Alfred North Whitehead and Bertrand Russell’s attempt to deduce mathematics from logic in Principia Mathematica gave the logistic movement in mathematics its definitive expression.

At the end of the nineteenth century, several new approaches to the foundations of mathematics were developing in response to a growing number of issues that challenged the stability of the previously accepted foundations of mathematics. By the first decade of the twentieth century, these new approaches divided many mathematicians into opposing schools of thought and provided grounds for disagreement as to the proper foundations of mathematics. These new approaches were the bases of the three principal contemporary philosophies of mathematics: the logistic school, of which Bertrand Russell and Alfred North Whitehead are the chief expositors; the intuitionist school, led by the Dutch mathematician L. E. J. Brouwer; and the formalist school, developed principally by the German mathematician and logician David Hilbert. Although there are contemporary philosophies of mathematics other than these three, none has been as widely followed or has developed as large a body of associated literature as these. Principia Mathematica (Whitehead and Russell)
Mathematics;logistic movement
Logistic movement
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Principia Mathematica (Whitehead and Russell)
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Russell, Bertrand
Whitehead, Alfred North
Frege, Gottlob
Peano, Giuseppe

The basic thesis of the logistic school is that mathematics is derivable from logic. The logistic school maintains that mathematics is a branch of logic rather than logic being merely a tool of mathematics. With the development of the logistic school, logic became the forefather of mathematics in that all mathematical concepts were to be developed as theorems of logic, and the distinction between mathematics and logic became merely one of practical convenience. The logicists argue that because the laws of logic are accepted as a body of truths (at least in the early 1900’s), mathematics must be accepted also as a body of truths. Furthermore, they contend that because truth is consistent, so is logic and mathematics.

The logicist thesis that mathematics is derivable from logic can be traced back to the German philosopher Gottfried Wilhelm Leibniz Leibniz, Gottfried Wilhelm (1646-1716). Leibniz made a distinction between necessary truths and contingent truths; a truth is called “necessary” when its opposite implies a contradiction (for example, all right angles are equal) and “contingent” when it is not necessary (for example, there are bodies in nature that possess angles of exactly 90 degrees). Hence Leibniz considered all mathematical truths to be necessary and, as such, derivable from logic whose principles are also necessary and hold true in all possible worlds. Nevertheless, Leibniz did not go on to derive mathematics from logic, nor did anyone else with similar beliefs for almost two hundred years.

It was not until the late nineteenth century that the German mathematician Gottlob Frege undertook to develop the logistic thesis. Frege thought that laws of mathematics say no more than what is implicit in the principles of logic, which are a priori truths. In his Die Grundlagen der Arithmetik (1884; The Foundations of Arithmetic, 1950) Foundations of Arithmetic, The (Frege) and in his two-volume Grundgesetze der Arithmetik (1893, 1903; the basic laws of arithmetic), Grundgesetze der Arithmetik (Frege) Frege proceeded to derive the concepts of arithmetic and the definitions and laws of number from logical premises. From the laws of number, it is possible to deduce algebra, analysis, and even geometry because analytic geometry expresses the concepts and properties of geometry in algebraic terms. Unfortunately, Frege’s symbolism was complex and strange to mathematicians, and he therefore had little influence on his contemporaries.

Another important forerunner of the logicist school was the Italian mathematician and logician Giuseppe Peano, who, between 1889 and 1908, had undertaken to state the theorems of mathematics by means of logical symbolization. Russell was greatly influenced by Peano’s work and met Peano at the Second International Congress of Philosophy in Paris in 1900. Russell carefully studied Peano’s work and adopted his notation as an instrument of analysis.

Although Russell conceived of the same program as Frege, he did so without any knowledge of Frege’s program. It was only while he was developing his own program that he ran across Frege’s work. In the early 1900’s, Russell believed with Frege that because logic is a body of truths, if the fundamental laws of mathematics could be derived from logic, then these laws would also be truths and consistent. In The Principles of Mathematics (1903), Principles of Mathematics, The (Russell) Russell writes, “The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.” The successful completion of the logistic school’s program would leave the foundations of mathematics beyond doubt.

