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Mathematics

"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it..." — Henri Poincaré, 1908

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"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it..." — Henri Poincaré, 1908 Key sections include: Mathematics An essay in nine chapters; The hierarchy of numbers natural · integer · rational · real · complex; Euclid's Elements and the geometries that followed; The infinitesimal, tamed . Newton, Leibniz, Cauchy, Weierstrass; Solving the unsolvable . From al-Khwārizmī to Galois; The trouble with everything . Cantor, Russell, Gödel; Rubber-sheet geometry . Continuity without distance; A small, eccentric guild.; Five beautiful equations.; A skeletal timeline..

Key sections

  • 01Mathematics An essay in nine chapters
  • 02The hierarchy of numbers natural · integer · rational · real · complex
  • 03Euclid's Elements and the geometries that followed
  • 04The infinitesimal, tamed . Newton, Leibniz, Cauchy, Weierstrass
  • 05Solving the unsolvable . From al-Khwārizmī to Galois
  • 06The trouble with everything . Cantor, Russell, Gödel
  • 07Rubber-sheet geometry . Continuity without distance
  • 08A small, eccentric guild.
  • 09Five beautiful equations.
  • 10A skeletal timeline.
  • 11The Millennium problems. and a few other ghosts
  • 12Where the field is moving.
  • 13Watch & read.
  • 14Q.E.D.

Topics covered

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Slide outline
  1. 01Mathematics An essay in nine chapters
  2. 02The hierarchy of numbers natural · integer · rational · real · complex
  3. 03Euclid's Elements and the geometries that followed
  4. 04The infinitesimal, tamed . Newton, Leibniz, Cauchy, Weierstrass
  5. 05Solving the unsolvable . From al-Khwārizmī to Galois
  6. 06The trouble with everything . Cantor, Russell, Gödel
  7. 07Rubber-sheet geometry . Continuity without distance
  8. 08A small, eccentric guild.
  9. 09Five beautiful equations.
  10. 10A skeletal timeline.
  11. 11The Millennium problems. and a few other ghosts
  12. 12Where the field is moving.
  13. 13Watch & read.
  14. 14Q.E.D.
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Slide 01

The hierarchy of numbersnatural · integer · rational · real · complex

  • Chapter I — Number
  • The first abstraction of mathematics is counting. Tally marks on a Lebombo bone (~43,000 BCE) suggest the impulse is older than writing. From counting we reach the natural numbers, ℕ = {0, 1, 2, 3, ...}; from inverses, the integers, ℤ; from ratios, the rationals, ℚ; by completion, the reals, ℝ; by closure under polynomial roots, the complex numbers, ℂ.
  • ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ ⊂ ℍ ⊂ 𝕆(I.1)
  • Each containment is a generalization with a price: ℂ loses ordering, ℍ (Hamilton's quaternions) loses commutativity of multiplication, 𝕆 (octonions) loses associativity. After 𝕆 — the Cayley–Dickson construction continues — but no normed division algebra exists in dimensions other than 1, 2, 4, 8 (Hurwitz, 1898).
  • √2 is irrational.
  • Suppose √2 = p/q in lowest terms. Then 2q² = p², so p is even, p = 2k. Then 2q² = 4k², q² = 2k², so q is even. But then p and q share factor 2, contradicting lowest terms. ∎
  • — First known proof of incommensurability.
Slide 02

Euclid's Elementsand the geometries that followed

  • Chapter II — Geometry
  • Euclid of Alexandria, around 300 BCE, organized centuries of geometric knowledge into thirteen books built on five postulates. The first four were obvious; the fifth — the parallel postulate — was suspect from antiquity. Two millennia of attempts to prove it from the others failed.
  • In the early 19th century, Bolyai, Lobachevsky, and Gauss independently realized why: the parallel postulate is independent of the others. Replacing it gives consistent non-Euclidean geometries. Riemann (1854) generalized further to manifolds of arbitrary curvature; Einstein later cast spacetime as one.
  • Figure II.1 — Three geometries differing only in the parallel postulate.
Slide 03

The infinitesimal, tamed.Newton, Leibniz, Cauchy, Weierstrass

  • Chapter III — Calculus
  • Newton (~1666, "fluxions") and Leibniz (1684, dy/dx notation) gave the world calculus, with rules for derivatives and integrals that immediately transformed physics and astronomy. Yet the underlying foundations — infinitesimals — were unrigorous. Bishop Berkeley's The Analyst (1734) called them "ghosts of departed quantities." It took until Cauchy (1820s) and Weierstrass (1860s) to put limits on solid (ε, δ) ground.
  • Differentiation and integration are inverse operations.
  • If f is continuous on [a, b] and F is any antiderivative, then
  • ∫ab f(x) dx = F(b) − F(a).
  • Conversely, the function F(x) = ∫ax f(t) dt is differentiable and F′(x) = f(x).
  • From this single hinge unfold ordinary differential equations, partial differential equations, the calculus of variations, real and complex analysis, measure theory, functional analysis. Most of physics translates into PDEs.
Slide 04

