Search Results

Russell discovered his paradox while writing The Principles of Mathematics. When this book appeared in 1903, it contained a discussion of his paradox but no more than a sketch of a solution. He then began to work on a second volume, in which the formal development of mathematics from logic was to be presented. When it eventually appeared almost ten years later, this second volume had metamorphosed into a separate three-volume work called Principia Mathematica, which was intended to provide the complete logical derivation promised in the Principles of the foundations, not just of arithmetic, but of the whole of pure mathematics. The programme Russell was engaged in was thus in a sense a direct continuation of Frege's: his task was to repair the error in Frege's system and hence establish the logicist thesis.

This chapter explores Gottlob Frege's contribution to logic. Frege has been called the greatest logician since Aristotle, but he failed to gain influence on the mathematical community of his time and the depth and pioneering character of his work was acknowledged only after the collapse of his logicist program due to the Zermelo–Russell antinomy in 1902. Frege, by proving his theorem χ without recourse to Wertverläufe, exhibited an inconsistency (or at least an incoherence) in the traditional notion of the extension of a concept. He prompted our awareness of a situation the future analyses of which will hopefully not only deepen our systematic control of the interplay of concepts and their extensions but also improve our understanding of the historical development of the notion of “extension of a concept” and its historiographical assessment.

The first sustained, rigorous development of a structuralist view of science appeared in the writings of Bertrand Russell, where the philosophical motivation precedes a precise formulation drawing on mathematical logic. He founded theoretical physics in a mathematics constructed along logicist lines, which is also what proved his undoing at the hands of a famous review by Newman that set the pattern for later objections to structuralist views.

Chapter 2 investigates the moment in 1917 when the philosophy of mathematics revealed to Gershom Scholem the symbolic potential of privation. Mathematics—in particular, the translation of logic into the symbols and operations of mathematics known as mathematical logic—produced novel results by discarding the conventional representational and meaning-making functions of language. Drawing on these mathematical insights, Scholem’s theorization of the poetic genre of lament and his translations of the biblical book of Lamentations employed erasure on the level of literary form to symbolize experiences, such as the Jewish diaspora, that exceed the limits of linguistic and historical representation. For Scholem, both poetry and history can mobilize deprivation as a means of retaining in language a symbol of experiences and ideas that remain unsayable in language and inexpressible in history—accounting for the erasure of exile and finding historical continuity in moments of silence, rupture, and catastrophe.

The first chapter gives a summary of the methodological and epistemological underpinnings of computable economics. There are, in addition, concise chapter summaries and a brief excursion into a discussion of the mathematical method in the formalization of economic theory.

This chapter considers the twentieth century response to the developments in mathematical practice in the nineteenth century. This response, we see, is essentially Kantian. Although Kant’s account of the practice of mathematics in terms of constructions in pure intuition must be jettisoned, philosophers assumed that Kant’s account of deductive reasoning as merely explicative and of logic as merely formal, without content and truth, remained viable. Three ideas central to mathematical logic are subjected to critical scrutiny: that generality is to be understood by appeal to quantifiers, that language is to be understood model theoretically, and that meaning can be adequately understood in terms of truth. The question of the role of writing in mathematics is also considered, and various answers are rehearsed. It is shown what an adequate written language for the current mathematical practice of reasoning deductively from concepts would have to be able to do.

How did critical theory, at least as it was first envisioned by Max Horkheimer and Theodor W. Adorno, come to be so opposed to mathematics? Chapter 1 examines the transformation of Horkheimer, Adorno, and Walter Benjamin’s prewar confrontation with Logical Positivism into a history of thinking that equated mathematics with the downfall of Enlightenment. According to the first generation of critical theorists, the reduction of philosophy to the operations and symbols of mathematics, as proposed by Logical Positivists such as Otto Neurath and Rudolph Carnap, rendered modern philosophy politically impotent and acquiesced to the powers of industry and authoritarian government. This initial phase of critical theory defined itself against the Logical Positivists’ equation of thought and mathematics, subsuming mathematics in their interpretation of reason’s return to myth and barbarism. Horkheimer and Adorno’s postwar texts and the work of second-generation critical theorists perpetuated this image of mathematics, canonizing it as an archetype of instrumental reason, reification, and social domination.

