Definitions of mathematics
Encyclopedia
Mathematics
has no generally accepted definition. Different schools of thought, particularly in philosophy
, have put forth radically different definitions. All are controversial.
defined mathematics as:
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 "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.
, projective geometry
, and non-Euclidean geometry
. As mathematicians pursued greater rigor and more-abstract foundations
, some proposed definitions purely in terms of logic
:
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:
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:
Still other approaches emphasize pattern, order, or structure. For example:
Yet another approach makes abstraction the defining criterion:
Many other attempts to characterize mathematics have led to humor or poetic prose:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
has no generally accepted definition. Different schools of thought, particularly in philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
, have put forth radically different definitions. All are controversial.
Early definitions
AristotleAristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
defined mathematics as:
The science of quantity.
In Aristotle's classification of the sciences, discrete quantities were studied by arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
, continuous quantities by geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
.
Auguste Comte
Auguste Comte
Isidore Auguste Marie François Xavier Comte , better known as Auguste Comte , was a French philosopher, a founder of the discipline of sociology and of the doctrine of positivism...
's definition tried to explain the role of mathematics in coordinating phenomena in all other fields:
The science of indirect measurement. Auguste ComteAuguste ComteIsidore Auguste Marie François Xavier Comte , better known as Auguste Comte , was a French philosopher, a founder of the discipline of sociology and of the doctrine of positivism...
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 theoryGroup theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, and non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
. As mathematicians pursued greater rigor and more-abstract foundations
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
, some proposed definitions purely in terms of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
:
Mathematics is the science that draws necessary conclusions. Benjamin PeirceBenjamin PeirceBenjamin Peirce 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 philosophy of mathematics....
All Mathematics is Symbolic Logic. Bertrand RussellBertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
1903
Peirce did not think that mathematics is the same as logic, since he thought mathematics makes only hypothetical assertions, not categorical
Categorical proposition
A categorical proposition contains two categorical terms, the subject and the predicate, and affirms or denies the latter of the former. Categorical propositions occur in categorical syllogisms and both are discussed in Aristotle's Prior Analytics....
ones. Russell's definition, on the other hand, expresses the logicist philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
without reservation. Competing philosophies of mathematics put forth different definitions.
Opposing the completely deductive character of logicism, intuitionism
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
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
Formalism (mathematics)
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....
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 SawyerWalter Warwick SawyerWalter Warwick Sawyer, or W. W. Sawyer, was a mathematician,mathematics educator and author, who taught on several continents -Life and career:Walter Warwick Sawyer was born in London, England on April 5, 1911. He attended...
, 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 DictionaryOxford English DictionaryThe Oxford English Dictionary , published by the Oxford University Press, is the self-styled premier dictionary of the English language. Two fully bound print editions of the OED have been published under its current name, in 1928 and 1989. The first edition was published in twelve volumes , and...
, 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 RussellBertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
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 DarwinCharles DarwinCharles Robert Darwin FRS was an English naturalist. He established that all species of life have descended over time from common ancestry, and proposed the scientific theory that this branching pattern of evolution resulted from a process that he called natural selection.He published his theory...
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. HardyG. H. HardyGodfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
, 1940
Mathematics is the art of giving the same name to different things. Henri PoincaréHenri PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
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 SylvesterJames Joseph SylvesterJames Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...
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 StewartIan Stewart (mathematician)Ian Nicholas Stewart FRS is a professor of mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer. He is the first recipient of the , awarded jointly by the LMS and the IMA for his work on promoting mathematics.-Biography:Stewart was born...