VIDEO SUMMARY

1. I.PROOF BEYOND DOUBT
   1. Newton's Light Experiment
      1. The Light Spectrum
      2. Science requires experiment.
      3. Mathematics requires proof.
   2. Kinds of Math
      1. Arithmetic - The study of number
      2. Geometry - The study of shape
      3. Calculus - The study of infinity
   3. Numbers
      1. Prime Numbers
      2. The Infinitude of Primes - Euclid
      3. Twin Primes
   4. Goldbach's Conjecture - All even numbers may be written as the sun of two primes.
   5. Mertens' Conjecture
      1. Demonstrated true up to
      2. Proven to fail before
   6. Fermat's Last Theorem
   7. Dieudonné - "It's tedious to write every step So we write that it is obvious that and if there is an error that is where it is likely to be"
2. The Foundations of Mathematics
   1. Chinese/Egyptian - non proof mathematics.
   2. Euclid (Axioms)
   3. Klein Bottle (Möbius Strip)
   4. Abstraction is piled upon abstraction
3. Invented or Discovered
   1. Platonism
   2. The Golden Rectangle
      1. Architecture
      2. Art
   3. Symmetry
      1. Rotational
      2. Group Theory
      3. Évariste Galois
4. A Question of Infinity
   1. Irrational Numbers
   2. Infinite Series
   3. Prime Numbers
   4. Mersenne Primes
   5. Cantor's Infinities-Transfinite Numbers
5. Cracks in the Foundation
   1. Frege - Reduce Arithmetic to Logic
   2. Syllogism
   3. Bertrand Russell
      1. Catalog of Catalogs
      2. 1+1 = 2 (362 pages) in Principia Mathematica
      3. Certainty and religious faith
6. Back to Basics
   1. Bourbaki
   2. Hilbert's Program concerning Paradoxes in Mathematics
   3. Mathematics as Rule Driven Game
   4. Rigor removes Meaning - What is Rigorous is Meaningless
   5. Gödel's Incompleteness Theorem
   6. The Pragmatic Approach to Mathematics
   7. "Just as Einstein transformed Physics, Godel transformed Mathematics
   8. The Computer and the Four Color Problem
   9. Mathematicians don't want answers, we want to understand.
   10. Mathematics is a human enterprise, we set the rules, we set the goals