With the Prior Analytics, Aristotle is credited with the earliest study of formal logic, and his conception of it was the dominant form of Western logic until 19th century advances in mathematical logic.

Introduction to mathematical logic.

Russell and Whitehead thought they could derive all mathematical truth using axioms and inference rules of formal logic, in principle opening up the process to automatisation.

The actual mathematical operation for each instruction is performed by a subunit of the CPU known as the arithmetic logic unit or ALU.

In any case, this article follows ISO 31-11 and the standard convention in mathematical logic, which make 0 a natural number .</ ref >

In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.

Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic, and co-authored with him the important book Grundlagen der Mathematik ( which eventually appeared in two volumes, in 1934 and 1939 ).

Hilbert's work had started logic on this course of clarification ; the need to understand Gödel's work then led to the development of recursion theory and then mathematical logic as an autonomous discipline in the 1930s.

The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

In 1970, a novel result in mathematical logic known as Matiyasevich's theorem settled the problem negatively: in general Diophantine problems are unsolvable.

In mathematical logic, there are two quantifiers, " some " and " all ", though as Brentano ( 1838 – 1917 ) pointed out, we can make do with just one quantifier and negation.

According to Husserl, this view of logic and mathematics accounted for the objectivity of a series of mathematical developments of his time, such as n-dimensional manifolds ( both Euclidean and non-Euclidean ), Hermann Grassmann's theory of extensions, William Rowan Hamilton's Hamiltonians, Sophus Lie's theory of transformation groups, and Cantor's set theory.

This is the case of the Mycin and Dendral expert systems, and of, for example, fuzzy logic, predicate logic ( Prolog ), symbolic logic and mathematical logic.

Logical empiricism ( aka logical positivism or neopositivism ) was an early 20th century attempt to synthesize the essential ideas of British empiricism ( e. g. a strong emphasis on sensory experience as the basis for knowledge ) with certain insights from mathematical logic that had been developed by Gottlob Frege and Ludwig Wittgenstein.

A finite-state machine ( FSM ) or finite-state automaton ( plural: automata ), or simply a state machine, is a mathematical model of computation used to design both computer programs and sequential logic circuits.

In logic and the foundations of mathematics, formal languages are used to represent the syntax of axiomatic systems, and mathematical formalism is the philosophy that all of mathematics can be reduced to the syntactic manipulation of formal languages in this way.

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation — and in the setting-out — has been exactly restated as the conclusion of the demonstration.

Tree structures are used to depict all kinds of taxonomic knowledge, such as family trees, the biological evolutionary tree, the evolutionary tree of a language family, the grammatical structure of a language ( a key example being S → NP VP, meaning a sentence is a noun phrase and a verb phrase, with each in turn having other components which have other components ), the way web pages are logically ordered in a web site, mathematical trees of integer sets, et cetera.

In mathematics, the phrase up to is useful for modeling fundamental concepts within a realm of mathematical inquiry, and can be compared with the phrase " all other things being equal " in other disciplines.

* Chaterism Where the length of words in a phrase or sentence increase or decrease in a uniform, mathematical way as in " I am the best Greek bowler running ", or " hindering whatever tactics appear ".

Here, the phrase " for all " implicitly requires a universe of discourse to specify which mathematical objects are " all " the possibilities for x.

Such mathematical formulas can be a part of speech in a natural-language phrase, or even assume the role of a full-fledged sentence.

: This book ... is written for people who have no training in mathematics and who may not have actively thought about mathematics since high school, or even during it, but who may wish to experience an act of mathematical imagining and to consider how such an experience compares with the imaginative work involved in reading and understanding a phrase in a poem.

In older mathematical terminology, the phrase " universal graph " was sometimes used to denote a complete graph.

In the mathematical discipline known as group theory, the phrase Suzuki group refers to:

The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly encountered in the mathematical literature.

During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday.

In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle.

