Definitions of mathematics

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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.[1][2]

Survey of leading definitions[edit]

Early definitions[edit]

Aristotle defined mathematics as:[3]

The science of quantity.

In Aristotle's classification of the sciences, discrete quantities were studied by arithmetic, continuous quantities by geometry.[4]

Auguste Comte's definition tried to explain the role of mathematics in coordinating phenomena in all other fields:[5]

The science of indirect measurement.[3] 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.[6]

Greater abstraction and competing philosophical schools[edit]

The preceding kind of definition, which had prevailed since Aristotle's time,[4] were abandoned in the 19th century as new branches of mathematics — such as group theory,[7] analysis,[8] projective geometry[3] and non-Euclidean geometry.[9] — were developed and which bore no obvious relation to measurement or the physical world. As mathematicians pursued greater rigor and more-abstract foundations, some proposed new definitions of mathematics which are purely based on logic:

Mathematics is the science that draws necessary conclusions.[10] Benjamin Peirce 1870

All Mathematics is Symbolic Logic.[9] 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.[11] Russell's definition, on the other hand, expresses the logicist philosophy of mathematics[12] without reservation. Competing philosophies of mathematics hence put forth different definitions of mathematics.

Opposing the completely deductive character of logicism, intuitionism is another school of thought which emphasizes mathematics as the construction of ideas in the mind:[12]

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

In other words, by combining fundamental ideas together, one reaches a definite result in mathematics.

On the other hand, formalism denies both physical and mental meaning to mathematics, and instead makes the symbols and rules themselves the object of study.[12] For a typical formalist:

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

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.[13] 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.[14]

Definitions in general reference works[edit]

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. [15]Oxford English Dictionary, 1933

The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.[16] American Heritage Dictionary, 2000

The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.[17] Encyclopædia Britannica, 2006

Playful, metaphorical, and poetic definitions[edit]

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.[18] 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.[19] Charles Darwin[20]

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.[21] G. H. Hardy, 1940

Mathematics is the art of giving the same name to different things.[10] Henri Poincaré

Mathematics is the science of skillful operations with concepts and rules invented just for this purpose. [this purpose being the skillful operation ....][22] 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.[23] 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.[23] Ian Stewart

See also[edit]


  1. ^ Mura, Robert (December 1993), "Images of Mathematics Held by University Teachers of Mathematical Sciences", Educational Studies in Mathematics, 25 (4): 375–385, doi:10.1007/BF01273907, JSTOR 3482762, S2CID 122351146
  2. ^ Tobies, Renate; Neunzert, Helmut (2012), Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry, Springer, p. 9, ISBN 978-3-0348-0229-1, It is first necessary to ask what is meant by mathematics in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.
  3. ^ a b c Florian Cajori et al., A History of Mathematics, 5th ed., p. 285–6. American Mathematical Society (1991).
  4. ^ a b James Franklin, "Aristotelian Realism" in Philosophy of Mathematics", ed. A.D. Irvine, p. 104. Elsevier (2009).
  5. ^ Arline Reilein Standley, Auguste Comte, p. 61. Twayne Publishers (1981).
  6. ^ Auguste Comte, The Philosophy of Mathematics, tr. W.M. Gillespie, pp. 17–25. Harper & Brothers, New York (1851).
  7. ^ See History of Group Theory for more.
  8. ^ "The Definitive Glossary of Higher Mathematical Jargon". Math Vault. 2019-08-01. Retrieved 2019-10-18.
  9. ^ a b Bertrand Russell, The Principles of Mathematics, p. 5. University Press, Cambridge (1903)
  10. ^ a b Foundations and fundamental concepts of mathematics By Howard Eves page 150
  11. ^ Carl Boyer, Uta Merzbach, A History of Mathematics, p. 426. John Wiley & Sons (2011).
  12. ^ a b c Snapper, Ernst (September 1979), "The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism", Mathematics Magazine, 52 (4): 207–16, doi:10.2307/2689412, JSTOR 2689412
  13. ^ Sawyer, W.W. (1955). Prelude to Mathematics. Penguin Books. p. 12. ISBN 978-0486244013.
  14. ^ Weisstein, Eric W. "Mathematics". Retrieved 2019-10-18.
  15. ^ "mathematics". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.) mathematics
  16. ^ "mathematics". The American Heritage Dictionary of the English Language (5th ed.). Boston: Houghton Mifflin Harcourt.
  17. ^ Mathematics at the Encyclopædia Britannica
  18. ^ Russell, Bertrand (1901), "Recent Work on the Principles of Mathematics", International Monthly, 4
  19. ^ "Pi in the Sky", John Barrow
  20. ^ Schwartz, Gary E. (2007). The G.O.D. Experiments: How Science Is Discovering God In Everything, Including Us (illustrated ed.). Simon and Schuster. p. 209. ISBN 978-0-7434-7741-3. Extract of page 209
  21. ^ "Quotations by Hardy". Retrieved 2019-10-18.
  22. ^ Wigner, Eugene P. (1960). "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," 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
  23. ^ a b "From Here to Infinity", Ian Stewart

Further reading[edit]