15 August 2019

John L Casti - Collected Quotes

"[…] a complex system is incomprehensible unless we can simplify it by using alternative levels of description." (John L Casti, "On System Complexity: Identification, Measurement, and Management" [in "Complexity, Language, and Life: Mathematical Approaches"] 1986)

"Coping with complexity involves the creation of faithful models of not only the system to be managed. but also of the management system itself." (John L Casti, "On System Complexity: Identification, Measurement, and Management" [in "Complexity, Language, and Life: Mathematical Approaches"] 1986)

"[…] complexity emerges from simplicity when alternative descriptions of a system are not reducible to each other. For a given observer, the more such inequivalent descriptions he or she generates, the more complex the system appears. Conversely, a complex system can be simplified in one of two ways: reduce the number of potential descriptions (by restricting the observer's means of interaction with the system) and/or use a coarser notion of system equivalence, thus reducing the number of equivalence classes." (John L Casti, "On System Complexity: Identification, Measurement, and Management" [in "Complexity, Language, and Life: Mathematical Approaches"] 1986)

"[…] error (or surprise) always involves a discrepancy between the objects (systems) open to interaction and the abstractions (models, descriptions) closed. to those same interactions. The remedy is equally clear, in principle: just supplement the description by adding more observables to account for the unmodeled interactions. In this sense, error and surprise are indistinguishable from bifurcations. A particular description is inadequate to account for uncontrollable variability in equivalent states and we need a new description to remove the error." (John L Casti, "On System Complexity: Identification, Measurement, and Management" [in "Complexity, Language, and Life: Mathematical Approaches"] 1986)

"Simple systems generally involve a small number of components. with self-interaction dominating the mutual interaction of the variables. […] Besides involving only a few variables. simple systems generally have very few feedback/feedforward loops. Such loops enable the system to restructure. or at least modify. the interaction pattern of its variables. thereby opening-up the possibility of a wider range of potential behavior patterns." (John L Casti, "On System Complexity: Identification, Measurement, and Management" [in "Complexity, Language, and Life: Mathematical Approaches"] 1986)

"Since most understanding and virtually all control is based upon a model (mental, mathematical, physical, or otherwise) of the system under study, the simplification imperative translates into a desire to obtain an equivalent, but reduced, representation of the original model of the system. This may involve omitting some of the original variables, aggregating others, ignoring weak couplings, regarding slowly changing variables as constants, and a variety of other subterfuges. All of these simplification techniques are aimed at reducing the degrees of freedom that the system has at its disposal to interact with its environment. A theory of system complexity would give us knowledge as to the limitations of the reduction process." (John L Casti, "On System Complexity: Identification, Measurement, and Management" [in "Complexity, Language, and Life: Mathematical Approaches"] 1986)

"The failure of individual subsystems to be sufficiently adaptive to changing environments results in the subsystems forming a collective association that, as a unit, is better able to function in new circumstances. Formation of such an association is a structural change; the behavioral role of the new conglomerate is a junctional change; both types of change are characteristic of the formation of hierarchies." (John L Casti, "On System Complexity: Identification, Measurement, and Management" [in "Complexity, Language, and Life: Mathematical Approaches"] 1986)

"A law explains a set of observations; a theory explains a set of laws. […] a law applies to observed phenomena in one domain (e.g., planetary bodies and their movements), while a theory is intended to unify phenomena in many domains. […] Unlike laws, theories often postulate unobservable objects as part of their explanatory mechanism." (John L Casti, "Searching for Certainty: How Scientists Predict the Future", 1990)

