Terms and Propositions in Russell's Principles of Mathematics

Terms and Propositions in Russell's Principles ofMathematics LEONARD LINSKY 1. WRITING IN 19OO, Russell declared: "That all sound philosophy should begin with an analysis of propositions, is a truth too evident, perhaps, to demand a proof."' Russell's Principles of Mathematics appeared in 19o3.2 This is the work which concerns Jules VuiUemin in his La Premikre Philosophie de Russell, and he says of it "The Principles inaugurate contemporary philosophy."~ In accordance with his declaration, Russell begins the Principles in Part 1, entitled "The Indefinables of Mathematics," with an analysis of propositions. Russell had already published three books, but the Principles is a new beginning for him. Russell writes, "The most important year in my intellectual life was the year I9OO, and the most important event in this year was my visit to the International Congress of Philosophy in Paris."4 The importance of the congress for Russell was that here he met Giuseppe Peano and became aware of his pioneering work in the renaissance of logic which was taking place during the last quarter of the nineteenth century. Russell was converted by Peano to a belief in the supreme importance of logic as an instrument for philosophical progress in the philosophy of mathematics in the first place and in epistemology and metaphysics as well, the areas of philosophy of most concern to him. It is Bertrand Russell, The Philosophyof Leibniz (first edition 19oo, London: George Allen and Unwin Ltd., 1964),8. ' Bertrand Russell, The Principles of Mathematics (first edition 19o3, second edition, New York: W. W. Norton &Company, Inc., 1938). s Jules Vuillemin, Lemonssur la premierephilosophiede Russell (Paris, Colin, 1968), 333. My quotations from Jules Vuillemin, and their English translations are taken from W. V. O. Quine, "Remarks for a Memorial Symposium" in David Pears, editor, Bertrand Russell:A Collectionof CriticalEssays(Garden City:Anchor Books, 1972), 1-5. 4 Bertrand Russell, "My Philosophical Development," in P. A. Schilpp, editor, ThePhilosophy of Bertrand Russell, The Library of Living Philosophers (Evanston and Chicago: Northwestern UniversityPress, 1944), 12. [6~1] 622 JOURNAL OF THE HISTORY OF PHILOSOPHY 26:4 OCTOBER 1988 the central role of the new logic in Principles which separates this work from Russell's earlier books and indeed from all earlier British philosophy. It is because of the central role of the new logic in it that Vuillemin can characterize this book as the work which inaugurates contemporary philosophy. In his Autobiography, Russell vividly describes the excitement associated with his discovery of logic: "The time was one of intellectual intoxication .... For years I had been endeavoring to analyze the fundamental notions of mathematics, such as order and cardinal numbers. Suddenly, in the space of a few weeks, I discovered what appeared to be definitive answers to the problems which had baffled me for years. And in the course of discovering these answers, I was introducing a new mathematical technique, by which regions formerly abandoned to the vaguenesses of philosophers were conquered for the precision of exact formulae. Intellectually, the month of September 1900 was the highest point of my life."5 One month later, in October 19oo, in the full tide of this intellectual enthusiasm, intoxication, and excitement, Russell began to write The Principles of Mathematics. During the months of October, November, and December, he wrote the first draft of the whole huge work which runs to nearly five hundred printed pages. Parts 3, 4, 5, and 6 of the finished book remain as they were written that autumn; Parts l, 2, and 7 underwent revision and were not in final form until May 19o2.6 The book conveys the high optimism and excitement with which it was written, and that excitement has been communicated directly and even more indirectly down through our century, "The Age of Analysis." There are indications that that age is now coming to an end. Vuillemin remarks that the Principles "was never in fashion." Quine correctly characterizes it as "this prelogical logic of Russell 's," and a "morass of half-formulated problems."7 "The whole of AngloSaxon philosophy has undergone Russell's influence," writes Vuillemin, "for better or worse. This is, moreover, the only living philosophy today. It is sufficient praise of an...