The logistic school received its definitive expression in the monumental three-volume Principia Mathematica (1910-1913), written by Russell in collaboration with his former teacher, the English philosopher Alfred North Whitehead. Whitehead and Russell spent the period from 1900 to 1911 developing what ultimately became Principia Mathematica—the definitive version of the logistic school’s position. This complex work purports to be a detailed reduction of the whole of mathematics to logic.

Even though the contents of Principia Mathematica elude summary—to do so would be as difficult as summarizing a dictionary—a brief sketch of the contents is in order. The work begins with the development of logic itself. Axioms of logic (for example, if q is true, then p or q is true) are carefully stated from which theorems are deduced to be used in subsequent reasoning. The development starts with primitive (or undefined) ideas and propositions. These primitive ideas and propositions are taken as descriptions and hypotheses concerning the real world, and although they are explained, the explanations are not part of the logical development. The aim of Principia Mathematica is to develop mathematical concepts and theorems from these primitive ideas and propositions, starting with a calculus of propositions, proceeding up through the theory of sets and relations to the establishment of the natural number system, and then to all mathematics derivable from the natural system. In this development, the natural numbers emerge with the unique meanings ordinarily assigned to them.

After having built up the logic of propositions, Whitehead and Russell proceed to propositional functions. Propositional functions These, in effect, represent sets (or classes), for instead of naming the members of a set, a propositional function describes them by a property. For example, the propositional function “x is red” denotes the set of all red objects. This method of defining a set enables one to define infinite sets as readily as one can define finite sets of objects. Whitehead and Russell wanted to avoid the paradoxes that arise when a collection of objects is defined that contains itself as a member. Although most sets are not members of themselves, some are. For example, whereas the set of all frogs is not a frog, the set of all comprehensible things is itself a comprehensible thing. Set theory To avoid such paradoxes, Whitehead and Russell require that no set is a member of itself and introduce the theory of types to carry out this restriction. The basic idea of their theory of types is that a set is on a higher level than its members; the set of which this set is a member is on a still higher level; and so on. Hence individuals, such as a particular frog, are type 0. An assertion about a property of individuals is of type 1, and so on.

Whitehead and Russell then go on to address the theory of relations, stating that relations are expressed by means of propositional functions of two or more variables (“x loves y” expresses a relation). Next is an explicit theory of sets defined in terms of propositional functions. On this basis, Whitehead and Russell introduce the notion of natural numbers. Given the natural numbers, it is possible to build up the real and complex number systems, functions, and all of analysis. To accomplish their objective, however, Whitehead and Russell had to introduce two more axioms: the axiom that infinite sets exist and the axiom of choice. Axiom of choice These axioms were to become the focus of much criticism.

In 1959, Russell wrote that he “used to know of only six people who had read the later parts” of Principia Mathematica. Despite Russell’s reservations, however, this work has attracted much careful study and has received extensive critical attention. In fact, Whitehead and Russell’s logical investigations into the foundations of mathematics opened up a world of possibilities not only among mathematicians but also among logicians and philosophers, and the study of the philosophy of logic became a central concern for philosophy itself.

Nevertheless, the great achievement of Principia Mathematica was not enough to shield the logistic approach to mathematics from a barrage of criticism. One point of attack has been directed toward the axioms of reducibility, choice, and infinity Axiom of infinity used by Whitehead and Russell. Controversy and discussion regarding these axioms has focused on the purity of the logic used in the work. For example, the axiom of reducibility Axiom of reducibility has been said to be arbitrary and lacking evidence. Critics have gone so far as to question whether it is an axiom of logic.

