The subsequent development of category theory was powered first by the computational needs of homological algebra, and later by the axiomatic needs of algebraic geometry, the field most resistant to being grounded in either axiomatic set theory or the Russell-Whitehead view of united foundations.

* Mathematical systems have an " essence ," namely their axiomatic algebraic structure ;

These were then re-formulated by Grigore Moisil in an axiomatic algebraic form, and also extended to n-valued logics in 1945.

Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.

* Algebraic semantics is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program semantics in a formal manner ;

Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities.

In 1917, Bernstein suggested the first axiomatic foundation of probability theory, based on the underlying algebraic structure.

* 1987 Samuel Eilenberg for his fundamental contributions to topology and algebra, in particular for his classic papers on singular homology and his work on axiomatic homology theory which had a profound influence on the development of algebraic toplogy.

It is not an axiomatic algebraic idea ; rather it defines a set of closure conditions on sets of homeomorphisms defined on open sets U of a given Euclidean space E or more generally of a fixed topological space S. The groupoid condition on those is fulfilled, in that homeomorphisms

He is best known for his contributions to the algebraic formulation of axiomatic quantum field theory, namely the Haag-Kastler axioms

Recently, it has been seen that they play an important role in the algebraic treatment and axiomatic foundation of topological quantum field theory.

Euclid's axiomatic approach and constructive methods were widely influential.

The title was adapted by Raymond F. Streater and Arthur S. Wightman for their ( serious ) textbook on axiomatic quantum field theory, < cite > PCT, Spin and Statistics, and All That </ cite >.

Subsequently Res Jost gave a more general proof in the framework of axiomatic quantum field theory.

He is one of the founders of the axiomatic approach to quantum field theory, and originated the set of Wightman axioms.

The Haag-Kastler axiomatic framework for quantum field theory, introduced by, is an application to local quantum physics of C *- algebra theory.

Hilbert, with the assistance of Johann von Neumann, L. Nordheim, and E. P. Wigner, worked on the axiomatic basis of quantum mechanics ( see Hilbert space ).

At the same time, but independently, Dirac formulated quantum mechanics in a way that is close to an axiomatic system, as did Hermann Weyl with the assistance of Erwin Schrödinger.

Since the 1960s, following the work of Arthur Wightman and Rudolf Haag, also modern quantum field theory can be considered close to an axiomatic description.

Most practitioners of QFT ignore Haag's theorem entirely, and it is currently unknown why QFT, and quantum electrodynamics in particular, produces accurate numbers given the lack of any axiomatic basis.

* Rudolf Haag ( born 1922 ), German physicist, known for his contributions to the axiomatic formulation of quantum field theory

The scope of his research also includes model theory, generalized Galois theory, axiomatic foundations of quantum theory and relativity, complexity theory, and abstract logics.

This means that if such a QFT is well-defined at all scales, as it has to be to satisfy the axioms of axiomatic quantum field theory, it would have to be trivial ( i. e. a free field theory ).

* 1955 Developed an axiomatic theory for scattering matrix ( S — matrix ) in quantum field theory and introduced the causality condition for S — matrix in terms of variational derivatives.

Combinatorics is an example of a field of mathematics which does not, in general, follow the axiomatic method.

Today, when mathematicians talk about " set theory " as a field, they usually mean axiomatic set theory.

Nearly simultaneously David Hilbert published " The Foundations of Physics ", an axiomatic derivation of the field equations ( see Einstein – Hilbert action ).

The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logic, but category theory is not ordinarily considered a subfield of mathematical logic.

He founded the now-mainstream field of positive political theory, which introduced game theory and the axiomatic method of social choice theory to political science.

In 1933, Kolmogorov published his book, the Foundations of the Theory of Probability, laying the modern axiomatic foundations of probability theory and establishing his reputation as the world's leading expert in this field.

# A real closed field has all the first-order properties of the real number system, regardless of whether they are usually taken as axiomatic, for statements involving the basic ordered-field relations +, *, and ≤.

In the Posterior Analytics, Aristotle ( 384 BC – 322 BC ) laid down the axiomatic method, to organize a field of knowledge logically by means of primitive concepts, axioms, postulates, definitions, and theorems, taking a majority of his examples from arithmetic and geometry.

Thus, for example, the studies of " hypercomplex numbers ", such as considered by the Quaternion Society, were put onto an axiomatic footing as branches of ring theory ( in this case, with the specific meaning of associative algebras over the field of complex numbers.

* Arthur Wightman: What is the point of so-called " axiomatic field theory "?.

Another field that emerged from axiomatic studies of projective geometry is finite geometry.

* the various ( but equivalent ) constructions of the real numbers by Dedekind and Cantor resulting in the modern axiomatic definition of the real number field ;

Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in ZFC, the standard form of axiomatic set theory.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets.

Informal applications of set theory in other fields are sometimes referred to as applications of " naive set theory ", but usually are understood to be justifiable in terms of an axiomatic system ( normally the Zermelo – Fraenkel set theory ).

It can be done by systematically making explicit all the axioms, as in the case of the well-known book Naive Set Theory by Paul Halmos, which is actually a somewhat ( not all that ) informal presentation of the usual axiomatic Zermelo – Fraenkel set theory.

However, the term naive set theory is also used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory ; care is required to tell which sense is intended.

Their work was an important part of the transition from intuitive and geometric homology to axiomatic homology theory.

Certain categories called topoi ( singular topos ) can even serve as an alternative to axiomatic set theory as a foundation of mathematics.

The related concept of " standard " numbers, which can only be defined within a finite time and space, is used to motivate axiomatic internal set theory, and provide a workable formulation for illimited and infinitesimal number.

In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.

Many common axiomatic systems, such as first-order Peano arithmetic and axiomatic set theory, including the canonical Zermelo – Fraenkel set theory ( ZF ), can be formalized as first-order theories.