Transcript Thomas Filk Slides

Physikalisches Institut
Albert-Ludwigs-Universität Freiburg
Shades of Uncertainty
Thomas Filk
Physics Institute, University of Freiburg
Parmenides Center for Conceptual Foundations of Science, Munich
medx GmbH, Berlin
Cortona
9.9.2016
Do Schools Kill Creativity?
If you are not prepared to be
wrong, you will never come up with
anything original. ... We are running
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national education systems where
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mistakes are the worst thing you
can make. ... The result is that we
are educating people out of their
Sir Ken Robinson
creativity. ...
Education dislocates many people from their natural
talents.... [citing Picasso] All children are born artists,
the problem is to remain an artist as we grow up.
“Do Schools Kill Creativity”, TED Talk 2006 (2010,
2013)
The World Today?
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Juan Enriquez
Content
- The Story of Edward Lorenz – A story of
serendipity, uncertainty and the creation of a new
field in physics and mathematics
- A second story about serendipity ...
- The shades (Uncertainty in Physics): From
Predictability to Ontic Uncertainty – With an
Emphasis on Deterministic Chaos
- The Art of Cheating with Uncertainty:
Simpson’s Paradox.
The Story of Edward Norton Lorenz
Edward Lorenz was a mathematician and meteorologist.
Around 1960 he became interested in
using computers for testing weather
models. His model reduced the NavierStokes field equations (describing the
behavior of fluids) to 12 ordinary
differential equations.
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Edward Norton Lorenz
(1917-2008)
LGP-30
“A Biographical Memoir”
“At one point, in 1961, Ed had wanted to examine
one of the solutions in greater detail, so he
stopped the computer and typed in the 12
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numbers from a row that the computer had printed
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earlier in the integration. He started the machine
again and stepped out for a cup of coffee. When
he returned about an hour later, he found that the
Kerry Emanuel
new solution did not agree with the original one.
At first he suspected trouble with the machine, a
common occurrence, but on closer examination of the output, he
noticed that the new solution was the same as the original for the
first few time steps but then gradually diverged .... He saw that
the divergence originated in the fact that he had printed the
output to three decimal places, whereas the internal numbers
were accurate to six decimal places.”
Chaos Theory: The Result of Serendipity
Deterministic Nonperiodic Flow, in Journal of the Atmospheric Sciences
(1963).
“Two states differing by imperceptible amounts may
eventually evolve into two considerably different states ... In
view of the inevitable inaccuracy and incompleteness of
weather observations, precise very-long-range forecasting
would seem to be nonexistent.”
Edward Lorenz had discovered
“deterministic chaos” and the
“butterfly effect”: the flapping of the
wings of a distant butterfly may
influence the details of a hurricane
several weeks later.
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Lorenz Attractor and Chaos Theory
Lorenz was later able to
reduce the relevant number
of parameters to 3.
Chaotic systems: Minimal changes in the initial conditions
(or the parameters of the system) increase exponentially.
Serendipity
- Having the “lucky finding”
- recognizing the value of this finding
- Capturing the finding to ones advantage
Pasteur: In the fields of observation chance favors
only the prepared mind
Albert Szent-Györgyi: Research (Discovery) is to
see what everybody else has seen, and to think
what nobody else has thought.
A Second Story of Serendipity
Around 1967, as part of her
PhD thesis, she was
working in the group of
Antony Hewish and
constructing a radio
telescope to investigate
quasars.
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Jocelyn Bell
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Antony Hewish
Between July and November 1967 she
analyzed strange signals in her data.
A Second Story of Serendipity
The charts were analyzed by hand by me... Six or eight weeks after
starting the survey I became aware that on occasions there was a bit
of "scruff" on the records... I started going out to the observatory
each day to make fast recordings. They were useless. For weeks I
recorded nothing but receiver noise. The "source" had apparently
gone. Then one day I skipped the observations to go to a lecture, and
next day on my normal recording I saw the scruff had been there. A
few days after that at the end of November '67 I got it on the fast
recording. As the chart flowed under the pen I could see that the
signal was a series of pulses. ... They were 1 1/3 seconds apart. I
contacted Tony Hewish ... and his first reaction was that they must be
man-made. This was a very sensible response in the circumstances,
but due to a truly remarkable depth of ignorance I did not see why
they could not be from a star....