This chapter re-examines Dodgson’s achievements during his lifetime, and how some of them resurfaced in the 20th century. The chapter is divided into subject areas: geometry (including his alternative version of Euclid’s parallel postulate), trigonometry (for which he devised a new set of symbols), algebra (his work on determinants), logic (where the work of Bartley and others has led to a significant re-evaluation of Dodgson’s involvement), voting theory (on which he was in contact with a number of senior politicians), and probability, and concludes with a section on ciphers and cryptology. There has been a growing interest by scholars in Dodgson’s serious mathematical work since the third quarter of the 20th century, particularly as more of it has been published.

This chapter focuses on Kant’s account of the possibility of mathematics in relation to the transcendental expositions of space and time in the Transcendental Aesthetic, that is, the role of sensibility therein rather than understanding (which is discussed in Chapter 15). It is argued that Kant was nothing like the Euclidean dogmatist he is commonly portrayed as being by showing that and how his account of sensibility is perfectly capable of accommodating mathematical spaces of any curvature and number of dimensions. It is also shown how this account enabled him to explain the possibility not only of geometry, but even the most abstract, purely symbolic varieties of mathematics, among which post-Fregean mathematical logic should probably be included.

This chapter discusses Dodgson’s work on syllogisms (a topic that can be traced back to Aristotle and Ancient Greece) and how to solve them systematically using a marked board and some counters. His method is explained in detail in this chapter. Dodgson introduced it in his Game of Logic, which he used to teach syllogisms to children, and which he then developed in his Symbolic Logic, Part I. The rest of the chapter is concerned with further work that Dodgson carried out, but which was not published at the time because of his premature death at the age of 65.

This chapter discusses the foundations of philosophical semantics, covering the work of Gottlob Frege and Bertrand Russell. Frege, along with Russell, did more than anyone else to create the subject. The development of symbolic logic, the analysis of quantification, the application of logical ideas and techniques to the semantics of natural language, the distinction between sense and reference, the linking of representational content to truth conditions, and the compositional calculation of the contents of compound expressions from the semantic properties of their parts are all due to Frege and Russell. Philosophy of language, as we know it today, would not exist without them.

This introductory chapter considers the work of mathematician George Boole (1815–1864), whose book An Investigation of the Laws of Thought (1854) would have a huge impact on humanity. Boole's mathematics, the basis for what is now called Boolean algebra, is the subject of this book. It is also called mathematical logic, and today it is a routine analytical tool of the logic-design engineers who create the electronic circuitry that we now cannot live without, from computers to automobiles to home appliances. Boolean algebra is not traditional or classical Aristotelian logic, a subject generally taught in college by the philosophy department. Boolean algebra, by contrast, is generally in the hands of electrical engineering professors and/or the mathematics faculty.

