Lorenz, "Father of Chaos Theory" died at 90

Edward Lorenz, Professor at MIT died, he was 90. Lorenz, a meteorologist, was known for many of his contributions in Chaos theory, hence nicknamed "the father of Chaos theory".

Among some of his famous findings which have now become popular were:

1. Discovery of deterministic chaos: " His discovery of "deterministic chaos" brought about "one of the most dramatic changes in mankind's view of nature since Sir Isaac Newton," said the committee that awarded Lorenz the 1991 Kyoto Prize for basic sciences. It was one of many scientific awards that Lorenz won." (physorg.com)

2. Lorenz attractor is a relatively simple attractor with complex behavior. This becomes the typical characteristic of chaos, complexity out of simplicity. (see image from "The Lorenz Attractor in 3D", the site has many other images of the Lorenz attractor)


3. Butterfly effect, the scientific concept that small effects lead to big changes, is illustrated by of the Lorenz attractor, see a Java animation here. The butterfly was originally a seagull, here is the story:

In a paper in 1963 given to the New York Academy of Sciences he remarks:

One meteorologist remarked that if the theory were correct, one flap of a seagull's wings would be enough to alter the course of the weather forever.

By the time of his talk at the December 1972 meeting of the American Association for the Advancement of Science in Washington, D.C. the sea gull had evolved into the more poetic butterfly - the title of his talk was^* Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?

4. Every day things are chaotic, chaos leads to creativity and life. The "edge of chaos" is where creativity "happens".

Lorenz's discovery shocked the scientific world. Chaotic systems soon began to be recognised in all branches of science. As mathematicians started to unravel its mysteries, science reeled before the implications of an uncertain world intricately bound up with chance. The human heartbeat is chaotic, the stock market, the solar system and of course the weather. In fact the more we learn about chaos the more closely it seems to be bound up with nature. Fractal structures seem to be everywhere we look: in ferns, cauliflowers, the coral reef, kidneys… Rather than turn its back on chaos, nature appears to use it and science is beginning to do the same.

Recently mathematicians have shown that you can control chaos. For instance here in the Mathematics and Physics Departments at The University of Queensland theoretical and experimental work with lasers shows that the rich structure inherent in chaos can be harnessed to expand the capabilities of lasers. Perhaps in the future single systems, which are capable of multi-tasking, such as the brain, will be modelled by chaotic systems. We still have a lot to learn about how nature uses chaos, but perhaps unpredictable behaviour is not undesirable.

As Henry Adams said "Chaos often breeds life, when order breeds habit" (maths.uq.edu.au)

Other Links:

• Wikipedia on Chaos
• Chaos on the Web
• Sci.Nonlinear Faq
  
• Other posts on Chaos

The Mandelbort Set Was a Discovery of a 13th Century Monk

A retired Math Professor at Harvard University Schipke discovered that the Mandelbrot Set, named after Benoit Mandelbrot, who until now is recognized as the discoverer of the fractal object in 1976 when he was working at IBM, was actually discovered in the 13th century by a mediaeval Benedictine monk, Udo of Aachen. Udo is now nicknamed the Mandelbrot Monk.

On a holiday visit to Aachen cathedral, the burial place of Charlemagne, Schipke saw something that amazed him. In a tiny nativity scene illuminating the manuscript of a 13th century carol, O froehliche Weihnacht, he noticed that the Star of Bethlehem looked odd. On examining it in detail, he saw that the gilded image seemed to be a representation of the Mandelbrot set, one of the icons of the computer age.


Udo's original motivation for the fractal was to calculate who would go to heaven. In so doing he came up with an iteration z -> z*z + c in the complex plane.

Related:

• Fractal FAQ
• Schipke, R.J. and Eberhardt, A. "The forgotten genius of Udo von Aachen", Harvard Journal of Historical Mathematics, 32, 3 (March 1999), pp 34-77.

What Makes Decision Making Hard

We make decisions all the time, some of them we call easy, hard and not so hard. What makes a decision hard?
The following looks at some conditions of a hard decision, and some of the tools available to help us.

Broadly, there are three categories of reasons why decision making can be hard.

Firstly, there is uncertainty.

This includes non-determinism as well as insufficiency of knowledge and information. It also includes complexity of the computing resources to obtain information.
For example, risk of a stock is associated with standard deviation, which is a statistical parameter of a probabilistic variable. However, the standard deviation is only approximated by looking at the historical data of the stock. Data may not be available, and computing resources may limit what can be calculated.
We have already seen this in relation to bounded rationality .
Another aspect of uncertainty is volatility, the uncertainty with respect to changes associated with the passage of time. Historical data may be useless if the system is changing very fast.

Secondly, structural complexity.

