Chaos: making a new science​​

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Chaos, by James Gleick, is a true masterpiece that makes a complex topic accessible. However, this book isn't for readers seeking in-depth, up-to-date details on chaos and complexity. Instead, it's a compelling narrative of chaos theory's origins and the individuals who shaped it, portraying scientists as heroes. This dual focus—on both people and science—is both a strength and a weakness.

 

To enhance your reading experience, consider using the cheat sheets (below), which provide overviews of the key players and concepts discussed in the book.

 

Strengths

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Gleick's writing is a major strength. For a work on such an arcane scientific topic, it's beautifully written and a pleasure to read, not just for its content but also for its style and flow. In fact, Gleick surpasses even himself; the section on information theory is more elegant (though abbreviated) than the corresponding section in his book The Information.

 

Another strength is the compelling character development. The book traces the discovery of chaos through the stories of key figures, beginning with mathematical physicist Mitchell Feigenbaum and mathematical meteorologist Edward Lorenz. Feigenbaum, with near-manic energy and determination, discovered patterns in the transition from order to chaos. Lorenz's keen observation of an anomaly in a weather computer model led to the discovery of the butterfly effect and the mathematical conditions describing deterministic chaos—systems that are unpredictable and never repeat themselves exactly. You can't help but admire the insights and persistence of these two and the other scientific pioneers. Telling the story of chaos through individual journeys makes the book relatable and captivating.

 

Weaknesses

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While highly meritorious, the book does have some weaknesses. Written for a general audience interested in science but not necessarily scientists, it lacks the depth some readers (especially scientists) might desire. While still valuable for scientists, its historical perspective and narrative are more significant than its scientific depth. Furthermore, being nearly 40 years old, it can't be entirely up to date, although it remains remarkably current.

 

A significant challenge is keeping track of all the characters. There are at least eight main figures (Mitchell Feigenbaum, Albert Libchaber, J. Edward Lorenz, Benoit Mandelbrot, Robert May, Stephen Smale, Robert Shaw, James Yorke), plus dozens of others mentioned. The interwoven and somewhat disjointed narratives compound the confusion. Focusing on the few main characters can help mitigate this.

 

Finally, the book's initial focus on the physical sciences might lose the attention of biologically or medically oriented readers before reaching the more relevant sections.

 

Key Takeaways

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Before reading, consider these key takeaways, which are discussed throughout the book:

1. The development of chaos theory, like many scientific revolutions, depended on a multidisciplinary effort and scientists who noticed and pursued anomalies in their data.

2. Similar patterns of complexity appear across disciplines and scales, from anatomy and physiology to meteorology and economics.

3. One manifestation of chaos is strange, non-repeating patterns that are not random. Nature favors a small number of these patterns.

4. Standard education emphasizes reductionism and exact solutions. However, real-life systems with multiple, continuously changing, interacting non-linear variables often lack exact mathematical solutions.

5. Deterministic long-range forecasting is practically impossible for complex systems like the weather.

6. Chaos involves interactions between different scales. A small-scale change might fizzle out or, alternatively, induce a large-scale butterfly effect.

 

The Story of a Scientific Revolution

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Above all, Chaos tells the story of a scientific revolution, illustrating many features described by Thomas S. Kuhn in The Structure of Scientific Revolutions. Lorenz's pursuit of a computer anomaly exemplifies this. We often overlook anomalies that contradict dominant paradigms. Those with the curiosity to see and the courage to pursue such anomalies can spark scientific revolutions. Another example is William Roentgen, who discovered x-rays while investigating a cathode ray tube anomaly.

 

Scientific revolutions often occur during scientific crises. While no single crisis triggered the chaos revolution, mounting dysfunction from hyper-specialization and disciplinary silos created a need for unifying principles. Chaos theory provided these principles. Furthermore, scientific revolutions are often multidisciplinary. Chaos theory arose from insights in theoretical mathematics, physics, meteorology, biology, geology, astronomy, engineering, and more. Many pioneers had dual expertise, combining math with other sciences. For example, Feigenbaum applied renormalization theory (from quantum field theory) to understand turbulence.

