In “Complexity: A Guided Tour,” Melanie Mitchell provides a comprehensive overview of the field of complex systems science, exploring its key concepts, historical development, and potential implications for our understanding of the world. The book’s central argument is that complex systems science offers a new way of thinking about complex phenomena that cannot be fully understood using traditional reductionist approaches.
Mitchell begins by defining the fundamental properties of complex systems, which include complex collective behavior emerging from simple rules, signaling and information processing, and adaptation through learning or evolution. She illustrates these properties using a wide range of examples, from insect colonies and the immune system to economies and the World Wide Web.
One of the key ideas explored in the book is chaos theory, which deals with systems in which small changes in initial conditions can lead to large, unpredictable consequences. Mitchell explains how simple deterministic equations, such as the logistic map, can generate apparently random chaotic behavior. She also discusses universal features of chaotic systems, such as the period-doubling route to chaos and Feigenbaum’s constant.
In her exploration of chaos theory, Mitchell delves into the surprising idea that simple, deterministic systems can give rise to complex, seemingly random behavior. Chaos theory, which emerged in the latter half of the 20th century, challenges the traditional notion that deterministic systems are always predictable and that randomness arises only from external noise or uncertainty.
To illustrate the principles of chaos theory, Mitchell focuses on the logistic map, a simple mathematical equation that models population growth with limited resources. Despite its simplicity, the logistic map exhibits a wide range of behaviors depending on the value of its growth rate parameter. As this parameter is increased, the system undergoes a series of period-doubling bifurcations, where the number of equilibrium points doubles repeatedly until the system eventually descends into chaos.
In the chaotic regime, the behavior of the logistic map becomes incredibly sensitive to initial conditions. Two trajectories that start at nearly identical points will quickly diverge, making long-term prediction practically impossible. This sensitivity to initial conditions is often referred to as the “butterfly effect,” a term coined by Edward Lorenz to describe how a butterfly flapping its wings in Brazil could set off a chain of events leading to a tornado in Texas.
Mitchell emphasizes that the unpredictability of chaotic systems does not imply that they are random or lack structure. In fact, chaotic systems often exhibit universal features that are independent of the specific details of the system. One such feature is the period-doubling route to chaos, where a system undergoes a sequence of period-doubling bifurcations en route to chaos. This route to chaos has been observed in a wide variety of physical, biological, and economic systems, suggesting that it is a fundamental property of nonlinear dynamical systems.
Another universal feature of chaotic systems is Feigenbaum’s constant, named after physicist Mitchell Feigenbaum. Feigenbaum discovered that the ratio of the differences between successive bifurcation points in the period-doubling sequence converges to a universal constant of approximately 4.669. This constant appears in many different chaotic systems, from the logistic map to fluid turbulence, indicating that there are deep mathematical regularities underlying the apparent randomness of chaotic behavior.
Mitchell’s discussion of chaos theory highlights the importance of nonlinearity and feedback in complex systems. In linear systems, the whole is always equal to the sum of its parts, and small changes in inputs lead to proportionally small changes in outputs. However, in nonlinear systems, such as those studied in chaos theory, the whole can be greater (or less) than the sum of its parts, and small changes can be amplified through positive feedback loops to produce large, unexpected outcomes.
The insights from chaos theory have had far-reaching implications across many fields, from physics and biology to economics and social science. They have challenged the traditional reductionist approach to science, which seeks to understand complex phenomena by breaking them down into simpler components. Chaos theory suggests that some complex behaviors may be irreducible and that a holistic approach that takes into account the nonlinear interactions between components may be necessary.
Moreover, the sensitivity of chaotic systems to initial conditions has important implications for the limits of prediction in complex systems. While short-term prediction may be possible in some cases, long-term prediction is often impossible due to the exponential growth of small uncertainties. This has led to a shift in focus from precise prediction to understanding the qualitative properties and statistical regularities of complex systems.