"[…] a model is a mathematical representation of the modeler's reality, a way of capturing some aspects of a particular reality within the framework of a mathematical apparatus that provides us with a means for exploring the properties of the reality mirrored in the model." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Basically, the point of making models is to be able to bring a measure of order to our experiences and observations, as well as to make specific predictions about certain aspects of the world we experience." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Experiencing the world ultimately comes down to the recognition of boundaries: self/non-self, past/future, inside/outside, subject/object, and so forth. And so it is in mathematics, too, where we are continually called upon to make distinctions: solvable/unsolvable, computable/uncomputable, linear/nonlinear and other categorical distinctions involving the identification of boundaries. In particular, in geometry we characterize the boundaries of especially important figures by giving them names like circles, triangles, ellipses, and polygons. But when it comes to using these kinds of boundaries to describe the natural world, these simple geometrical shapes fail us completely: mountains are not cones, clouds are not spheres, and rivers are not straight lines." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Great mathematics seldom comes from idle speculation about abstract spaces and symbols. More often than not it is motivated by definite questions arising in the worlds of nature and humans." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Mathematical modeling is about rules - the rules of reality. What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of ‘model’, is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors. These symbols are then strung together in accordance with a set of rules expressed in a special language, the language of mathematics." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Reliable information processing requires the existence of a good code or language, i.e., a set of rules that generate information at a given hierarchical level, and then compress it for use at a higher cognitive level. To accomplish this, a language should strike an optimum balance between variety (stochasticity) and the ability to detect and correct errors (memory)."(John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Skewness is a measure of symmetry. For example, it's zero for the bell-shaped normal curve, which is perfectly symmetric about its mean. Kurtosis is a measure of the peakedness, or fat-tailedness, of a distribution. Thus, it measures the likelihood of extreme values." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"[…] the complexity of a given system is always determined relative to another system with which the given system interacts. Only in extremely special cases, where one of these reciprocal interactions is so much weaker than the other that it can be ignored, can we justify the traditional attitude regarding complexity as an intrinsic property of the system itself." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"The idea of one description of a system bifurcating from another also provides the key to begin unlocking one of the most important, and at the same time perplexing, problems of system theory: characterization of the complexity of a system." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"[…] the study of natural systems begins and ends with the specification of observables describing such a system, and a characterization of the manner in which these observables are linked." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"The key to making discontinuity emerge from smoothness is the observation that the overall behavior of both static and dynamical systems is governed by what's happening near the critical points. These are the points at which the gradient of the function vanishes. Away from the critical points, the Implicit Function Theorem tells us that the behavior is boring and predictable, linear, in fact. So it's only at the critical points that the system has the possibility of breaking out of this mold to enter a new mode of operation. It's at the critical points that we have the opportunity to effect dramatic shifts in the system's behavior by 'nudging' lightly the system dynamics, one type of nudge leading to a limit cycle, another to a stable equilibrium, and yet a third type resulting in the system's moving into the domain of a 'strange attractor'. It's by these nudges in the equations of motion that the germ of the idea of discontinuity from smoothness blossoms forth into the modern theory of singularities, catastrophes and bifurcations, wherein we see how to make discontinuous outputs emerge from smooth inputs." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"To function effectively, the system scientist must know a considerable amount about the natural world AND about mathematics, without being an expert in either field. This is clearly a prescription for career disaster in today's world of ultra-high specialization." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Virtually all mathematical theorems are assertions about the existence or nonexistence of certain entities. For example, theorems assert the existence of a solution to a differential equation, a root of a polynomial, or the nonexistence of an algorithm for the Halting Problem. A platonist is one who believes that these objects enjoy a real existence in some mystical realm beyond space and time. To such a person, a mathematician is like an explorer who discovers already existing things. On the other hand, a formalist is one who feels we construct these objects by our rules of logical inference, and that until we actually produce a chain of reasoning leading to one of these objects they have no meaningful existence, at all." (John L Casti, "Reality Rules: Picturing the world in mathematics" Vol. II, 1992)

"We can think of the evolution of a cellular automaton as a pattern-recognition process, in which all initial configurations in the basin of attraction of a particular attractor are thought of as instances of some pattern with the attractor being the 'archetype' of this pattern. Thus, the evolution of the different state trajectories toward this attractor constitutes recognition of the pattern." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"We see an ever-increasing move toward inter and trans-disciplinary attacks upon problems in the real world […]. The system scientist has a central role to play in this new order, and that role is to first of all understand ways and means of how to encode the natural world into 'good"' formal structures." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"What is usually left unsaid is an account of the equally great failures of science, failures that most scientists fervently wish would simply curl up into a little ball, roll off into a corner and disappear, much like the now mythical ether. Perhaps the greatest failure of this sort in classical physics is the inability to give any sort of coherent account of the puzzling phenomenon of turbulence. […] The central difficulty in giving a mathematical account of turbulence is the lack of any single scale of length appropriate to the description of the phenomenon. Intuitively - and by observation - turbulent flow involves nested eddies of all scales, ranging from the macroscopic down to the molecular. So any mathematical description of the process must take all these different scales into account. This situation is rather similar to the problem of phase transitions, where length scales ranging from the correlation length, which approaches infinity at the transition temperature, down to the atomic scale all play an important role in the overall transition process." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"When all the mathematical smoke clears away, Godel's message is that mankind will never know the final secret of the universe by rational thought alone. It's impossible for human beings to ever formulate a complete description of the natural numbers. There will always be arithmetic truths that escape our ability to fence them in using the tools, tricks and subterfuges of rational analysis." (John L Casti, "Reality Rules: Picturing the world in mathematics" Vol. II, 1992)

"[…] a rule for choosing an action is termed a strategy. If the rule says to always take the same action, it's called a pure strategy; otherwise, the strategy is called mixed. A solution to a game is simply a strategy for each player that gives each of them the best possible payoff, in the sense of being a regret-free choice." (John L Casti, "Five Golden Rules", 1995)

