On Exponential Growth (2020-)

"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)

"With cloud data lakes you typically pay for what you use, so your costs always align with your data volumes. Since there is only a single storage layer, less data movement across different systems, availability settings, and decoupled storage versus compute, you have isolated and minimized costs for just data storage. For greater cost allocation, most cloud data lakes offer buckets, or containers (filesystems, not to be confused with application containers), to store different layers of the data (e.g., raw versus transformed data). These containers allow you to have finer-grained cost allocation for different areas of your organization. Since data sources and volumes are growing exponentially, it is extremely important to allocate and optimize costs without limiting the volume or variety of data that can be stored." (Bennie Haelen & Dan Davis, "Delta Lake: Up and Running - Modern Data Lakehouse Architectures with Delta Lake", 2023)

"In effect, AI is transforming the entire globe into a gigantic meta-brain – a vast, interconnected web of intelligence, continuously fed by humans, AI models, robots, self-driving cars, and autonomous systems of every kind. As this process accelerates, critical data density reaches unprecedented levels, driving AI into a self-reinforcing spiral of perpetual improvement, where each new insight compounds into an unstoppable force of exponential progress." (Lars Tvede et al, "Hyperintelligence How the Universe Engineers Its Own Mind", 2025

"When we explore the mathematics behind the natural world, we encounter a fascinating phenomenon: the principle of combinatorial complexity. This principle explains the exponential increase in possibilities that arise when we combine even a small number of elements." (Lars Tvede et al, "Hyperintelligence How the Universe Engineers Its Own Mind", 2025)

"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)

On Exponential Growth (-1989)

"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)

"The greatest shortcoming of the human race is our inability to understand the exponential function." (Albert A Bartlett, "Arithmetic, Population and Energy", 1969)

"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)

"In fact, in all those cases in which the initial state is given with limited precision (if we assume that the space-time is continuous this is always the case because a generic point turns out to be completely specified only by an infinite amount of information, for example by an infinite string of numbers), we can observe a situation in which, when time becomes large, two trajectories emerge from the 'same' initial point. So, even though there is a deterministic situation from a mathematical point of view (the uniqueness theorem for ordinary differential equations is not in question), nevertheless the exponential growth of errors makes the time evolution self-independent from its past history and then nondeterministic in any practical sense." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)

📡On Technology I

"Unlike art, science is genuinely progressive. Achievement in the fields of research and technology is cumulative; each generation begins at the point where its predecessor left off." (Aldous Huxley, "Science, Liberty and Peace", 1946)

"Doing engineering is practicing the art of the organized forcing of technological change." (George Spencer-Brown, Electronics, Vol. 32 (47),  1959)

"Science is the reduction of the bewildering diversity of unique events to manageable uniformity within one of a number of symbol systems, and technology is the art of using these symbol systems so as to control and organize unique events. Scientific observation is always a viewing of things through the refracting medium of a symbol system, and technological praxis is always handling of things in ways that some symbol system has dictated. Education in science and technology is essentially education on the symbol level." (Aldous L Huxley, "Essay", Daedalus, 1962)

"Engineering is the art of skillful approximation; the practice of gamesmanship in the highest form. In the end it is a method broad enough to tame the unknown, a means of combing disciplined judgment with intuition, courage with responsibility, and scientific competence within the practical aspects of time, of cost, and of talent. This is the exciting view of modern-day engineering that a vigorous profession can insist be the theme for education and training of its youth. It is an outlook that generates its strength and its grandeur not in the discovery of facts but in their application; not in receiving, but in giving. It is an outlook that requires many tools of science and the ability to manipulate them intelligently In the end, it is a welding of theory and practice to build an early, strong, and useful result. Except as a valuable discipline of the mind, a formal education in technology is sterile until it is applied." (Ronald B Smith, "Professional Responsibility of Engineering", Mechanical Engineering Vol. 86 (1), 1964)

"It is a commonplace of modern technology that there is a high measure of certainty that problems have solutions before there is knowledge of how they are to be solved." (John K Galbraith, "The New Industrial State", 1967)

"Technological invention and innovation are the business of engineering. They are embodied in engineering change." (Daniel V DeSimone & Hardy Cross, "Education for Innovation", 1968)

"The future masters of technology will have to be lighthearted and intelligent. The machine easily masters the grim and the dumb." (Marshall McLuhan, "Counterblast", 1969)