Another important concept covered in the book is information theory, which arose from Claude Shannon’s work on quantifying information in terms of message probabilities. Mitchell traces the origins of information theory to Maxwell’s demon thought experiment and Boltzmann’s statistical mechanics, showing how these ideas laid the foundation for modern understandings of information and computation.
In her discussion of information theory, Mitchell highlights the fundamental role that information plays in complex systems. She traces the origins of information theory to the work of Claude Shannon, who developed a mathematical framework for quantifying information in terms of the probabilities of different messages.
Shannon’s key insight was that information is related to the amount of uncertainty or surprise associated with a message. In his framework, the information content of a message is measured in bits, with each bit corresponding to a binary choice that reduces uncertainty by half. A message that is highly predictable, such as a string of identical letters, contains little information, while a message that is highly unpredictable, such as a string of random letters, contains a lot of information.
Mitchell shows how Shannon’s work on information theory was deeply influenced by earlier ideas in physics, particularly Maxwell’s demon thought experiment and Boltzmann’s statistical mechanics. Maxwell’s demon is a hypothetical creature that can observe and sort molecules in a gas, creating a temperature difference that could be used to perform work. The demon appears to violate the second law of thermodynamics, which states that entropy (disorder) always increases in a closed system.
However, as Mitchell points out, the resolution to this paradox lies in the realization that the demon must have information about the molecules in order to sort them. Acquiring this information requires an expenditure of energy, which increases the entropy of the system and preserves the second law. This insight, later formalized by Léon Brillouin and Rolf Landauer, established a fundamental link between information and thermodynamics.
Mitchell also discusses the contributions of Ludwig Boltzmann, who developed the statistical foundations of thermodynamics in the late 19th century. Boltzmann showed that the macroscopic properties of a system, such as temperature and pressure, could be understood in terms of the statistical properties of its microscopic components. In particular, he related the entropy of a system to the number of microstates (configurations) that are consistent with its macroscopic properties.
Boltzmann’s statistical view of entropy provided a natural way to think about information in physical systems. The more microstates a system can occupy, the higher its entropy and the more information is required to specify its exact state. This idea was later formalized by Shannon, who defined the entropy of a message source in terms of the probabilities of different messages.
Mitchell emphasizes that Shannon’s information theory provided a powerful framework for understanding communication, coding, and computation. By quantifying information in terms of probabilities, Shannon was able to develop efficient codes for transmitting messages over noisy channels, such as telephone lines. His work laid the foundation for modern digital communication and data compression techniques.
Moreover, Shannon’s theory had important implications for the study of computation. In the 1940s, Alan Turing and Alonzo Church independently developed formal models of computation (Turing machines and lambda calculus) that were equivalent to Shannon’s information theory. These models showed that any computation could be encoded as a string of symbols and processed by a simple machine, providing a universal framework for understanding computation.
Mitchell argues that the confluence of information theory, statistical mechanics, and computation theory in the mid-20th century laid the foundation for the modern understanding of complex systems. By providing a way to quantify and process information, these theories opened up new possibilities for modeling and analyzing complex phenomena.
For example, the concept of algorithmic information theory, developed by Andrey Kolmogorov and Gregory Chaitin in the 1960s, provides a way to measure the complexity of a system in terms of the length of the shortest program that can generate its description. This idea has been applied to a wide range of complex systems, from DNA sequences to financial markets, and has provided new insights into the nature of complexity.
Similarly, the development of computational mechanics by James Crutchfield and Karl Young in the 1980s provided a way to extract the causal structure of a complex system from its behavior. By identifying the minimal set of states and transitions required to reproduce a system’s dynamics, computational mechanics provides a powerful tool for modeling and predicting complex systems.
Mitchell’s discussion of information theory in “Complexity: A Guided Tour” highlights the fundamental role that information plays in complex systems. By tracing the historical developments that led to Shannon’s formalization of information theory, she shows how ideas from physics, mathematics, and computation converged to provide a powerful framework for understanding complex phenomena.
