Transcript Mathematical Beauty

Beauty as a Guiding Principle
In Search of Scientific Truth
Avshalom C. Elitzur
© Everyone 2007
Permission is granted to everyone to copy and/or use this work or any part of it.
Contents
1. Preface: The Scientific Ideal of Harmony
2. Classical Physics:
Two Harmonies give rise to Disharmony
4. Quantum Theory:
Opposites Unite
3. Special and General Relativity:
Less Assumptions, More Explanations
5. A New Disharmony
What Comes Next?
Are Truth and Beauty Related?
Pythagoras of Samos (ca. 560 – ca. 480 BC(
Mathematical Beauty
Theorem: Every sum of consecutive odd
numbers beginning with 1 is quadratic.
1+3=4=22
1+3+5=9=32
1+3+5+7=16=42
1+3+5+7+9=25=52
…
Human Beauty
Ideas existed prior
to objects!
Come on!
Plato, 427-347 BC
The Beauty in the Axiomatic System:
Simplicity, Parsimony, Unity
Explanations
Assumptions
Euclides, 365-300 AC
The Beauty in the Axiomatic System:
Simplicity, Parsimony, Unity
Euclides, 365-300 AC
Unity Underlying Plurality
God must be a
harmony freak…
J. Kepler, 1571-1630
TWO WORLD VIEWS
Geocentric
(Ptolemy, 85-165)
Heliocentric
(Copernicus, 1473-1543 )
Tetrahedron
Octahedron
Cube
Icosahedron
Dodecahedron
The Insight:
r1/r2 = 1/2
There are 6 planets
Hence 5 spaces between their spheres
There are also 5 perfect solids!
Mysterium Cosmographicum
I’m a genius alright !!!
O yeah?
T. Brahe (1546-1601)
Mars deviates in 8 minutes!
?
I. The orbits of the planets are ellipses,
with the Sun at one focus of the ellipse.
A=+B=Constant
II. The line joining the planet to the Sun
sweeps out equal areas in equal times
as the planet travels around the ellipse.
III. The ratio of the squares of the
revolutionary periods for two planets is
equal to the ratio of the cubes of their
semimajor axes.
Newton Simplifies
“To this purpose the philosophers say, that Nature
does nothing in vain, and more is in vain, when less
will serve; for Nature is pleased with simplicity, and
affects not the pomp of superfluous causes.”
Philosophiae Naturalis Principia Mathematica )1687( Book III, Rule I
I. Newton, 1643-1727
First Law
An object at rest tends to stay in rest and an object in motion tends to stay in motion in a
straight line at constant speed unless acted upon by an external, unbalanced force.
Second Law
The rate of change of momentum of a body is proportional to the resultant force acting on the
body and is in the same direction.
Third Law
To every action (force applied) there is an equal and opposite reaction (equal force applied in
the opposite direction).
Newton Unifies
Newton Unifies
Analyzing

