* Primitive recursive arithmetic, one proposed formalization of finitary methods in mathematics

** Dual ( category theory ), a formalization of mathematical duality

In computability theory, primitive recursive functions are a class of functions that form an important building block on the way to a full formalization of computability.

The analysis of logical concepts and the machinery of formalization that is essential to Principia Mathematica ( 3 vols., 1910 – 1913 ) ( by Bertrand Russell, 1872 – 1970, and Alfred North Whitehead, 1861 – 1947 ), to Russell's theory of descriptions, to Kurt Gödel's ( 1906 – 1978 ) incompleteness theorems, and to Alfred Tarski's ( 1901 – 1983 ) theory of truth, is ultimately due to Frege.

But a paradox may either refer to a surprising but true result in a given formal theory, or to an informal argument leading to a contradiction, so that a candidate theory where a formalization of the argument might be attempted must disallow at least one of its steps ; in this case the problem is to find a satisfying theory without contradiction.

In most of mathematics as it is practiced, the incompleteness and paradoxes of the underlying formal theories never played a role anyway, and in those branches in which they do or whose formalization attempts would run the risk of forming inconsistent theories ( such as logic and category theory ), they may be treated carefully.

** Dual ( category theory ), a formalization of mathematical duality

Furthermore, just as the later-proposed minimum description length principle in algorithmic information theory ( AIT ), a. k. a. the theory of Kolmogorov complexity, it can be seen as a formalization of Occam's Razor in which the best hypothesis for a given set of data is the one that leads to the largest compression of the data.

The above intuition of domains being information orderings will be emphasized to motivate the mathematical formalization of the theory.

In the formalization of order theory, this is just the least upper bound of the directed set.

One such formalization, referred to as “ behavioral dynamics ”, treats the agent and the environment as a pair of coupled dynamical systems based on classical dynamical systems theory.

This idea is found in Douglas Hofstadter's book, Gödel, Escher, Bach, in a discussion of the relationship between formal languages and number theory: “... it is in the nature of any formalization of number theory that its metalanguage is embedded within it .”.

The bureaucracy theory is implemented in Flat organization structures and is suited for larger organizations that require formalization of roles.

Probability theory is the formalization and study of the mathematics of uncertain events or knowledge.

Zermelo's further work on the foundations of set theory after Skolem's paper led to his discovery of the cumulative hierarchy and formalization of infinitary logic ( van Dalen and Ebbinghaus, 2000, note 11 ).

It is used for logic programming and for the formalization of programming language theory.

** Information Flow: The Logic of Distributed Systems an influential handbook ( ISBN 0-521-58386-1 ) by Jon Barwise and Jerry Seligman for the analysis of theories using its framework based on a melange of topics from information, model, and discourse theory which is applied to give a formalization of the logic of Quantum Mechanics.

Later, Kleene and Kreisel would study formalized versions of intuitionistic logic ( Brouwer rejected formalization, and presented his work in unformalized natural language ).

An explicit formalization of natural languages semantics without confusions with implicit assumptions such as closed world assumption ( CWA ) vs. open world assumption, or subjective Yes / No vs. objective True / False is expected for the construction of a basis of semantics formalization.

Philosophical logic is a term introduced by Bertrand Russell to represent his idea that the workings of natural language and thought can only be adequately represented by an artificial language ; essentially it was his formalization program for the natural language.

A successful formalization of natural language semantics usually can not avoid an explicit revelation of the implicit logical backgrounds based on whether CWA or OWA.

In order to achieve this understanding, Eilenberg and Mac Lane proposed an axiomatic formalization of the relation between structures and the processes preserving them.

Each of these can give a complete and axiomatic formalization of propositional or predicate logic of either the classical or intuitionistic flavour, almost any modal logic, and many substructural logics, such as relevance logic or

A partial formalization of the concept began with attempts to solve the Entscheidungsproblem ( the " decision problem ") posed by David Hilbert in 1928.

However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects.

The formalization of the struggles over taxation between the monarch and other elements of society ( especially the nobility and the cities ) gave rise to what is now called the Standestaat, or the state of Estates, characterized by parliaments in which key social groups negotiated with the king about legal and economic matters.

Like many other patronymics in other languages, with the formalization of naming conventions by laws in the late modern contemporary age many turned into surnames.

Wiener is regarded as the originator of cybernetics, a formalization of the notion of feedback, with many implications for engineering, systems control, computer science, biology, philosophy, and the organization of society.

Wiener is regarded as the originator of cybernetics, a formalization of the notion of feedback, with many implications for engineering, systems control, computer science, biology, philosophy, and the organization of society.

The meeting, held in Cambridge, United Kingdom, concluded with the formalization of the CIE 1931 XYZ color space and definitions of the 1931 CIE 2 ° standard observer with the corresponding color matching functions, and standard illuminants A, B, and C.

The Commercial Revolution is also marked by the formalization of pre-existing, informal methods of dealing with trade and commerce.

In 1984, Jean Morlet introduced Gabor's work to the seismology community and, with Goupillaud and Grossmann, modified it to keep the same wavelet shape over equal octave intervals, resulting in the first formalization of the continuous wavelet transform.

found that individual face-recognizing neurons in the inferotemporal cortex respond more strongly to caricatured faces than to the veridical representations of the same face, and suggest that the visual brain may code faces relative to a prototypical face, consistent with Brennan's formalization.

A crucial difference with respect to the traditional, so-called transformational, formalization of visual regularity is that, holographically, mirror symmetry is composed of many relationships between symmetry pairs rather than one relationship between symmetry halfs.

The formalization of the division between small powers and great powers came about with the signing of the Treaty of Chaumont in 1814.

Taxicab geometry satisfies all of Hilbert's axioms ( a formalization of Euclidean geometry ) except for the side-angle-side axiom, as one can generate two triangles each with two sides and the angle between them the same, and have them not be congruent.

It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then deriving B from this assumption conjoined with known results.

Colloquially, " lease " and " leasing " are often a formalization of a longer, specific period as compared with a " rental " that created a tenancy at will, terminable or renewable at the end of a short period.

According to scholars, Prem Rawat's desire to consolidate his power and authority over the U. S. movement led to increasing formalization, with rules and regulations for ashram living, standards for recruited " candidates ," and pressure towards certification of the movement's teachers.

#* The notion of institution has been developed as an abstract formalization of the notion of logical system, with the goal of handling the " population explosion " of logics used in formal methods.

In Italy in 13 – 15th centuries bill of exchange and promissory note obtain their main features and further phases of its development have been associated with France ( 16 – 18th centuries, where the endorsement had appeared ) and Germany ( 19th century, formalization of Exchange Law ).

These crews were considered trainees and were the first formalization of sail trainers with crew drawn from members of the public who just went for the adventure, as opposed to a career.

His National Security Advisor, Dr. Henry Kissinger, convinced the president to reconsider, since he did not want to jeopardize the formalization of the Strategic Arms Limitation Treaty ( SALT I ) with the Soviets, that was due to be formalized in May.