0.) A Timeline of the Mathematics of Human History

Prehistory & Antiquity (c. 35,000 BCE – 500 CE)

• c. 35,000–20,000 BCE: Tally bones such as the Lebombo and Ishango bones are carved in Africa, marking humanity's earliest evidence of counting, tracking lunar cycles, and basic arithmetic sequences.
• c. 3000 BCE: Sumerians and Babylonians in Mesopotamia develop a sexagesimal (base-60) positional numeral system, facilitating trade, astronomy, and quadratic algebraic calculations.
• c. 1800 BCE: The Babylonian Plimpton 322 tablet records Pythagorean triples centuries before Pythagoras. In Egypt, texts like the Rhind Mathematical Papyrus outline practical arithmetic, unit fractions, and area calculations.
• c. 800–500 BCE: The Indian Sulba Sutras detail geometric constructions for altar building, including early approximations of $\sqrt{2}$ and explicit uses of the Pythagorean theorem.
• c. 500 BCE: Pythagoras and his followers establish deductive reasoning and formal mathematical proofs in Greece, discovering irrational numbers along the way.
• c. 300 BCE: Euclid writes the Elements, gathering and organizing geometry into an axiomatic system that remains the standard textbook format for two millennia.
• c. 250 BCE: Archimedes of Syracuse uses the method of exhaustion (a precursor to integral calculus) to approximate $\pi$, compute areas, and calculate volumes of revolution.
• c. 200 BCE – 100 CE: Chinese scholars compile The Nine Chapters on the Mathematical Art, detailing Gaussian elimination for systems of linear equations and operations with negative numbers.
• c. 250 CE: Diophantus of Alexandria writes Arithmetica, introducing algebraic notation for powers and unknown quantities.

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The Middle Ages & Islamic Golden Age (500 – 1400 CE)

• 628 CE: Indian mathematician Brahmagupta writes the Brahmasphutasiddhanta, defining zero as a number in its own right, establishing rules for operations with negative numbers, and providing general solutions to quadratic equations.
• c. 820 CE: Muḥammad ibn Mūsā al-Khwārizmī publishes Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala in Baghdad, systematizing algebra as an independent field (from which the word algebra originates).
• c. 1070 CE: Omar Khayyam classifies and solves cubic equations geometrically using intersecting conic sections.
• 1202 CE: Fibonacci introduces the Hindu-Arabic numeral system and place-value notation to Western Europe through his book Liber Abaci.

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Renaissance & The Scientific Revolution (1400 – 1800)

• 1545: Girolamo Cardano publishes Ars Magna, disclosing algebraic solutions for cubic and quartic equations and laying the groundwork for complex numbers.
• 1637: René Descartes publishes La Géométrie, linking algebra and geometry through the Cartesian coordinate system (analytic geometry).
• 1654: Pierre de Fermat and Blaise Pascal establish the foundations of probability theory while analyzing games of chance in personal correspondence.
• 1665–1684: Isaac Newton and Gottfried Wilhelm Leibniz independently develop infinitesimal calculus, creating mathematical tools to model continuous change and physical motion.
• 1736–1748: Leonhard Euler standardizes modern mathematical notation, solves the Seven Bridges of Königsberg problem (founding graph theory), and connects analysis and trigonometry through Euler's identity: $$e^{i\pi} + 1 = 0$$

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The 19th Century: Rigor & Expansion (1800 – 1900)

• 1801: Carl Friedrich Gauss publishes Disquisitiones Arithmeticae, consolidating modern number theory and modular arithmetic.
• 1820s–1830s: Nikolai Lobachevsky and János Bolyai independently invent non-Euclidean hyperbolic geometry, proving that Euclid's parallel postulate is not an absolute rule.
• 1832: Évariste Galois establishes group theory, using abstract symmetry to prove that fifth-degree (quartic) polynomials cannot be solved by general radical formulas.
• 1854: Bernhard Riemann introduces Riemannian geometry, expanding non-Euclidean spaces to higher dimensions—providing the geometric language Einstein would later use for general relativity.
• 1874: Georg Cantor creates set theory and demonstrates that infinite sets come in different sizes, showing that the set of real numbers is strictly larger than the set of integers.

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The 20th Century to Present: Abstraction & Computing (1900 – Present)

• 1900: David Hilbert presents a list of 23 major unsolved problems at the International Congress of Mathematicians, shaping research priorities for the entire century.
• 1931: Kurt Gödel publishes his Incompleteness Theorems, proving that no formal axiomatic system can be both complete and consistent, setting hard limits on mathematical certainty.
• 1936: Alan Turing defines the formal concept of computation using theoretical "Turing machines," founding theoretical computer science.
• 1976: Kenneth Appel and Wolfgang Haken prove the Four Color Theorem using computer software, introducing computer-assisted proofs to mathematics.
• 1994: Andrew Wiles completes the proof of Fermat’s Last Theorem, resolving a problem left open since 1637 by synthesizing modular forms and elliptic curves.
• 2002–2003: Grigori Perelman proves the Poincaré Conjecture, solving the first of the seven Millennium Prize Problems.
• Present: Mathematics heavily intersects with artificial intelligence, cryptography, interactive theorem provers (e.g., Lean), topology, and high-dimensional data analysis.

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What started as practical tally marks for tracking seasons evolved into abstract languages capable of modeling the universe's geometry and the boundaries of computation itself.