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.

In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness.

Gromov's compactness theorem can refer to either of two mathematical theorems:

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

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.

* Automated theorem proving, the proving of mathematical theorems by a computer program

Automated theorem proving ( also known as ATP or automated deduction ) is the proving of mathematical theorems by a computer program.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

His mathematical specialties were noncommutative ring theory and computational algebra and its applications, including automated theorem proving in geometry.

A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation.

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups ( and their action on other mathematical objects ) can sometimes be reduced to questions about finite simple groups.

Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics.

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.

All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.

In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle.

He was awarded the Bôcher Memorial Prize in mathematical analysis in 1964 for his paper " On a conjecture by Littlewood and idempotent measures ", and lends his name to the Cohen-Hewitt factorization theorem.

Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem.

* Ptolemy's theorem – mathematical theorem described by Ptolemaeus

When using an LR parser within some larger program, you can usually ignore all the mathematical details about states, tables, and generators.

All of the possible consistent states of the measured system and the measuring apparatus ( including the observer ) are present in a real physical-not just formally mathematical, as in other interpretations-quantum superposition.

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function.

The claim of " Five users is enough " was later described by a mathematical model which states for the proportion of uncovered problems U

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process ( or " experiment ") consisting of states that occur randomly.

Abstract objects, such as mathematical objects, are ideas, which in turn exist as electrical and chemical states of the billions of neurons in the human brain.

A new generation is created ( advancing t by 1 ), according to some fixed rule ( generally, a mathematical function ) that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood.

A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states.

In mathematical form, Faraday's law states that:

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. showed that it is unprovable in Peano arithmetic ( but it can be proven in stronger systems, such as second order arithmetic ).

A conservation law states that some quantity X in the mathematical description of a system's evolution remains constant throughout its motion — it is an invariant.

In the mathematical areas of order and lattice theory, the Knaster – Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following:

In the mathematical theory of queues, Little's result, theorem, lemma, law or formula is a theorem by John Little which states:

* Functional programming, a programming paradigm based on mathematical functions rather than changes in variable states

One approach and formation is model checking, which consists of a systematically exhaustive exploration of the mathematical model ( this is possible for finite models, but also for some infinite models where infinite sets of states can be effectively represented finitely using abstraction ).

A more mathematical definition is that Fock states are those elements of a Fock space which are eigenstates of the particle number operator.

However, the concept of coherent states has been considerably generalized, to the extent that it has become a major topic in mathematical physics and in applied mathematics, with applications ranging from quantization to signal processing and image processing ( see Coherent states in mathematical physics ).

The state is called a canonical coherent state in the literature, since there are many other types of coherent states, as can be seen in the companion article Coherent states in mathematical physics.

Schrödinger found minimum uncertainty states for the linear harmonic oscillator to be the eigenstates of, and using the notation for multi-photon states, Glauber found the state of complete coherence to all orders in the electromagnetic field to be the right eigenstate of the annihilation operator — formally, in a mathematical sense, the same state.