Annex A. Mathematical Timeline: 40 Milestones in 4000 Years

A world measured in thought: from clay tokens to code.

Organized into four great eras, each marking a transformation in how humanity conceived number, space, and truth.

A1. Ancient Foundations (c. 2000 BCE – 300 BCE)

No.	Date	Milestone	Description
1	c. 2000 BCE	Babylonian Place Value System	The Sumerians and Babylonians devised a base-60 positional system using cuneiform wedges - enabling large-scale accounting, geometry, and astronomy.
2	c. 1800 BCE	Egyptian Unit Fractions	Egyptian scribes expressed fractions as sums of unit fractions (1/n), revealing algorithmic reasoning in practical computation.
3	c. 1600 BCE	Rhind Mathematical Papyrus	A compilation of 84 problems in arithmetic, geometry, and algebra - documenting early mathematical pedagogy.
4	c. 1000 BCE	Chinese Counting Rods	Movable rods on counting boards introduced positional notation and negative numbers, anticipating the decimal system.
5	c. 600 BCE	Greek Geometric Proofs	Thales and Pythagoras transformed measurement into deduction - founding mathematics as a logical discipline.
6	c. 500 BCE	Pythagorean Theorem Formalized	The relation a² + b² = c² unified number and form, inaugurating mathematical universality.
7	c. 450 BCE	Zeno’s Paradoxes	Logical dilemmas of motion and infinity spurred inquiry into continuity and limit.
8	c. 400 BCE	Indian Sulba Sutras	Geometric constructions for ritual altars, revealing sophisticated approximations of π and √2.
9	c. 370 BCE	Plato’s Academy and Ideal Forms	Geometry elevated to philosophy - mathematics as pathway to eternal truths.
10	c. 300 BCE	Euclid’s Elements	Axiomatic geometry systematized; proof became the standard of certainty for all rational thought.

A2. Classical Transformations (c. 250 BCE – 1200 CE)

No.	Date	Milestone	Description
11	c. 250 BCE	Archimedes’ Method of Exhaustion	Measured curves and volumes via limiting processes - precursor to integration.
12	c. 200 BCE	Indian Decimal Place System Emerges	Positional base-10 notation solidified; foundation for modern numerals.
13	3rd cent. CE	Diophantus’ Arithmetica	Systematic study of equations in integers - proto-algebraic reasoning.
14	5th cent. CE	Chinese Remainder Theorem	Solving congruences across moduli - early modular arithmetic.
15	628 CE	Brahmagupta’s Rules for Zero	Formal arithmetic with zero and negatives; quadratic solutions generalized.
16	820 CE	Al-Khwarizmi’s Al-Jabr	Equation solving codified; algebra and algorithm named.
17	9th cent. CE	House of Wisdom, Baghdad	Translation and synthesis of Greek, Indian, Persian mathematics; algebra and trigonometry flourished.
18	10th cent. CE	Arabic Numerals Spread West	Through trade and scholarship, positional notation reached Europe.
19	11th cent. CE	Omar Khayyam’s Cubic Equations	Intersection of conics used to solve cubics - blending algebra and geometry.
20	1202 CE	Fibonacci’s Liber Abaci	Introduced Hindu-Arabic numerals and commercial arithmetic to Latin Europe.

A3. Early Modern Revolution (1200 – 1800 CE)

No.	Date	Milestone	Description
21	14th cent.	Oxford Calculators’ Kinematics	Quantified velocity and acceleration; seeds of analytic mechanics.
22	1543 CE	Copernican Cosmology	Mathematics re-centered the universe; geometry became cosmic law.
23	1637 CE	Descartes’ Analytic Geometry	Unified algebra and geometry; curves became equations.
24	1654 CE	Pascal–Fermat Correspondence	Probability theory born from games of chance.
25	1665 CE	Newton–Leibniz Calculus	Independent creation of differential and integral calculus.
26	1687 CE	Newton’s Principia	Mathematical physics achieves universality; calculus validated in nature.
27	1713 CE	Bernoulli’s Ars Conjectandi	Foundations of combinatorics and expectation.
28	1748 CE	d’Alembert’s Wave Equation	Differential equations formalize motion and vibration.
29	1755 CE	Euler’s Introductio	Function concept, infinite series, notation; analysis unified.
30	1799 CE	Gauss’ Fundamental Theorem of Algebra	Every polynomial has a complex root; ℂ made complete.

A4. Modern and Digital Age (1800 CE – 2000 CE)

No.	Date	Milestone	Description
31	1821 CE	Cauchy’s Rigorous Limits	Precision replaces intuition; calculus becomes analysis.
32	1830 CE	Galois Theory of Groups	Symmetry structures unify algebraic solutions.
33	1854 CE	Boole’s Algebra of Logic	Thought rendered algebraic; logic mechanized.
34	1872 CE	Dedekind’s Real Numbers	Continuum constructed from rationals via cuts.
35	1890 CE	Cantor’s Set Theory	Infinite hierarchies defined; mathematics re-founded.
36	1931 CE	Gödel’s Incompleteness Theorems	Limits of formal proof exposed.
37	1936 CE	Turing’s Machine Model	Computability formalized; algorithm meets mechanism.
38	1948 CE	Shannon’s Information Theory	Communication and entropy quantified; bits as measures of knowledge.
39	1976 CE	Four-Color Theorem (Computer Proof)	First theorem proved with computational aid; new epistemology of proof.
40	2000 CE	Millennium Prize Problems	Seven unsolved questions define frontiers of 21st-century mathematics.