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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

The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects
#Early definitions
Aristotle defined mathematics as:
The science of quantity.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 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.
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