My public lecture on randomness from 2013.

1. Does God play dice?
On probability and randomness.
Adam Kleczkowski
A random walk through
Mathematics and Computing Science
21st February 2013
Computing Science and Mathematics

2. Plan
• Determinism and randomness
• Mechanics as deterministic science
• Randomness
• Deterministic chaos
• Quantum weirdness
• Summary

3. What is Mathematics about
• Not really about remembering big numbers or being able to
multiply/divide quickly
• Real-world problems
• Describe and synthesise knowledge
• Predict
• Generalisation of the assumptions
• How general can we make the theory?
• Can we question its fundamental assumptions?
• Back to real-world
• Does the new theory make it possible to answer even more complicated
questions?

4. Mathematics
How to describe, analyse and predict such different
phenomena like:
• pendulum and clock movements
• planet movements
• movement of dust in air
• weather
• light and electron emission and interference

5. Deterministic and random
• These phenomena can be broadly described as being either
• Deterministic:
• any state is completely determined by prior states
• If we know the history and all parameters, we can
exactly determine the state of the system
or
• Random:
• states cannot be exactly predicted
• but there are regularities in the occurrences of events
even those which outcomes are not certain

6. Deterministic approach
Deterministic approach has a very long history
It is very intuitive and usually associated with Mathematics
2+2=4 (usually)
In a plane, through a point not on a given straight line, at
most one line can be drawn that never meets the given
line. (5th Euclidean postulate)

7. Euclid
of
Alexandria
• Greek mathematician
• ca. 300BC
• wrote Elements, a treatise in 13 books
• a single, logically coherent framework, including
a system of rigorous mathematical proofs
• it remains the basis of mathematics 23
centuries later
Built upon work of Thales (ca 600BC)
and Pythagoras (ca 500BC)

8. Galileo Galilei (1564-1642)
Galileo was one of the first modern thinkers
to clearly state that the laws of nature are
mathematical.
Philosophy is written in this grand book, the
universe ... It is written in the language of
mathematics, and its characters are triangles,
circles, and other geometric figures;....
Put forward the basic principle of relativity:
All laws of mechanics are the same in any system moving with
constant speed

9. Example: Pendulum
• Galileo was the first one to describe
regular movement of pendulum
• For small swings, period is independent
of the different size of swings
• To double the period, the length needs
to be quadrupled
The movement is highly predictable

10. Isaac Newton (1642-1727)
1st law: If no forces act on an object, it
moves with constant velocity and in the
straight line
2nd law:Application of a force results in
acceleration
3rd law:Action of a force results in
counteraction
Newton’s laws of mechanics are a foundation for our view of
the natural world – and they are deterministic!

11. Newton’s laws of gravity
In 1687 Newton published the law of gravity:
• the higher the mass, the stronger the force
• the further away the two masses, the weaker the force

12. Example: Planetary motion
Very regular motion, at least on the scale of many thousands years
Kepler (1571-1630)

13. Planetary orbits…
z-axis direction) at the initial and final parts of the integrations N±1. The axes units are au.
ular momentum. (a) The initial part of N+1 (t = 0 to 0.0547 × 109 yr). (b) The final part
t of N−1 (t = 0 to −0.0547 × 109 yr). (d) The final part of N−1 (t = −3.9180 × 109 to
7
Newton’s laws of gravity
and mechanics can be used
to predict how the Solar
system will look like 109
years from now, that is:
1 000 002 013 AD
Ito & Tanikawa, 2002

14. n-body problem
Henri Poincare (1854-1912)
When more than two bodies interact
by gravitational force, the movement
becomes very complex…
… and unpredictable
For planets and moons, this might take a very long time!
But not for comets or asteroids…

15. Crazy planets

16. Crazy
planets

17. Determinism and randomness
Deterministic approach has been very successful,
but many real-world phenomena can be very unpredictable
Is random outcome completely determined, and random only
by virtue of our ignorance of the most minute contributing
factors? – randomness due to ignorance
Or, is the system deterministic but we cannot control its initial
conditions? – randomness due to uncertainty in initial state

18. Randomness
Means ‘lack of predictability or pattern’
In ancient times associated with ‘fate’
Fortuna: a Roman goddess associated with
‘luck’
Ancients also played games where
‘luck’ was important

19. Randomness
Ordinary people face an inherent difficulty in understanding
randomness, although the concept is often taken as being
obvious and self-evident.
Bennett (1999)
Scientific – rigorous – study came late even though games
of chance played since antiquity

20. Randomness
Democritus, ca. 400 BC
• The world is governed by the unambiguous
laws of order;
• Randomness is a subjective concept
• originates from the inability of humans to
understand the nature of events.
Epicurus, ca. 300 BC
• Everything that occurs is the result of the
atoms colliding, rebounding, and becoming
entangled with one another,
• There is no purpose or plan behind their
motions.