Structural complexity refers to the degree of entanglement of the situation. Structural complexity may imply computational complexity as well.
Many cases involve quantitative as well as non-quantitative relations among variables. One variable is likely to influence another, e.g. decrease in one variable will increase the other, but the relation cannot be quantified (see systems thinking and systems dynamics).
The number of variables and the relation and interactions among variables determine structural complexity.
We have seen how in Buddhist Economics , many more considerations such as Ethics and Ecology must be considered in addition to traditional Economics. This increases the structural complexity of the problem.

Finally, conflict.

Conflict refers to conflicting goals, interests and opinions.
Decision making with multiple goals is easy if the different goals can be weighted numerically and thus reduced to a single goal. In general, conflicting goals cannot be so treated.
Conflict can come from internal as well as external sources. It is interesting, that according to Minsky's "society of minds" theory , there are multiple minds inside us, including thinking and emotional minds, beliefs and desires, mostly in conflict with each other.

People found that our decisions are not rational, but instead we make decisions according to which mind happens to be strongest, and afterwards construct narrative rationalizations around them.

The combination of uncertainty, structural complexity and conflict makes decision making hard.
Some tools have been developed to assist us. For example Robert Clemens in his book Making Hard Decisions: An Introduction to Decision Analysis (Business Statistics) listed decision trees, cash flow discounting, probability and statistics, sensitivity analysis, Monte Carlo simulation, and utility theory.
These are useful, but often inadequate.

Systems Dynamics would be good for modeling, if everything is numerical. The system could be run under various what-if settings to produce simulations.
When numerical values are not available, Systems Thinking can be used for modeling. But the model cannot be "executed" by a computer program.
An alternative is FCM (fuzzy cognitive maps ), which uses fuzzy logic to express relations. Fuzzy logic has a natural way of resolving conflicts, allowing various "expert" opinions to be combined.
All three modeling methods have disadvantages. What is needed is perhaps a modeling method where the relations are can be partly numerical, partly fuzzy, and partly qualitative.
The model must be executable, at least for some of the sub-models.
Models must also be decomposable into top-down hierarchies to allow different levels of details.

Econophysics And The Theory Of Games

What has Econophysics to do with the Theory of Games?
Econophysics is the study of Economics by using analogies and methodologies from Physics, particularly that of Statistical Mechanics. The latter uses probability and statistics to derive aggregate properties (Thermodynamics) from a population of interacting particles/molecules. The macro properties of the population is then related to the microscopic properties of the particles.

-------------------------------------------------------------------------------------------------------------
Economics Nobel laureate Markowitz: "I believe that microscopic market simulations have an important role to play in economics and finance. If it takes people from outside economics and finance -- perhaps physicists -- to demonstrate this role, it won't be for the first time that outsiders have made substational contributions to these fields.''
-------------------------------------------------------------------------------------------------------------


In a way this was inevitable, since treating the system as an N-particle system is intractable. We know that the N-body problem is impossible to solve for N>=3, let alone for N a large number.

Now apply this analogy in Economics, but replace the particles with economic agents, and we have Econophysics. The economic agents are interacting in transactions, such as order, bid, ask, buy and sell.

Of course since the interactions are quite different from the interactions of particles in Statistical Mechanics, we expect Econophysics to have very different results. Some objects to Econophysics, because physics deals mostly with equilibrium and linear systems, whereas the new Economics is nonlinear and far from equilibrium in distinction to classical Economics which deals with equilibrium conditions in close analogy with mechanical equilibriums, such as the power of supply and demand, but could not tackle the fluctuations in a stock market, which sometimes lead to chaos and crashes. This is why Econophysics is also considered as a Complexity Science studying dynamical systems.

The model of a population of agents interacting in economic transactions can be studied theoretically or experimentally.
Theoretical studies are possible with simple systems only, such as applying the Ising lattice model to a the case where agents at the lattice nodes buy or sell or stay idle randomly with some probabilities.
Experimental means studying data from the real world, or making computer simulations of models.

The study of Econophysics described overlaps greatly with social algorithms (see "Defining Social Algorithm") which deals with the algorithmic aspects of a population of agents. Social algorithms does have to do with Economics.

Game theory enters the scene as the rules of engagement of the agents. So Game Theory is the microscopic model, and Econophysics tries to derive the global properties of the population of agents. In most cases, the agents will have different behaviors, leading to a multi-agent system.
The minority game (MG) is a famous setting used in the study by Challet and Chang.
The game itself is derived from the "El-Farol Bar" game invented at Santa Fe. Players in the "El-Farol Bar" decide to go or not to go to the bar. If they decide to go, and the bar is crowded, the payoff is low.
The minority game involves agents who buy or sell. If there are more agents buying than selling, the price goes up, and if there are more agents selling, the price goes down. Thus the minority always win (just like in the "El-Farol Bar"), hence the name MG.
It is worth looking at MG's setting, partial information and bounded rationality. Agents only have memory of the last M histories, for a finite M. It does not satisfy the Efficient Market Hypothesis, one of its premises being that the whole history is reflected in the present situation (Markov property). Bounded rationality asserts that people make decisions based only on partial information, even if more complete information is available (but may be too hard to find), and does not always look for optimal solutions (uses rules of thumb instead).