Moreover, her discussion emphasizes the deep connections between information, entropy, and complexity. The more information a system contains, the more complex it is and the more difficult it is to describe or predict. This insight has important implications for the study of complex systems, from biological organisms to social networks.
At the same time, Mitchell’s exploration of information theory underscores the importance of interdisciplinary thinking in the study of complex systems. The ideas that led to the development of information theory emerged from the intersection of physics, mathematics, and engineering, and their applications have spanned a wide range of fields, from computer science to neuroscience.
The book also delves into the history of computation, from Hilbert’s early questions about the foundations of mathematics to Gödel’s incompleteness theorems and Turing’s work on computability. Mitchell emphasizes the importance of understanding the limits of computation and the existence of uncomputable problems.
Hilbert’s program sought to establish mathematics on a firm logical foundation by showing that all mathematical statements could be derived from a set of axioms using formal rules of inference. Hilbert believed that this could be done in a way that was both consistent (free from contradictions) and complete (able to prove or disprove any mathematical statement).
However, as Mitchell points out, Hilbert’s program was challenged by the work of Kurt Gödel, an Austrian-American mathematician who proved two groundbreaking theorems in 1931. Gödel’s first incompleteness theorem showed that any consistent formal system that includes arithmetic must contain statements that are true but unprovable within the system. In other words, there are mathematical truths that cannot be derived from any set of axioms.
Gödel’s second incompleteness theorem went even further, showing that no consistent formal system can prove its own consistency. This means that the consistency of mathematics cannot be established using mathematical methods alone, but must rely on meta-mathematical arguments.
Mitchell emphasizes the profound implications of Gödel’s theorems for the foundations of mathematics and the philosophy of mind. They showed that Hilbert’s program of establishing mathematics on a firm logical foundation was impossible, and that there were inherent limits to what could be proved or computed.
Mitchell then turns to the work of Alan Turing, a British mathematician and computer scientist who made seminal contributions to the theory of computation. In the 1930s, Turing developed a formal model of computation, known as the Turing machine, which provided a precise definition of what it means to compute a function.
A Turing machine consists of a tape divided into cells, a read-write head that can move along the tape, and a set of rules that specify how the head should move and what symbols it should write based on its current state and the symbol it reads. Turing showed that any function that can be computed by a Turing machine can also be computed by a human following a set of rules, and vice versa.
Moreover, Turing proved that there are some functions that cannot be computed by any Turing machine, no matter how much time or memory it has. These are known as uncomputable functions, and they include problems such as the halting problem (determining whether a given Turing machine will halt on a given input) and the Entscheidungsproblem (determining whether a given mathematical statement is provable).
Mitchell emphasizes the importance of understanding the limits of computation, both for practical reasons (knowing what can and cannot be computed) and for philosophical reasons (understanding the nature of the human mind and the limits of formal systems). She notes that the existence of uncomputable functions has important implications for fields such as artificial intelligence and the philosophy of mind.
For example, some philosophers have argued that the human mind cannot be a purely computational system, because it can solve problems that are uncomputable by Turing machines. Others have argued that the existence of uncomputable functions places limits on what can be achieved by artificial intelligence, and that there may be tasks that are inherently beyond the reach of computers.
Mitchell also discusses the work of other pioneers in the theory of computation, such as Alonzo Church and Stephen Kleene, who developed alternative models of computation that were equivalent to Turing’s. She notes that the Church-Turing thesis, which states that any function that can be computed by a human following a set of rules can also be computed by a Turing machine, is widely accepted in the field of computer science.
However, Mitchell also points out that the Church-Turing thesis is not a mathematical theorem, but rather a hypothesis about the nature of computation. She notes that there are some models of computation, such as quantum computers and analog computers, that may be able to solve problems that are uncomputable by classical Turing machines.
Mitchell’s discussion of the history of computation in “Complexity: A Guided Tour” provides a clear and engaging introduction to some of the key ideas and figures in the development of computer science. By tracing the development of ideas from Hilbert’s program to Gödel’s incompleteness theorems and Turing’s work on computability, she shows how the limits of computation were gradually uncovered and how they continue to shape our understanding of the nature of the mind and the capabilities of artificial intelligence.
