Transcript 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 tortoise)

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 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)
1/2 + 1/4 + 1/8 + 1/16 + . . .= 1
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-y=y
1 = 2y
y = 1/2
Set Theory
•
•
•
•
Georg Cantor
Paradoxes: Bertrand Russell
Axiomatics
Bourbaki School
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 !!
Bertrand Russell’s Paradox
• Let’s define the set
A={C| C
C }
A, C entities
• The paradox:
If
A
But, if
 A,
A
 A,
then
A
then
A
A
A
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
A1
1
2
A2
2
3
A3
3
4
A4
4
.....
.....
The cardinals (Alephs)
Natural numbers:
N
Integer numbers:
Z
Rational numbers:
Q
Real numbers:
R
‫א‬0
‫א‬0
‫א‬0
‫א‬1
Cantor
Does it exist? X: Q < X < R Gödel
Paul Cohen
Mathematics
► Alan Turing’s machine
Gödel’s
Incompleteness Theorem ► Complexity
► Crisis of axiomatics
► Cryptography
► Quantum Computation
► Peter Shor’s theorem
► Kurt
Roger Penrose, The Emperor’s New Mind
Douglas R. Hofstadter, Gödel, Escher, Bach
Physics
Isaac
Newton
Mm
F G 2
r
Albert
Einstein
E  mc
2
tot  r  m  k  
Recent ideas & trends
• 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
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?
EL ALEPH
JORGE LUIS BORGES
O God, I could be bounded in a nutshell 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 ...
• "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.
Thanks so much
for your attention