"It follows from this that man's most urgent and pre-emptive need is maximally to utilize cybernetic science and computer technology within a general systems framework, to build a meta-systemic reality which is now only dimly envisaged. Intelligent and purposeful application of rapidly developing telecommunications and teleprocessing technology should make possible a degree of worldwide value consensus heretofore unrealizable." (Richard F Ericson, "Visions of Cybernetic Organizations", 1972)

"The march of science and technology does not imply growing intellectual complexity in the lives of most people. It often means the opposite." (Thomas Sowell, "Knowledge And Decisions", 1980)

"A chipped pebble is almost part of the hand it never leaves. A thrown spear declares a sort of independence the moment it is released. [...] The whole trend in technology has been to devise machines that are less and less under direct control and more and more seem to have the beginning of a will of their own." (Isaac Asimov, "Past, Present, and Future", 1987)

📡On Technology IV

"If you think technology can solve your security problems, then you don't understand the problems and you don't understand the technology." (Bruce Schneier, "Secrets and Lies: Digital Security in a Networked World", 2000)

"Ultimately, progress in applications is not deterministic, but opportunistic, exploiting for new applications whatever new science and technology happen to be coming along." (Herbert Kroemer, "Quasi-Electric Fields and Band Offsets: Teaching Electrons New Tricks", [Nobel Lecture], 2000)

"Technology can relieve the symptoms of a problem without affecting the underlying causes. Faith in technology as the ultimate solution to all problems can thus divert our attention from the most fundamental problem - the problem of growth in a finite system - and prevent us from taking effective action to solve it." (Donella H Meadows & Dennis L Meadows, "The Limits to Growth: The 30 Year Update", 2004)

"Although the Singularity has many faces, its most important implication is this: our technology will match and then vastly exceed the refinement and suppleness of what we regard as the best of human traits."  (Ray Kurzweil, "The Singularity is Near", 2005)

"The Singularity will represent the culmination of the merger of our biological thinking and existence with our technology, resulting in a world that is still human but that transcends our biological roots. There will be no distinction, post-Singularity, between human and machine or between physical and virtual reality. If you wonder what will remain unequivocally human in such a world, it’s simply this quality: ours is the species that inherently seeks to extend its physical and mental reach beyond current limitations." (Ray Kurzweil, "The Singularity is Near", 2005)

"Chance is just as real as causation; both are modes of becoming.  The way to model a random process is to enrich the mathematical theory of probability with a model of a random mechanism. In the sciences, probabilities are never made up or 'elicited' by observing the choices people make, or the bets they are willing to place.  The reason is that, in science and technology, interpreted probability exactifies objective chance, not gut feeling or intuition. No randomness, no probability." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"Synergy occurs when organizational parts interact to produce a joint effect that is greater than the sum of the parts acting alone. As a result the organization may attain a special advantage with respect to cost, market power, technology, or employee." (Richard L Daft, "The Leadership Experience" 4th Ed., 2008)

"What’s next for technology and design? A lot less thinking about technology for technology’s sake, and a lot more thinking about design. Art humanizes technology and makes it understandable. Design is needed to make sense of information overload. It is why art and design will rise in importance during this century as we try to make sense of all the possibilities that digital technology now affords." (John Maeda, "Why Apple Leads the Way in Design", 2010) 

"Today, technology has lowered the barrier for others to share their opinion about what we should be focusing on. It is not just information overload; it is opinion overload." (Greg McKeown, "Essentialism: The Disciplined Pursuit of Less", 2014)

"For a successful technology, reality must take precedence over public relations, for nature cannot be fooled." (Richard Feynman)

