Introduction to Mathematical Philosophy (Barnes & Noble Library of Essential Reading)

Bertrand Russell's Introduction to Mathematical Philosophy (1919) is one of the most challenging books ever written about the philosophy of mathematics. In it the great British mathematician and philosopher gives a thoughtful exposition of mathematical concepts, their relation to logical and set theoretical notions, and the general background necessary for any serious discussion on the foundational crisis of mathematics in the beginning of the twentieth century. A seminal work in the field for more than nine decades now, Introduction to Mathematical Philosophy is skilfully written with an accessible lucidity by a brilliant scholar. Requiring neither prior knowledge of mathematics nor aptitude for mathematical symbolism, the book serves as essential reading for anyone interested in the intersection of mathematics and logic and in the development of analytic philosophy in the twentieth century. Rather than an exhaustive treatment, Russell offers to his readers a penetrating discussion on certain issues of mathematical logic that, apart from demonstrating the immense pedagogical talent of its author, is regarded by many as embodying the dawn of modern analytic philosophy.

A Nobel laureate in literature (1950) and a fierce pacifist, Bertrand Russell (1872-1970)-one of the greatest philosophers of the twentieth century-made noteworthy contributions not just to logic and philosophy but to a broad range of other subjects including education, history, political theory, and religious studies. The influential thinker who joined Cambridge Trinity College in 1890 led a life marked by controversy. After the First World War broke out, he took an active part in the No Conscription fellowship and was fined £100 as the author of a leaflet criticizing a sentence of two years on a conscientious objector. As a result of his activities, Trinity College deprived him of his lectureship in 1916, and when he was offered a post at Harvard University he was refused a passport. Russell also intended to give a course of lectures at the City University of New York (afterwards published in America as Political Ideals, 1918) but was prevented by the military authorities. In the same year he was sentenced to six months in prison for a pacifistic article; Introduction to Mathematical Philosophy was written during his imprisonment. Noted also for his many spirited antinuclear protests in the years to come, Russell remained a prominent public figure until his death at the age of ninety-seven.

Russell's work in mathematical logic combines rigor with sharp philosophical insights. In much the same way that he used logic in an attempt to clarify issues in the foundations of mathematics, Russell also used logic in an attempt to clarify issues in philosophy. As one of the founders of analytic philosophy, Russell made significant contributions to a wide variety of areas, including metaphysics, epistemology, ethics and political theory, as well as to the history of philosophy. Underlying these various projects was Russell's use of logical analysis. According to Andrew Irvine, a contemporary Russell expert from the University of British Columbia, it was Russell's belief that by using the new logic of his day, philosophers would be able to exhibit the underlying "logical form" of natural language statements. A statement's logical form, in turn, would help philosophers resolve problems of reference associated with the ambiguity and vagueness of natural language. Thus, just as we distinguish three separate sense of is (the is of predication, the is of identity, and the is of existence) and exhibit these three senses by using three separate logical notations (Px, x=y, and x respectively) we will also discover other ontologically significant distinctions by being aware of a sentence's correct logical form.

In Russell's view, the subject matter of philosophy is then distinguished from that of the sciences only by the generality and the a priori character of philosophical statements, not by the underlying methodology of the discipline. In philosophy, as in mathematics, Russell believed that it was by applying logical machinery and insights that advances would be made. Faithful to this view, Introduction to Mathematical Philosophy is a tour de force in the logical foundations of mathematics-a domain in which Russell made significant breakthroughs by discovering Russell's paradox, defending logicism (the view that mathematics is, in some significant sense, reducible to formal logic), developing the theory of types, and refining Frege's first-order predicate calculus.

Russell discovered the paradox that bears his name in 1901, while working on his Principles of Mathematics (published in 1903). The paradox is a version of the famous "liar paradox" and arises in connection with the set of all sets that are not members of themselves. Such a set, if it exists, will be a member of itself if and only if it is not a member of itself. The discovery of the paradox marks the beginning of the famous crisis in the foundations of mathematics since, using classical logic, it means that all sentences are entailed by a contradiction. Russell's paradox is also regarded by many as a fatal blow to Gottlob Frege's attempt to reduce mathematics to set theory and logic. Indeed, although Russell later suggested a possible way out of his paradox and advocated Frege's logicism, Frege himself stopped publishing his work on the foundations of mathematics and abandoned his life project after receiving Russell's letter on the contradictions that threaten it.

