VI.71 Bertrand Arthur William Russell
b. Trelleck, Wales, 1872; d. Plas Penrhyn, Wales, 1970
Mathematical logic and set theory; philosophy of mathematics
Russell’s training at Cambridge University in the early 1890s inspired the part of his long and varied life that relates to mathematics. He divided his Tripos into Part 1 (Mathematics) and Part 2 (Philosophy), and then united these two trainings to seek a general philosophy of mathematics, especially its epistemological foundations, with geometry as the first test case (1897). But over the next few years he changed his philosophical stance, especially when he recognized the significance of CANTOR’S [VI.54] set theory from 1896 onward, and also discovered in 1900 a group of mathematicians around PEANO [VI.62] in Turin. Wishing to raise the level of axiomatization and rigor in mathematics, the followers of Peano formalized theories as much as possible, including the “mathematical logic” of propositions and predicates with set theory, but they kept mathematical and logical notions separate. After learning their system and adding to it a logic of relations, Russell decided in 1901 that their distinction of notions was not necessary: all notions lay in that logic. This is the philosophical position that has become known as “logicism,” and Russell wrote a largely nonsymbolic account of it in The Principles of Mathematics (1903). In an appendix to this book he publicized the work of FREGE [VI.56], who had anticipated logicism (but advocated it only for arithmetic and some analysis); Russell read him in detail after forming his own position, which continued to be influenced more by Peano.
Now the job was to expound logicism in Peanesque detail—a daunting task, made even harder by Russell’s discovery in 1901 that set theory was susceptible to paradoxes, which would have to be avoided or even solved. He was joined in the effort by his former Cambridge tutor, A. N. Whitehead; eventually three volumes of Principia Mathematica appeared between 1910 and 1913. After the basic logic and set theory, the arithmetic of real numbers and also the arithmetic of transfinite numbers were worked out in detail; a fourth volume on geometry was due to be written by Whitehead, but he abandoned it around 1920.
The paradoxes were solved by a “theory of types,” which formed a hierarchy of individuals, sets of individuals, sets of sets of individuals, and so on. A set or individual could only be a member of a set immediately above it in the hierarchy; thus, a set could not belong to itself. Comparable restrictions were laid on relations and predicates. While this avoided the paradoxes, it also ruled out a great deal of good mathematics, since different kinds of numbers lay in different types and so could not be brought together for arithmetic operations: for example, 
was not even definable. The authors proposed the “axiom of reducibility” to allow such definitions to be made; but this was, frankly, just a fudge.
Among the various features of Russell’s theory was a form of THE AXIOM OF CHOICE [III.1], called the “multiplicative axiom,” that he had found in 1904, just before Ernst Zermelo. It had a curious role within logicism, partly because its logicist status was suspect.
While there was discussion of Principia Mathematica, concerning both its logic and its logicism, it tended to be too mathematical for the philosophers and too philosophical for the mathematicians. However, the program influenced some kinds of philosophy, including Russell’s own; and as an example of high-level axiomatization it served as a model for foundational studies, including GÖDEL’S INCOMPLETENESS THEOREMS [V.15] of 1931, which showed that logicism as Russell had conceived it could not be achieved.
Further Reading
Grattan-Guinness, I. 2000. The Search for Mathematical Roots. Princeton, NJ: Princeton University Press.
Russell, B. 1983-. Collected Papers, thirty volumes. London: Routledge.