Euclid to Godel: The Arc of Proof

c. 300 BCE. The Elements founds the axiomatic method: build everything from a few definitions and postulates by strict deduction.

c. 820 CE. Algebra gets its name and its first systematic treatment as an independent subject.

1202. The Hindu-Arabic positional numerals, and zero with them, reach Europe.

1637. Analytic geometry unites algebra and geometry through the coordinate plane.

1660s-1687. The calculus gives mathematics its language for continuous change.

1730s-1780s. Notation and analysis mature; the most prolific mathematician in history standardizes how the subject is written down.

1796-1801. A nineteen year old proves the heptadecagon constructible, then founds modern number theory outright.

1854-1859. Curved space gets its geometry; the primes get their still-unsolved hypothesis.

1874-1891. Infinity itself becomes an object of proof, and comes in more than one size.

1900. Twenty three problems set the agenda for the coming century, and a program to prove mathematics consistent from the ground up.

1931. Hilbert's program meets its permanent limit: no sufficiently powerful consistent system can prove its own consistency.