A MATHEMATICAL MYSTERY TOUR
This is an outline of the topics covered in the 1985 NOVA episode A Mathematical Mystery Tour. People interviewed in the show are only mentioned the first time they appear.
Opening
Bertrand Russell and Principia Mathematica
Kurt Godel
Fermat's Last Theorem
The Goldbach Conjecture
The Riemann Hypothesis
Classification Problem for 4-D manifolds
P not equal to NP Problem
Invariant Subspace Problem for Hilbert Spaces
Jean Dieudonne
Michael Atiyah
Greg Moore
1. Proof Beyond a Doubt
Isaac Newton
Proof
Arithmetic, Geometry, Analysis
"Numbers are the fabric of mathematics."
Prime numbers
Euclid proves there are an infinite number of primes.
Twin primes
The Goldbach Conjecture. All even numbers are the sum of two primes.
Merten's Conjecture (false)
Fermat's Last Theorem
Rene Thom
Paul Erdos
2. Foundations of Mathematics
Aristotle
Euclid and the Elements
Non-Euclidean geometries
Klein bottle
Dimensions
"Time is an example of the fourth dimension."
3. Discovery or Invention
Platonism (mathematical objects are "real")
The Golden Rectangle
Symmetry in nature
Group theory and Galois
4. A Question of Infinity
Irrational numbers
pi
Mersenne primes (2n-1)
Raymond Smullyan
George Cantor and cardinality
Different infinities (alephs)
5. Cracks in the Foundation
Gottlieb Frege
Bertrand Russell
Russell's Paradox: catalogues and metacatalogues
Alfred North Whitehead
Principia Mathematica
Ivor Grattan-Guiness
6. Back to Basics
Nicholas Bourbaki
David Hilbert
"He hoped by taking the meaning out of mathematics... he would free it once and for all from paradox and contradiction."
7. The Uncertain Future
Kurt Godel
Godel's Incompleteness Theorem (no complete system of axioms is complete)
Continuum Hypothesis
Symbolic logic and computers
The 4-Color Problem
"We are not simply, as mathematicians, in the business to get answers, we want to understand."
"Mathematics has lead to the most certain body of knowledge we have."
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