Definitions (Mathematics)

Mathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions. All are controversial.

Survey of leading definitions

Early definitions

Aristotle defined mathematics as:

The science of quantity.

In Aristotle's classification of the sciences, discrete quantities were studied by arithmetic, continuous quantities by geometry.
Auguste Comte's definition tried to explain the role of mathematics in coordinating phenomena in all other fields:

The science of indirect measurement. Auguste Comte 1851

The "indirectness" in Comte's definition refers to determining quantities that cannot be measured directly, such as the distance to planets or the size of atoms, by means of their relations to quantities that can be measured directly.

Greater abstraction and competing philosophical schools

The preceding kinds of definitions, which had prevailed since Aristotle's time, were abandoned in the 19th century as new branches of mathematics were developed, which bore no obvious relation to measurement or the physical world, such as group theory, projective geometry, and non-Euclidean geometry. As mathematicians pursued greater rigor and more-abstract foundations, some proposed definitions purely in terms of logic:

Mathematics is the science that draws necessary conclusions. Benjamin Peirce 1870

All Mathematics is Symbolic Logic. Bertrand Russell 1903

Peirce did not think that mathematics is the same as logic, since he thought mathematics makes only hypothetical assertions, not categorical ones. Russell's definition, on the other hand, expresses the logicist philosophy of mathematics without reservation. Competing philosophies of mathematics put forth different definitions.
Opposing the completely deductive character of logicism, intuitionism emphasizes the construction of ideas in the mind. Here is an inituitionist definition:

Mathematics is mental activity which consists in carrying out, one after the other, those mental constructions which are inductive and effective.

meaning that by combining fundamental ideas, one reaches a definite result.
Formalism denies both physical and mental meaning to mathematics, making the symbols and rules themselves the object of study. A formalist definition:

Mathematics is the manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules.

Still other approaches emphasize pattern, order, or structure. For example:

Mathematics is the classification and study of all possible patterns. Walter Warwick Sawyer, 1955

Yet another approach makes abstraction the defining criterion:

Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined. Wolfram MathWorld

Definitions in nonspecialist reference works

Most contemporary reference works define mathematics mainly by summarizing its main topics and methods:

The abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra. Oxford English Dictionary, 1933

The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. American Heritage Dictionary, 2000

[Mathematics is] the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Encyclopaedia Britannica

Playful, metaphorical, and poetic definitions

Bertrand Russell wrote this famous tongue-in-cheek definition, describing the way all terms in mathematics are ultimately defined by reference to undefined terms:

The subject in which we never know what we are talking about, nor whether what we are saying is true. Bertrand Russell 1901

Many other attempts to characterize mathematics have led to humor or poetic prose:

A mathematician is a blind man in a dark room looking for a black cat which isn't there. Charles Darwin

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. G. H. Hardy, 1940

Mathematics is the art of giving the same name to different things. Henri Poincaré

Mathematics is the science of skilful operations with concepts and rules invented just for this purpose. [this purpose being the skilful operation ....] Eugene Wigner

Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud cell, and is forever ready to burst forth into new forms of vegetable and animal existence. James Joseph Sylvester

What is mathematics? What is it for? What are mathematicians doing nowadays? Wasn't it all finished long ago? How many new numbers can you invent anyway? Is today's mathematics just a matter of huge calculations, with the mathematician as a kind of zookeeper, making sure the precious computers are fed and watered? If it's not, what is it other than the incomprehensible outpourings of superpowered brainboxes with their heads in the clouds and their feet dangling from the lofty balconies of their ivory towers? Mathematics is all of these, and none. Mostly, it's just different. It's not what you expect it to be, you turn your back for a moment and it's changed. It's certainly not just a fixed body of knowledge, its growth is not confined to inventing new numbers, and its hidden tendrils pervade every aspect of modern life. Ian Stewart