MATHEMATICS, FOUNDATIONS OF

the study of the logical and philosophical basis of mathematics, including whether the axioms of a given system ensure its completeness and its consistency. Because mathematics has served as a model for rational inquiry in the West and is used extensively in the sciences, foundational studies have far-reaching consequences for the reliability and extensibility of rational thought itself. For 2,000 years the foundations of mathematics seemed perfectly solid. Euclid's Elements (c. 300 BC), which presented a set of formal logical arguments based on a few basic terms and axioms, provided a systematic method of rational exploration that guided mathematicians, philosophers, and scientists well into the 19th century. Even serious objections to the lack of rigour in Sir Isaac Newton's notion of fluxions (derivatives) in the calculus, raised by the Anglo-Irish empiricist George Berkeley (among others), did not call into question the basic foundations of mathematics. The discovery in the 19th century of consistent alternative geometries, however, precipitated a crisis, for it showed that Euclidean geometry, based on seemingly the most intuitively obvious axiomatic assumptions, did not correspond with reality as mathematicians had believed. This, together with the bold discoveries of the German mathematician Georg Cantor in set theory, made it clear that, to avoid further confusion and satisfactorily answer paradoxical results, a new and more rigorous foundation for mathematics was necessary. Thus began the 20th-century quest to rebuild mathematics on a new basis independent of geometric intuitions. Early efforts included those of the logicist school of the British mathematicians Bertrand Russell and Alfred North Whitehead, the formalist school of the German mathematician David Hilbert, the intuitionist school of the Dutch mathematician L.E.J. Brouwer, and the French set theory school of mathematicians collectively writing under the pseudonym of Nicolas Bourbaki. Some of the most promising current research is based on the development of category theory by the American mathematician Saunders Mac Lane and the Polish-born American mathematician Samuel Eilenberg following World War II. This article presents the historical background of foundational questions and 20th-century efforts to construct a new foundational basis for mathematics. Additional reading W.S. Anglin and J. Lambek, The Heritage of Thales (1995), a textbook aimed primarily at undergraduate mathematics students, deals with the history, philosophy, and foundations of mathematics and includes an elementary introduction to category theory. Collections of important readings and original articles include Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings, 2nd ed. (1983), treating the foundations of mathematics, the existence of mathematical objects, the notion of mathematical truth, and the concept of set; Jaako Hintikka (ed.), The Philosophy of Mathematics (1969), which includes articles by Henkin on completeness, by Feferman on predicativity, by Robinson on the calculus, and by Tarski on elementary geometry; and Jean Van Heijenoort (compiler), From Frege to Gdel: A Source Book in Mathematical Logic, 18791931 (1967, reissued 1977). Bertrand Russell, A History of Western Philosophy and Its Connection with Political and Social Circumstances from the Earliest Times to the Present Day, 2nd ed. (1961, reprinted 1991), an extremely readable work, portrays the relevant views of the pre-Socratics, Plato, Aristotle, Leibniz, and Kant. Mario Bunge, Treatise on Basic Philosophy, vol. 7, Epistemology & Methodology III: Philosophy of Science and Technology, part 1, Formal and Physical Sciences (1985), contains a discussion by a philosopher of the different philosophical schools in the foundations of mathematics. William Kneale and Martha Kneale, The Development of Logic (1962, reprinted 1984), offers a thorough scholarly account of the growth of logic from ancient times to the contributions by Frege, Russell, Brouwer, Hilbert, and Gdel. Saunders Mac Lane, Mathematics, Form and Function (1986), records the author's personal views on the form and function of mathematics as a background to the philosophy of mathematics, touching on many branches of mathematics. Michael Hallett, Cantorian Set Theory and Limitation of Size (1984), provides a scholarly account of Cantor's set theory and its further development by Fraenkel, Zermelo, and von Neumann. William S. Hatcher, Foundations of Mathematics (1968), surveys different systems, including those of Frege, of Russell, of von Neumann, Bernays, and Gdel, and of Quine as well as Lawvere's category of categories. Y.I. Manin (Iu.I. Manin), A Course in Mathematical Logic, trans. from Russian (1977), is addressed to mathematicians at a sophisticated level and presents the most significant discoveries up to 1977 concerning the continuum hypothesis, the nonexistence of algorithmic solutions, and other topics. George S. Boolos and Richard C. Jeffrey, Computability and Logic, 3rd ed. (1989), for graduate and advanced undergraduate philosophy or mathematics students, deals with computability, Gdel's theorems, and the definability of truth, among other topics. J. Lambek and P.J. Scott, Introduction to Higher Order Categorical Logic (1986), is an advanced textbook addressed to graduate students in mathematics and computer science in which the relationship between topoi and type theories is explored in detail and some of the metatheorems cited in this article are proved. Joachim Lambek