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.

Differential geometry is a mathematical discipline that uses the techniques of differential calculus and integral calculus, as well as linear algebra and multilinear algebra, to study problems in geometry.

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.

Due to a perceived lack of scientific rigour in and overly descriptive nature of the discipline, and a continued separation of geography from geology and the two subfields of physical and human geography, geographers in the mid-20th century began to apply statistical and mathematical model methods to solving spatial problems.

The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry ; thus, " applied mathematics " is a mathematical science with specialized knowledge.

Modern group theory — a very active mathematical discipline — studies groups in their own right.

Science fiction author and scientist / science writer Isaac Asimov popularized the term in his famous Foundation series of novels, though in his works the term is used fictionally for a mathematical discipline that can be used to predict the general course of future flow.

There is also a discipline called mathematical statistics that studies statistics mathematically.

The concept of discrete-time signal processing also refers to a theoretical discipline that establishes a mathematical basis for digital signal processing, without taking quantization error into consideration.

However, although such external forces may change the direction of some mathematical research, there are strong internal constraints — the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated — that work to conserve the historically defined discipline.

In the mathematical discipline of general topology, Stone – Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX.

Khayyám was very careful to respect the boundaries of each discipline, and criticized ibn al-Haytham in his proof of the parallel postulate precisely because he had broken this rule and had brought a subject belonging to natural philosophy, that is, motion, which belongs to natural bodies, into the domain of geometry, which deals with mathematical bodies.

Since its crystallization in the 1930s, computation has been primarily approached from two traditions: engineering, which seeks to build practical systems using computations ; and mathematics, which seeks to prove theorems about computation ( albeit already in the 1970s computing as a discipline was described as being at the intersection of mathematical, engineering, and empirical / scientific traditions ).

In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results.

The systematic search for the foundations of mathematics started at the end of the 19th century, and formed a new mathematical discipline called mathematical logic, with strong links to theoretical computer science.

The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing ( CAD / CAM ), but many problems in computational geometry are classical in nature, and may come from mathematical visualization.

Norman applied his training as an engineer and computer scientist, and as an experimental and mathematical psychologist, to the emerging discipline of cognitive science.

2003 US mortality ( life table | life ) table, Table 1, Page 1Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries.

Actuarial science became a formal mathematical discipline in the late 17th century with the increased demand for long-term insurance coverages such as Burial, Life insurance, and Annuities.

Independently from each other, compound interest was studied and probability theory emerged as a well understood mathematical discipline.

Traditionally a highly mathematical discipline, modern population genetics encompasses theoretical, lab and field work.

* Number theory, the mathematical discipline which studies whole numbers

However, it is essential that the various mathematical symbols used in the equations be understood so that the mathematical processes can be done properly and in their correct order.

As his main mathematical achievements (" maître-thèmes "), he chose this collection of 12 topics ( his chronological order ):

Category theory is also, in some sense, a continuation of the work of Emmy Noether ( one of Mac Lane's teachers ) in formalizing abstract processes ; Noether realized that in order to understand a type of mathematical structure, one needs to understand the processes preserving that structure.

Lattice theory captures the mathematical structure of order relations.

The origin of the goes back to Gottfried Leibniz, who in the seventeenth century, after having constructed a successful mechanical calculating machine, dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements.

( b ) defining mathematical rules such as ( a + b ) = ( b + a ) stating that when adding two numbers, the order of numbers does not matter ( see commutativity ).

Perhaps this assumption entailed a mathematical relationship that would restore astronomical order.

TeX concentrated on detailed layout of text and font descriptions in order to typeset mathematical books in professional quality.

This philosophy, poorly understood at first, stated that in order for a mathematical statement to be true to a mathematician, that person must be able to intuit the statement, to not only believe its truth but understand the reason for its truth.

In a mathematical expression, the order of operation is carried out from left to right.

In his 1944 Russell's mathematical logic, Godel offers a " critical but sympathetic discussion of the logicistic order of ideas ":

The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers.

For example, Spearman is a non-parametric test as it is computed from the order of the data regardless of the actual values, whereas Pearson is a parametric test as it is computed directly from the data and can be used to derive a mathematical relationship.

The players of the game may not need to use mathematics in order to play mathematical games.

For example, Mancala is a mathematical game, because mathematicians can study it using combinatorial game theory, even though no mathematics is necessary in order to play it.

In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation.

In the mathematical theories of the planetary orbits that form the basis of ephemerides, quadratic and higher order secular terms do occur, but these are mostly Taylor expansions of very long time periodic terms.

This mathematical combination creates a problem for crawlers, as they must sort through endless combinations of relatively minor scripted changes in order to retrieve unique content.

These studies used mathematical models of several cities and their emission sources in order to compare the cost and effectiveness of various control strategies.

Friedrich Wilhelm Joseph Schelling ( 1775 – 1854 ), his student Johann Baptist von Spix ( 1781-1826 ) and several others followed the idea that was a hidden and innate mathematical order in the forms of birds.

In FM screening, dot size remains constant and dots are placed in random order to create darker or lighter areas of the image ; dot placement is precisely controlled by sophisticated mathematical algorithms.

In the mathematical field of group theory, the Monster group M or F < sub > 1 </ sub > ( also known as the Fischer-Griess Monster, or the Friendly Giant ) is a group of finite order:

This issue arises in first order theories of arithmetic, such as Peano arithmetic, because the principle of mathematical induction is expressed as an infinite set of axioms ( an axiom schema ).