The reduction of a body of propositions to a particular collection of axioms underlies
mathematical research. This dependence was very prominent, and contentious, in the mathematics of the first half of the twentieth century, a period to which some major landmarks of the axiomatic method belong. The
probability axioms of
Andrey Kolmogorov, from 1933, are a salient example. The approach was sometimes attacked as "formalism", because it cut away parts of the working intuitions of mathematicians, and those applying mathematics. In historical context, this alleged formalism is now discussed as
deductivism, still a widespread philosophical approach to mathematics.
Timeline of axiomatic systems to 1900 Major axiomatic systems were developed in the nineteenth century. They included
non-Euclidean geometry,
Georg Cantor's abstract
set theory, and Hilbert's revisionist axioms for
Euclidean geometry.
Situation at the beginning of the 20th century David Hilbert "was the first who explicitly adopted the axiomatic method as an investigative framework for the study of the
foundations of mathematics". For Hilbert, a major foundational issue was the logical status of
Cantor's set theory. In his
list of 23 unsolved problems in mathematics from 1900, Hilbert made the
continuum hypothesis the first problem on the list.
Hilbert's sixth problem asked for "axiomatization of all branches of science, in which mathematics plays an important part". He had in mind at least major areas in
mathematical physics and probability. Of the effect on science, Giorgio Israel has written: Founded by mathematician
Felix Klein ... the Göttingen School, under the influence of David Hilbert, turned its efforts towards ... set theory, functional analysis, quantum mechanics and mathematical logic. It did so by taking on as its methodical principle the axiomatic method that was to revolutionise the science of [the twentieth century], from the theory of probabilities to theoretical physics. Israel comments also on cultural resistance, at least in France and Italy, to this "German model" and its international scope. The
Italian school of algebraic geometry took a different attitude to axiomatic work in theory building and pedagogy.
Timeline of axiomatic systems from 1901 In the period to 1950, much of
pure mathematics received widely-accepted axiomatic foundations. Multiple systems coexisted in
axiomatic set theory. Mathematics began to be written in a tighter, less discursive if still informal style. On the other hand, the approach associated with Hilbert of regarding the axiomatic method as fundamental came under criticism. Part of
L. E. J. Brouwer's critique of Hilbert's entire program resulted in an axiomatisation of
intuitionistic propositional logic by
Arend Heyting. It allowed
constructivism in mathematics to be reconciled with "deductivism", by an exchange of logical calculus, under the title of the
Brouwer–Heyting–Kolmogorov interpretation.
Situation at mid-20th century Three prominent features of mathematics in 1950 were: • The continuing publication in France by the
Bourbaki group of the book series
Éléments de mathématique. It aimed at an encyclopedic treatment of foundational concepts. • A dynamic situation in the foundations of
algebraic geometry, following the publication of
Foundations of Algebraic Geometry by
André Weil. •
Quantum field theory (QFT), which lacked a satisfactory axiomatic foundation.
Axiomatics à la Bourbaki The aims of Bourbaki were for a treatment in the large of mathematics, which would be: (a) axiomatic, based down on a stripped-down logical foundation in set theory; (b) in the tradition of Hilbert and the Göttingen School, though excluding the needs of physics and computation; (c) a French reception of current developments. The initial work was carried out in a sharp
young Turk reaction against the ''Cours d'analyse mathématique
, a standard text on classical analysis from the beginning of the 20th century, by Édouard Goursat, and in favour of the text Moderne Algebra'' from the early 1930s on
abstract algebra, by
Bartel Leendert van der Waerden. A pseudonymous paper from 1950, in fact the work of
Jean Dieudonné, explained the attitude of Bourbaki to the axiomatic method. The principal advantage of working axiomatically is asserted to lie in "elaboration" of mathematical "forms", or
structures; this takes precedence over the foundational work and the clarification of
inference. What Dieudonné wrote was of his time, as a departure from Hilbert's approaches, and not yet an arrival at structure in the sense implied by the
morphisms of
category theory.
Axiomatic QFT Plausible axioms for QFT, the
Wightman axioms, were introduced by
Arthur Wightman. The need for non-trivial examples for these axioms led to
constructive quantum field theory, launched by work of
Arthur Jaffe and
Oscar Lanford, in doctoral dissertations supervised by Wightman in the mid-1960s. ==Discussion of axiomatic systems==