Definitions of Mathematics

Introduction

     Many contemporary references define mathematics by summarizing its main topics:
     "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
These definitions all include relations and other abstractions, and so these definitions are broader than the Aristotelian definition of mathematics as "the science of quantity."
   
     Some mathematicians have attempted to define mathematics under a single principle, such as the study of patterns:
     "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 classification and study of all possible patterns." Walter Warwick Sawyer, 1955.

     Sawyer goes on to explain in his Prelude to Mathematics that patterns are "any kind of regularity that can be recognized by the mind."
Instead of pattern, some see the primary principle as that of order and structure.
     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
   
     Others see the defining criterion as that of abstraction:
     "Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined." Wolfram MathWorld

     Many people understand mathematics as the study of theorems and axiomatic systems. However, this can be problematic as a definition because it does not describe initial mathematical explorations of new topics, including most of the mathematics done in the 17th and 18th centuries. In their book What is Mathematics?, Courant & Robbins caution against emphasizing the deductive character of mathematics over the driving forces of intuition and construction.
     "There is more to mathematics than proof. Indeed, the vast majority of people who earn their living 'doing math' are not engaged in finding proofs at all; their goal is to solve problems to whatever degree of accuracy or certainty is required. While proof remains the ultimate, 'gold standard' for mathematical truth, conclusions reached on the basis of assessing the available evidence have always been a valid part of the mathematical enterprise." Keith Devlin, 2009

Philosophical positions

Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one sort of mathematics that can be discovered

Logicist definitions try to reduce mathematics to logic, especially deductive logic, or set theory, for example: 
                             "all propositions that can be deduced from Zermelo–Fraenkel set theory"

Intuitionist definitions regard mental activity as the essence of mathematics:
     "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 constructs a definite result.

Formalist definitions deny both physical and mental meaning to mathematics, making the symbols and rules themselves the object of study:
     "the manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules" 

Quotations

     Some definitions emphasize the study of patterns and relationships in the abstract:
     "The subject in which we never know what we are talking about, nor whether what we are saying is true.” Bertrand Russell, giving a tongue-in-cheek definition meant to describe the way all terms in mathematics are ultimately defined by reference to undefined terms
     "A mathematician is a blind man in a dark room looking for a black cat which isn't there." Charles Darwin
     "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 the science that draws necessary conclusions" Benjamin Peirce
     "Mathematics is the giving of the same name to different things" Henri Poincaré
     "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