21. Gerolamo Cardano (1501-1576)
Liber de ludo aleae – Book on Games of Chance
is the first known book on probability
also contains a section on cheating…

22. Andrey Kolmogorov (1903-1987)
In 1933 Andrey Kolmogorov
defined probability in an axiomatic way,
finally producing a systematic approach to
the probability theory; as Euclid did 2300
years earlier for geometry
In fact, there are still many unsolved problems in probability
and its relationship to ‘real-world’:
a subject of another talk…

23. Deterministic child with a pie
A walker moving left or right
according to rules
Case 1: going straight
Case 2: moving periodically

24. Deterministic child with a pie
• Keep moving in one direction, or
• Moves some steps to the right and then some
steps to the left
• but we always know where he will be at a
given time…
0 200 400 600 800 1000
0
200
400
600
800
1000
Time
Position
0 200 400 600 800 1000
0
200
400
600
800
1000
Time
Position

25. A random child
Case 3: a child in a crowd

26. A random child
Case 3: a child in a crowd
We do not know exactly
where he will be at a
given time.
0 200 400 600 800 1000
-100-50050100
Time
Position

27. A random child
But we can roughly say:
• study a lot of children and see how far they spread from the
point of the origin…
0 200 400 600 800 1000
-100-50050100
0 2 4 6 8 10-100-50050100
Time
Position
+1

28. A random child
But we can roughly say:
• study a lot of children and see how far they spread from the
point of the origin…
0 200 400 600 800 1000
-100-50050100
0 200 400 600 800 1000
-100-50050100
0 2 4 6 8 10-100-50050100
+1

29. A random child
But we can roughly say:
• study a lot of children and see how far they spread from the
point of the origin…
0 200 400 600 800 1000
-100-50050100
0 200 400 600 800 1000
-100-50050100
0 200 400 600 800 1000
-100-50050100
0 2 4 6 8 10-100-50050100
+1

30. 0 2 4 6 8 10-100-50050100
A random child
But we can roughly say:
• study a lot of children and see how far they spread from the
point of the origin…
0 200 400 600 800 1000
-100-50050100
0 200 400 600 800 1000
-100-50050100
0 200 400 600 800 1000
-100-50050100
0 200 400 600 800 1000
-100-50050100
+1

31. A random child
But we can roughly say:
• study a lot of children and see how far they spread from the
point of the origin…
0 50 100 150 200
-100-50050100
0 200 400 600 800 1000
-100-50050100

32. A random child
This requires a different way to describe the position of a child:
what is a chance of meeting him at a given position?
-100 -50 0 50 100
050100150200
Position
Chance/probability
Probability!

33. Brownian motion
Motion of pollen grains suspended in water (1827)
Robert Brown (1773-1858)

34. Brownian motion
The same mechanism is responsible for dispersion of
colour in water

35. Brownian motion
Albert Einstein (1879-1955)
Marian Smoluchowski (1872-1917)
Theory of Brownian motion (1905, 1906)
Fundamental for random processes
A large particle is being acted upon by many
small, randomly moving particles:
This results in a random motion of the large
particle

36. Quantum cloud
The steel sections were arranged
using a computer model with a
random walk algorithm starting
from points on the surface of an
enlarged figure based on
Gormley's body that forms a
residual outline at the centre of
the sculpture.
Antony Gormley (1950-)

37. Turbulence
• Water and air motion can in principle be
described in terms of deterministic
equations
• These equations are deterministic, so that
the future behaviour can be fully
predicted by laws of motion
• But the behaviour can be very
unpredictable due to sensitivity to initial
conditions and perturbations
CL Navier
(1785-1836)
G Stokes
(1819-1903)