Before Cantor, there were only finite sets ( which are easy to understand ) and " the infinite " ( which was considered a topic for philosophical, rather than mathematical, discussion ).

One of the most vigorous and fruitful branches of mathematics [...] a paradise created by Cantor from which nobody shall ever expel us [...] the most admirable blossom of the mathematical mind and altogether one of the outstanding achievements of man's purely intellectual activity.

Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.

* Cantor – Bernstein – Schroeder theorem, a mathematical theorem in set theory

Whole chapters are devoted to the emergence of algebraic logic in the 19th century UK, Cantor and the emergence of set theory, the emergence of mathematical logic in Germany told in a way that downplays Frege's importance, and to Peano and his followers.

Hoffman does give some relatively simple examples of mathematical problems throughout the book ( like Cantor diagonalization argument ) to illustrate some of the ideas in modern mathematics.

In mathematical logic, the theory of infinite sets was first developed by Georg Cantor.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

The Copernican theory of the solar system – that the Earth revolved annually about the Sun – had received confirmation by the observations of Galileo and Tycho Brahe ( who, however, never accepted heliocentrism ), and the mathematical investigations of Kepler and Newton.

* John Crank ( 1916 – 2006 ), British mathematical physicist

* Hanging With Galileo – mathematical derivation of formula for suspended and free-hanging chains ; interactive graphical demo of parabolic versus hyperbolic suspensions.

Isaac Newton's ( 1642 – 1727 ) mathematical explanation of universal gravitation explained the behavior both of objects here on earth and of objects in the heavens in a way that promoted a worldview in which the natural universe is controlled by laws of nature.

In this way we get a proof of the Euler – Maclaurin summation formula by mathematical induction, in which the induction step relies on integration by parts and on the identities for periodic Bernoulli functions.

Engineering – discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of people.

Stated in mathematical terms, for the Cassie – Baxter state to exist, the following inequality must be true.

From 1950 to 1955, Simon studied mathematical economics and during this time, together with David Hawkins, discovered and proved the Hawkins – Simon theorem on the “ conditions for the existence of positive solution vectors for input-output matrices.

* 1822 – Charles Babbage proposes a difference engine in a paper to the Royal Astronomical Society entitled " Note on the application of machinery to the computation of astronomical and mathematical tables ".

It is not known how the name " knapsack problem " originated, though the problem was referred to as such in the early works of mathematician Tobias Dantzig ( 1884 – 1956 ), suggesting that the name could have existed in folklore before a mathematical problem had been fully defined.

" Optimal welfare " usually takes on a Paretian norm, which in its mathematical application of Kaldor – Hicks method.

Although the existence of molecules has been accepted by many chemists since the early 19th century as a result of Dalton's laws of Definite and Multiple Proportions ( 1803 – 1808 ) and Avogadro's law ( 1811 ), there was some resistance among positivists and physicists such as Mach, Boltzmann, Maxwell, and Gibbs, who saw molecules merely as convenient mathematical constructs.

* 1998 – José Enrique Moyal, Israeli mathematical physicist and engineer ( b. 1910 )

The situation changed rapidly in the years 1925 – 1930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrödinger, Werner Heisenberg, Max Born, Pascual Jordan, and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas.

* C. L. Bouton: Nim, a game with a complete mathematical theory, Annals of Mathematics 3 ( 1901 – 02 ), 35 – 39.

Oliver Heaviside FRS ( ( 18 May 1850 – 3 February 1925 ) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations ( later found to be equivalent to Laplace transforms ), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis.

Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither the continuum hypothesis ( CH ), nor the axiom of choice, can be proved from the standard Zermelo – Fraenkel axioms ( ZF ) of set theory.

* Ptolemy's theorem – mathematical theorem described by Ptolemaeus

In mathematical logic, the Peano axioms, also known as the Dedekind – Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.

The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled ; i. e. the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two dimensions.

The object of a stimulus – response model is to establish a mathematical function that describes the relation f between the stimulus x and the expected value ( or other measure of location ) of the response Y: