Calculus worked brilliantly for two centuries before anyone could justify it
Newton and Leibniz built calculus on infinitesimals, numbers somehow infinitely close to zero, and used it to explain why planets move in ellipses. The philosopher George Berkeley mocked them as ghosts of departed quantities. It took until the 19th century to give derivatives and limits a rigorous footing.
For most of history mathematics rested on an ancient Greek model. Aristotle set out how to organise knowledge from axioms and definitions, and Euclid's Elements, around 300 BC, proved every proposition by chains of logical steps. For Aristotle a proved theorem was simply true. Modern mathematics is more modest: a proof shows only that the axioms imply the result.
Cracks appeared early. Zeno of Elea's paradoxes leaned on infinity, a concept nobody handled well for over two thousand years. The Pythagoreans, who believed all numbers were whole numbers or their ratios, were shaken to learn that a square's diagonal and side have no such ratio, which is why such numbers are still called irrational, originally meaning beyond reason. Eudoxus of Cnidus, a student of Plato, found a workaround that anticipated modern definitions of the real numbers.
Descartes' La Géométrie in 1637 turned geometry into algebra with coordinates, paving the way for calculus. Calculus then ran ahead of its logic. In the 19th century Cauchy began making it rigorous, Bolzano in 1817 and later Karl Weierstrass developed the precise definition of a limit, and Weierstrass unsettled everyone with functions continuous everywhere yet smooth nowhere. In 1872 Richard Dedekind and Georg Cantor independently published complete definitions of the real numbers, but both relied on infinite sets, and the ensuing paradoxes triggered what became known as the foundational crisis of mathematics.
The way out was mathematical logic, a new field spanning set theory, proof theory and computability. By the 20th century mathematics had settled on the axiomatic method built on set theory, in the version called ZFC, short for Zermelo–Fraenkel plus the axiom of choice, while type theory now underpins many computer proof assistants. Numbers, points and lines are no longer abstractions from the physical world but objects defined by their axioms, though reality still guides which axioms and theorems mathematicians find worth pursuing.
Source: Foundations of mathematics