🅶Malcolm T Gladwell - Collected Quotes

"But the world of the Tipping Point is a place where the unexpected becomes expected, where radical change is more than possibility. It is - contrary to all our expectations - a certainty." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"Epidemics are sensitive to the conditions and circumstances of the times and places in which they occur." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"Epidemics tip because of the extraordinary efforts of a few select carriers. But they also sometimes tip when something happens to transform the epidemic agent itself. This is a well-known principle in virology. The strains of flu that circulate at the beginning of each winter's flu epidemic are quite different from the strains of flu that circulate at the end." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"Epidemics are another example of geometric progression: when a virus spreads through a population, it doubles and doubles again, until it has (figuratively) grown from a single sheet of paper all the way to the sun in fifty steps. As human beings we have a hard time with this kind of progression, because the end result - the effect - seems far out of proportion to the cause. To appreciate the power of epidemics, we have to abandon this expectation about proportionality. We need to prepare ourselves for the possibility that sometimes big changes follow from small events, and that sometimes these changes can happen very quickly." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"It takes only the smallest of changes to shatter an epidemic's equilibrium. [...] There is more than one way to tip an epidemic, in other words. Epidemics are a function of the people who transmit infectious agents, the infectious agent itself, and the environment in which the infectious agent is operating. And when an epidemic tips, when it is jolted out of equilibrium, it tips because something has happened, some change has occurred in one (or two or three) of those areas. These three agents of change I call the Law of the Few, the Stickiness Factor, and the Power of Context."  (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"Simply by finding and reaching those few special people who hold so much social power, we can shape the course of social epidemics. In the end, Tipping Points are a reaffirmation of the potential for change and the power of intelligent action." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"That is the paradox of the epidemic: that in order to create one contagious movement, you often have to create many small movements first." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"The theory of Tipping Points requires, however, that we reframe the way we think about the world. [...] We have trouble estimating dramatic, exponential change. [...] There are abrupt limits to the number of cognitive categories we can make and the number of people we can truly love and the number of acquaintances we can truly know. We throw up our hands at a problem phrased in an abstract way, but have no difficulty at all solving the same problem rephrased as a social dilemma. All of these things are expressions of the peculiarities of the human mind and heart, a refutation of the notion that the way we function and communicate and process information is straightforward and transparent. It is not. It is messy and opaque." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"The three rules of the Tipping Point - the Law of the Few, the Stickiness Factor, the Power of Context - offer a way of making sense of epidemics. They provide us with direction for how to go about reaching a Tipping Point." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"The tipping point is that magic moment when an idea, trend, or social behavior crosses a threshold, tips, and spreads like wildfire." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"The world - much as we want it to - does not accord with our intuition. This is the second lesson of the Tipping Point. Those who are successful at creating social epidemics do not just do what they think is right, They deliberately test their intuitions." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"There are times when we need a convenient shortcut, a way to make a lot out of a little, and that is what Tipping Points, in the end, are all about." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"These three characteristics - one, contagiousness; two, the fact that little causes can have big effects; and three, that change happens not gradually but at one dramatic moment - are the same three principles that define how measles moves through a grade-school classroom or the flu attacks every winter. Of the three, the third trait - the idea that epidemics can rise or fall in one dramatic moment - is the most important, because it is the principle that makes sense of the first two and that permits the greatest insight into why modern change happens the way it does. The name given to that one dramatic moment in an epidemic when everything can change all at once is the Tipping Point." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"This is the first lesson of the Tipping Point. Starting epidemics requires concentrating resources on a few key areas. The Law of the Few says that Connectors, Mavens, and Salesmen are responsible for starting word-of-mouth epidemics, which means that if you are interested in starting a word-of-mouth epidemic, your resources ought to be solely concentrated on those three groups." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"This possibility of sudden change is at the center of the idea of the Tipping Point and might well be the hardest of all to accept. [...] The Tipping Point is the moment of critical mass, the threshold, the boiling point." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"Tipping Points are moments of great sensitivity. Changes made right at the Tipping Point can have enormous consequences." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"To make sense of social epidemics, we must first understand that human communication has its own set of very unusual and counterintuitive rules." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

Tipping Point I

"For any given population of susceptibles, there is some critical combination of contact frequency, infectivity, and disease duration just great enough for the positive loop to dominate the negative loops. That threshold is known as the tipping point. Below the tipping point, the system is stable: if the disease is introduced into the community, there may be a few new cases, but on average, people will recover faster than new cases are generated. Negative feedback dominates and the population is resistant to an epidemic. Past the tipping point, the positive loop dominates .The system is unstable and once a disease arrives, it can spread like wildfire that is, by positive feedback-limited only by the depletion of the susceptible population." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"If the contact rate, infectivity, and duration of infection are small enough, the system is below the tipping point and stable. Such a situation is known as herd immunity because the arrival of an infected individual does not produce an epidemic (though a few unlucky individuals may come in contact with any infectious arrivals and contract the disease, the group as a community is protected). However, changes in the contact rate, infectivity, or duration of illness can push a system past the tipping point." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"The existence of the tipping point means it is theoretically possible to completely eradicate a disease. Eradication does not require a perfect vaccine and universal immunization but only the weaker condition that the reproduction rate of the disease fall and remain below one so that new cases arise at a lower rate than old cases are resolved." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world. ", 2000)

