different definitions. All are controversial. Aristotle defined mathematics as: The science of quantity. In Aristotle's classification of the sciences...

two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that...

definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. This became the foundational crisis of mathematics....

Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly...

definition. These definitions are equivalent in the context of a given mathematical structure (Euclidean space, in this case). Second, a mathematical...

to zero to the power of zero as undefined, indefinite, or equal to 1. Controversy exists as to which definitions are mathematically rigorous, and under...

problems were left open by these definitions, which contributed to the foundational crisis of mathematics. Firstly both definitions suppose that rational numbers...

In logic, extensional and intensional definitions are two key ways in which the objects, concepts, or referents a term refers to can be defined. They...

These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics. Irrational numbers can also...

Mathematical physics is the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the...

isomorphism) and the recursive definitions of addition and multiplication from the successor function and mathematical induction. In the mid-19th century...

it into a Lie group, a type of topological group. Abstract structure Isomorphism Equivalent definitions of mathematical structures Forgetful functor...

information on mathematical structures see Wikipedia: mathematical structure, equivalent definitions of mathematical structures, and transport of structure...

have a rigorous definition in mathematics. It is subjective, and often determined in a given situation by the knowledge and experience of those considering...

is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be...

useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved. A classic example of a pathology...

ridge of the hip bone (see ). There are three main types of technical definitions.[definition needed] Power definitions Secondary definitions Extended...

In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects...

cognitive science of mathematics, a theory of embodied mathematics based on conceptual metaphor. Mathematics makes up that part of the human conceptual...

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that...

In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes....

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function...

In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements...

modern mathematics, the foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus...

other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers...

wide Denjoy integral – is one of a number of inequivalent definitions of the integral of a function. It is a generalization of the Riemann integral, and in...

In mathematics, a matrix (pl.: matrices) is a rectangular array or table of numbers or other mathematical objects with elements or entries arranged in...

formal mathematical system based on lambda abstraction and function application. Two definitions of the language are given here: a standard definition, and...

Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. Mathematical proof was revolutionized...

a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic...