❄️Systems Thinking: On Economic Systems (Quotes)

"The usefulness of observation and measurement in testing economic theories arises because the theorems of economics are supposed to relate to the actual world. [...] Any economic theorem rigorously deduced from given postulates may be regarded as a hypothesis about the actual world which experience may show to be false." (Richard Stone, "The Role of Measurement in Economics", 1951)

"An economic system is not a linear system, and [...] this fact stands in the way of the determination of the parameters of the system by methods that presume linearity, and [...] it introduces great difficulties in the extrapolation from past behaviour for purposes of prediction. [...] Actual economic systems are constantly subjected to change and disturbances, which would result in irregularity." (Arnold Tustin, "The Mechanism of Economic System", 1953)

"Dynamic theory [...] shows how certain changes in the variables can be explained on the basis of [...] structural characteristics of the system. [...] The economy, of course, does not necessarily find an equilibrium position." (Wassily Leontief, "Studies in the Structure of the American Economy", 1953)

"The analysis of engineering systems and the understanding of economic structure have advanced since then, and the time is now more ripe to bring these topics into a potentially fruitful marriage." (Arnold Tustin, "The Mechanism of Economic Systems", 1953)

"The striking parallel between the economic models that are currently under discussion and some engineering systems suggests the hope that in some way the rapid progress in the development of the theory and practice of automatic control in the world of engineering may contribute to the solution of the economic problems." (Arnold Tustin, "The Mechanism of Economic Systems", 1953) 

"The ability to work with systems of general equilibrium is perhaps one of the most important skills of the economist - a skill which he shares with many other scientists, but in which he has perhaps a certain comparative advantage." (Kenneth Boulding, "The Skills of the Economist", Journal of Political Economy 67 (1), 1959)

"The treatment of the economy as a single system, to be controlled toward a consistent goal, allowed the efficient systematization of enormous information material, its deep analysis for valid decision-making. It is interesting that many inferences remain valid even in cases when this consistent goal could not be formulated, either for the reason that it was not quite clear or for the reason that it was made up of multiple goals, each of which to be taken into account." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture] 1975)

"The world is a complex, interconnected, finite, ecological–social–psychological–economic system. We treat it as if it were not, as if it were divisible, separable, simple, and infinite. Our persistent, intractable global problems arise directly from this mismatch." (Donella Meadows, "Whole Earth Models and Systems", 1982)

"When it comes to very highly organized systems, such as a living cell, the task of modeling by approximation to simple, continuous and smoothly varying quantities is hopeless. It is for this reason that attempts by sociologists and economists to imitate physicists and describe their subject matter by simple mathematical equations is rarely convincing." (Paul C W Davies, "The Cosmic Blueprint: New Discoveries in Nature’s Creative Ability to Order the Universe", 1987)

"In nonlinear systems - and the economy is most certainly nonlinear - chaos theory tells you that the slightest uncertainty in your knowledge of the initial conditions will often grow inexorably. After a while, your predictions are nonsense." (M Mitchell Waldrop, "Complexity: The Emerging Science at the Edge of Order and Chaos", 1992)

"In many parts of the economy, stabilizing forces appear not to operate. Instead, positive feedback magnifies the effects of small economic shifts; the economic models that describe such effects differ vastly from the conventional ones. Diminishing returns imply a single equilibrium point for the economy, but positive feedback - increasing returns - makes for many possible equilibrium points. There is no guarantee that the particular economic outcome selected from among the many alternatives will be the 'best' one." (W Brian Arthur, "Increasing Returns and Path Dependence in the Economy", 1994)

"We need to abandon the economist's notion of the economy as a machine, with its attendant concept of equilibrium. A more helpful way of thinking about the economy is to imagine it as a living organism." (Paul Ormerod, "The Death of Economics", 1994)

"A major clash between economics and ecology derives from the fact that nature is cyclical, whereas our industrial systems are linear. Our businesses take resources, transform them into products plus waste, and sell the products to consumers, who discard more waste […]" (Fritjof Capra, "The Web of Life", 1996)

"Optimization by individual agents, often used to derive competitive equilibria, are unnecessary for an actual economy to approximately attain such equilibria. From the failure of humans to optimize in complex tasks, one need not conclude that the equilibria derived from the competitive model are descriptively irrelevant. We show that even in complex economic systems, such equilibria can be attained under a range of surprisingly weak assumptions about agent behavior." (Antoni Bosch-Domènech & Shyam Sunder, "Tracking the Invisible Hand", 2000)

"Most systems displaying a high degree of tolerance against failures are a common feature: Their functionality is guaranteed by a highly interconnected complex network. A cell's robustness is hidden in its intricate regulatory and metabolic network; society's resilience is rooted in the interwoven social web; the economy's stability is maintained by a delicate network of financial and regulator organizations; an ecosystem's survivability is encoded in a carefully crafted web of species interactions. It seems that nature strives to achieve robustness through interconnectivity. Such universal choice of a network architecture is perhaps more than mere coincidences." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"The diversity of networks in business and the economy is mindboggling. There are policy networks, ownership networks, collaboration networks, organizational networks, network marketing-you name it. It would be impossible to integrate these diverse interactions into a single all-encompassing web. Yet no matter what organizational level we look at, the same robust and universal laws that govern nature's webs seem to greet us. The challenge is for economic and network research alike to put these laws into practice."  (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"The butterfly effect demonstrates that complex dynamical systems are highly responsive and interconnected webs of feedback loops. It reminds us that we live in a highly interconnected world. Thus our actions within an organization can lead to a range of unpredicted responses and unexpected outcomes. This seriously calls into doubt the wisdom of believing that a major organizational change intervention will necessarily achieve its pre-planned and highly desired outcomes. Small changes in the social, technological, political, ecological or economic conditions can have major implications over time for organizations, communities, societies and even nations." (Elizabeth McMillan, "Complexity, Management and the Dynamics of Change: Challenges for practice", 2008)