At the same time, Mitchell’s exploration of the history of computation underscores the importance of understanding the limitations of formal systems and the inherent complexity of the world. The existence of uncomputable functions and the incompleteness of mathematics suggest that there are aspects of reality that may be beyond the reach of purely computational methods.
Moreover, the development of alternative models of computation, such as quantum computers and analog computers, suggests that there may be ways to push beyond the limits of classical computation and solve problems that are currently intractable.
A significant portion of the book is devoted to exploring the role of evolution in complex systems. Mitchell discusses Darwin’s theory of evolution by natural selection, which explains how complex adaptations can arise through the accumulation of small, inherited variations shaped by competition for survival. She also examines the contributions of Mendel’s work on genetic inheritance and the development of the Modern Synthesis in the early 20th century.
Mitchell then turns to more recent challenges to the Modern Synthesis, such as Stephen Jay Gould and Niles Eldredge’s theory of punctuated equilibrium and the role of contingency and constraints in shaping evolutionary outcomes. She argues that these debates highlight the ongoing challenges in understanding the complex dynamics of evolutionary processes.
The book also grapples with the difficulty of defining and measuring complexity, surveying a range of proposed measures that capture different aspects of complex systems, such as size, entropy, algorithmic information content, and hierarchy. Mitchell suggests that while no single measure is likely to be universally applicable, these different approaches can provide complementary insights into the nature of complexity.
One of the key themes running throughout the book is the potential of computer modeling and simulation to advance our understanding of complex systems. Mitchell introduces the field of artificial life, which seeks to simulate lifelike properties such as self-reproduction and evolution in computational systems. She discusses how cellular automata, such as John von Neumann’s self-reproducing automaton and John Conway’s Game of Life, can exhibit complex behavior and support universal computation.
As Mitchell explains, artificial life seeks to understand the fundamental principles of living systems by creating computational models that exhibit lifelike properties such as self-reproduction, adaptation, and evolution.
The field of artificial life can be traced back to the work of John von Neumann, a Hungarian-American mathematician and computer scientist who, in the 1940s, developed a theoretical model of a self-reproducing automaton. Von Neumann’s automaton consisted of a two-dimensional grid of cells, each of which could be in one of several states. By carefully designing the rules governing the behavior of the cells, von Neumann showed that it was possible to create a configuration of cells that could not only perform computations but also create copies of itself.
Von Neumann’s work laid the foundation for the study of cellular automata, which are simple computational models consisting of a grid of cells that change state based on the states of their neighbors. Mitchell discusses several examples of cellular automata that have been studied extensively in the field of artificial life, including John Conway’s Game of Life.
The Game of Life, which was developed in the 1970s, consists of a two-dimensional grid of cells that can be in one of two states: alive or dead. The state of each cell in the next generation is determined by the states of its eight neighbors in the current generation, according to a simple set of rules. Despite the simplicity of these rules, the Game of Life can exhibit remarkably complex behavior, including the emergence of stable patterns, oscillators, and even self-replicating structures.
Mitchell points out that the Game of Life and other cellular automata have been used to study a wide range of phenomena in complex systems, from the formation of patterns in nature to the evolution of cooperation in social systems. By providing a simplified model of the key features of these systems, cellular automata allow researchers to explore the emergent properties that arise from the interactions of simple components.
Moreover, as Mitchell notes, some cellular automata have been shown to be capable of universal computation, meaning that they can perform any computation that can be performed by a Turing machine. This has important implications for the study of complex systems, as it suggests that even simple rules can give rise to highly complex and unpredictable behavior.
Mitchell also discusses other examples of computer modeling and simulation in the study of complex systems, such as agent-based models and genetic algorithms. Agent-based models are computational models in which individual agents interact with each other and with their environment according to a set of rules. These models have been used to study a wide range of phenomena, from the behavior of financial markets to the spread of epidemics.