"Allowing more than two players into the game and/or postulating payoff structures in which one player's gain does not necessarily equal the other player's loss brings us much closer to the type of games played in real life. Unfortunately, it's generally the case that the closer you get to the messiness of the real world, the farther you move from the stylized and structured world of mathematics. Game theory is no exception." (John L Casti, "Five Golden Rules", 1995)

"Mathematics is about theorems: how to find them; how to prove them; how to generalize them; how to use them; how to understand them. […] But great theorems do not stand in isolation; they lead to great theories. […] And great theories in mathematics are like great poems, great paintings, or great literature: it takes time for them to mature and be recognized as being 'great'." (John L Casti, "Five Golden Rules", 1995)

"Catastrophe theory is a local theory, telling us what a function looks like  in a small neighborhood of a critical point; it says nothing about what the function may be doing far away from the singularity. Yet most of the applications of the theory [...]  involve extrapolating these rock-solid, local results to regions that may  well be distant in time and space from the singularity."  (John L Casti, "Five Golden Rules", 1995)

"Probably the most important reason that catastrophe theory received as much popular press as it did in the mid-1970s is not because of its unchallenged mathematical elegance, but because it appears to offer a coherent mathematical framework within which to talk about how discontinuous behaviors - stock market booms and busts or cellular differentiation, for instance - might emerge as the result of smooth changes in the inputs to a system, things like interest rates in a speculative market or the diffusion rate of chemicals in a developing embryo. These kinds of changes are often termed bifurcations, and playa central role in applied mathematical modeling. Catastrophe theory enables us to understand more clearly how - and why - they occur." (John L Casti, "Five Golden Rules", 1995)

"Since geometry is the mathematical idealization of space, a natural way to organize its study is by dimension. First we have points, objects of dimension O. Then come lines and curves, which are one-dimensional objects, followed by two-dimensional surfaces, and so on. A collection of such objects from a given dimension forms what mathematicians call a 'space'. And if there is some notion enabling us to say when two objects are 'nearby' in such a space, then it's called a topological space." (John L Casti, "Five Golden Rules", 1995)

"[...] there is no area of mathematics where thinking abstractly has paid more handsome dividends than in topology, the study of those properties of geometrical objects that remain unchanged when we deform or distort them in a continuous fashion without tearing, cutting, or breaking them." (John L Casti, "Five Golden Rules", 1995)


"So the strategy of mixing the choices with equal likelihood is an equilibrium point for the game, in the same sense that the minimax point is an equilibrium for a game having a saddle point. Thus, using a strategy that randomizes their choices, Max and Min can each announce his or her strategy to the other without the opponent being able to exploit this information to get a larger average payoff for himself or herself." (John L Casti, "Five Golden Rules", 1995)

"The goal of catastrophe theory is to classify smooth functions with degenerate critical points, just as Morse's Theorem gives us a complete classification for Morse functions. The difficulty, of course, is that there are a lot more ways for critical points to 'go bad' than there are for them to stay 'nice'. Thus, the classification problem is much harder for functions having degenerate critical points, and has not yet been fully carried out for all possible types of degeneracies. Fortunately, though, we can obtain a partial classification for those functions having critical points that are not too bad. And this classification turns out to be sufficient to apply the results to a wide range of phenomena like the predator-prey situation sketched above, in which 'jumps' in the system's biomass can occur when parameters describing the process change only slightly." (John L Casti, "Five Golden Rules", 1995)

"The Minimax Theorem applies to games in which there are just two players and for which the total payoff to both parties is zero, regardless of what actions the players choose. The advantage of these two properties is that with two players whose interests are directly opposed we have a game of pure competition, which allows us to define a clear-cut mathematical notion of rational behavior that leads to a single, unambiguous rule as to how each player should behave." (John L Casti, "Five Golden Rules", 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)

"The fruitful generalization in mathematics often involves starting from a commonsense concept such as a point on a line. A mathematical framework is then developed within which the particular example of a point in space is seen to be just a very special case of a much broader structure, say a point in three-dimensional space. Further generalizations then show this new structure itself to be only a special case of an even broader framework, the notion of a point in a space of n dimensions. And so it goes, one generalization piled atop another, each element leading to a deeper understanding of how the original object fits into a bigger picture." (John L Casti, "Five More Golden Rules : Knots, Codes, Chaos, and Other Great Theories of 20th Century Mathematics", 2000)

