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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 picture national education systems where removed 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? picture removed 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. picture removed picture removed picture removed 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 picture numbers from a row that the computer had printed removed 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. picture removed 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. picture removed picture removed Jocelyn Bell picture removed 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 picture removed The Sad Part of the Story Antony Hewish picture removed picture removed 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”. picture removed picture removed 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 picture removed picture removed 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. picture removed 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) picture removed picture removed picture removed Mandelbrot & Julia Sets 5 pictures removed Uncertainty in Quantum Theory 5 pictures removed Uncertainty in Quantum Theory picture removed picture removed 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? picture removed Uncertainty in Quantum Theory picture removed 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. picture removed 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)