Genetic algorithms, on the other hand, are computational models inspired by the process of natural selection in biology. In a genetic algorithm, a population of candidate solutions to a problem is encoded as a set of strings, which are then subjected to a process of selection, mutation, and recombination over many generations. Over time, the population evolves to find better solutions to the problem, mimicking the process of adaptation in natural systems.
Mitchell argues that computer modeling and simulation provide a valuable complement to traditional mathematical analysis and physical experimentation in the study of complex systems. By allowing researchers to explore the behavior of these systems in a controlled and repeatable way, computational tools can help to generate new hypotheses and insights that might be difficult to obtain through other means.
At the same time, Mitchell cautions against overreliance on computer models and simulations, noting that they are only as good as the assumptions and simplifications that go into them. She emphasizes the importance of validating models against empirical data and of using multiple approaches to study complex systems, including mathematical analysis, physical experimentation, and computer simulation.
Mitchell’s discussion of computer modeling and simulation in “Complexity: A Guided Tour” highlights the significant contributions that these tools have made to the study of complex systems. By providing concrete examples of how cellular automata, agent-based models, and genetic algorithms have been used to study a wide range of phenomena, she demonstrates the power and flexibility of these approaches.
Moreover, her emphasis on the importance of validating models against empirical data and of using multiple approaches to study complex systems underscores the need for a holistic and interdisciplinary approach to the study of complexity. As Mitchell argues, computer modeling and simulation are valuable tools in the toolkit of complex systems science, but they must be used in conjunction with other approaches to gain a full understanding of these systems.
Mitchell also explores Stephen Wolfram’s work on cellular automata, including his classification of one-dimensional cellular automata into four classes of behavior and his discovery of the computationally universal Rule 110. She discusses Wolfram’s controversial “Principle of Computational Equivalence,” which suggests that universal computation is common in nature and represents an upper limit on the complexity of natural processes.
The book also examines the role of idea models, such as the Prisoner’s Dilemma, in shedding light on the evolution of cooperation in populations of self-interested agents. Mitchell stresses the importance of replicating and carefully evaluating these models to ensure their reliability and avoid overgeneralization.
Another key argument made in the book is the importance of network thinking in understanding complex systems. Mitchell illustrates this point using the example of metabolic scaling in biology, showing how a network model based on space-filling fractal networks can explain the 3/4 power scaling law observed in the relationship between an organism’s metabolic rate and its body mass.
One of the most famous idea models in the study of cooperation is the Prisoner’s Dilemma, which was originally formulated by Merrill Flood and Melvin Dresher in the 1950s. The Prisoner’s Dilemma is a two-player game in which each player has two options: cooperate or defect. The payoff for each player depends on the choices made by both players, with the highest payoff going to a player who defects while the other cooperates, and the lowest payoff going to a player who cooperates while the other defects.
The Prisoner’s Dilemma captures the tension between individual self-interest and collective welfare that lies at the heart of many social dilemmas. In a single round of the game, the rational choice for each player is to defect, regardless of what the other player does. However, if both players defect, they both receive a lower payoff than if they had both cooperated.
Mitchell discusses how the Prisoner’s Dilemma has been used to study the evolution of cooperation in populations of self-interested agents. She describes the work of Robert Axelrod, a political scientist who, in the 1980s, organized a series of computer tournaments in which different strategies for playing the Prisoner’s Dilemma competed against each other. Axelrod found that the most successful strategy was a simple one called “tit for tat,” in which a player starts by cooperating and then copies the other player’s previous move on subsequent rounds.
Mitchell points out that the success of the tit-for-tat strategy in Axelrod’s tournaments led to a surge of interest in the Prisoner’s Dilemma and its implications for the evolution of cooperation. However, she also stresses the importance of replicating and carefully evaluating these models to ensure their reliability and avoid overgeneralization. She notes that subsequent research has shown that the success of tit-for-tat depends on the specific parameters of the model, such as the number of players, the length of the game, and the structure of the population.
