Definitions of mathematics
Aristotle defined mathematics as:
The science of quantity.
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:
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.
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.
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
General, nonspecialist perspectives
Most contemporary reference works define mathematics mainly by summarizing its main topics and methods and referencing its history:
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
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:
Many other attempts to characterize mathematics have led to humor or poetic prose:
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 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
- Mura, Robert (Dec 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics 25 (4): 375–385.
- Tobies, Renate and Helmut Neunzert (2012). Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. pp. 9. ISBN 3-0348-0229-3.
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.
- James Franklin, "Aristotelian Realism" in Philosophy of Mathematics", ed. A.D. Irvine, p. 104. Elsevier (2009).
- Arline Reilein Standley, Auguste Comte, p. 61. Twayne Publishers (1981).
- Florian Cajori et al., A History of Mathematics, 5th ed., p. 285–6. American Mathematical Society (1991).
- Auguste Comte, The Philosophy of Mathematics, tr. W.M. Gillespie, pp. 17–25. Harper & Brothers, New York (1851).
- Bertrand Russell, The Principles of Mathematics, p. 5. University Press, Cambridge (1903)
- Foundations and fundamental concepts of mathematics By Howard Eves page 150
- Carl Boyer, Uta Merzbach, A History of Mathematics, p. 426. John Wiley and Sones (2011).
- 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.
- "Mathematics. Encyclopaedia Britannica from Encyclopaedia Britannica 2006 Ultimate reference Suite DVD.
- Russell, Bertrand (1901), "Recent Work on the Principles of Mathematics", International Monthly 4.
-  9.999...reasons that .999...=1, Vi Hart
- "Pi in the Sky", John Barrow
- What is mathematics?
- "From Here to Infinity", Ian Stewart
- Courant, Richard; Robbins, Herbert (1996), What is Mathematics? (2nd ed.), Oxford University Press, ISBN 978-0-19-510519-3.
- Gowers, Timothy; Barrow-Green, June; Leader, Imre, eds. (2008), The Princeton Companion to Mathematics, Princeton University Press, ISBN 978-0-691-11880-2.
- Hersh, Reuben (1999), What is Mathematics, Really?, Oxford University Press, ISBN 978-0-19-513087-4.
- Paulos, John Allen (1991), Beyond Numeracy, Viking, ISBN 978-0-670-83654-3.
- Stewart, Ian (1996), From Here to Infinity, Oxford University Press, ISBN 0-19-283202-6.