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

Early definitions

Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politi ...

stated "All is number. Number rules the universe", paraphrased by

Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...

, from which

Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at l ...

historically was the main mathematics school of thought and is still large. His student

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...

defined mathematics as "the science of quantity", and this definition prevailed until the 18th century. In his classification of the sciences, he further distinguished between arithmetic, which studies discrete quantities, and

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

that studies continuous quantities. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart. Peripatetic/ Aristotelian Realism influenced most modern Realism. Auguste Comte's definition tried to explain the role of mathematics in coordinating phenomena in all other

fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...

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

In the 19th century, as mathematics branched out into abstract topics such as

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

and

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...

, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. Three leading types of definition of mathematics today are called logicist,

intuitionist In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...

, and formalist, each reflecting a different

philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in peop ...

. However, each has its own flaws, none have achieved mainstream consensus, and all three appear irreconcilable.

Logicism

With mathematicians pursuing greater rigor and more abstract foundations, some proposed defining mathematics purely in terms of deduction and

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

Mathematics is the science that draws necessary conclusions.Foundations and fundamental concepts of mathematics By Howard Eves
page 150
Benjamin Peirce Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philoso ...
1870

All Mathematics is Symbolic Logic.Bertrand Russell, ''The Principles of Mathematics,'
p. 5
University Press, Cambridge (1903)
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
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 view without reservation.

Intuitionism

Rather than characterize mathematics by deductive logic, intuitionism views mathematics as primarily about the construction of ideas in the mind:

The only possible foundation of mathematics must be sought in this construction under the obligation carefully to watch which constructions intuition allows and which not. L. E. J. Brouwer 1907

... intuitionist mathematics is nothing more nor less than an investigation of the utmost limits which the intellect can attain in its self-unfolding.
Arend Heyting __NOTOC__ Arend Heyting (; 9 May 1898 – 9 July 1980) was a Dutch mathematician and logician. Biography Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a foot ...
1968

Intuitionism sprang from the philosophy of mathematician L. E. J. Brouwer and also led to the development of a modified

intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...

. As a result, intuitionism has generated some genuinely different results that, while coherent and valid, differ from some theorems grounded in classical logic.

Formalism

One other perspective,

formalism Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scie ...

, de-emphasizes logical or intuitive meanings altogether, grounding mathematics instead in its symbols and syntax rules for manipulating them:

Mathematics is the science of formal systems.
Haskell Curry Haskell Brooks Curry (; September 12, 1900 – September 1, 1982) was an American mathematician and logician. Curry is best known for his work in combinatory logic. While the initial concept of combinatory logic was based on a single paper by ...
1951

Other views

Aside from the definitions above, other definitions approach mathematics by emphasizing the element of pattern, order or structure. For example:

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

Yet another approach is to make abstraction the defining criterion:

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

Contemporary general reference works

Most contemporary reference works define mathematics 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 The ''Oxford English Dictionary'' (''OED'') is the first and foundational historical dictionary of the English language, published by Oxford University Press (OUP). It traces the historical development of the English language, providing a co ...
, 1933

The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.
American Heritage Dictionary American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, p ...
, 2000

The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.
Encyclopædia Britannica The (Latin for "British Encyclopædia") is a general knowledge English-language encyclopaedia. It is published by Encyclopædia Britannica, Inc.; the company has existed since the 18th century, although it has changed ownership various t ...
, 2006

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 Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
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 Charles Robert Darwin ( ; 12 February 1809 – 19 April 1882) was an English naturalist, geologist, and biologist, widely known for his contributions to evolutionary biology. His proposition that all species of life have descended ...

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 Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
, 1940

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

Mathematics is the science of skillful operations with concepts and rules invented just for this purpose. his purpose being the skillful operation ....ref>Wigner, Eugene P. (1960). "
The Unreasonable Effectiveness of Mathematics in the Natural Sciences "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a 1960 article by the physicist Eugene Wigner. In the paper, Wigner observes that a physical theory's mathematical structure often points the way to further advances in that ...
," ''Communications in Pure and Applied Sciences'', 13(1960):1–14. Reprinted in ''Mathematics: People, Problems, Results,'' vol. 3, ed. Douglas M. Campbell and John C. Higgins
p. 116
/ref>
Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his co ...

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."From Here to Infinity", Ian Stewart 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