"The sharp boundary between an epidemic and stability defined by the tipping point in the deterministic models becomes a probability distribution characterizing the chance an epidemic will occur for any given average rates of interaction, infectivity, and recovery. Likewise, the SI and SIR models assume a homogeneous and well-mixed population, while in reality it is often important to represent subpopulations and the spatial diffusion of an epidemic." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"In the real world, advertising is expensive and does not persist indefinitely. The marketing plan for most new products includes a certain amount for a kickoff ad campaign and other initial marketing efforts. If the product is successful, further advertising can be supported out of the revenues the product generates. If, however, the product does not take off, the marketing budget is soon exhausted and external sources of adoption fall. Advertising is not exogenous, as in the Bass model, but is part of the feedback structure of the system. There is a tipping point for ideas and new products no less than for diseases." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"The tipping point is that magic moment when an idea, trend, or social behavior crosses a threshold, tips, and spreads like wildfire." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"This possibility of sudden change is at the center of the idea of the Tipping Point and might well be the hardest of all to accept. [...] The Tipping Point is the moment of critical mass, the threshold, the boiling point." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"But in mathematics there is a kind of threshold effect, an intellectual tipping point. If a student can just get over the first few humps, negotiate the notational peculiarities of the subject, and grasp that the best way to make progress is to understand the ideas, not just learn them by rote, he or she can sail off merrily down the highway, heading for ever more abstruse and challenging ideas, while an only slightly duller student gets stuck at the geometry of isosceles triangles." (Ian Stewart, "Why Beauty is Truth: A history of symmetry", 2007)

"The product that first gets over its own tipping point attracts many consumers and this may make the competing product less attractive. Being the first to reach this tipping point is very important - more important than being the 'best' in an abstract sense." (David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)

"Stochastic variability and tipping points in the catch are two different dynamical phenomena. Yet they are both compatible with real-world data [...]" (John D W Morecroft, "Strategic Modelling and Business Dynamics: A Feedback Systems Approach", 2015)

💹On Economics I: Models I

"Economics is a science of thinking in terms of models joined to the art of choosing models which are relevant to the contemporary world. It is compelled to be this, because, unlike the typical natural science, the material to which it is applied is, in too many respects, not homogeneous through time. The object of a model is to segregate the semi-permanent or relatively constant factors from those which are transitory or fluctuating so as to develop a logical way of thinking about the latter, and of understanding the time sequences to which they give rise in particular cases." (John M Keynes, [letter to Roy Harrod] 1938)

"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 construction of an economic model, or of any model or theory for that matter (or the writing of a novel, a short story, or a play) consists of snatching from the enormous and complex mass of facts called reality, a few simple, easily-managed key points which, when put together in some cunning way, become for certain purposes a substitute for reality itself." (Evsey Domar, "Essays in the Theory of Economic Growth", 1957)

"One of the most important skills of the economist, therefore, is that of simplification of the model." (Kenneth Boulding, "The Skills of the Economist", Journal of Political Economy 67 (1), 1959)

"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)

"What is a mathematical model? One basic answer is that it is the formulation in mathematical terms of the assumptions and their consequences believed to underlie a particular ‘real world’ problem. The aim of mathematical modeling is the practical application of mathematics to help unravel the underlying mechanisms involved in, for example, economic, physical, biological, or other systems and processes." (John A Adam, "Mathematics in Nature", 2003)

"The long term solution to the financial crisis is to move beyond the ‘growth at all costs’ economic model to a model that recognizes the real costs and benefits of growth." (Robert Costanza, "Toward a New Sustainable Economy", 2008)

"Real economic efficiency implies including all resources that affect sustainable human well-being in the allocation system, not just marketed goods and services. Our current market allocation system excludes most non-marketed natural and social capital assets and services that are critical contributors to human well-being. The current economic model ignores this and therefore does not achieve real economic efficiency. A new, sustainable ecological economic model would measure and include the contributions of natural and social capital and could better approximate real economic efficiency." (Robert Costanza, "Toward a New Sustainable Economy", 2008)