Moreover, Mitchell argues that the Prisoner’s Dilemma is just one of many idea models that have been used to study the evolution of cooperation in complex systems. She discusses other models, such as the Stag Hunt and the Hawk-Dove game, which capture different aspects of social dilemmas and have generated new insights into the conditions that favor the emergence of cooperation.
Another key argument that Mitchell makes in “Complexity: A Guided Tour” is the importance of network thinking in understanding complex systems. She argues that many complex systems, from metabolic networks in biology to social networks in human societies, can be understood in terms of the interactions between their components, which are often organized in complex networks.
To illustrate this point, Mitchell discusses the example of metabolic scaling in biology. Metabolic scaling refers to the relationship between an organism’s metabolic rate (the rate at which it consumes energy) and its body mass. Empirical studies have shown that metabolic rate scales with body mass to the power of 3/4, meaning that larger organisms have lower metabolic rates per unit of body mass than smaller organisms.
Mitchell describes how a network model based on space-filling fractal networks can explain this 3/4 power scaling law. The model, which was developed by Geoffrey West and his colleagues at the Santa Fe Institute, assumes that the metabolic network of an organism is organized as a fractal, with a self-similar structure that repeats at different scales. The model predicts that the metabolic rate of an organism should scale with its body mass to the power of 3/4, which is consistent with empirical observations.
Mitchell argues that the success of the metabolic scaling model demonstrates the power of network thinking in understanding complex systems. By focusing on the interactions between the components of a system, rather than on the components themselves, network models can capture the emergent properties and dynamics of complex systems in a way that traditional reductionist approaches cannot.
Moreover, Mitchell points out that network thinking has applications beyond biology, in fields such as social science, economics, and computer science. She discusses how network models have been used to study the spread of information and influence in social networks, the robustness and resilience of financial networks, and the structure and function of the Internet and other complex technological systems.
At the same time, Mitchell cautions against overreliance on network models, noting that they are only as good as the assumptions and simplifications that go into them. She emphasizes the importance of validating network models against empirical data and of using multiple approaches to study complex systems, including mathematical analysis, physical experimentation, and computer simulation.
Mitchell’s discussion of idea models and network thinking in “Complexity: A Guided Tour” highlights the importance of these conceptual tools in understanding complex systems. By providing concrete examples of how idea models such as the Prisoner’s Dilemma and network models such as the metabolic scaling model have been used to generate new insights into the behavior of complex systems, she demonstrates the power and flexibility of these approaches.
Moreover, her emphasis on the importance of replicating and carefully evaluating these models, and of using multiple approaches to study complex systems, underscores the need for a rigorous and interdisciplinary approach to the study of complexity. As Mitchell argues, idea models and network thinking are valuable tools in the toolkit of complex systems science, but they must be used in conjunction with other approaches to gain a full understanding of these systems.
In the final chapters of the book, Mitchell reflects on the current state and future prospects of complex systems science. She acknowledges the criticisms leveled against the field, such as its lack of rigor and overreliance on computer models, but argues that complex systems science is making important contributions by questioning long-held assumptions, developing new conceptual frameworks, and fostering interdisciplinary collaboration.
Mitchell suggests that the field is unlikely to develop a “unified theory” of complex systems in the near future, but that progress can be made by identifying common principles that provide new insights into related classes of complex systems. She also emphasizes the need for a new vocabulary and mathematical framework to rigorously define and analyze the phenomena studied by complex systems science.
In conclusion, “Complexity: A Guided Tour” presents a compelling case for the importance of complex systems science as a new approach to understanding the world. Through a wide-ranging survey of key concepts, historical developments, and current challenges, Mitchell argues that complex systems science offers a powerful set of tools and ideas for grappling with the emergent properties and nonlinear dynamics of complex phenomena. While acknowledging the limitations and criticisms of the field, she suggests that complex systems science has the potential to transform our understanding of the world and guide us in navigating the complexities of the 21st century.