 

Scientific revolutions require a shift in worldview. Chaos shows how the field shifted viewpoints from reductionism to holism, helping scientists find common patterns in seemingly random processes. This relates to the idea that science is theory-laden—we perceive and interpret science with some subjectivity and background assumptions, creating a kind of tunnel vision. A revolution requires a change in these assumptions, often preceded by resistance.

 

Revolutions are rarely attributable to a single person or time, even if textbooks portray them that way. While Lorenz's weather experiments mark the start of modern chaos exploration, earlier hints existed, such as those by Henri Poincaré. James Yorke coined the term "chaos," but this wasn't the beginning of the field's study. There's no single point when chaos theory was discovered. Unlike most revolutions confined to a single discipline (e.g., Einstein and physics, Copernicus and astronomy), chaos is a multidisciplinary revolution—a fundamental approach to complex situations.

 

Additional characteristic features of a revolution found in the development of chaos theory include a tendency to be sparked by young investigators who are less entrenched in the old ways, a new theory that simpler than the last (reductionism is incalculable), quantitative backing, and successful prediction of future outcomes.

 

Other Messages

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Beyond the key takeaways and the narrative of chaos theory's revolution and its pioneers, Gleick's Chaos effectively highlights several broader points:

 

Simplicity as a Starting Point: Understanding complexity begins with surprisingly simple concepts. The book uses examples such as idealized pendulums and oscillators, fluid convection, and the logistic difference equation to illustrate this. Even strange attractors can be described with just three non-linear differential equations.

 

Conceptualizing Mathematics: Chaos excels at conceptually explaining dynamical systems using differential and difference equations. Differential equations model smooth, continuous change, while difference equations track discrete jumps. The book also emphasizes that many non-linear differential equations, which often best represent natural phenomena, lack exact solutions.

 

The Concept of Infinity: Chaos involves different conceptions of infinity. Fractals exhibit infinite scales, while strange attractors generate infinitely non-repeating patterns. Fractal infinity contrasts with the infinity of calculus, where partitioning lines into smaller segments approximates straight lines from curves. Fractals, however, reveal new jagged structures at every level. Similarly, a strange attractor, like a continuously refined figure-eight pattern, can trace infinitely within a finite space.

 

Implications for Medicine: The book suggests that chaos holds a key to understanding the link between the physical and biological worlds—a clue to life itself. Chaos underpins many aspects of anatomy and physiology, enabling strong structures, efficient genetic encoding, effective spatial relationships in branching structures, physiological adaptability, and biological regulation. It may also help explain the self-organization of life.

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Cheat Sheet for the Characters

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While reading Chaos, I initially struggled to keep track of the many individuals mentioned. To address this, I've separated the characters into major and minor players. The minor players are important, but the book focuses more extensively on the major figures whose stories are revisited throughout the narrative.

 

There are too many minor characters to list exhaustively. While these individuals are given their due, remembering them all isn't essential to appreciate the book. Note that some individuals considered minor players in this book (like Isaac Newton) are major figures in other fields.

 

Major Players

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• Mitchell Feigenbaum (1944-2019): American mathematical physicist whose study of turbulence led to the discovery of the Feigenbaum constants, which describe bifurcations in the transition from order to chaos and are found in many systems, illustrating the concept of universality.

• Albert Libchaber (born 1934): French physicist who conducted an experiment providing physical evidence of bifurcations (later found to follow the Feigenbaum constants) in the transition from orderly flow to chaos.

• Edward Lorenz (1917-2008): American mathematician and meteorologist who serendipitously discovered the butterfly effect while modeling weather patterns. This led to the concept of deterministic chaos and the Lorenz attractor. He's often credited as the founder of modern chaos theory.

• Benoit Mandelbrot (1924-2010): French-American mathematician who described fractal geometry and created the Mandelbrot set.

• Robert May (1936-2020): Australian biologist and ecologist who advanced mathematical modeling in epidemiology and infectious disease, finding clues to chaos through simple mathematical equations.

• Stephen Smale (born 1930): American mathematician who studied the connections between topology and dynamical systems.

• Robert Shaw (born 1946): One of a group of Santa Cruz graduate students who pursued chaos in the 1970s, finding connections between chaos, topology, and information theory, and mapping the topology of strange attractors.