"Economists also use models to learn about the world, but instead of being made of plastic, they are most often composed of diagrams and equations. Like a biology teacher’s plastic model, economic models omit many details to allow us to see what is truly important. Just as the biology teacher’s model does not include all the body’s muscles and capillaries, an economist’s model does not include every feature of the economy." (N Gregory Mankiw, "Principle of Economics" 6th ed., 2012)

"Many of the stories economists tell take the form of models - for whatever else they are, economic models are stories about how the world works." (Paul Krugman & Robin Wells, "Economics" 3rd Ed., 2013)

On Nonlinearity III

"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)

"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)

"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)

"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) 

"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)

"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)

"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)

"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)

On Exponential Growth (1990-1999)

"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) 

"Small changes in the initial conditions in a chaotic system produce dramatically different evolutionary histories. It is because of this sensitivity to initial conditions that chaotic systems are inherently unpredictable. To predict a future state of a system, one has to be able to rely on numerical calculations and initial measurements of the state variables. Yet slight errors in measurement combined with extremely small computational errors (from roundoff or truncation) make prediction impossible from a practical perspective. Moreover, small initial errors in prediction grow exponentially in chaotic systems as the trajectories evolve. Thus, theoretically, prediction may be possible with some chaotic processes if one is interested only in the movement between two relatively close points on a trajectory. When longer time intervals are involved, the situation becomes hopeless." (Courtney Brown, "Chaos and Catastrophe Theories", 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)

"One of the main reasons that large numbers grow so explosively is that multiplication is a powerful engine for growth - even when the only number you happen to be multiplying is insignificantly puny, like the number two."  (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

"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) 

"The Law of Accelerating Returns applies equally to the evolutionary process of computation, which inherently will grow exponentially and essentially without limit. The two resources it needs - the growing order of the evolving technology itself and the chaos from which an evolutionary process draws its options for further diversity - are unbounded. Ultimately, the innovation needed for further turns of the screw will come from the machines themselves." (Ray Kurzweil, "The Age of Spiritual Machines: When Computers Exceed Human Intelligence", 1999)

On Exponential Growth (2000-2009)

"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)

"Nature normally hates power laws. In ordinary systems all quantities follow bell curves, and correlations decay rapidly, obeying exponential laws. But all that changes if the system is forced to undergo a phase transition. Then power laws emerge-nature's unmistakable sign that chaos is departing in favor of order. The theory of phase transitions told us loud and clear that the road from disorder to order is maintained by the powerful forces of self-organization and is paved by power laws. It told us that power laws are not just another way of characterizing a system's behavior. They are the patent signatures of self-organization in complex systems." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"[…] real networks not only are connected but are well beyond the threshold of one. Random network theory tells us that as the average number of links per node increases beyond the critical one, the number of nodes left out of the giant cluster decreases exponentially. That is, the more links we add, the harder it is to find a node that remains isolated. Nature does not take risks by staying close to the threshold. It well surpasses it."  (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"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)

"The butterfly effect came to be the most familiar icon of the new science, and appropriately so, for it is the signature of chaos. […] The idea is that in a chaotic system, small disturbances grow exponentially fast, rendering long-term prediction impossible." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Natural laws, and for that matter determinism, do not exclude the possibility of chaos. In other words, determinism and predictability are not equivalent. And what is an even more surprising rinding of recent chaos theory has been the discovery that these effects are observable in many systems which are much simpler than the weather. [...] Moreover, chaos and order (i.e., the causality principle) can be observed in juxtaposition within the same system. There may be a linear progression of errors characterizing a deterministic system which is governed by the causality principle, while (in the same system) there can also be an exponential progression of errors (i.e., the butterfly effect) indicating that the causality principle breaks down." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"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)

"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)

"A population that grows logistically, initially increases exponentially; then the growth lows down and eventually approaches an upper bound or limit. The most well-known form of the model is the logistic differential equation." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)

"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)

"The addition of new elements or agents to a particular system multiplies exponentially the number of connections or potential interactions among those elements or agents, and hence the number of possible outcomes. This is an important attribute of complexity theory." (Mark Marson, "What Are Its Implications for Educational Change?", 2008)

On Exponential Growth (2010-2019)

"A chain reaction is a string of interactions, each of which creates a by-product that starts another reaction. When the number of reactions grows exponentially, you get a cascade as one reaction begets two, and each of these begets two more, and so on. […] Just so, when you create an interaction that then generates other interactions, you produce amazing impact. As one person talks to two and two talk to four, the number of conversations grows exponentially, creating powerful increases in the speed and spread of change." (Seth Kahan, "Getting Change Right", 2010)

