable for the time, that mathematics was based on logic and ‘derivable from a number of fundamental principles which were themselves logical.’46 He planned to set out his own philosophy of logic in the first volume and then in the second explain in detail the mathematical consequences. The first volume was well received, but Russell had hit a snag, or as it came to be called, a paradox of logic. In Principles he was particularly concerned with ‘classes.’ To use his own example, all teaspoons belong to the class of teaspoons. However, the class of teaspoons is not itself a teaspoon and therefore does not belong to the class. That much is straightforward. But then Russell took the argument one step further: take the class of all classes that do not belong to themselves – this might include the class of elephants, which is not an elephant, or the class of doors, which is not a door. Does the class of all classes that do not belong to themselves belong to itself? Whether you answer yes or no, you encounter a contradiction.47 Neither Russell nor Whitehead, his mentor, could see a way around this, and Russell let publication of Principles go ahead without tackling the paradox. ‘Then, and only then,’ writes one of his biographers, ‘did there take place an event which gives the story of mathematics one of its moments of high drama.’ In the 1890s Russell had read Begriffsschrift (‘Concept-Script’), by the German mathematician Gottlob Frege, but had failed to understand it. Late in 1900 he bought the first volume of the same author’s Grundgesetze der Arithmetik (Fundamental Laws of Arithmetic) and realised to his shame and horror that Frege had anticipated the paradox, and also failed to find a solution. Despite these problems, when Principles appeared in 1903 – all 500 pages of it – the book was the first comprehensive treatise on the logical foundation of mathematics to be written in English.48
The manuscript for Principles was finished on the last day of 1900. In the final weeks, as Russell began to think about the second volume, he became aware that Whitehead, his former examiner and now his close friend and colleague, was working on the second volume of his book Universal Algebra. In conversation, it soon became clear that they were both interested in the same problems, so they decided to collaborate. No one knows exactly when this began, because Russell’s memory later in his life was a good deal less than perfect, and Whitehead’s papers were destroyed by his widow, Evelyn. Her behaviour was not as unthinking or shocking as it may appear. There are strong grounds for believing that Russell had fallen in love with the wife of his collaborator, after his marriage to Alys Pearsall Smith collapsed in 1900.49
The collaboration between Russell and Whitehead was a monumental affair. As well as tackling the very foundations of mathematics, they were building on the work of Giuseppe Peano, professor of mathematics at Turin University, who had recently composed a new set of symbols designed to extend existing algebra and explore a greater range of logical relationships than had hitherto been specifiable. In 1900 Whitehead thought the project with Russell would take a year.50 In fact, it took ten. Whitehead, by general consent, was the cleverer mathematician; he thought up the structure of the book and designed most of the symbols. But it was Russell who spent between seven and ten hours a day, six days a week, working on it.51 Indeed, the mental wear and tear was on occasions dangerous. ‘At the time,’ Russell wrote later, ‘I often wondered whether I should ever come out at the other end of the tunnel in which I seemed to be…. I used to stand on the footbridge at Kennington, near Oxford, watching the trains go by, and determining that tomorrow I would place myself under one of them. But when the morrow came I always found myself hoping that perhaps “Principia Mathematica” would be finished some day.’52 Even on Christmas Day 1907, he worked seven and a half hours on the book. Throughout the decade, the work dominated both men’s lives, with the Russells and the Whiteheads visiting each other so the men could discuss progress, each staying as a paying guest in the other’s house. Along the way, in 1906, Russell finally solved the paradox with his theory of types. This was in fact a logicophilosophical rather than a purely logical solution. There are two ways of knowing the world, Russell said: acquaintance (spoons) and description (the class of spoons), a sort of secondhand knowledge. From this, it follows that a description about a description is of a higher order than the description it is about. On this analysis, the paradox simply disappears.53
Slowly the manuscript was compiled. By May 1908 it had grown to ‘about 6,000 or 8,000 pages.’54 In October, Russell wrote to a friend that he expected it to be ready for publication in another year. ‘It will be a very big book,’ he said, and ‘no one will read it.’55 On another occasion he wrote, ‘Every time I went for a walk I used to be afraid that the house would catch fire and the manuscript get burnt up.’56 By the summer of 1909 they were on the last lap, and in the autumn Whitehead began negotiations for publication. ‘Land in sight at last,’ he wrote, announcing that he was seeing the Syndics of the Cambridge University Press (the authors carried the manuscript to the printers on a four-wheeled cart). The optimism was premature. Not only was the book very long (the final manuscript was 4,500 pages, almost the same size as Newton’s book of the same title), but the alphabet of symbolic logic in which it was half written was unavailable in any existing printing font. Worse, when the Syndics considered the market for the book, they came to the conclusion that it would lose money – around £600. The press agreed to meet 50 percent of the loss, but said they could publish the book only if the Royal Society put up the other £300. In the event, the Royal Society agreed to only £200, and so Russell and Whitehead between them provided the balance. ‘We thus earned minus £50 each by ten years’ work,’ Russell commented. ‘This beats “Paradise Lost.” ‘57
Volume I of Principia Mathematica appeared in December 1910, volume 2 in 1912, volume 3 in 1913. General reviews were flattering, the Spectator concluding that the book marked ‘an epoch in the history of speculative thought’ in the attempt to make mathematics ‘more solid’ than the universe itself.58 However, only 320 copies had been sold by the end of 1911. The reaction of colleagues both at home and abroad was awe rather than enthusiasm. The theory of logic explored in volume I is still a live issue among philosophers, but the rest of the book, with its hundreds of pages of formal proofs (page 86 proves that 1 + 1=2), is rarely consulted. ‘I used to know of only six people who had read the later parts of the book,’ Russell wrote in the 1950s. ‘Three of these were Poles, subsequently (I believe) liquidated by Hitler. The other three were Texans, subsequently successfully assimilated.’59