A Second Story of Serendipity
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The Sad Part of the Story
Antony Hewish
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Jocelyn Bell
Antony Hewish was rewarded the Nobel price
in 1974.
Shades of Uncertainty
lack of knowledge
deterministic
epistemic uncertainty
principle “unknowability”
non-deterministic
ontic uncertainty
These distinctions are
- very convenient on a superficial level
- (almost) meaningless on a fundamental level
They depend on the model/theory which is used to
describe “the world”.
Shades of Uncertainty
Lack of knowledge may refer to
- not knowing the laws
- not knowing the parameters in these laws with
sufficient precision
- not knowing the initial conditions with sufficient
precision
- not being able to solve the equations with
sufficient precision in order to make predictions
with sufficient precision.
Degrees of Uncertainty
- Predictability almost independent of uncertainty
(fixed point attractors)
- Predictability “proportional” to lack of uncertainty
(global predictability; planetary motion)
- Predictability “proportional” to logarithmic
uncertainty (deterministic chaos; local
predictability; exponential growth of uncertainty)
- Predictability “proportional” to ...
- “Retro”-dictability of outcome and “not otherwise”
(Leibniz principle of sufficient reason)
- “Retro”-dictability of outcome only
- Principle (ontic) uncertainty
Degrees of Uncertainty
- Predictability almost independent of uncertainty
(fixed point attractors)
- Predictability “proportional” to lack of uncertainty
(global predictability; planetary motion)
- Predictability “proportional” to logarithmic
uncertainty (deterministic chaos; local
predictability; exponential growth of uncertainty)
- Predictability “proportional” to ...
- “Retro”-dictability of outcome and “not otherwise”
(Leibniz principle of sufficient reason)
- “Retro”-dictability of outcome only
- Principle (ontic) uncertainty
Predictability Proportional to Knowledge Uncertainty Proportional to Lack of Knowledge
If the story of Newton’s apple
were true, it would be an
example of serendipity.
Newton’s theory of gravity
shaped the image of physics as
an “exact science”.
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An increase in the
precision of the initial
data by a factor of 2
also increases the time
scale of predictability
by a factor of 2.
Deterministic Chaos
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Deterministic Chaos:
- The fundamental laws are deterministic.
- Any minor uncertainty in the initial conditions leads to an
exponential growth of the uncertainty as a function of time.
Dx(t) = Dx(0)× exp(lt)
(λ>0 Lyapunov exponent)
The Logistic Map
The logistic map describes an iterative
mathematical algorithm:
xn+1 = f(xn)=r⋅xn⋅ (1−xn)
x0  x1 = f(x0)  x2 = f(x1)  x3 = f(x2)  ...
It is a “growth”-map for which the growth factor
r·(1–xn) depends on the value of xn (it gets smaller for
large xn).
Example ( r = 4 ):
x0 = 0.4
x1 = 4 * 0.4 * (1 − 0.4) = 0.96
x2 = 4 * 0.96 * (1 − 0.96) = 0.1536
Logistic Map as Notes
0.0
x
1.0
x → y=5x
0.0
5.0
x
y → z=55.0×2y
55.0
1760.0
5 Octaves
x
Logistic Map as Sound
Do you hear a difference?
Version 1
Version 2
Butterfly Effect
Compare two „time“-series for the same value of r (=3.9) which
initially are very close together: x0=0.65
x0‘=0.650000001
The difference
of the two
series of data.
Volatility
Phase transitions denote structural changes of a system as a
function of external parameters (temperature, pressure, ...).
Typical indicators of a
phase transition:
• large sensitivity of the
system with respect to
minor perturbations,
• large
fluctuations(volatility)
• long-ranged (temporal
and/or spatial)
correlations.
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Lessons for decision-makers
• Already very simple non-linear dynamics can exhibit
“chaotic” and “complex” behavior.