Even Whitehead and Russell were uneasy about this axiom in the first edition of Principia Mathematica; in the second edition, they attempted to rephrase the axiom, but in doing so they only created new difficulties. Although they thought that the axiom was justified on the pragmatic basis of leading to the desired conclusion, it was not a justification with which they could rest content. They had to determine how essential this axiom was to the logistic program. So far, all efforts to reduce mathematics to logic without the axiom have failed.

The use of the axioms of reducibility, infinity, and choice has challenged the entire logistic program and has raised questions as to where the line between logic and mathematics is to be drawn. On one hand, if logic actually contains these three controversial axioms—as proponents of the logistic program maintain—then the logic of Principia Mathematica is pure. On the other hand, if the logic of Principia Mathematica is not pure, opponents of the logistic program deny that mathematics, or even any important branch of mathematics, has yet been reduced to logic. Others, while arguing for the impurity of the logic of Principia Mathematica, are willing to extend the meaning of the term “logic” so that it includes these axioms.

Other critics have charged that Principia Mathematica reduced only arithmetic, algebra, and analysis to logic; it did not reduce the nonarithmetical parts of mathematics (such as geometry, topology, and abstract algebra) to logic. Those who hold that all mathematics can be reduced to logic claim that it is possible to reduce geometry, topology, and abstract algebra to logic.

Although both Principia Mathematica in particular and the logistic program in general have had a long, complicated development and have been criticized on various grounds, many of which have not been mentioned here, whether or not the logistic thesis has been established seems to be a matter of opinion. Although some accept the program as satisfactory, even though they might be critical of its present state, others have found its shortcomings insurmountable, charging that it produces conclusions formed in advance from unwarranted assumptions. Principia Mathematica (Whitehead and Russell)
Mathematics;logistic movement
Logistic movement

• Bell, E. T. The Development of Mathematics. 2d ed. New York: McGraw-Hill, 1945. A broad account of the general development of mathematics, with particular emphasis on main concepts and methods. Highly accessible to the general reader. Chapter 23, “Uncertainties and Probabilities,” is particularly informative about the state of mathematics around the time Principia Mathematica was published.
• Jager, Ronald. The Development of Bertrand Russell’s Philosophy. 1972. Reprint. London: Routledge, 2004. A fine survey of the development of Russell’s philosophy. Two sections of the part devoted to Russell’s philosophy of mathematics are particularly relevant: “The Poetry and Essence of Mathematics” and “Logicism.” Accessible to the diligent general reader.
• Kneebone, G. T. Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. New York: D. Van Nostrand, 1963. Even though this book is based on a series of lectures given at the University of London to advanced undergraduates and graduate students, it has much material for the general reader interested in an introduction to mathematical logic and the philosophy of logic. Generous amounts of space are given to nontechnical discussions of Principia Mathematica.
• Russell, Bertrand. Introduction to Mathematical Philosophy. 1919. Reprint. New York: Longman, 2005. Presents results “hitherto only available to those who have mastered logical symbolism, in a form offering the minimum of difficulty to the beginner.” An excellent nontechnical introduction to Russell’s work on mathematics and logic, and a highly recommended source for this subject.
• _______. My Philosophical Development. 1959. Rev. ed. New York: Routledge, 1995. This is the place to read Russell’s thoughts on the mathematical and philosophical aspects of Principia Mathematica. The discussion is addressed to a general audience and is informative and enjoyable reading. Cites influences on Russell’s work and provides a good general account of Principia Mathematica.
• Schoenman, Ralph, ed. Bertrand Russell: Philosopher of the Century. London: George Allen & Unwin, 1967. An excellent selection of essays. Section 4, “Mathematician and Logician,” is especially useful. Hilary Putnam’s essay titled “The Thesis That Mathematics Is Logic” is particularly relevant.
• Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica. 3 vols. Cambridge, England: Cambridge University Press, 1910-1913. A monumental work, the culmination of ten years of development, and the founding work of the logistic school. Virtually incomprehensible to those without a knowledge of formal logic, but the introduction is somewhat accessible to the diligent general reader.

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