"Standard economists don't seem to understand exponential growth. Ecological economics recognizes that the economy, like any other subsystem on the planet, cannot grow forever. And if you think of an organism as an analogy, organisms grow for a period and then they stop growing. They can still continue to improve and develop, but without physically growing, because if organisms did that you’d end up with nine-billion-ton hamsters." (Robert Costanza, "What is Ecological economics", 2010)

"The laws of thermodynamics tell us something quite different. Economic activity is merely borrowing low-entropy energy inputs from the environment and transforming them into temporary products and services of value. In the transformation process, often more energy is expended and lost to the environment than is embedded in the particular good or service being produced." (Jeremy Rifkin, "The Third Industrial Revolution", 2011)

"Defining an indicator as lagging, coincident, or leading is connected to another vital notion: the business cycle. Indicators are lagging or leading based on where economists believe we are in the business cycle: whether we are heading into a recession or emerging from one." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

"[…] economics is a profession grounded in the belief that 'the economy' is a machine and a closed system. The more clearly that machine is understood, the more its variables are precisely measured, the more we will be able to manage and steer it as we choose, avoiding the frenetic expansions and sharp contractions. With better indicators would come better policy, and with better policy, states would be less likely to fall into depression and risk collapse." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

"The concept that an economy (1) is characterized by regular cycles that (2) follow familiar patterns (3) illuminated by a series of statistics that (4) determine where we are in that cycle has become part and parcel of how we view the world." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

❄️Systems Thinking: On Nonlinearity vs. Linearity (Quotes)

"Any organism must be treated as-a-whole; in other words, that an organism is not an algebraic sum, a linear function of its elements, but always more than that. It is seemingly little realized, at present, that this simple and innocent-looking statement involves a full structural revision of our language […]" (Alfred Korzybski, "Science and Sanity", 1933)

"Beauty had been born, not, as we so often conceive it nowadays, as an ideal of humanity, but as measure, as the reduction of the chaos of appearances to the precision of linear symbols. Symmetry, balance, harmonic division, mated and mensurated intervals - such were its abstract characteristics." (Herbert E Read, "Icon and Idea", 1955)

"Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. (Edward N Lorenz, "Deterministic Nonperiodic Flow", Journal of the Atmospheric Science 20, 1963)

"Up until now most economists have concerned themselves with linear systems, not because of any belief that the facts were so simple, but rather because of the mathematical difficulties involved in nonlinear systems [... Linear systems are] mathematically simple, and exact solutions are known. But a high price is paid for this simplicity in terms of special assumptions which must be made." (Paul A Samuelson, "Foundations of Economic Analysis", 1966)

"We've seen that even in the simplest situations nonlinearities can interfere with a linear approach to aggregates. That point holds in general: nonlinear interactions almost always make the behavior of the aggregate more complicated than would be predicted by summing or averaging." (Lewis Mumford, "The Myth of the Machine" Vol 1, 1967)

"The structure of a complex system is not a simple feedback loop where one system state dominates the behavior. The complex system has a multiplicity of interacting feedback loops. Its internal rates of flow are controlled by non‐linear relationships. The complex system is of high order, meaning that there are many system states (or levels). It usually contains positive‐feedback loops describing growth processes as well as negative, goal‐seeking loops." (Jay F Forrester, "Urban Dynamics", 1969)

"In complex systems cause and effect are often not closely related in either time or space. The structure of a complex system is not a simple feedback loop where one system state dominates the behavior. The complex system has a multiplicity of interacting feedback loops. Its internal rates of flow are controlled by nonlinear relationships. The complex system is of high order, meaning that there are many system states (or levels). It usually contains positive-feedback loops describing growth processes as well as negative, goal-seeking loops. In the complex system the cause of a difficulty may lie far back in time from the symptoms, or in a completely different and remote part of the system. In fact, causes are usually found, not in prior events, but in the structure and policies of the system." (Jay Wright Forrester, "Urban dynamics", 1969)

"Self-organization can be defined as the spontaneous creation of a globally coherent pattern out of local interactions. Because of its distributed character, this organization tends to be robust, resisting perturbations. The dynamics of a self-organizing system is typically non-linear, because of circular or feedback relations between the components. Positive feedback leads to an explosive growth, which ends when all components have been absorbed into the new configuration, leaving the system in a stable, negative feedback state. Non-linear systems have in general several stable states, and this number tends to increase (bifurcate) as an increasing input of energy pushes the system farther from its thermodynamic equilibrium. " (Francis Heylighen, "The Science Of Self-Organization And Adaptivity", 1970)

"[The] system may evolve through a whole succession of transitions leading to a hierarchy of more and more complex and organized states. Such transitions can arise in nonlinear systems that are maintained far from equilibrium: that is, beyond a certain critical threshold the steady-state regime become unstable and the system evolves into a new configuration." (Ilya Prigogine, Gregoire Micolis & Agnes Babloyantz, "Thermodynamics of Evolution", Physics Today 25 (11), 1972)

"I would therefore urge that people be introduced to [the logistic equation] early in their mathematical education. This equation can be studied phenomenologically by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student’s intuition about nonlinear systems. Not only in research but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties." (Robert M May, "Simple Mathematical Models with Very Complicated Dynamics", Nature Vol. 261 (5560), 1976)

"A nonlinear relationship causes the feedback loop of which it is a part to vary in strength, depending on the state of the system. Linked nonlinear feedback loops thus form patterns of shifting loop dominance- under some conditions one part of the system is very active, and under other conditions another set of relationships takes control and shifts the entire system behavior. A model composed of several feedback loops linked nonlinearly can produce a wide variety of complex behavior patterns." (Jørgen Randers, "Elements of the System Dynamics Method", 1980)

"When one combines the new insights gained from studying far-from-equilibrium states and nonlinear processes, along with these complicated feedback systems, a whole new approach is opened that makes it possible to relate the so-called hard sciences to the softer sciences of life - and perhaps even to social processes as well. […] It is these panoramic vistas that are opened to us by Order Out of Chaos." (Ilya Prigogine, "Order Out of Chaos: Man's New Dialogue with Nature", 1984)

"Linear relationships are easy to think about: the more the merrier. Linear equations are solvable, which makes them suitable for textbooks. Linear systems have an important modular virtue: you can take them apart and put them together again - the pieces add up. Nonlinear systems generally cannot be solved and cannot be added together. [...] Nonlinearity means that the act of playing the game has a way of changing the rules. [...] That twisted changeability makes nonlinearity hard to calculate, but it also creates rich kinds of behavior that never occur in linear systems." (James Gleick, "Chaos: Making a New Science", 1987)

"Algorithmic complexity theory and nonlinear dynamics together establish the fact that determinism reigns only over a quite finite domain; outside this small haven of order lies a largely uncharted, vast wasteland of chaos." (Joseph Ford, "Progress in Chaotic Dynamics: Essays in Honor of Joseph Ford's 60th Birthday", 1988)

"It is sometimes said that the great discovery of the nineteenth century was that the equations of nature were linear, and the great discovery of the twentieth century is that they are not." (Thomas W Körner, "Fourier Analysis", 1988)

"Never in the annals of science and engineering has there been a phenomenon so ubiquitous‚ a paradigm so universal‚ or a discipline so multidisciplinary as that of chaos. Yet chaos represents only the tip of an awesome iceberg‚ for beneath it lies a much finer structure of immense complexity‚ a geometric labyrinth of endless convolutions‚ and a surreal landscape of enchanting beauty. The bedrock which anchors these local and global bifurcation terrains is the omnipresent nonlinearity that was once wantonly linearized by the engineers and applied scientists of yore‚ thereby forfeiting their only chance to grapple with reality." (Leon O Chua, "Editorial", International Journal of Bifurcation and Chaos, Vol. l (1), 1991) 

"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic (that is fixed) rules (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order (a pattern) within disorder (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)

"Indeed, except for the very simplest physical systems, virtually everything and everybody in the world is caught up in a vast, nonlinear web of incentives and constraints and connections. The slightest change in one place causes tremors everywhere else. We can't help but disturb the universe, as T.S. Eliot almost said. The whole is almost always equal to a good deal more than the sum of its parts. And the mathematical expression of that property - to the extent that such systems can be described by mathematics at all - is a nonlinear equation: one whose graph is curvy." (M Mitchell Waldrop, "Complexity: The Emerging Science at the Edge of Order and Chaos", 1992)

"Just as few concrete physical systems are strictly deterministic in their behavior, so very few are strictly linear. The great importance of linearity lies in a combination of two circumstances. First, many tangible phenomena behave approximately linearly over restricted periods of time or restricted ranges of the variables, so that useful linear mathematical models can simulate their behavior. A pendulum swinging through a small angle is a nearly linear system. Second, linear equations can be handled by a wide variety of techniques that do not work with nonlinear equations." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Scientists try to make things simple. That is in good part why we are stuck with bivalence. Scientists' first instinct is to fit a linear model to a nonlinear world. This creates another mismatch problem, the math modeler's dilemma: linear math, nonlinear world." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"An artificial neural network is an information-processing system that has certain performance characteristics in common with biological neural networks. Artificial neural networks have been developed as generalizations of mathematical models of human cognition or neural biology, based on the assumptions that: 1. Information processing occurs at many simple elements called neurons. 2. Signals are passed between neurons over connection links. 3. Each connection link has an associated weight, which, in a typical neural net, multiplies the signal transmitted. 4. Each neuron applies an activation function (usually nonlinear) to its net input (sum of weighted input signals) to determine its output signal." (Laurene Fausett, "Fundamentals of Neural Networks", 1994)

"Symmetry breaking in psychology is governed by the nonlinear causality of complex systems (the 'butterfly effect'), which roughly means that a small cause can have a big effect. Tiny details of initial individual perspectives, but also cognitive prejudices, may 'enslave' the other modes and lead to one dominant view." (Klaus Mainzer, "Thinking in Complexity", 1994)

"[…] nonlinear interactions almost always make the behavior of the aggregate more complicated than would be predicted by summing or averaging."  (John H Holland," Hidden Order: How Adaptation Builds Complexity", 1995)

"When it comes to modeling processes that are manifestly governed by nonlinear relationships among the system components, we can appeal to the same general idea. Calculus tells us that we should expect most systems to be 'locally' flat; that is, locally linear. So a conservative modeler would try to extend the word 'local' to hold for the region of interest and would take this extension seriously until it was shown to be no longer valid." (John L Casti, "Five Golden Rules", 1995)

"When we examine the modeling literature, its most striking aspect is the predominance of 'flat' linear models. Why is this the case? After all, from a singularity theory viewpoint these linear objects are mathematical rarities. On mathematical grounds we should certainly not expect to see them put forth as credible representations of reality. Yet they are. And the reason is simple: linearity is a neutral assumption that leads to mathematically tractable models. So unless there is good reason to do otherwise, why not use a linear model?"  (John L Casti, "Five Golden Rules", 1995)

“[…] self-organization is the spontaneous emergence of new structures and new forms of behavior in open systems far from equilibrium, characterized by internal feedback loops and described mathematically by nonlinear equations.” (Fritjof  Capra, “The web of life: a new scientific understanding of living  systems”, 1996)

"Today the network of relationships linking the human race to itself and to the rest of the biosphere is so complex that all aspects affect all others to an extraordinary degree. Someone should be studying the whole system, however crudely that has to be done, because no gluing together of partial studies of a complex nonlinear system can give a good idea of the behaviour of the whole." (Murray Gell-Mann, 1997)

"There is a new science of complexity which says that the link between cause and effect is increasingly difficult to trace; that change (planned or otherwise) unfolds in non-linear ways; that paradoxes and contradictions abound; and that creative solutions arise out of diversity, uncertainty and chaos." (Andy P Hargreaves & Michael Fullan, "What’s Worth Fighting for Out There?", 1998)

"Bounded rationality simultaneously constrains the complexity of our cognitive maps and our ability to use them to anticipate the system dynamics. Mental models in which the world is seen as a sequence of events and in which feedback, nonlinearity, time delays, and multiple consequences are lacking lead to poor performance when these elements of dynamic complexity are present." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Even if our cognitive maps of causal structure were perfect, learning, especially double-loop learning, would still be difficult. To use a mental model to design a new strategy or organization we must make inferences about the consequences of decision rules that have never been tried and for which we have no data. To do so requires intuitive solution of high-order nonlinear differential equations, a task far exceeding human cognitive capabilities in all but the simplest systems."  (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Much of the art of system dynamics modeling is discovering and representing the feedback processes, which, along with stock and flow structures, time delays, and nonlinearities, determine the dynamics of a system. […] the most complex behaviors usually arise from the interactions (feedbacks) among the components of the system, not from the complexity of the components themselves." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"The mental models people use to guide their decisions are dynamically deficient. […] people generally adopt an event-based, open-loop view of causality, ignore feedback processes, fail to appreciate time delays between action and response and in the reporting of information, do not understand stocks and flows and are insensitive to nonlinearities that may alter the strengths of different feedback loops as a system evolves." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Most physical systems, particularly those complex ones, are extremely difficult to model by an accurate and precise mathematical formula or equation due to the complexity of the system structure, nonlinearity, uncertainty, randomness, etc. Therefore, approximate modeling is often necessary and practical in real-world applications. Intuitively, approximate modeling is always possible. However, the key questions are what kind of approximation is good, where the sense of 'goodness' has to be first defined, of course, and how to formulate such a good approximation in modeling a system such that it is mathematically rigorous and can produce satisfactory results in both theory and applications." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"In a linear system a tiny push produces a small effect, so that cause and effect are always proportional to each other. If one plotted on a graph the cause against the effect, the result would be a straight line. In nonlinear systems, however, a small push may produce a small effect, a slightly larger push produces a proportionately larger effect, but increase that push by a hair’s breadth and suddenly the system does something radically different." (F David Peat, "From Certainty to Uncertainty", 2002)

"Most physical processes in the real world are nonlinear. It is our abstraction of the real world that leads us to the use of linear systems in modeling these processes. These linear systems are simple, understandable, and, in many situations, provide acceptable simulations of the actual processes. Unfortunately, only the simplest of linear processes and only a very small fraction of the nonlinear having verifiable solutions can be modeled with linear systems theory. The bulk of the physical processes that we must address are, unfortunately, too complex to reduce to algorithmic form - linear or nonlinear. Most observable processes have only a small amount of information available with which to develop an algorithmic understanding. The vast majority of information that we have on most processes tends to be nonnumeric and nonalgorithmic. Most of the information is fuzzy and linguistic in form." (Timothy J Ross & W Jerry Parkinson, "Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems", 2002)

"Linearity means that the rule that determines what a piece of a system is going to do next is not influenced by what it is doing now. More precisely, this is intended in a differential or incremental sense: For a linear spring, the increase of its tension is proportional to the increment whereby it is stretched, with the ratio of these increments exactly independent of how much it has already been stretched. Such a spring can be stretched arbitrarily far, and in particular will never snap or break. Accordingly, no real spring is linear." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"Swarm intelligence can be effective when applied to highly complicated problems with many nonlinear factors, although it is often less effective than the genetic algorithm approach [...]. Swarm intelligence is related to swarm optimization […]. As with swarm intelligence, there is some evidence that at least some of the time swarm optimization can produce solutions that are more robust than genetic algorithms. Robustness here is defined as a solution’s resistance to performance degradation when the underlying variables are changed. (Michael J North & Charles M Macal, Managing Business Complexity: Discovering Strategic Solutions with Agent-Based Modeling and Simulation, 2007) 

"Thus, nonlinearity can be understood as the effect of a causal loop, where effects or outputs are fed back into the causes or inputs of the process. Complex systems are characterized by networks of such causal loops. In a complex, the interdependencies are such that a component A will affect a component B, but B will in general also affect A, directly or indirectly.  A single feedback loop can be positive or negative. A positive feedback will amplify any variation in A, making it grow exponentially. The result is that the tiniest, microscopic difference between initial states can grow into macroscopically observable distinctions." (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"Let's face it, the universe is messy. It is nonlinear, turbulent, and chaotic. It is dynamic. It spends its time in transient behavior on its way to somewhere else, not in mathematically neat equilibria. It self-organizes and evolves. It creates diversity, not uniformity. That's what makes the world interesting, that's what makes it beautiful, and that's what makes it work." (Donella H Meadow, "Thinking in Systems: A Primer", 2008)

"[…] our mental models fail to take into account the complications of the real world - at least those ways that one can see from a systems perspective. It is a warning list. Here is where hidden snags lie. You can’t navigate well in an interconnected, feedback-dominated world unless you take your eyes off short-term events and look for long-term behavior and structure; unless you are aware of false boundaries and bounded rationality; unless you take into account limiting factors, nonlinearities and delays. You are likely to mistreat, misdesign, or misread systems if you don’t respect their properties of resilience, self-organization, and hierarchy." (Donella H Meadows, "Thinking in Systems: A Primer", 2008)

"You can’t navigate well in an interconnected, feedback-dominated world unless you take your eyes off short-term events and look for long term behavior and structure; unless you are aware of false boundaries and bounded rationality; unless you take into account limiting factors, nonlinearities and delays." (Donella H Meadow, "Thinking in Systems: A Primer", 2008)

"A network of many simple processors ('units' or 'neurons') that imitates a biological neural network. The units are connected by unidirectional communication channels, which carry numeric data. Neural networks can be trained to find nonlinear relationships in data, and are used in various applications such as robotics, speech recognition, signal processing, medical diagnosis, or power systems." (Adnan Khashman et al, "Voltage Instability Detection Using Neural Networks", 2009)

"Complexity theory can be defined broadly as the study of how order, structure, pattern, and novelty arise from extremely complicated, apparently chaotic systems and conversely, how complex behavior and structure emerges from simple underlying rules. As such, it includes those other areas of study that are collectively known as chaos theory, and nonlinear dynamical theory." (Terry Cooke-Davies et al, "Exploring the Complexity of Projects", 2009)

"Linearity is a reductionist’s dream, and nonlinearity can sometimes be a reductionist’s nightmare. Understanding the distinction between linearity and nonlinearity is very important and worthwhile." (Melanie Mitchell, "Complexity: A Guided Tour", 2009)

"All forms of complex causation, and especially nonlinear transformations, admittedly stack the deck against prediction. Linear describes an outcome produced by one or more variables where the effect is additive. Any other interaction is nonlinear. This would include outcomes that involve step functions or phase transitions. The hard sciences routinely describe nonlinear phenomena. Making predictions about them becomes increasingly problematic when multiple variables are involved that have complex interactions. Some simple nonlinear systems can quickly become unpredictable when small variations in their inputs are introduced." (Richard N Lebow, "Forbidden Fruit: Counterfactuals and International Relations", 2010)

"Most systems in nature are inherently nonlinear and can only be described by nonlinear equations, which are difficult to solve in a closed form. Non-linear systems give rise to interesting phenomena such as chaos, complexity, emergence and self-organization. One of the characteristics of non-linear systems is that a small change in the initial conditions can give rise to complex and significant changes throughout the system. This property of a non-linear system such as the weather is known as the butterfly effect where it is purported that a butterfly flapping its wings in Japan can give rise to a tornado in Kansas. This unpredictable behaviour of nonlinear dynamical systems, i.e. its extreme sensitivity to initial conditions, seems to be random and is therefore referred to as chaos. This chaotic and seemingly random behaviour occurs for non-linear deterministic system in which effects can be linked to causes but cannot be predicted ahead of time." (Robert K Logan, "The Poetry of Physics and The Physics of Poetry", 2010)

"Complexity is a relative term. It depends on the number and the nature of interactions among the variables involved. Open loop systems with linear, independent variables are considered simpler than interdependent variables forming nonlinear closed loops with a delayed response." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Nature's tendency for iteration, pattern formation, and creation of order out of chaos creates expectations of predictability. It seems, however, that nature, because of varying degrees of interaction between chance and choice, and the nonlinearity of systems, escapes the boredom of predictability." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Remember that to map the dynamic behavior of a system is to capture the interaction of positive and negative feedback loops. These interactions, in essence, define the interdependencies, which in turn are responsible for nonlinearity in the system. It is the interdependency that poses the major challenge to our cognitive abilities. It is this challenge that we need to overcome by using operational modeling. Pattern recognition is critical for understanding and changing undesirable behavior. This leads us to the need for development of interactive operational representation of the phenomenon under investigation." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Complex systems are full of interdependencies - hard to detect - and nonlinear responses." (Nassim N Taleb, "Antifragile: Things That Gain from Disorder", 2012)

"Where simplifications fail, causing the most damage, is when something nonlinear is simplified with the linear as a substitute. That is the most common Procrustean bed." (Nassim N Taleb, "Antifragile: Things that Gain from Disorder", 2012)

"Complex systems defy intuitive solutions. Even a third-order, linear differential equation is unsolvable by inspection. Yet, important situations in management, economics, medicine, and social behavior usually lose reality if simplified to less than fifth-order nonlinear dynamic systems. Attempts to deal with nonlinear dynamic systems using ordinary processes of description and debate lead to internal inconsistencies. Underlying assumptions may have been left unclear and contradictory, and mental models are often logically incomplete. Resulting behavior is likely to be contrary to that implied by the assumptions being made about' underlying system structure and governing policies." (Jay W Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"Without precise predictability, control is impotent and almost meaningless. In other words, the lesser the predictability, the harder the entity or system is to control, and vice versa. If our universe actually operated on linear causality, with no surprises, uncertainty, or abrupt changes, all future events would be absolutely predictable in a sort of waveless orderliness." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

"Even more important is the way complex systems seem to strike a balance between the need for order and the imperative for change. Complex systems tend to locate themselves at a place we call 'the edge of chaos'. We imagine the edge of chaos as a place where there is enough innovation to keep a living system vibrant, and enough stability to keep it from collapsing into anarchy. It is a zone of conflict and upheaval, where the old and new are constantly at war. Finding the balance point must be a delicate matter - if a living system drifts too close, it risks falling over into incoherence and dissolution; but if the system moves too far away from the edge, it becomes rigid, frozen, totalitarian. Both conditions lead to extinction. […] Only at the edge of chaos can complex systems flourish. This threshold line, that edge between anarchy and frozen rigidity, is not a like a fence line, it is a fractal line; it possesses nonlinearity."(Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)

"There is no linear additive process that, if all the parts are taken together, can be understood to create the total system that occurs at the moment of self-organization; it is not a quantity that comes into being. It is not predictable in its shape or subsequent behavior or its subsequent qualities. There is a nonlinear quality that comes into being at the moment of synchronicity." (Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)

"To remedy chaotic situations requires a chaotic approach, one that is non-linear, constantly morphing, and continually sharpening its competitive edge with recurring feedback loops that build upon past experiences and lessons learned. Improvement cannot be sustained without reflection. Chaos arises from myriad sources that stem from two origins: internal chaos rising within you, and external chaos being imposed upon you by the environment. The result of this push/pull effect is the disequilibrium [...]." (Jeff Boss, "Navigating Chaos: How to Find Certainty in Uncertain Situations", 2015)

"[...] perhaps one of the most important features of complex systems, which is a key differentiator when comparing with chaotic systems, is the concept of emergence. Emergence 'breaks' the notion of determinism and linearity because it means that the outcome of these interactions is naturally unpredictable. In large systems, macro features often emerge in ways that cannot be traced back to any particular event or agent. Therefore, complexity theory is based on interaction, emergence and iterations." (Luis Tomé & Şuay Nilhan Açıkalın, "Complexity Theory as a New Lens in IR: System and Change" [in "Chaos, Complexity and Leadership 2017", Şefika Şule Erçetin & Nihan Potas], 2019)

"Exponentially growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems. How exponential growth emerges in nonlinear systems is mathematically unclear. […] The emergence of exponential growth from a multivariable nonlinear network is not mathematically intuitive. This indicates that the network structure and the flux functions of the modeled system must be subjected to constraints to result in long-term exponential dynamics." (Wei-Hsiang Lin et al, "Origin of exponential growth in nonlinear reaction networks", PNAS 117 (45), 2020)

❄️Systems Thinking: On Exponential Growth (Quotes)

"Anyone who believes that exponential growth can go on forever in a finite world is either a madman or an economist." (Kenneth E Boulding, "General Systens Theory - The skeleton of science", Management Science Vol.2 (3), 1956)

"Negative feedback is the form normally encountered in the control of physical systems. Yet, positive feedback dominates in the growth and decline patterns of social systems." (Jay W Forrester, "Modeling the Dynamic Processes of Corporate Growth", 1964)

"However, and conversely, our models fall far short of representing the world fully. That is why we make mistakes and why we are regularly surprised. In our heads, we can keep track of only a few variables at one time. We often draw illogical conclusions from accurate assumptions, or logical conclusions from inaccurate assumptions. Most of us, for instance, are surprised by the amount of growth an exponential process can generate. Few of us can intuit how to damp oscillations in a complex system." (Donella H Meadows, "Limits to Growth", 1972) 

"Taking no action to solve these problems is equivalent of taking strong action. Every day of continued exponential growth brings the world system closer to the ultimate limits of that growth. A decision to do nothing is a decision to increase the risk of collapse." (Donella Meadows et al, "The Limits to Growth", 1972) 

"Every day of continued exponential growth brings the world system closer to the ultimate limits of that growth." (Mihajlo D Mesarovic, "Mankind at the Turning Point", 1974)

"The world's present industrial civilization is handicapped by the coexistence of two universal, overlapping, and incompatible intellectual systems: the accumulated knowledge of the last four centuries of the properties and interrelationships of matter and energy; and the associated monetary culture which has evolved from folkways of prehistoric origin. […] Despite their inherent incompatibilities, these two systems during the last two centuries have had one fundamental characteristic in common, namely exponential growth, which has made a reasonably stable coexistence possible. But, for various reasons, it is impossible for the matter-energy system to sustain exponential growth for more than a few tens of doublings, and this phase is by now almost over. The monetary system has no such constraints, and according to one of its most fundamental rules, it must continue to grow by compound interest." (Marion K Hubbert, "Two Intellectual Systems: Matter-energy and the Monetary Culture", [seminar] 1981)

"It has long been appreciated by science that large numbers behave differently than small numbers. Mobs breed a requisite measure of complexity for emergent entities. The total number of possible interactions between two or more members accumulates exponentially as the number of members increases. At a high level of connectivity, and a high number of members, the dynamics of mobs takes hold. " (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"Mathematics says the sum value of a network increases as the square of the number of members. In other words, as the number of nodes in a network increases arithmetically, the value of the network increases exponentially. Adding a few more members can dramatically increase the value of the network." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"Swarm systems generate novelty for three reasons: (1) They are 'sensitive to initial conditions' - a scientific shorthand for saying that the size of the effect is not proportional to the size of the cause - so they can make a surprising mountain out of a molehill. (2) They hide countless novel possibilities in the exponential combinations of many interlinked individuals. (3) They don’t reckon individuals, so therefore individual variation and imperfection can be allowed. In swarm systems with heritability, individual variation and imperfection will lead to perpetual novelty, or what we call evolution." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"The change from atoms to bits is irrevocable and unstoppable. Why now? Because the change is also exponential - small differences of yesterday can have suddenly shocking consequences tomorrow." (Nicholas Negroponte, "Being Digital", 1995)

"It is in the nature of exponential growth that events develop extremely slowly for extremely long periods of time, but as one glides through the knee of the curve, events erupt at an increasingly furious pace. And that is what we will experience as we enter the twenty-first century." (Ray Kurzweil, "The Age of Spiritual Machines: When Computers Exceed Human Intelligence", 1999)

"The Law of Accelerating Returns: As order exponentially increases, time exponentially speeds up (that is, the time interval between salient events grows shorter as time passes)." (Ray Kurzweil, "The Age of Spiritual Machines: When Computers Exceed Human Intelligence", 1999)

"A more extreme form of exponential growth was probably responsible for the start of the universe. Astronomer and physicists now generally accept the Big Bang theory, according to which the universe started at an unimaginably small size and then doubled in a split second 100 times, enough to make it the size of a small grapefruit. This period of 'inflation' or exponential growth then ended, and linear growth took over, with an expanding fireball creating the universe that we know today." (Richar Koch, "The Power Laws", 2000)

"Mathematics has given us dazzling insights into the power of exponential growth and how the same patterns recur in numbers, regardless of the phenomena being observed." (Richar Koch, "The Power Laws", 2000)

"Periods of rapid change and high exponential growth do not, typically, last long. A new equilibrium with a new dominant technology and/or competitor is likely to be established before long. Periods of punctuation are therefore exciting and exhibit unusual uncertainty. The payoff from establishing a dominant position in this short time is therefore extraordinarily high. Dominance is more likely to come from skill in marketing and positioning than from superior technology itself." (Richar Koch, "The Power Laws", 2000)

"There is a strong tendency today to narrow specialization. Because of the exponential growth of information, we can afford (in terms of both economics and time) preparation of specialists in extremely narrow fields, the various branches of science and engineering having their own particular realms. As the knowledge in these fields grows deeper and broader, the individual's field of expertise has necessarily become narrower. One result is that handling information has become more difficult and even ineffective." (Semyon D Savransky, "Engineering of Creativity", 2000)

"In order to have a continuing influence, the stock market has to continue rising at an accelerating pace faster than exponential. Only a faster-than-exponential stock market growth makes private investors feel richer." (Didier Sornette, "Why Stock Markets Crash" , 2003)

"Evolution moves towards greater complexity, greater elegance, greater knowledge, greater intelligence, greater beauty, greater creativity, and greater levels of subtle attributes such as love. […] Of course, even the accelerating growth of evolution never achieves an infinite level, but as it explodes exponentially it certainly moves rapidly in that direction." (Ray Kurzweil, "The Singularity is Near", 2005)

"Limiting factors in population dynamics play the role in ecology that friction does in physics. They stop exponential growth, not unlike the way in which friction stops uniform motion. Whether or not ecology is more like physics in a viscous liquid, when the growth-rate-based traditional view is sufficient, is an open question. We argue that this limit is an oversimplification, that populations do exhibit inertial properties that are noticeable. Note that the inclusion of inertia is a generalization - it does not exclude the regular rate-based, first-order theories. They may still be widely applicable under a strong immediate density dependence, acting like friction in physics." (Lev Ginzburg & Mark Colyvan, "Ecological Orbits: How Planets Move and Populations Grow", 2004)

"Most long-range forecasts of what is technically feasible in future time periods dramatically underestimate the power of future developments because they are based on what I call the 'intuitive linear' view of history rather than the 'historical exponential'.” view'." (Ray Kurzweil, "The Singularity is Near", 2005)