• Beyond a certain point, predictability becomes
almost impossible (the ratio of „increase of effort“ to
„increase of predictability“ becomes immense).
• Complex systems can have stable states of
equilibria as well as chaotic regimes (depending on
details of environmental parameters).
• “Volatility” can be a measure for the degree of
“criticality” – tendency for a phase transition − of a
state.
• The “directions of fluctuations” are indicators for the
critical dimensions.
Complex Systems
zn+1= r zn(1−zn)
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Mandelbrot & Julia Sets
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Uncertainty in Quantum Theory
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Uncertainty in Quantum Theory
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According to the standard interpretation, the
attribute of “having a position” or “having a
momentum” cannot be assigned to a
microscopic particle in all circumstances.
Is the moon still there when nobody looks?
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Uncertainty in Quantum Theory
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If quantum theory is a
fundamental theory, the way we
experience the world will be
fundamentally nondeterministic.
According to the standard
interpretation of quantum theory,
the world is fundamentally
nondeterministic.
In order to predict the state of the pendulum
for more than 2-3 minutes, the accuracy in
the initial conditions has to be larger than is
allowed by the uncertainty principles.
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Simpson’s Paradox
There is a city with 500.000 inhabitants
250.000 are white and 250.000 are black
White
Black
Total
250.000
250.000
Criminals
100.000
(40%)
150.000
(60%)
It turns out that amongst white people there are
100.000 criminals (40%) while amongst black
people there are 150.000 (60%) criminals. But!
poor
White
Black
rich
White
Black
Total
50.000
200.000
Total
200.000
50.000
Criminals
40.000
(80%)
140.000
(70%)
Criminals
60.000
(30%)
10.000
(20%)
Simpson’s Paradox
Simpson’s Paradox:
There are partitions of a total set such that the
correlations in any subset of this partition are
opposite to the correlations in the total set.
White
Black

poor
rich
Criminal
Non Criminal
lurking variable
Simpson’s Paradox
x1 y1
<
X1 Y1
and
x2 y2
<
X2 Y2
14 4
1 6
<
and
<
20 5
5 20
15
14
10
6
4
1
5
20
25
but
x1 + x2 y1 + y2
>
X1 + X2 Y1 +Y2
15 10
but
>
25 25
Simpson’s Paradox
Evaluation of medical treatment
Med. 1
Med. 2
Total
250
250
survivals
rate
100
(40%)
150
(60%)
In total, treatment with drug 2 seems to be better.
However, there are two forms of this disease: Form A
(which is less severe) and Form B (more severe).
Form A
Med. 1
Med. 2
Form B
Med. 1
Med. 2
Total
50
200
Total
200
50
survivals
rate
40
(80%)
140
(70%)
survivals
rate
60
(30%)
10
(20%)
Simpson’s Paradox
Applicants for Berkeley University (Fall 1973)
(actual numbers were different)
Women
Men
Total
2500
2500
accepted
1000
(40%)
1500
(60%)
In total there seems to be a discrimination of women.
Department group 1: high acceptance rate
Department group 2: low acceptance rate
Dept. 1
Women
Men
Dept. 2
Women
Men
Total
500
2000
Total
2000
500
accepted
400
(80%)
1400
(70%)
accepted
600
(30%)
100
(20%)
Simpson’s Paradox
Applicants for Berkeley University (Fall 1973)
(actual numbers were different)
Women
Men
Total
2500
2500
accepted
1000
(40%)
1500
(60%)
“Women are shunted by their socialization and education toward fields
of graduate study that are generally more crowded, less productive of
completed degrees, and less well funded, and that frequently offer
poorer professional employment prospects.” (from final report)
Dept. 1
Women
Men
Dept. 2
Women
Men
Total
500
2000
Total
2000
500
accepted
400
(80%)
1400
(70%)
accepted
600
(30%)
100
(20%)
Simpson’s Paradox
Thank you!
“Women are shunted by their socialization and education toward fields
of graduate study that are generally more crowded, less productive of
completed degrees, and less well funded, and that frequently offer
poorer professional employment prospects.” (from final report)