Update: There were many interesting comments here and on FB. Itay Ben-Dan wrote a very interesting : Alternative to Gil Kalai & Nati Linial 10 Milestones in the History of Mathematics.
In 2006, the popular science magazine “Galileo” prepared a special issue devoted to milestones in the History of several areas of science and Nati Linial and me wrote the article about mathematics Ten milestones in the history of mathematics (in Hebrew). Our article had 10 sections highlighting one or two discoveries in each section.
Here are our choices. What would you add? what would you delete?
1) Numbers and Number Systems – The Irrationality of the square root of 2
Discovery No.1: the square root of 2 is not a rational number.
2) Geometry, the Discovery of Non-Euclidean Geometry, and Topology
Discovery no.2(A): Euclidean Geometry
Discovery no.2(B): Non-Euclidean Geometry
3) Algebra, Equations and Mathematical Formulas. Galois Theory.
Discovery no.3: Abel-Galois Theorem: there is no solution with radicals to the general equation of the fifth degree and above.
4) Analysis and the Connection to Physics
Discovery no. 4(A): Differential and integral calculus (Isaac Newton, Gottfried Leibniz, 17th Century).
Discovery no. 4(B): The analysis of complex functions (Augustin-Louis Cauchy, Bernhard Riemann, 19th century).
5) Proofs and their Limitations: Logic, Set Theory, the Infinity, and Gödel’s Incompleteness Theorem.
Discovery no. 5(A): There are various kinds of infinity. For example, there are more real numbers than natural numbers.
Discovery no. 5(B): Gödel’s Incompleteness theorem: A mathematical theory broad enough includes true statements that cannot be proven.
6) Linear Algebra, Linear Programming and Optimization
Discovery no. 6(A): The Gauss elimination method for solving systems of linear equations.
Discovery no. 6(B): Linear programming and the Simplex algorithm for solving it.
7) Probability Theory and the Bell curve
Discovery no. 7: The Bell Curve and the Central Limit Theorem
8) Prime Numbers and their Density
Discovery no. 8: The Prime Number Theorem.
9) Algorithms, Digital Computers and their Limitations
Discovery no. 9 (a): The theory of computability. Undecidable problems.
Discovery no. 9 (b): Computational Complexity theory. The theory of NP-complete problems.
10) Applied Mathematics
Discovery no. 10: Additional paradigms of mathematical research beyond the paradigm of theorem/proof. Numerical methods, simulations, scientific computation and the development of mathematical models.