38. Weather
• Weather prediction is probably one
of the hardest problems in physics
• Forecasts are usually phrased in
terms of probability
Latest 15 days ensemble forecast wind for London,
meteogroup.co.uk

39. Weather
EN Lorenz
(1917-2008)
• Lorenz was studying weather models.
• In 1950s he discovered that small
rounding-off errors in some equations can
produce completely different results in the
future.
• This has been called a ‘butterfly effect’

40. Chaos theory
Poincare and later Lorenz discovery opened a new area of
research, called
Deterministic ChaosTheory
In this theory, the rules are deterministic and each trajectory
is well described and predictable, but uncertainty in initial
conditions means a big and increasing uncertainty –
randomness – later on.
M Feigenbaum (1944-)

41. Double pendulum
Double pendulum motion is highly unpredictable, even though it
can be described by the same equations of motion as the single
one!

42. Strange number π
Are the digits ‘predictable’?
Is every digit 0-9 equally represented?
There is also randomness even where we normally do
not expect it…

43. A child with a pi
Move up or down by:
di − mean di( )
e.g.:
3 – move left by 2
1 – move left by 4
4 – move left by 1
1 – move left by 4
5 – move right by 1
9 – move right by 4
3.14159265358979

44. A child with a pi
Move up or down by:
di − mean di( )
e.g.:
3 – move left by 2
1 – move left by 4
4 – move left by 1
1 – move left by 4
5 – move right by 1
9 – move right by 4
0 200 400 600 800 1000
-40
-20
0
20
40

45. A boy with a pi – or rather 1/2013
0 200 400 600 800 1000
-40
-20
0
20
40
• 1/2013 is a rational
number
• There is a pattern to
the digits – they repeat
about every 59 steps
• 1/2013 is not a
‘normal’ number

46. Normal numbers
Emile Borel (1871-1956)
A number is ‘normal’ if in its sequence of
digits, all digits and all subsequences of
digits are equally represented
• Many non-rational numbers appear ‘normal’:
• It has not been proven that π is indeed ‘normal’
but it looks very likely
2
He also stated the ‘infinite monkey
theorem’: a monkey typing long
enough would eventually produce
the complete works of
Shakespeare!

47. Quantum theory
So far we looked at systems
where randomness comes
because we cannot describe
all that is going on
• Like throwing a coin
so in principle, if we could
describe the movement of
hand, fingers and coin , we
could exactly predict heads
or tails
But in quantum world,
unpredictability is
fundamental!

48. Quantum view of light
Light is a wave, caused by electric and magnetic forces
pushing back and forth against each other, like water
waves that are caused by gravity and water pressure
pushing against each other. All waves carry energy.

49. Light also comes in packets, photons!
• When you shine light onto a metal it can
cause an electrical current to flow which
is how solar cells work.
• No matter how faint the light is, electrons
will flow as long as it is the right color.
• This means that atoms receive light
energy in packages, in chunks: You either
get one, or you do not.
This idea that light is a wave of energy that can only be received in
certain specific quantities laid the foundation for the quantum
theory.

50. Randomness
of photons and electrons
• On the atomic level, things
behave in a fundamentally
random way. The location of
a photon or an electron
cannot be predicted
precisely
• Randomness is woven into
the fabric of our world!
N Bohr
(1885-1962)
E Schrödinger
(1887-1961)

51. “God does not play dice.”
• Einstein hated the idea that our world behaves in ways that are
fundamentally random.
• He thought that the quantum theory was just an
approximation to a deeper theory that would allow us to
predict exactly where electrons would be.
• No one has found the deeper theory that Einstein was looking
for…
• Experimental evidence suggests that such a theory does not exist

52. Summary
• Our understanding of the world is largely coloured by
determinism: cause and effect are very closely linked.
• But the real world is often not deterministic, but random.
• Even deterministic equations can often produce a complicated,
apparently random pattern.
• Quantum theory says that the world is fundamentally random
at small scales.
• We need to learn to deal with such random and apparently
complex systems.

53. Thank
you
for
your
a5en6on!
A
Random
Walk
through
Mathema3cs
and
Compu3ng
Science
Spring
2013
h5p://www.maths.s6r.ac.uk/lectures/
The
next
talk
is
by
Ken
Turner
28th
February
2013
7pm