Definitions of mathematics

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. Pythagoras stated "All is number. Number rules the universe", paraphrased by Plato, from which Platonism historically was the main mathematics school of thought and is still large. His student Aristotle defined mathematics as "the science of quantity", and this definition prevailed until the 18th century. In his classification of the sciences, he further distinguished between arithmetic, which studies discrete quantities, and geometry that studies continuous quantities. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart. Peripatetic/Aristotelian Realism influenced most modern Realism. 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. In the 19th century, as mathematics branched out into abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. Three leading types of definition of mathematics today are called logicist, intuitionist, and formalist, each reflecting a different philosophy of mathematics. However, each has its own flaws, none have achieved mainstream consensus, and all three appear irreconcilable. With mathematicians pursuing greater rigor and more abstract foundations, some proposed defining mathematics purely in terms of deduction and logic: Mathematics is the science that draws necessary conclusions.