Revealing the underlying unity

Predicting new Phenomena

A Scientific Theory is Beautiful
when its Laws are
•
•
•
•
•
•
•
•
Few
“It’s the simple things that
take your breath away”
Simple
Invariant
Comprehensive
Reveal Order beneath Randomness
Unify Different and even Opposite Phenomena
Incorporate the laws of Earlier Theories
Predict New Phenomena
Classical Physics:
Observational Data gives rise to Mathematical Formalism
J. C. Maxwell, 1831-1879
M. Faraday , 1791-1867
I. Newton, 1643-1727
R. Descartes, 1596-1650
G. Galilei, 1564-1642
J. Kepler, 1571-1630
Contents
1. Preface: The Scientific Ideal of Harmony
2. Classical Physics:
Two Harmonies give rise to Disharmony
4. Quantum Theory:
Opposites Unite
3. Special and General Relativity:
Less Assumptions, More Explanations
5. A New Disharmony
What Comes Next?
Electromagnetism
Mechanics
Galilean Invariance: Natural Law
should appear the Same for every Observer
Galilean Invariance: Natural Law
should appear the Same for every Observer
Galilean Invariance: Natural Law
should appear the Same for every Observer
Galilean Invariance: Natural Law
should appear the Same for every Observer
As long as motion is inertial, David can
• Build a house of cards
• Perform a brain surgery
• Drink tea with old English ladies
• Etc.
Maxwell gets more than he bargained for
J. C. Maxwell, 1831-1879
Maxwell gets more than he bargained for
light
magnetism
radiation
300,000 Km/sec
electromagnetism
electricity
“What would you say have you been told
that the man you go out with has
discovered the nature of the light coming
from these stars?”
What’s so special about the speed of light?
E=elasticity, ρ=viscosity
• The velocity of sound can be derived from the properties of air
• The velocity of water waves can be derived from the properties of water
• The velocity of light can be derived from the properties of aether?
• But no one knows what are the properties of aether!
Einstein demands harmony (& gets it)
A. Einstein, 1879-1955
If Maxwell’s equations are always valid,
What about Galilean invariance?
And if Galilean invariance is always valid,
would Maxwell’s equation lose their
validity?
Contents
1. Preface: The Scientific Ideal of Harmony
2. Classical Physics:
Two Harmonies give rise to Disharmony
4. Quantum Theory:
Opposites Unite
3. Special and General Relativity:
Less Assumptions, More Explanations
5. A New Disharmony
What Comes Next?
Einstein asks (at 16):
• What will I see if I ride on a light ray?
In our case:
• What happens if David, in the moving
room, will play with a light ray rather
than with a rock?
Einstein replies (at 26):
Let’s replace two assumptions
with one:
Theisspeed
of light
is absolute
Space
absolute
Time
is absolute
How can any speed be absolute
for all observers?!
Galileo’s velocity addition rule: V1+V2
Galileo’s velocity addition rule: V1+V2
Galileo’s velocity addition rule: V1+V2
Galileo’s velocity addition rule: V1+V2
How can any speed be absolute for all observers?!
Assumption: Light velocity is constant for all observers

Conclusion: rulers’ length and clock rates are not equal
for all observers
When a frame moves at a near-light velocity, its rulers contract
and its clocks slow down relative to one another, such that light
velocity remains 300,000 km/sec.
Galileo’s velocity addition rule:
Einstein’s velocity addition rule:
v1+v2
v1  v2
v1v2
1 2
c
How can any speed be absolute for all observers?!
Assumption: Light velocity is constant for all observers

Conclusion: Nothing can go faster than light
When an accelerated body approaches light velocity, the
addition of energy cannot turn into velocity, hence it
turns into mass.

E=Mc2
WhyAisfew
Einstein
unhappy?
years later…
(please, pleeeaaase…)
I want my relativity
theory to hold also for
non-inertial frames!
“The happiest thought of my life…”
The Equivalence Principle
David in space
The Equivalence Principle
David freely falling
Mere coincidence of profound principle?
The Equivalence Principle
David resisting gravity
The Equivalence Principle
David accelerates
upwards in space
The Equivalence Principle
Free fall = no gravity
Resisting gravity = acceleration
Acceleration=Gravity
An incoming rock
draws a parabola
David accelerates
upwards in space
The Equivalence Principle
An incoming rock
draws a parabola
David resisting gravity
Einstein’s new question:
• What would David find if he
studies the acceleration=gravity
equivalence with a light ray
rather than with a rock? Would
the principle then turn invalid?
The Equivalence Principle
An incoming rock
draws a parabola
David accelerates
upwards in space
The Equivalence Principle:
Einstein’s new prediction
An incoming light
draws a Parabola!
_______?
David resisting gravity
The Equivalence Principle:
Einstein’s new prediction
"Raffiniert ist der Hergott, aber
boshaft ist er nicht !"
Subtle is the Lord, but not vicious!
!‫ אבל לא מניאק‬,‫פיקח הוא אלוהים‬
Spacetime
inertia
t
H. Minkowski, 1864-1909
y
x
Gravitational
curvature
Mass=Energy
Space=Time
Gravity=Inertia
Pygmalion (2002) Mark Dennis
‫‪Canzone di Leonardo‬‬
‫מילים‪ :‬לאונארדו דא וינצ'י‬
‫לחן‪? :‬‬
‫ביצוע‪ :‬אורנלה ואנוני‬