Axiomatic system

In mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science. It consists of a set of formal statements known as axioms that are used for the logical deduction of other statements. In mathematics these logical consequences of the axioms may be known as lemmas or theorems. A mathematical theory is an expression used to refer to an axiomatic system and all its derived theorems.
A proof within an axiomatic system is a sequence of deductive steps that establishes a new statement as a consequence of the axioms. By itself, the system of axioms is, intentionally, a syntactic construct: when axioms are expressed in natural language, which is normal in books and technical papers, the nouns are intended as placeholder words. The use of an axiomatic approach is a move away from informal reasoning, in which nouns may carry real-world semantic values, and towards formal proof. In a fully formal setting, a logical system such as predicate calculus must be used in the proofs. The contemporary application of formal axiomatic reasoning differs from traditional methods both in the exclusion of semantic considerations, and in the specification of the system of logic in use.

The axiomatic method in mathematics

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.

Date	Author	Work	Comments
Fourth century BCE to third century BCE	Euclid of Alexandria	The Elements	Known as the earliest extant axiomatic presentation of Euclidean plane geometry, covering also parts of number theory.
published 1677	Baruch Spinoza	Ethica, ordine geometrico demonstrata	Just as for Principia philosophiae cartesianae of 1663, Spinoza in his Ethics claimed to be using the "geometric method" of Euclid. A modern view is "the contrast is glaring between the aspiration to prove points by way of deductive argument from self-evident axioms and the obvious source of those points from experience of life and at best some mix of theory and intuition."
1829	Nikolai Lobachevsky	О началах геометрии	Lobachevsky's paper is now recognised as the first publication on axiomatic plane geometry developed without the parallel axiom of Euclid, so founding the subject of non-Euclidean geometry.
1879	Gottlob Frege	Begriffschrift	Frege published a formal system for the foundations of mathematics. In modern parlance, it was a second-order logic, with identity relation. It was expressed in a linear notation for parse trees.
1882/3 to 1890s	Walther von Dyck	Axioms for abstract group theory	Von Dyck is credited with the now-standard group theory axioms. It is clear from von Dyck's introduction of free groups that he was working with the standard concept of abstract group. It is not, however, evident whether the existence of inverse elements was axiomatic: it would follow from the semantic assumption that groups were permutation groups or geometric transformations with the same property. The discursive style of the period did not labour such points. James Pierpont, one of the American "postulate theorists", did have by 1896 a set of axioms for groups. It is of the modern type, though uniqueness of the identity element was not assumed.
1888	Richard Dedekind	construction of the real numbers	When Dedekind introduced his construction of real numbers by Dedekind cuts, axioms for the reals were already mathematical folklore; a subset of those would, later, define ordered field. The further requirement was a theory of mathematical limits. For example, to capture the idea that the real number line forms a linear continuum means dealing with the historical Zeno's paradoxes; and also clarifying the issue of decimal representations not being unique, so that 0.999...=1, by subjecting it to a mathematical proof. Dedekind's modelling of axioms of the reals put these matters on a firm footing. In practice, the theorems proved using Dedekind cuts that were fundamental results in real analysis could also be proved for other constructions, for example using Cauchy sequences of rational numbers. In other words, they were verifiable axioms, an example being the Archimedean property.
1889	Giuseppe Peano	Arithmetices principia, nova methodo exposita	After some earlier work of others, the Peano axioms provided an axiomatic basis for the arithmetical operations on natural numbers, and mathematical induction, that gained wide acceptance.
1898	Alfred North Whitehead	Treatise on Universal Algebra	Whitehead gave the first axiomatic system for Boolean algebra, as introduced by George Boole in fundamental work on logic and probability.
1899	David Hilbert	Grundlagen der Geometrie	Presented what are now known as Hilbert's axioms, a revised axiomatization of solid geometry.

Situation at the beginning of the 20th century

"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 , 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 initial International Congress of Mathematicians had heard the views of Henri Poincaré from France on mathematical physics; Hilbert's list was a submission to the second Congress. 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.

Date	Author	Work	Comments
1908 to 1922	Ernst Zermelo and Abraham Fraenkel	Zermelo-Fraenkel set theory	Building on Zermelo set theory from 1908, the Zermelo-Fraenkel theory provided an axiomatic basis for set theory with a clarified axiom system. With the addition of the axiom of choice, the ZFC theory provided a working foundation for much of classical mathematics.
1910	Ernst Steinitz	Algebraische Theorie der Körper	Steinitz, under the influence of the introduction by Kurt Hensel of the p-adic numbers, gave an axiomatic theory of the field concept in abstract algebra.
1911 to 1913	Alfred North Whitehead and Bertrand Russell	Principia Mathematica	A work devoted to the principle of axiomatic formalization of mathematics, that addressed the set theory paradoxes by an idiosyncratic version of type theory. The system does not involve the axiom of extensionality.
1913	Hermann Weyl	Die Idee der Riemannschen Fläche	Weyl gave the Riemann surface concept of complex analysis an axiomatic treatment, defining it as a complex manifold of dimension one in terms of neighbourhood systems.
1914	Felix Hausdorff	Grundzüge der Mengenlehre	The book included axioms for what is now called a Hausdorff topological space, building on Weyl's use of neighbourhoods.
1915	Maurice Fréchet	abstract measures on measure spaces	The ideas of Lebesgue measure and associated integral, introduced firstly on the real line and Euclidean spaces, were handled axiomatically on set systems.
1920	Stefan Banach	complete normed vector space	Known now as Banach space, it is the classic setting for functional analysis; initially a real vector space was assumed.
1921	John Maynard Keynes	A Treatise on Probability	Keynes's work subordinated probability to logic, under the influence of Principia Mathematica. It gave an axiomatic treatment of probability interpretations.
1921	Emmy Noether	Idealtheorie in Ringbereichen	Noether's paper introduced the ascending chain condition on ideals as an axiom in commutative rings, giving a subclass now called Noetherian rings. It allowed a straightforward inductive proof of Hilbert's basis theorem. It is also considered the beginning of an "epoch" in abstract algebra.
1923	Norbert Wiener	Wiener process	Wiener constructed a measure defining a stochastic process model of Brownian motion.
1932	Oswald Veblen and J. H. C. Whitehead	The Foundations of Differential Geometry	The work gave the accepted axiomatic definition of differential manifold, apart from certain issues with separation axioms.
1932	John von Neumann	Mathematische Grundlagen der Quantenmechanik, Dirac–von Neumann axioms	A contribution to the mathematical formulation of quantum mechanics, dating back to a 1927 paper by von Neumann, proposing an axiomatisation of the founding works of quantum mechanics, modelled formally on the notations of Paul Dirac. It used abstract Hilbert space methods and unbounded operators.
1933	Andrey Kolmogorov	probability axioms	Kolmogorov's work subordinated, in effect, mathematical probability to measure theory, while leaving its interpretation open. It built therefore expected values on the Lebesgue integral. From Georg Bohlmann at the beginning of the century onwards, there had been numerous axiomatic formulations. Making probability a sigma-additive set function by fiat was decisive.
1945	Samuel Eilenberg and Norman Steenrod	Eilenberg–Steenrod axioms	An axiomatic system for homology theory in algebraic topology, it reflected developments since Noether advocated that homology classes be organised on abstract algebra principles.
1945–1950	Laurent Schwartz	theory of distributions	Using duality for topological vector spaces of test functions, Schwartz gave a unified axiomatic treatment of the Dirac delta-function and a number of other formal operator methods, and the geometric theory of currents.