Solving the unsolvable.From al-Khwārizmī to Galois

  • Chapter IV — Algebra
  • Around 820 CE, al-Khwārizmī's al-Jabr gave us both the discipline's name and a systematic treatment of linear and quadratic equations. The cubic and quartic fell to del Ferro, Tartaglia, Cardano, Ferrari in the 1500s. The quintic did not yield — and Évariste Galois, before dying in a duel at 20, proved why.
  • The general polynomial of degree ≥ 5 has no solution in radicals.
  • The roots of a polynomial form a field extension of ℚ; its automorphism group (the Galois group) acts on them. The polynomial is solvable in radicals if and only if this group is solvable. The general degree-5 polynomial has Galois group S5, which contains the simple non-abelian group A5. S5 is therefore not solvable. ∎
  • Galois's idea — that algebraic structure governs solvability — opened modern algebra: groups, rings, fields, modules, categories.
Slide 05

The trouble with everything.Cantor, Russell, Gödel

  • Chapter V — Foundations
  • Georg Cantor showed in the 1870s that the infinite is not one but many: |ℕ| < |ℝ|. He gave us cardinals, ordinals, the diagonal argument. He died in a sanatorium; the suspicion that he had broken something was widespread.
  • The set of real numbers is uncountable.
  • Suppose to the contrary that ℝ ∩ [0, 1] could be enumerated r1, r2, ... . Construct d by changing the n-th decimal of rn. Then d differs from every rn. ∎
  • Russell's paradox (1901): "the set of all sets that are not members of themselves" leads to contradiction in naïve set theory. Hilbert hoped a complete and consistent axiomatization could rescue mathematics. Gödel (1931) proved otherwise.
  • Any sufficiently powerful, consistent formal system contains true statements it cannot prove.
  • By arithmetizing syntax and constructing a self-referential statement G: "G is not provable in this system," Gödel showed that G is true if and only if it is unprovable. ∎
  • The Continuum Hypothesis — is there a cardinality strictly between ℕ and ℝ? — was shown by Gödel (1940) and Cohen (1963) to be independent of ZFC.
Slide 06

Rubber-sheet geometry.Continuity without distance

  • Chapter VI — Topology
  • A topology specifies which subsets of a space are "open." That is enough to define continuity, convergence, and connectedness, without measuring distance. Two spaces are topologically equivalent (homeomorphic) if one can be deformed into the other without tearing or gluing.
  • Fig. VI.1 — Disk ≅ square (homeomorphic); torus ≇ disk (different genus).
  • The genus — number of holes — is a topological invariant. The Euler characteristic χ = V − E + F links combinatorics to topology. Poincaré conjectured (1904) that any simply connected, closed 3-manifold is homeomorphic to S³; Grigori Perelman proved it (2003) using Hamilton's Ricci flow, declined the Fields Medal and the $1 M Clay Millennium Prize.
Slide 07

Slide 7

  • Plate VII — A working blackboard. Chalk, like proof, leaves traces.
Slide 08

A small, eccentric guild.

  • Chapter VIII — Mathematicians
  • Euclid
  • ~300 BCE. Elements; the axiomatic method.
  • Archimedes
  • ~287–212 BCE. Method of exhaustion; π bounds.
  • al-Khwārizmī
  • ~780–850. Algebra; the algorithm.
  • Newton
  • 1643–1727. Calculus; mechanics.
  • Euler
  • 1707–83. Wrote half of 18th-century mathematics.
  • Gauss
  • 1777–1855. "Princeps mathematicorum."
  • Galois
  • 1811–32. Group theory in 60 pages, age 20.
  • Riemann
  • 1826–66. Manifolds; ζ-function; complex analysis.
  • Cantor
  • 1845–1918. Transfinite arithmetic.
  • Hilbert
  • 1862–1943. 23 problems; foundations.
  • Noether
  • 1882–1935. Abstract algebra; symmetries ↔ conservation laws.
  • Gödel
  • 1906–78. Incompleteness theorems.
  • Erdős
  • 1913–96. ~1,500 papers, no fixed address.
  • Mirzakhani
  • 1977–2017. Riemann surfaces; first female Fields Medalist.
  • Tao
  • b. 1975. Combinatorics, harmonic analysis, PDEs.
  • Witten
  • b. 1951. Physicist Fields Medalist (1990).
Slide 09

Five beautiful equations.

