Mathematics proves by reason what science checks by measurement
Math studies abstract objects—numbers, shapes, sets, functions, probabilities—using logic and proof rather than experiment. It models the world constantly, yet its theorems stand on deduction. That double life, applied and austere, is older than Euclid and newer than every fresh axiom system.
Major branches include number theory, algebra, geometry, analysis, and set theory used as a modern foundation. Objects may be abstracted from nature or stipulated by axioms; proofs string deductive steps onto earlier results to establish theorems. Engineering, economics and everyday data work lean on those results even when users never see the proof tree underneath.
Written mathematical records appear in ancient Egypt and Mesopotamia, but Greek practice, crystallized in Euclid's Elements, made proof and rigor central. For centuries the subject split mainly into arithmetic and geometry, and astrology and numerology were not always cleanly separated from respectable calculation. Renaissance notation then powered modern algebra, while calculus, differential and integral, grew from geometric problems into the study of continuous change among varying quantities.
That four-part map of arithmetic, geometry, algebra and calculus held until the end of the nineteenth century. Some fields once worked by mathematicians, such as celestial and solid mechanics, now count as physics, and combinatorics, studied for most of recorded history, only became its own branch in the seventeenth century. Since then advances in mathematics and science have fed each other's growth.
A foundational crisis late in the nineteenth century pushed mathematicians toward systematic use of axioms, which triggered an explosion of new areas; today's Mathematics Subject Classification lists more than sixty at the top level. Some, like game theory, grow alongside their applications, while others develop in isolation and only later find practical uses. How mathematical truth relates to logic and to reality remains a live philosophical question.
Source: Mathematics