"[...] 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)

"Strange attractors, unlike regular ones, are geometrically very complicated, as revealed by the evolution of a small phase-space volume. For instance, if the attractor is a limit cycle, a small two-dimensional volume does not change too much its shape: in a direction it maintains its size, while in the other it shrinks till becoming a 'very thin strand' with an almost constant length. In chaotic systems, instead, the dynamics continuously stretches and folds an initial small volume transforming it into a thinner and thinner 'ribbon' with an exponentially increasing length." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010

"The main defining feature of chaos is the sensitive dependence on initial conditions. Two nearby initial conditions on the attractor or in the chaotic sea separate by a distance that grows exponentially in time when averaged along the trajectory, leading to long-term unpredictability. The Lyapunov exponent is the average rate of growth of this distance, with a positive value signifying sensitive dependence (chaos), a zero value signifying periodicity (or quasiperiodicity), and a negative value signifying a stable equilibrium." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 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)

"Some forecasting methods directly model these components by making assumptions about their structure. For example, a popular assumption about trend is that it is linear or exponential over parts, or all, of the given time period. Another common assumption is about the noise structure: many statistical methods assume that the noise follows a normal distribution. The advantage of methods that rely on such assumptions is that when the assumptions are reasonably met, the resulting forecasts will be more robust and the models more understandable." (Galit Shmueli, "Practical Time Series Forecasting: A Hands-On Guide", 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)

"The first question: Even if the participant is aware of being modeled, or what the model is used for, is the model opaque, or even invisible? […] the second question: Does the model work against the subject’s interest? In short, is it unfair? Does it damage or destroy lives? […] The third question is whether a model has the capacity to grow exponentially. As a statistician would put it, can it scale?" (Cathy O'Neil, "Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy", 2016)

"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)

"Although base e is uniquely distinguished, other exponential functions obey a similar principle of growth. The only difference is that the rate of exponential growth is proportional to the function’s current level, not strictly equal to it. Still, that proportionality is sufficient to generate the explosiveness we associate with exponential growth." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"Chaotic systems are finicky. A little change in how they’re started can make a big difference in where they end up. That’s because small changes in their initial conditions get magnified exponentially fast. Any tiny error or disturbance snowballs so rapidly that in the long term, the system becomes unpredictable. Chaotic systems are not random - they’re deterministic and hence predictable in the short run - but in the long run, they’re so sensitive to tiny disturbances that they look effectively random in many respects." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"The important point about e is that an exponential function with this base grows at a rate precisely equal to the function itself. Let me say that again. The rate of growth of ex is ex itself. This marvelous property simplifies all calculations about exponential functions when they are expressed in base e. No other base enjoys this simplicity. Whether we are working with derivatives, integrals, differential equations, or any of the other tools of calculus, exponential functions expressed in base e are always the cleanest, most elegant, and most beautiful." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"With a linear growth of errors, improving the measurements could always keep pace with the desire for longer prediction. But when errors grow exponentially fast, a system is said to have sensitive dependence on its initial conditions. Then long-term prediction becomes impossible. This is the philosophically disturbing message of chaos." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

🅼Donella H Meadows - Collected Quotes

"Everything we think we know about the world is a model. Every word and every language is a model. All maps and statistics, books and databases, equations and computer programs are models. So are the ways I picture the world in my head - my mental models. None of these is or ever will be the real world. […] Our models usually have a strong congruence with the world. That is why we are such a successful species in the biosphere. Especially complex and sophisticated are the mental models we develop from direct, intimate experience of nature, people, and organizations immediately around us." (Donella H Meadows, "Limits to Growth", 1972)

"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)

"Technology can relieve the symptoms of a problem without affecting the underlying causes. Faith in technology as the ultimate solution to all problems can thus divert our attention from the most fundamental problem - the problem of growth in a finite system - and prevent us from taking effective action to solve it." (Donella H Meadows, "The Limits to Growth", 1972)

"Models can easily become so complex that they are impenetrable, unexaminable, and virtually unalterable." (Donella H Meadows, "The unavoidable a priori", 1980)

"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 H Meadows, "Whole Earth Models and System", 1982)

