"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)
"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)
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