"The first idea is that human progress is exponential (that is, it expands by repeatedly multiplying by a constant) rather than linear (that is, expanding by repeatedly adding a constant). Linear versus exponential: Linear growth is steady; exponential growth becomes explosive." (Ray Kurzweil, "The Singularity is Near", 2005)

"The standard big bang model doesn’t explain the smoothness and flatness of the universe, so it’s been embellished by an additional component: inflation. A minuscule fraction of a second after the big bang, the universe was propelled into an exponential expansion that increased its size from a proton to a grapefruit. […] Moreover, if we accept the theory that the universe emerged from a quantum seed and exponentially expanded in the big bang, there is the possibility that other regions of space-time exist, remote in time or space from our universe. Due to the random nature of quantum processes, these parallel universes could have wildly different properties. This extravagant concept is called the multiverse." (Chris Impey, "The Living Cosmos: Our search for life in the universe", 2007)

"Thus, nonlinearity can be understood as the effect of a causal loop, where effects or outputs are fed back into the causes or inputs of the process. Complex systems are characterized by networks of such causal loops. In a complex, the interdependencies are such that a component A will affect a component B, but B will in general also affect A, directly or indirectly.  A single feedback loop can be positive or negative. A positive feedback will amplify any variation in A, making it grow exponentially. The result is that the tiniest, microscopic difference between initial states can grow into macroscopically observable distinctions." (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"A characteristic of such chaotic dynamics is an extreme sensitivity to initial conditions (exponential separation of neighboring trajectories), which puts severe limitations on any forecast of the future fate of a particular trajectory. This sensitivity is known as the ‘butterfly effect’: the state of the system at time t can be entirely different even if the initial conditions are only slightly changed, i.e., by a butterfly flapping its wings." (Hans J Korsch et al, "Chaos: A Program Collection for the PC", 2008)

"A quantity growing exponentially toward a limit reaches that limit in a surprisingly short time." (Donella Meadows, "Thinking in systems: A Primer", 2008)

"In physical, exponentially growing systems, there must be at least one reinforcing loop driving growth and at least one balancing feedback loop constraining growth, because no system can grow forever in a finite environment." (Donella H Meadows, “Thinking in Systems: A Primer”, 2008)

"[...] a high degree of unpredictability is associated with erratic trajectories. This not only because they look random but mostly because infinitesimally small uncertainties on the initial state of the system grow very quickly - actually exponentially fast. In real world, this error amplification translates into our inability to predict the system behavior from the unavoidable imperfect knowledge of its initial state." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010)

"In chaotic deterministic systems, the probabilistic description is not linked to the number of degrees of freedom (which can be just one as for the logistic map) but stems from the intrinsic erraticism of chaotic trajectories and the exponential amplification of small uncertainties, reducing the control on the system behavior." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010)

"Standard economists don't seem to understand exponential growth. Ecological economics recognizes that the economy, like any other subsystem on the planet, cannot grow forever. And if you think of an organism as an analogy, organisms grow for a period and then they stop growing. They can still continue to improve and develop, but without physically growing, because if organisms did that you’d end up with nine-billion-ton hamsters." (Robert Costanza, "What is Ecological economics", 2010)

"Cyberneticists argue that positive feedback may be useful, but it is inherently unstable, capable of causing loss of control and runaway. A higher level of control must therefore be imposed upon any positive feedback mechanism: self-stabilising properties of a negative feedback loop constrain the explosive tendencies of positive feedback. This is the starting point of our journey to explore the role of cybernetics in the control of biological growth. That is the assumption that the evolution of self-limitation has been an absolute necessity for life forms with exponential growth." (Tony Stebbing, "A Cybernetic View of Biological Growth: The Maia Hypothesis", 2011)

"We can draw several general conclusions. First, because populations of living organisms tend to grow exponentially, numbers can rise very rapidly. This explains the inevitable population pressure that helped Darwin realize the role of natural selection, Second, exponential growth must always be a short-term, temporary phenomenon; for living organisms, the growth typically stops because of predation or a lack of sufficient nutrients or energy. Third, these laws about growth apply to all species- our intelligence cannot make us immune to simple mathematical laws. This is a critical lesson, because human population has been growing exponentially for the past few centuries. Of course, our intelligence gives us one option not available to bacteria. Exponential growth can stop only through some combination of an increase in the death rate and a decrease in the birth rate." (Jeffrey O Bennett & Seth Shostak, "Life in the universe" 3rd Ed., 2012)

"Informally, people say things are growing exponentially just to mean they’re growing a lot, which is sort of true, but the formal mathematical meaning is that it’s growing at the same proportional rate all the time." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Exponentially growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems. How exponential growth emerges in nonlinear systems is mathematically unclear. […] The emergence of exponential growth from a multivariable nonlinear network is not mathematically intuitive. This indicates that the network structure and the flux functions of the modeled system must be subjected to constraints to result in long-term exponential dynamics." (Wei-Hsiang Lin et al, "Origin of exponential growth in nonlinear reaction networks", PNAS 117 (45), 2020)

"While the universe as a whole tends towards greater disorder, there are local, pinched pockets where complexity spontaneously increases. And such processes can, as we have seen, continue even over billions of years and come a very long way. We live in such a pocket, and the complexity in it is growing exponentially – and on some fronts hyperexponentially." (Lars Tvede et al, "Hyperintelligence How the Universe Engineers Its Own Mind", 2025)