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Infinity in Mathematics and Physics

Infinity in Mathematics and Physics. Emilio Elizalde ICE/CSIC & IEEC, Barcelona Trento, June 13, 2006. Infinities. The Bible: stars in heaven , sand grains, 70x7 Zeno ’s paradox (Achilles t ortois e ) & other Euclide’s axioms Euler: infinite series ; zeta z

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Infinity in Mathematics and Physics

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  1. Infinityin Mathematics and Physics Emilio Elizalde ICE/CSIC & IEEC, Barcelona Trento, June 13, 2006

  2. Infinities • The Bible: stars in heaven, sand grains, 70x7 • Zeno’s paradox (Achilles tortoise) & other • Euclide’s axioms • Euler: infinite series; zeta z • Riemann:higher dimensions; zeta z • Cantor: cardinals; paradoxes • QFT: Regul./Renorm. (Einstein, Dirac) • “El Aleph” (Jorge Luis Borges)

  3. 1/2 + 1/4 + 1/8 + 1/16 + . . .= 1

  4. 1/2 +1/4 + 1/8 + 1/16 + … = x 1+ 1/2 +1/4 + 1/8 + 1/16 + … = 2x 1 + x = 2x X = 1 1 – 1 +1 – 1 + 1 – 1 + … = y 1 – (1 – 1 + 1 – 1 + 1 – 1 + … ) = y 1 = 2y 1 - y = y y = 1/2

  5. Set Theory • Georg Cantor • Paradoxes: Bertrand Russell • Axiomatics • Bourbaki School

  6. Barber paradox • In a village there is a barber who shaves every person in the village who does not shave itself And the question is: who shaves the barber ?? Since, if he shaves himself he’ll be a person from the place shaving itself, but he is the barber and, as such, he shouldn’t shave this person! But, if he does not shave himself, he’ll be a person in the village who doesn’t shave itself, but he is the barber and must shave such person! Thus:he can neither shave nor remain unshaved !!

  7. Bertrand Russell’s Paradox • Let’s define the set A = { C | C C } A, C entities • The paradox: If A A, then A A But, if A A, then A A

  8. Hilbert’s Grand Hotel: has infinite rooms, is full! … and still infinite new hosts arrive… WHAT CAN WE DO!? 1 2 3 4 5 6 7 8 . . . . . A1 A2 A3 A4 A5 A6 A7 A8 . . . . . 1 2 3 4 . . . . . A1 1 A2 2 A3 3 A4 4 . . . . .

  9. The cardinals (Alephs) Natural numbers: N א0 Integer numbers: Z א0 Rational numbers: Qא0 Real numbers:R א1 Cantor Does it exist? X: Q < X < RGödel Paul Cohen

  10. Kurt Gödel’s Incompleteness Theorem Crisisof axiomatics Alan Turing’s machine Complexity Cryptography Quantum Computation Peter Shor’s theorem Mathematics Roger Penrose, The Emperor’s New Mind Douglas R. Hofstadter, Gödel, Escher, Bach

  11. Isaac Newton Albert Einstein Physics

  12. Inflation (A. Guth, A. Linde, P. Steinhard, A. Starobinski) Strings, Branes, M Theories The vacuum energy (H.G.B. Casimir) Obs. Cosmology DNA & Genome Codes &Cryptography Computational Biology Quantum Computation Nanotechnology Recentideas & trends

  13. Understanding the Universe • Presocratics:substance, number, power, infinity, movement, being, atom, space, time, ... • Pythagorean School:“all things are numbers” • Emmanuel Kant:“the problem is to make inteligible the idea itself of an inteligible Universe” • Albert Einstein:“the eternal mystery of the Universe is its comprehensibility”; “the fact that the Universe is so comprehensible is a miracle” • Eugene Wigner:“the unreasonable effectiveness of mathematics in the natural sciences” Did you ever think about that?

  14. EL ALEPH JORGE LUIS BORGES O God, I could be bounded in anutshell and count myself a King of infinite space. Hamlet, II, 2. … En la parte inferior del escalón, hacia la derecha, vi una pequeña esfera tornasolada, de casi intolerable fulgor. Al principio la creí giratoria; luego comprendí que ese movimiento era una ilusión producida por los vertiginosos espectáculos que encerraba. El diámetro del Aleph sería de dos o tres centímetros, pero el espacio cósmico estaba ahí, sin disminución de tamaño. Cada cosa (la luna del espejo, digamos) era infinitas cosas, porque yo claramente la veía desde todos los puntos del universo ...

  15. • "It is said that there is no such thing as a free lunch. But the universe is the ultimate free lunch". A. Guth. • The fundamentals of the Universe were created in"the first three minutes”. S. Weinberg. • How does our Universe evolve? And how did structures like stars and galaxies form? Contemporary cosmology for the general reader. T. Padmanabhan.

  16. Thanks so much for your attention

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