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

"A system is a set of things - people, cells, molecules, or whatever - interconnected in such a way that they produce their own pattern of behavior over time. […] The system, to a large extent, causes its own behavior." (Donella H Meadows, "Thinking in Systems: A Primer", 2008) 

"In fact, one of the most frustrating aspects of systems is that the purposes of subunits may add up to an overall behavior that no one wants." (Donella H 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)

"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)

"Like resilience, self-organizazion is often sacrificed for purposes of short-term productivity and stability." (Donella H Meadows, "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)

"The world is nonlinear. Trying to make it linear for our mathematical or administrative convenience is not usually a good idea even when feasible, and it is rarely feasible." (Donella H Meadow, "Thinking in Systems: A Primer", 2008)

"The bounded rationality of each actor in a system may not lead to decisions that further the welfare of the system as a whole." (Donella H Meadows, "Thinking in Systems: A Primer", 2008)

"When there are long delays in feedback loops, some sort of foresight is essential." (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)

From Parts to Wholes (2000-2009)

"Each part in itself constitutes the whole to which it belongs." (José Saramago, "The Cave The Cave", 2000)

"In the existing sciences much of the emphasis over the past century or so has been on breaking systems down to find their underlying parts, then trying to analyze these parts in as much detail as possible. [...] But just how these components act together to produce even some of the most obvious features of the overall behavior we see has in the past remained an almost complete mystery." (Stephen Wolfram, "A New Kind of Science", 2002)

"Systems thinking means the ability to see the synergy of the whole rather than just the separate elements of a system and to learn to reinforce or change whole system patterns. Many people have been trained to solve problems by breaking a complex system, such as an organization, into discrete parts and working to make each part perform as well as possible. However, the success of each piece does not add up to the success of the whole. to the success of the whole. In fact, sometimes changing one part to make it better actually makes the whole system function less effectively." (Richard L Daft, "The Leadership Experience", 2002)

"A system is an open set of complementary, interacting parts, with properties, capabilities and behaviours of the set emerging both from the parts and from their interactions to synthesize a unified whole." (Derek Hitchins, "Advanced Systems Thinking, Engineering, and Management", 2003)

"Emergence is the phenomenon of properties, capabilities and behaviours evident in the whole system that are not exclusively ascribable to any of its parts." (Derek Hitchins, "Advanced Systems Thinking, Engineering and Management", 2003)

"Emergence is not really mysterious, although it may be complex. Emergence is brought about by the interactions between the parts of a system. The galloping horse illusion depends upon the persistence of the human retina/brain combination, for instance. Elemental gases bond in combination by sharing outer electrons, thereby altering the appearance and behavior of the combination. In every case of emergence, the source is interaction between the parts - sometimes, as with the brain, very many parts - so that the phenomenon defies simple explanation." (Derek Hitchins, "Advanced Systems Thinking, Engineering and Management", 2003)

"There exists an alternative to reductionism for studying systems. This alternative is known as holism. Holism considers systems to be more than the sum of their parts. It is of course interested in the parts and particularly the networks of relationships between the parts, but primarily in terms of how they give rise to and sustain in existence the new entity that is the whole whether it be a river system, an automobile, a philosophical system or a quality system." (Mike Jackson, "Systems Thinking: Creative Holism for Managers", 2003)

"The traditional, scientific method for studying such systems is known as reductionism. Reductionism sees the parts as paramount and seeks to identify the parts, understand the parts and work up from an understanding of the parts to an understanding of the whole. The problem with this is that the whole often seems to take on a form that is not recognizable from the parts. The whole emerges from the interactions between the parts, which affect each other through complex networks of relationships. Once it has emerged, it is the whole that seems to give meaning to the parts and their interactions." (Mike Jackson, "Systems Thinking: Creative Holism for Managers", 2003)

"The progress of science requires the growth of understanding in both directions, downward from the whole to the parts and upward from the parts to the whole." (Freeman J Dyson, "The Scientist As Rebel", 2006)

"This reduction principle - the reduction of the behavior of a complex system to the behavior of its parts - is valid only if the level of complexity of the system is rather low." (Andrzej P Wierzbicki & Yoshiteru Nakamori, "Creative Space: Models of Creative Processes for the Knowledge Civilization Age", Studies in Computational Intelligence Vol.10, 2006)

"The fabric of our complex society is woven too tightly to permit any part of it to be damaged without damaging the whole." (Joe Biden, "Promises to Keep", 2008)