• James Yorke (born 1941): American mathematician who coined the term "chaos."

 

 

Minor Players

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• William Burke (1941-1996): Santa Cruz professor who mentored a group of graduate students, including Robert Shaw.

• James Crutchfield (born 1955): A Santa Cruz graduate student who studied chaos in the 1970s.

• J. Doyne Farmer (born 1952): A Santa Cruz graduate student who studied chaos in the 1970s.

• Johann Wolfgang von Goethe (1749-1832): Challenged Newton's view of colors.

• Bernardo Huberman (born 1943): Physicist who contributed to chaos concepts, including phase transitions and universal properties of non-linear systems. Formerly at Xerox PARC.

• Thomas S. Kuhn (1922-1996): Historian of science who wrote The Structure of Scientific Revolutions, introducing concepts of normal science and paradigm shifts.

• Andrey Kolmogorov (1903-1987): Russian mathematician whose work found later applications in chaos theory.

• George Mines (1886-1914): Cardiac electrophysiologist who likely died from self-induced atrial fibrillation.

• Isaac Newton (1643-1727): Revolutionized mathematics and physics.

• Norman Packard (born 1954): A Santa Cruz graduate student who studied chaos in the 1970s.

• Henri Poincaré (1854-1912): French mathematician, physicist, and engineer whose work foreshadowed key concepts in chaos theory.

• David Ruelle (born 1935): French theoretical physicist, one of the discoverers of the strange attractor.

• William Schaffer (1945-2021): American ecologist who studied disease epidemiology.

• Erwin Schrödinger (1887-1961): Theoretical physicist who developed key aspects of quantum mechanics and later explored the complexity of biology.

• Claude Shannon (1916-2001): American mathematician and electrical engineer who formulated information theory.

• Edward Spiegel (1931-2020): American professor of astronomy who worked on convection theory and fluid flow.

• Paul Stein: American physicist and mathematician at Los Alamos National Laboratory.

• Harry Swinney (born 1939): American physicist who studied non-linear dynamics.

• Floris Takens (1940-2010): Dutch mathematician, one of the discoverers of the strange attractor, who predicted its role in the transition to turbulence.

• J. Tuzo Wilson (1908-1993): Canadian geophysicist who studied plate tectonics.

• Balthasar van der Pol (1889-1959): Dutch physicist and electrical engineer whose work on vacuum tube circuits and oscillations contained, in retrospect, clues to chaos.

• John von Neumann (1903-1957): Hungarian-American mathematician, physicist, and engineer who proposed predicting and controlling the weather.

• Arthur Winfree (1942-2002): American mathematical/theoretical biologist who studied circadian and heart rhythms.

 

Cheat Sheet for the Basic Concepts

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Having a cheat sheet of key chaos concepts is also helpful. The following provides layman's terms; more technical definitions exist.

• Attractors: States that a system is drawn towards (fixed point, limit cycle, or strange attractor).

• Strange Attractors: States that chaotic systems are drawn towards; they never repeat exactly. Limit cycle attractors are associated with periodic systems, and fixed point attractors pull a system to a single final state.

• Basin Boundary: In chaos, the boundary in phase space between basins of attraction for different attractors.

• Self-Organization: The emergence of structure and function in a system from the interactions of its components, without external control.

• Renormalization Group: Analytical techniques from quantum field theory applied by Feigenbaum to study the transition from order to chaos, allowing assessment of differences at different scales.

• Sensitive Dependence on Initial Conditions (Butterfly Effect): Small initial changes lead to drastic downstream effects, making long-term prediction impossible for complex systems.

• Bifurcation: A smooth change in a system parameter resulting in a sudden change in behavior.

• Intransitive System: A system with two or more co-existing attractors.

• Poincaré Map: A method to visualize a complex dynamical system by taking cross-sections in a lower dimension.

• Phase Space: A conceptual grid describing all physical states of a system with a single point (one dimension per degree of freedom). Chaotic systems following a strange attractor never occupy the same point twice.

• Fractal: A self-similar pattern at any scale.

• Lyapunov Exponent: Measures the rate of convergence or divergence of nearby trajectories.

• Entrainment (Mode Locking): Synchronization of two or more oscillating systems.​