Alan Turing’s mathematical interests were deep and wide-ranging. From the beginning of his career in Cambridge he was involved with probability theory, algebra (the theory of groups), mathematical logic, and number theory. Prime numbers and the celebrated Riemann hypothesis continued to preoccupy him until the end of his life. As a mathematician, and as a scientist generally, Turing was enthusiastically omnivorous. His collected mathematical works comprise thirteen papers, not all published during his lifetime, as well as the preface from his Cambridge Fellowship dissertation; these cover group theory, probability theory, number theory (analytic and elementary), and numerical analysis. This broad swathe of work is the focus of this chapter. But Turing did much else that was mathematical in nature, notably in the fields of logic, cryptanalysis, and biology, and that work is described in more detail elsewhere in this book. To be representative of Turing’s mathematical talents is a more realistic aim than to be encyclopaedic. Group theory and number theory were recurring preoccupations for Turing, even during wartime; they are represented in this chapter by his work on the word problem and the Riemann hypothesis, respectively. A third preoccupation was with methods of statistical analysis: Turing’s work in this area was integral to his wartime contribution to signals intelligence. I. J. Good, who worked with Turing at Bletchley Park, has provided an authoritative account of this work, updated in the Collected Works. By contrast, Turing’s proof of the central limit theorem from probability theory, which earned him his Cambridge Fellowship, is less well known: he quickly discovered that the theorem had already been demonstrated, the work was never published, and his interest in it was swiftly superseded by questions in mathematical logic. Nevertheless, this was Turing’s first substantial investigation, the first demonstration of his powers, and was certainly influential in his approach to codebreaking, so it makes a fitting first topic for this chapter. Turing’s single paper on numerical analysis, published in 1948, is not described in detail here. It concerned the potential for errors to propagate and accumulate during large-scale computations; as with everything that Turing wrote in relation to computation it was pioneering, forward-looking, and conceptually sound. There was also, incidentally, an appreciation in this paper of the need for statistical analysis, again harking back to Turing’s earliest work.

The interaction between mathematicians and mathematical logicians has always been much slighter than one might imagine. This chapter examines the case of Turing’s mentor, Maxwell Hermann Alexander Newman (1897–1984). The young Turing attended a course of lectures on logical matters that Newman gave at Cambridge University in 1935. After briefly discussing examples of the very limited contact between mathematicians and logicians in the period 1850–1930, I describe the rather surprising origins and development of Newman’s own interest in logic. One might expect that the importance to many mathematicians of means of proving theorems, and their desire in many contexts to improve the level of rigour of proofs, would motivate them to examine and refine the logic that they were using. However, inattention to logic has long been common among mathematicians. A very important source of the cleft between mathematics and logic during the 19th century was the founding, from the late 1810s onwards, of the ‘mathematical analysis’ of real variables, grounded on a theory of limits, by the French mathematician Augustin-Louis Cauchy. He and his followers extolled rigour—most especially, careful definitions of major concepts and detailed proofs of theorems. From the 1850s onwards, this project was enriched by the German mathematician Karl Weierstrass and his many followers, who introduced (for example) multiple limit theory, definitions of irrational numbers, and an increasing use of symbols, and then from the early 1870s by Georg Cantor with his set theory. However, absent from all these developments was explicit attention to any kind of logic. This silence continued among the many set theorists who participated in the inauguration of measure theory, functional analysis, and integral equations. The mathematicians Artur Schoenflies and Felix Hausdorff were particularly hostile to logic, targeting the famous 20th-century logician Bertrand Russell. (Even the extensive dispute over the axiom of choice focused mostly on its legitimacy as an assumption in set theory and its use of higher-order quantification: its ability to state an infinitude of independent choices within finitary logic constituted a special difficulty for ‘logicists’ such as Russell.) Russell, George Boole, and other creators of symbolic logics were exceptional among mathematicians in attending to logic, but they made little impact on their colleagues.

The introduction of recursion into linguistics was the result of applying some of the results of mathematical logic to the study of language. In particular, recursion was introduced in the 1950s as a general property of the mechanical procedure underlying the grammar, in order to account for language’s discrete infinity and expressive power—in the 1950s, this mechanical procedure was a production system, whereas more recently, of course, it is the set-operator merge. Unfortunately, the recent literature has confused the general recursive property of a grammar with specific instances of (recursive) rules/operations within a grammar; more worryingly still, there has been a general conflation of these recursive rules with some of the self-embedded structures these rules can generate, adding to the confusion. The conflation is manifold but always fallacious. Moreover, language manifests a much more generally recursive structure than is usually recognized: bundles of the universal (Specifier)-Head-Complement(s) geometry.