The “Bourbaki circle” of French mathematicians published the first of more than sixty monographs surveying and synthesizing the abstract structure of extant operational mathematics.
From the publications of Georg Cantor
Principia Mathematica (1910-1913), to the formal axiomatics of David Hilbert’s
Grundlagen der Geometrie (1899; The Foundations of Geometry, 1902) and Grundzüge der Theoretischen Logik (1928; Principles of Mathematical Logic, 1950), mathematics has evolved through several distinct phases in regard to its degree of abstraction, self-consistency, and unity. Notwithstanding the apparent limits of formalization attendant on Kurt Gödel’s
The modern ideas of Peano, Russell, Hilbert, and others have loosened many of the traditional connections of mathematics with specific ideas about “number” and “quantity,” instead underscoring the general roles of abstract structures and axioms. As Hermann Weyl
In contrast to the practically inapplicable logicism of Gottlob Frege
Notwithstanding a 1949 biographical note on “Professor Bourbaki” of the “Royal Poldavian Academy and Nancago University,” the name “Bourbaki” designates a largely anonymous group of principally French mathematicians, first organized shortly before World War II. In a list that was never confirmed, André Deleachet in his text on mathematical analysis includes as key Bourbaki circle members noted mathematicians André Weil, Jean Dieudonné, Henri Cartan,
Some have suggested that the pseudonym “Bourbaki” refers to the Greco-French general Charles Bourbaki, defeated in the Franco-Prussian War by unconventional German tactics, and thus obliquely to many French mathematicians responding to a perceived “germanic” onslaught of strict Hilbertian formalists. The nature of the Bourbaki group’s systematic approach to and interpretation of mathematics was first presented in the initial volumes of the Éléments de mathématique, in 1939, then continued in several annual installments under the aegis of the Séminaire Bourbaki. As recounted by noted mathematician Laurent Schwartz,
Although in many respects incorporating the spirit and some of the technologies of Hilbert’s formal axiomatics, as a prologue “Bourbaki” clearly distinguishes the latter’s logical formalism from its own “structural axiomatics.” In this structuralist approach, functions, operations, transformations, and substitutions are but different names for various types of fundamental relations. Bourbaki criticizes Hilbert’s overemphasis on logico-deductive reasoning as the sole basis and unifying principle for mathematical relations, calling it the only external form and vehicle that the mathematician gives to his thought. What the Bourbakian method sets as its aim is exactly what it believes Hilbert’s logical formalism cannot by itself supply—namely, the profound creativity and intelligibility of mathematics.
Taking a “naïve realist” approach, assuming mathematical theories as “given” and ignoring metamathematical questions on the nature and existence of mathematical objects or the connections between language and intuition, “structural relations” or structures are simply posited as the most fundamental points of common access for conceptually unifying the diversity of mathematical theories. Although never defined to the full satisfaction of many readers, mathematical structures are characterized by Bourbaki as abstract common or generic concepts that can be applied to different sets of elements whose nature has been specified, common properties expressible in the same way in different mathematical theories, and the form of a possible system of related objects that ignores specific material features of the objects not relevant to their abstract interrelations. In contrast to premathematical givens or assumptions of Russell, Hilbert, or Brouwer, Bourbaki structures are described as practical and useful tools from which the global aspects of a mathematical problem or theory can be reconstructed from its local aspects.
As outlined in a number of ancillary journal publications in the American Journal of Symbolic Logic and the American Mathematical Monthly, the main Bourbakian structural principles propose a hierarchy and network of interrelations between mathematical subdisciplines and theories. Particular structures are thought of as inhering in specific sets, the fundamental mathematical entity of the Bourbaki system. As understood by Bourbaki, set theory
The main structural features of the algebraic family are its forms of “reversibility,” as best characterized by inversion and negation operators. Prototypical properties of ordinal or order structures are those networks, or lattices, defined via the predecessor/successor relation. The most abstract fundamental structure—the topological—is characterized by the basic concepts of neighborhood, continuity, and limit. Beyond the first structural level of parent structures are so-called multiple structures, involving combinations of two or more fundamental structures simultaneously. Bourbaki cites as examples of multistructures topological algebra (including topological entities and properties and algebraic composition rules) and algebraic topology (including algebraic entities and properties together with topological construction rules). Finally, the level of particular or special structures corresponds to the different theories and branches of contemporary mathematics, seen not as independent and separate areas, but as crossroads where several general and multiple structures intersect and meet. The legitimacy and plausibility of this structurally integrative approach in mathematics is for Bourbaki based on the closer, but largely unperceived, functional unity between different mathematical theories and departments arising from the internal evolution of mathematics since about 1860. In contrast to Hilbert’s formal axiomatics, Bourbaki repeatedly states that the total number and interconnections between multiple and particular structures cannot be delimited or classified in advance. For Bourbaki, Hilbert’s original program of complete axiomatization is possible only for certain “univalent” mathematical theories (such as relational logic and geometry), which are determined entirely by a finite system of explicit axioms. Although easier said than shown, Bourbaki repeatedly argues that it is necessary to identify and explicate parent and multistructures by working with the rich fields of particular structures in which higher-level abstractions are embedded.