About MG, Challet wrote: "Funny enough is this game known in the French speaking part of Switzerland as "Zig-Zag-Zoug" : if three children must elect a leader for their games, they put their right foot close together, say the magical words, and at the end of "Zoug", they remove their right foot or not. The child who is in the minority wins."

Other than in Economics, Game Theory is also used in social and biological sciences.
The famous Iterated Prisoner's Dilemma has an interesting way of explaining how ethics/altruism develops, when each are supposed to act selfishly in their own interests.
Prisoner's Dilemma is also an example where the Nash equilibrium (Nash from "Beautiful Mind" ) does not give the best for players, which is why it is called a dilemma.

The above is a very elementary introduction to Econophysics and Game Theory.
It presents only one view of Econophysics, there are other people who look at Econophysics as everything where Physics and Mathematics are applied to Economics.
Econophysics in this wider sense, include stochastic processes, power laws, fractal market hypothesis, and "Physics of Finance".

The last is a title of a book "Physics of Finance, Gauge modeling in non-equilibrium pricing" by Kirill Ilinski , a physicist now working in finance. Ilinski starts with invariant properties (symmetry groups) of Quantum Electrodynamics, and applies the analogy to Finance. Both Quantum Electrodynamics and Finance become abstract mathematical theories of fibre bundles and differential geometry. Ilinksi derived Black Scholes purely from principles of invariance albeit under certain rather strong assumptions.

Econophysics has received much attention, see the following sample list of web sites dealing with Econophysics to get some idea of it:

• Econophysics.org ,

• Econophysics Forum ,

• Santa Fe Institute (SFI) ,

• Econophysics in Physica A ,

• Wikipedia Entry on Econophysics ,

• MoneyScience Econophysics Hub .

Some links on Game Theory:

• Minority Game's web page ,

• El-Farol Bar Problem ,

• Wikipedia Entry on Prisoner's Dilemma ,

• Wikipedia Entry on Game Theory ,

• Gametheory.net

Some of my favorite quotes: Einstein

Here are some of my favorite quotes,I will start with Einstein, and will continue later with Dogen, Shunryu Suzuki, Thich Nhat Hanh, Dhammapada, Tao Te Ching, Bhagavad Gita, Rumi, etc If you ask why Einstein, here is one possible answer.

"I like quoting Einstein. Know why? Because nobody dares contradict you." Studs Terkel, Guardian interview (March 1002)

Actually, Einstein was my hero when I was in school and at the university. Later I have other heroes, but I still admire Einstein.

Subject: Religion "I believe in Spinoza's God, Who reveals Himself in the lawful harmony of the world, not in a God Who concerns Himself with the fate and the doings of mankind." (In response the telegrammed question of New York's Rabbi Herbert S. Goldstein in (24 April 1929): "Do you believe in God? Stop. Answer paid 50 words." Einstein replied in only 25 (German) words. Spinoza's ideas of God are often characterized as being pantheistic.) "Science without religion is lame, religion without science is blind." "The religion of the future will be a cosmic religion. It should transcend personal God and avoid dogma and theology. Covering both the natural and the spiritual, it should be based on a religious sense arising from the experience of all things natural and spiritual as a meaningful unity. Buddhism answers this description. If there is any religion that could cope with modern scientific needs it would be Buddhism." "I cannot imagine a God who rewards and punishes the objects of his creation, whose purposes are modeled after our own -- a God, in short, who is but a reflection of human frailty. Neither can I believe that the individual survives the death of his body, although feeble souls harbor such thoughts through fear or ridiculous egotisms." "I am a deeply religious nonbeliever.... This is a somewhat new kind of religion."

Comment: For one like myself who comes from the Buddhist side, it is comforting to know that someone from the side of science came to the same conclusions.

Subject: Self "The true value of a human being is determined by the measure and the sense in which they have obtained liberation from the self." "The fact that man produces a concept "I" besides the totality of his mental and emotional experiences or perceptions does not prove that there must be any specific existence behind such a concept. We are succumbing to illusions produced by our self-created language, without reaching a better understanding of anything. Most of so-called philosophy is due to this kind of fallacy." "A person who is religiously enlightened appears to me to be one who has, to the best of his ability, liberated himself from the fetters of his selfish desires and is preoccupied with thoughts, feelings, and aspirations to which he clings because of their superpersonal value."

