Le Progrès de l'Esprit

Abstract

IN the introductory volume of his work on questions of structure and existence in mathematics, M. Lautman is mainly concerned with Prof. Weyl's distinction between the two types of modern mathematical systems in which the analysis of the infinite is contrasted with the synthetic methods of algebra and group theory. He shows that this distinction need not be conceived of as a fundamental opposition between two irreducible disciplines. On the contrary, among the modern theories of analysis, it is possible to discover points of view which are characteristic of algebra and group theory. For example, the theory of equations with an infinity of variables, the arithmetical theory of algebraic functions, the theory of non-Euclidean metrics, the theory of continuous groups, the analytic theory of numbers and the theory of Pfaffian forms, all occupy an equally intermediary position between algebra and analysis. For in all these, the methods are algebraic and the results extend to analysis; and although there is a certain analogy between contemporary physics and contemporary mathematics, in that both present a spectacle of facts which can be interpreted either by the calculus of the continuous or by that of the discrete, this duality of method, which is the principal source of the difficulties inherent in contemporary physics, is evidence of the profound unity of the mathematical sciences. Le Progres de l'Esprit Par Dr. Albert Lautman. 4: Essai sur l'unite des sciences mathematiques dans leur developpement actuel. Pp. 60. 15 francs. 5: Essai sur les notions de structure et d'existence en mathematiques, 1: Les schemas de structure. Pp. 82. 20 francs. 6: Essai sur les notions de structure et d'existence en mathematiques, 2: Les schemas de genese. Pp. 83–160. 20 francs. (Actualites scientifiques et industrielles, 589–591.) (Paris: Hermann et Cie., 1938.)