Shabbat:

Friday Candle Lighting: 5:45 PM

Shabbat Ends: 6:40 PM

Torah Message:

The Master of Chaos

“And Avraham expired and died at a good old age, mature and content…” (25:08)

A butterfly flapping its wings in Brazil can cause a tornado in Texas.

Chaos theory is an interdisciplinary theory and branch of mathematics focusing on the study of chaos: dynamical systems whose apparently random states of disorder and irregularities are actually governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as: Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate. It also occurs spontaneously in some systems with artificial components, such as the stock market and road traffic. This behavior can be studied through the analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in a variety of disciplines, including meteorology, anthropology, sociology, environmental science, computer science, engineering, economics, ecology, pandemic crisis management.

I’ve just finished reading a fascinating book called “Chaos: Making a New Science” by James Gleick. It’s a tantalizing book that made me regret not having applied myself with more seriousness to learning mathematics at school. “Chaos” turns much of classical physics on its head:

“The idea that all these classical deterministic systems we’d learned about could generate randomness was intriguing. We were driven to understand what made that tick. You can’t appreciate the kind of revelation that is unless you’ve been brainwashed by six or seven years of a typical physics curriculum. You’re taught that there are classical models where everything is determined by initial conditions, and then there are quantum mechanical models where things are determined but you have to contend with a limit on how much initial information you can gather. Nonlinear was a word that you only encountered in the back of the book. A physics student would take a math course and the last chapter would be on nonlinear equations. You would usually skip that, and, if you didn’t, all they would do is take these nonlinear equations and reduce them to linear equations, so you just get approximate solutions anyway. It was just an exercise in frustration. We had no concept of the real difference that nonlinearity makes in a model. The idea that an equation could bounce around in an apparently random way — that was pretty exciting. You would say, ‘Where is this random motion coming from?’”

And:

“It was a realization that here is a whole realm of physical experience that just doesn’t fit in the current framework. Why wasn’t that part of what we were taught? We had a chance to look around the immediate world—a world so mundane it was wonderful—and understand something. They enchanted themselves and dismayed their professors with leaps to questions of determinism, the nature of intelligence, the direction of biological evolution. The glue that held us together was a long-range vision… It was striking to us that if you take regular physical systems which have been analyzed to death in classical physics, but you take one little step away in parameter space, you end up with something to which all of this huge body of analysis does not apply. The phenomenon of chaos could have been discovered long, long ago. It wasn’t, in part because this huge body of work on the dynamics of regular motion didn’t lead in that direction. But if you just look, there it is. It brought home the point that one should allow oneself to be guided by the physics, by observations, to see what kind of theoretical picture one could develop. In the long run we saw the investigation of complicated dynamics as an entry point that might lead to an understanding of really, really complicated dynamics.”

People don’t know what they see. They see what they think they know.

“And Avraham expired and died at a good old age, mature and content…”

In what sense was Avraham “mature and content”? He could see the order in the “chaos” after looking into every aspect of Creation — higher and further than anyone before him. As a result, he could recognize his Creator. Avraham was indeed a very special soul who could see that “mother nature” has a Father.