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Definitions of mathematics


Mathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions. All are controversial.

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.

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:

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

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


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