Comment: not quite as strong as the doctrine of no-self, but definitely in right direction

Subject: Understanding & Science "The hardest thing to understand is why we can understand anything at all." "A clever person solves a problem. A wise person avoids it." "Education is what remains after one has forgotten everything he learned in school." "One thing I have learned in a long life: All our science, measured against reality, is primitive and childlike — and yet it is the most precious thing we have."

Comment: Science remains our precious tool for truth seekers

Subject: Simplicity and Complexity "Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius — and a lot of courage — to move in the opposite direction." "Everything should be made as simple as possible, but no simpler."

Comment: This is a little different from the discussion of simplicity and complexity in my post, "The Yin and Yang of simplicity and complexity". There I discuss seeing simplicity in complexity, and complexity in simplicity. Einstein talks about, making or fabricating systems, which should be as simple as possible as in Occam's razor. Another similar saying (stated first by Alan Kay?),is: "simple things should be simple, complex things should be possible". This last one is very useful when developing software systems, particularly with regard to user friendliness.

The Yin and Yang of Simplicity and Complexity

The subjects of simplicity and complexity, and how one appears in the other, have always fascinated our minds. In an earlier post, "The mystery of the ubiquity of the power law", we see how the simple power law appears in many complex phenomena. Growth and size of cities is complex, but it obeys the power law. Firm sizes obey power law. The frequencies of words or baby names obey power law. In the stock market, the chance of Google going up 1% is 4 times as much as the a 2% increase, and 16 times as a 4% increase, which shows a power law relation. In all these cases, the underlying phenomena is always complex, the power law is a simple way of understanding, though an incomplete one, of the complex happenings.

Another simple mathematical model is the "iterated prisoner's dilemma (IPD)". Very simple to formulate, but a very complex field of study. Would you believe it that the notion of altruism can be explained using IPD? The variations of IPD has produced an stream of thousands of research papers.

"Everything is simpler than you think and at the same time more complex than you imagine." Johann Wolfgang von Goethe

Stephen Wolfram, the inventor of the software Mathematica, wrote a book "A new kind of science", devoted to cellular automata. Cellular automata have simple rules of interaction with their neighbors, where all the rules are local. The choice of initial conditions and rules are the decisive factors for determining whether the configuration is boring, dead, chaotic, or interesting like the "game of life", and "the garden of eden". The latter is self-reproducing, it can generate itself. Wolfram's program is the find a minimal set of rules and initial conditions which can produce all or some of life. It is known that much of physics can be simulated as finite physics using cellular automata.

Other extreme examples of complexity out of simplicity is how all mathematics can be derived from the empty set, or from the natural numbers 1,2,3, etc.

The science of complexity is the study of how complex phenomena (including chaos) arise from the interactions of agents, who themselves have simple behavior. Traditionally, complexity studies are relatively recent, pre-complexity science looked at simple things only, because that's what science could handle at the time. The mathematics of pre-complexity dealt with simple objects like lines and sine waves, simple functions, functionals and integrals. Pre-complexity science followed the mathematics, and when the reality was too complex, they applied simplifying assumptions and approximations. Social sciences followed the physical analogies, like forces, equilibrium, energy, etc.

"Every decade or so, a grandiose theory comes along, bearing similar aspirations and often brandishing an ominous-sounding C-name. In the 1960 it was cybernetics. In the '70s it was catastrophe theory. Then came chaos theory in the '80s and complexity theory in the '90s." Steven Strogatz

These assumptions broke down when systems are not in equilibrium, such as in fluid turbulence, market crashes, phase transitions, heart attacks, tsunamis and earth quakes and general catastrophes. Complexity science look at complex phenomena from the interactions of components which themselves are defined using simple descriptions. The components are sometimes called agents, and the approach is called agent-based simulation. For example, in market studies, the agents have simple buying and selling rules, and their interactions exhibit properties as in real markets.

Returning to the general simplicity and complexity relations, it can be summarized that the interesting parts are always in seeing simplicity in complexity and in generating complexity out of simplicity. Simplicity and complexity interpenetrates like Yin and Yang.

In ordinary life, we have people who look at simple things and see simple things only, they could aptly be called simpletons. Then there are people who get into complexity, but see only perplexity. They are confused people. Interestingly, they fear complexity, and would try to find comfort in superstitions. The third category of people are those which can see simplicity in complex things, they are natural leaders for they have direction, or see the "through the trees". Capablance, the Cuban ex world chess champion, is a good example of a character who sees and follows through fundamental strategies in complex situations. The last category of people, are those who can see complex things in even the most trivial things. They are deep thinkers, innovators. A person who looks at the empty set and can see the whole mathematics developed from it, is one of this type.

"Chaos begets complexity, and complexity begets life" John Gribbin

References:

• Wikipedia on complexity
• Santa Fe Institute (SFI)
• Map of complexity and related fields
• Wikipedia on Chaos theory
• The Chaos hyperbook
• Complexity in Business