Russell's own response to his paradox came with the development of his theory of types in 1903. Rather than attacking another source of the paradox, namely, the law of excluded middle (according to which every statement is either true or false), it was clear to Russell that some restrictions needed to be placed upon the original comprehension (or abstraction) axiom of naive set theory, the axiom that formalizes the intuition that any coherent condition may be used to determine a set (or a class). In order to avoid the self-referential paradox that might otherwise emerge from such abstract notions as "the barber who shaves all but only those who do not shave themselves" or "the class of all classes that are not members of themselves," Russell's paradox claimed that all sentences should be arranged into a hierarchy, beginning with sentences about individuals at the lowest level, sentences about sets of individuals at the next lowest level, sentences about sets of sets of individuals at the next lowest level, and so on. Using a vicious circle principle similar to that adopted by the mathematician Henri Poincaré, and his own so-called "no class" theory of classes, Russell was able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function "x is a set," may not be applied to themselves since self-application would involve a vicious circle. In Russell's view, all objects for which a given condition (or predicate) holds must be at the same level or of the same "type."

Of equal significance during this period was Russell's defense of logicism, the theory that mathematics was in some important sense reducible to logic. Russell developed his logicism independently of Frege after attending Hilbert's famous address in Paris in 1900. Having become acquainted with the work of the Italian mathematician Peano, he first defended it in his 1901 article "Recent Work on the Principles of Mathematics," and then later in greater detail in his Principles of Mathematics and in Principia Mathematica (1910). Russell's logicism consisted of two main theses. The first was that all mathematical truths can be translated into logical truths or, in other words, that the vocabulary of mathematics constitutes a proper subset of that of logic. The second was that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of those of logic.

Like Gottlob Frege, Russell's basic idea for defending logicism was that numbers may be identified with classes of classes and that number-theoretic statements may be explained in terms of quantifiers and identity. Thus the number 1 would be identified with the class of all unit classes, the number 2 with the class of all two-membered classes, and so on. Statements such as "There are two books" would be recast as statements such as "There is a book, x, and there is a book, y, and x is not identical to y." It followed that number-theoretic operations could be explained in terms of set-theoretic operations such as intersection, union, and difference. In Principia Mathematica, Whitehead and Russell were able to provide many detailed derivations of major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory. A fourth volume was planned but never completed.

In many ways Introduction to Mathematical Philosophy is a philosophical, non-technical companion to Russell's monumental Principia, and from a literary perspective it contains many philosophical gems that could not be incorporated into his technical work with Whitehead. Russell, for example, states philosophy's role, and in particular the role of logicism, in the foundations of mathematics in the following way: ". . . it is the object of mathematical philosophy to put off saying it [what we mean by primitive mathematical notions such as 'number', 'zero', and 'successor' - AH] as long as possible. By the logical theory of arithmetic we are able to put it off for a very long time." Furthermore, "[b]ecause language is misleading, as well as because it is diffuse and inexact when applied to logic (for which it was never intended)," says Russell, "logical symbolism is absolutely necessary to any exact or thorough treatment of mathematical philosophy."

But the logical treatment of mathematical philosophy has proved far more consequential to Russell's career, indeed to the entire Western analytic tradition in philosophy, than it might have seemed when Introduction to Mathematical Philosophy was originally published. The book also contains an exposition of Russell's groundbreaking papers in analytic philosophy and includes Russell's most famous example of his "analytic" method that concerns denoting phrases such as descriptions and proper names. Earlier in his career Russell had adopted the view that every denoting phrase (for example, "Scott," "blue" "the number two," "the golden mountain") denoted, or referred to, an existing entity. In 1905, when his landmark article "On Denoting" was published, Russell had modified this extreme realism and had instead become convinced that denoting phrases need not possess a theoretical unity. While logically proper names (words such as "this" or "that" which refer to sensations of which an agent is immediately aware) do have referents associated with them, descriptive phrases (such as "the smallest number less than pi") should be viewed as a collection of quantifiers (such as "all" and "some") and propositional functions (such as "x is a number"). As such, they are not to be viewed as referring terms but, rather, as "incomplete symbols." In other words, they should be viewed as symbols that take on meaning within appropriate contexts, but that are meaningless in isolation.

This distinction between various logical forms allowed Russell to explain important logical puzzles and ambiguities in natural language, and, more important, had bearing on his metaphysics, epistemology, and even on his meta-philosophy. Ultimately, Russell saw the philosopher's task as discovering a logically ideal language that will exhibit the true nature of the world in such a way that the speaker will not be misled by the casual surface structure of natural language.

Russell's new methods cleared the way for a whole new generation of metaphysicians, epistemologists, and analytical philosophers. From a historical perspective, Introduction to Mathematical Philosophy puts Russell at the top of the distinguished and short list of mathematicians-philosophers-a list that includes giants such as Gottlob Frege, Edmond Husserl, and Kurt Gödel-whose work in the foundations of mathematics has, for better or worse, widened the gap between modern analytic philosophy and the traditional continental philosophy of the nineteenth century. As such, it serves as a first-rate exemplar of the character of Russell's work, forever remaining a true classic in the history of mathematics as well as of philosophy.

Amit Hagar is a philosopher of physics with a Ph.D. from the University of British Columbia, Vancouver. His area of specialization is the conceptual foundations of modern physics, especially in the domains of statistical and quantum mechanics.