  • Chapter IX — Identities
  • The relation among the five constants.
  • eiπ + 1 = 0
  • Connects 0, 1, π, e, i. "The most beautiful equation in mathematics" (Feynman called it "the most remarkable formula").
  • The right triangle.
  • a2 + b2 = c2
  • The 50-million-times-proved relation; metric of Euclidean space.
  • Sum of inverse squares.
  • Σn=1∞ 1/n2 = π2/6
  • Solved by Euler in 1734; rocketed him to fame.
  • Contour integral around poles.
  • ∮γ f(z) dz = 2πi Σ Res
  • The engine of complex analysis; computes integrals impossible on the real line.
Slide 10

A skeletal timeline.

  • Chapter X — Chronicle
  • ~3000 BCESumerian and Egyptian arithmetic; sexagesimal place value.
  • ~530 BCEPythagoreans; incommensurables.
  • ~300 BCEEuclid's Elements.
  • ~250 BCEArchimedes computes areas, volumes, π bounds.
  • ~250 CEDiophantus's Arithmetica: integer equations.
  • ~628Brahmagupta defines zero, negative numbers.
  • ~820al-Khwārizmī's al-Jabr.
  • ~1500sCardano, Tartaglia, Bombelli; cubics, complex numbers.
  • 1637Descartes: analytic geometry; Fermat's last theorem in margin.
  • 1666–1684Newton and Leibniz independently develop calculus.
  • 1736Euler solves the Königsberg bridges — graph theory.
  • ~1820–60sBolyai/Lobachevsky non-Euclidean geometry; Cauchy & Weierstrass rigor.
  • 1854Riemann's habilitation lecture: manifolds.
  • 1874Cantor on infinity.
  • 1900Hilbert's 23 problems.
  • 1931Gödel's incompleteness theorems.
  • 1995Wiles proves Fermat's Last Theorem.
  • 2003Perelman proves the Poincaré conjecture.
  • 2024AlphaProof / AlphaGeometry achieve IMO-silver-medal performance.
Slide 11

Slide 11

  • Pull quote
  • "Pure mathematics is, in its way, the poetry of logical ideas."— Albert Einstein, 1935
Slide 12

The Millennium problems.and a few other ghosts

  • Chapter XII — Open
  • P vs. NPIs every efficiently checkable problem efficiently solvable? Most expect P ≠ NP; nobody can prove it.
  • Riemann HypothesisAll non-trivial zeros of ζ(s) lie on Re s = ½. Verified for first ~10¹³ zeros; no proof.
  • Yang–Mills mass gapQuantum Yang–Mills has a positive mass gap. Physical "obvious", mathematical "obscure."
  • Navier–StokesSmooth solutions exist and are unique for all time in 3D.
  • Hodge conjectureHodge classes on smooth projective complex varieties are algebraic.
  • Birch & Swinnerton-DyerRank of an elliptic curve = order of vanishing of its L-function at s = 1.
  • Poincaré conjectureSolved by Perelman (2003) — the only Millennium problem so far resolved.
  • Plus: Twin primes infinitude, Goldbach, Collatz 3n+1, abc conjecture, Langlands program.
Slide 13

Where the field is moving.

  • Chapter XIII — Frontier
  • Computer-assisted proof
  • Lean, Coq, Isabelle. The Liquid Tensor Experiment (Scholze's challenge, 2020) succeeded; Kepler's sphere-packing was formally verified by Hales et al.
  • AI for mathematics
  • DeepMind's AlphaProof, AlphaGeometry, FunSearch. LLMs as proof assistants; lemma discovery; routine to brilliant problems.
  • Langlands program
  • A web of conjectures relating Galois groups, automorphic forms, geometry. Geometric Langlands proven 2024 (Gaitsgory et al., 1,000+ pages).
  • Topological data analysis
  • Persistent homology applied to neural data, materials, networks. Algebraic topology gone applied.
  • Quantum computing
  • Algorithms (Shor, Grover, HHL, VQE), error correction codes, post-quantum crypto. Mathematics shaping and shaped.
  • Number theory
  • Bounds on prime gaps (Yitang Zhang, 2013; Polymath); Mochizuki's IUT; Manjul Bhargava on elliptic curves.
Slide 14

Watch & read.

  • Chapter XIV — Go Deeper
  • 3Blue1Brown — Visual mathematics
  • Plus Numberphile, Mathologer, PBS Infinite Series. For long-form: Tao's blog "What's New."
  • Watch ↗
  • References
  • StewartFrom Here to Infinity
  • HardyA Mathematician's Apology (1940)
  • LakatosProofs and Refutations
  • SpivakCalculus
  • Dummit & FooteAbstract Algebra
  • MunkresTopology
  • Princeton Companion to Mathematics (Gowers ed.)
Slide 15

Q.E.D.

  • Finis
  • Quod erat demonstrandum.
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