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.

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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.

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

In computer science, an abstract data type ( ADT ) is a mathematical model for a certain class of data structures that have similar behavior ; or for certain data types of one or more programming languages that have similar semantics.

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

The use of aerodynamics through mathematical analysis, empirical approximations, wind tunnel experimentation, and computer simulations form the scientific basis for heavier-than-air flight and a number of other technologies.

Caltech received $ 144 million in federal funding for the physical sciences, $ 40. 8 million for the life sciences, $ 33. 5 million for engineering, $ 14. 4 million for environmental sciences, $ 7. 16 million for computer sciences, and $ 1. 97 million for mathematical sciences in 2008.

Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces.

The term theoretical chemistry may be defined as a mathematical description of chemistry, whereas computational chemistry is usually used when a mathematical method is sufficiently well developed that it can be automated for implementation on a computer.

If numerical iterative methods have to be employed, the aim is to iterate until full machine accuracy is obtained ( the best that is possible with a finite word length on the computer, and within the mathematical and / or physical approximations made ).

In this context, a computational problem is understood to be a task that is in principle amenable to being solved by a computer ( i. e. the problem can be stated by a set of mathematical instructions ).

Graph ( mathematics ) | Graphs like this are among the objects studied by discrete mathematics, for their interesting graph property | mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithm s.

Included within theoretical computer science is the study of algorithms for computing mathematical results.

It is distinguished from computer-generated art, which is produced by a computer using mathematical models created by the artist.

* Project Euler a series of challenging mathematical / computer programming problems

A filter implemented in a computer program ( or a so-called digital signal processor ) is a discrete-time system ; a different ( but parallel ) set of mathematical concepts defines the behavior of such systems.

One of Dijkstra's sidelines was serving as Chairman of the Board of the fictional Mathematics Inc., a company that he imagined having commercialized the production of mathematical theorems in the same way that software companies had commercialized the production of computer programs.

In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data.

The distinction between the two is subtle: " higher-order " describes a mathematical concept of functions that operate on other functions, while " first-class " is a computer science term that describes programming language entities that have no restriction on their use ( thus first-class functions can appear anywhere in the program that other first-class entities like numbers can, including as arguments to other functions and as their return values ).

In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects from a certain collection.

The word " computer " was a job title assigned to people who used these calculators to perform mathematical calculations.

The college offers four-year degrees in chemistry, mathematics, physics, computer science, biology, and engineering, as well as interdisciplinary degrees in mathematical biology, and a joint major in either computer science and mathematics ; or biology and chemistry.

Courtois was influenced by the work of Simon and Albert Ando on hierarchical nearly-decomposable systems in economic modelling as a criterion for computer systems design, and in this book he presents the mathematical theory of these nearly-decomposable systems in more detail than Simon and Ando do in their original papers.

In this connection, it might be noted that the theory of games was a mathematical discovery long before its uses in political science were exploited.

Computer science or computing science ( abbreviated CS or compsci ) is the scientific and mathematical approach in information technology and computing.

It has often been categorized as science fiction although it could more precisely be called " mathematical fiction ".

It fails to fully consider the structure and method of mathematical science, the products of which are arrived at through an internally consistent deductive set of procedures which do not, either today or at the time Mill wrote, fall under the agreed meaning of induction.

" Fixed point " has many meanings in science, most of them mathematical.

It was to be an " Encyclopedia of Mathematics ", containing all known formulae and theorems of mathematical science using a standard notation invented by Peano.

Dissatisfaction with older philosophical approaches had begun earlier and had produced other changes in society, such as the Protestant Reformation, but the revolution in science began when natural philosophers began to mount a sustained attack on the Scholastic philosophical program and supposed that mathematical descriptive schemes adopted from such fields as mechanics and astronomy could actually yield universally valid characterizations of motion and other concepts.

In computer science, an integer is a datum of integral data type, a data type which represents some finite subset of the mathematical integers.

To acknowledge outstanding research contributions to mathematics, through the awarding of scientific prizes and to encourage and support other international mathematical activities, considered likely to contribute to the development of mathematical science in any of its aspects, whether pure, applied, or educational.

** for his profound contributions to two major areas of operations research and management science: stochastic networks and mathematical finance.

According to Gödel mathematical logic was " a science prior to all others, which contains the ideas and principles underlying all sciences.

In mathematical logic and computer science, the Kleene star ( or Kleene operator or Kleene closure ) is a unary operation, either on sets of strings or on sets of symbols or characters.

However, a second definition and usage has historically been in practice in many fields of computer science and information technology, which defines the prefix kilo when used with byte or bit units of data as 1024 ( 2 < sup > 10 </ sup >); this is due to the mathematical coincidence that Thus, in these fields 1 kilobyte is equal to 1 kibibyte, a new unit standardized as part of the binary prefixes to resolve the ambiguity.