The subsequent impact and history of Bourbaki’s initial publications has largely been continued contributions to structurally explicating other domains of pure and applied mathematics. As an example of three intersecting multistructures discussed in later volumes, the Éléments considers the theory of real numbers. If considered with regard to compositional rules such as addition and multiplication, the real numbers form an algebraic “group,” which is a set or class of relations having a special characteristic property such as symmetry. If arranged according to their ordinal magnitudes, the real numbers encompass an ordered set. Finally, examining the theory of continuity and limits for real numbers in the most general fashion necessitates recourse to central connectivity and adjacency properties of a topological space. Thus in principle there are many advantages of a structuralist mathematics. Most notably, once a theorem is proven for a general abstract structure, it is applicable immediately for any specific realization of that structure. For example, developments in the theory of measure and integration can be structurally applicable to some aspects of probability theory, by virtue of the common parent set of structural axioms.
Bourbaki has subsequently endorsed the view that the most appropriate foundation for mathematics is a combination of axiomatic set theory and symbolic logic. In this view, mathematical entities (numbers, geometric figures, and the like) are never given in isolation, but only in and as part of parent and multistructures. Nevertheless, in contrast to Hilbert, Bourbaki replies that mathematicians cannot map out all structures by working mechanically with symbols, but require their own special “intuition” to inform but not eliminate symbolism, formalisms, and axiomatics. What counts is not formal limits in themselves but whether a domain of mathematics is enlarged permanently by a study of its structural axioms.
The efforts of the Bourbaki circle to generalize axiomatically those (multiple and particular) mathematical structures in overlapping theories has continued in the more than sixty volumes and numerous journal papers appearing since 1939. To date, the order of publication of the Éléments has reflected only multistructures corresponding to particular extant areas of algebra, topology, topological vector space, integration theory, group theory, and Lie algebras. Not all the originally proposed topics of the Bourbaki circle have as yet been examined. In many cases, the Bourbakian structuralist approach has made it considerably more simple to see the general structures than the many previous hypotheses in areas such as the theory of fields, groups, lattices, and numbers. It is also true that in some classical areas of mathematics, it has not proven so simple to perceive and formulate mathematical theories in terms of axiomatized structures.
As the process of structural axiomatization continues, Bourbaki employs sometimes colorful idiosyncratic modifications of ordinary language whenever formal accuracy can be preserved along with intuitive perspicacity. This, among other aspects of the Bourbaki movement, motivated several other new approaches during the 1960’s to reformulate elementary mathematics, in some cases contributing driving ideas behind the so-called new math. More particularly, the “structural” theory of mathematics by Bourbaki directly stimulated, and in some cases interacted with, the efforts of Jean Piaget
Some mathematicians and educators have reacted negatively to Bourbaki structuralism, fearing that its dream of all-inclusive axiomatics is too inflexible and too removed from specific content and examples. Many educators, in particular, emphasize that it is clearly possible to derive adequately and apply many aspects of classical mathematics without knowing their parent structures and their interconnections. Other more supportive developments in philosophy and critical theory, such as the diverse“structuralisms” of Claude Lévi-Strauss, Louis Althusser, Michel Foucault, Michel Serres, and others, bear a superficial resemblance to the jargon and rigor of the Bourbaki mathematics. Bourbaki structures should not be confused with Thomas S. Kuhn’s The Structure of Scientific Revolutions (1962) or with other American philosophers’ debates concerning how knowledge of abstract structures “matches” real structures of the physical world.
With the retirement of the Bourbaki group’s original founders, it remains to be seen whether the colossal task of the project will be continued and completed beyond the roughly 25 percent now extant. Nevertheless, even the initial efforts of Bourbaki’s structuralist mathematics have provided a stimulating and efficient method and model for organizing scientific as well as mathematical hypotheses in areas such as mathematical physics, linguistics, and economics, where there is adequate prior development and strong basic relationships that lend themselves to systematization.
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Aczel, Amir D. The Artist and the Mathematician: The Genius Mathematician Nicolas Bourbaki and How He Shaped Our World. New York: Thunder’s Mouth Press, 2006. Places the Bourbaki group’s work in the context of its times and explores the revolution in mathematics associated with Bourbaki. -
Beth, Evert Willem. Formal Methods: An Introduction to Symbolic Logic and to the Study of Effective Operations in Arithmetic and Logic. 1962. Reprint. New York: Springer-Verlag, 1970. Discusses typical influences of Bourbaki structuralism. -
Beth, Evert Willem, and Jean Piaget. Mathematical Epistemology and Psychology. Translated by W. Mays. 1966. Reprint. New York: Springer-Verlag, 1974. The primary source for Piaget’s structuralism. -
Cartier, Pierre. “The Continuing Silence of Bourbaki: An Interview with Pierre Cartier, June 18, 1997.” Interview by Marjorie Senechal. Mathematical Intelligencer 20, no. 1 (1998): 22-28. Discussion of the Bourbaki group by one of its members. -
Fang, J., ed. Towards a Philosophy of Modern Mathematics. Hauppauge, N.Y.: Paideia Press, 1970. Excellent source for Bourbaki papers. -
Geroch, Robert. Mathematical Physics. Chicago: University of Chicago Press, 1985. Discusses analyses influenced by Bourbaki and Piaget. -
Kneebone, G. T. Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. 1963. Reprint. Mineola, N.Y.: Dover, 2001. Provides a good introduction to the set theory, logical, and axiomatic methods of Bourbaki.
Levi Recognizes the Axiom of Choice in Set Theory
Russell Discovers the “Great Paradox”
Brouwer Develops Intuitionist Foundations of Mathematics
Zermelo Undertakes Comprehensive Axiomatization of Set Theory
Fréchet Introduces the Concept of Abstract Space
Gödel Proves Incompleteness-Inconsistency for Formal Systems