"A large part of Turing's genius was to show that the very primitive type of abstract computing machine he invented is actually the most general type of computer imaginable. In fact, every real-life computer that's ever been built is just a special case that materially embodies the machine that Turing dreamed up." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"By common consensus in the mathematical world, a good proof displays three essential characteristics: a good proof is (1) convincing, (2) surveyable, and (3) formalizable. The first requirement means simply that most mathematicians believe it when they see it. […] Most mathematicians and philosophers of mathematics demand more than mere plausibility, or even belief. A proof must be able to be understood, studied, communicated, and verified by rational analysis. In short, it must be surveyable. Finally, formalizability means we can always find a suitable formal system in which an informal proof can be embedded and fleshed out into a formal proof." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"Generally speaking, there are three grades of proof in mathematics. The first, or highest quality type of proof, is one that incorporates why and how the result is true, not simply that it is so. […] Second-grade proofs content themselves with showing that their   conclusion is true, by relying on the law of the excluded middle. Thus, they assume that the conclusion they want to demonstrate is false and then derive a contradiction from this assumption. In polite company, these are often termed "nonconstructive proofs," since they lack the how and why. […] Finally, there is the third order, or lowest grade, of proof. In these situations, the idea of proof degenerates into mere verification, in which a (usually) large number of cases are considered separately and verified, one by one, very often by a computer." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"Somehow mathematicians seem to long for more than just results from their proofs; they want insight." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"That a proof must be convincing is part of the anthropology of mathematics, providing the key to understanding mathematics as a human activity. We invoke the logic of mathematics when we demand that every informal proof be capable of being formalized within the confines of a definite formal system. Finally, the epistemology of mathematics comes into play with the requirement that a proof be surveyable. We can't really say that we have created a genuine piece of knowledge unless it can be examined and verified by others; there are no private truths in mathematics." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"The core of a decision problem is always to find a single method that can be applied to each question, and that will always give the correct answer for each individual problem." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"The double periodicity of the torus is fairly obvious: the circle that goes around the torus in the 'long' direction around the rim, together with the circle that goes around it through the hole in the center. And just as periodic functions can be defined on a circle, doubly periodic functions can be defined on a torus." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"The general idea of a model is to provide a concrete example of a mathematical framework that satisfies the axioms and relations of an abstract mathematical theory." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001) 

"The real raison d'etre for the mathematician's existence is simply to solve problems. So what mathematics really consists of is problems and solutions. And it is the "good" problems, the ones that challenge the greatest minds for decades, if not centuries, that eventually become enshrined as mathematical mountaintops." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"Traditionally, mathematical truths have been considered to be a priori truths, either in the sense that they are truths that would be true in any possible universe, or in the sense that they are truths whose validity is independent of our sensory impressions." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"[…] Turing machines are definitely not machines in the everyday sense of being material devices. Rather they are "paper computers," completely specified by their programs. Thus, when we use the term machine in what follows, the reader should read program or algorithm (i.e., software) and put all notions of hardware out of sight and out of mind." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"What's important about the Turing machine from a theoretical point of view is that it represents a formal mathematical object. So with the invention of the Turing machine, for the first time we had a well-defined notion of what it means to compute something." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"[…] accept that X-events will occur. That is simply a fact of life. So prepare for them as you’d prepare for any other life-changing, but inherently unpredictable, event. This means remaining adaptive and open to new possibilities, creating a life with as many degrees of freedom in it as possible by educating yourself to be as self-sufficient as you can, and not letting hope be replaced by fear and despair." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"[…] according to the bell-shaped curve the likelihood of a very-large-deviation event (a major outlier) located in the striped region appears to be very unlikely, essentially zero. The same event, though, is several thousand times more likely if it comes from a set of events obeying a fat-tailed distribution instead of the bell-shaped one." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"[…] both rarity and impact have to go into any meaningful characterization of how black any particular [black] swan happens to be." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"[...] complexity overload is the precipitating cause of X-events. That overload may show up as unmanageable stress or pressure in a single system, be it a society, a corporation, or even an individual. The X-event that reduces the pressure then ranges from a societal collapse to a corporate bankruptcy to a nervous breakdown. […] You must add and subtract complexity judiciously throughout the entire system in order to bring the imbalances back into line." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"Due to the problem of predicting outlier events, they are not usually factored into the design of systems." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"[…] events will always occur that cannot be foreseen by following a chain of logical deductive reasoning. Successful prediction requires intuitive leaps and/or information that is not part of the original data available." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"Forecasting models […] ordinarily are based only on past data, which is generally a tiny sample of the total range of possible outcomes. The problem is that those 'experts' who develop the models often come to believe they have mapped the entire space of possible system behaviors, which could not be further from the truth. Worse yet, when outliers do crop up, they are often discounted as 'once in a century' events and are all but ignored in planning for the future. […] the world is much more unpredictable than we’d like to believe."(John L Casti, "X-Events: The Collapse of Everything", 2012)

"Generally speaking, the best solution for solving a complexity mismatch is to simplify the system that’s too complex rather than 'complexify' the simpler system." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"If you want a system - economic, social, political, or otherwise - to operate at a high level of efficiency, then you have to optimize its operation in such a way that its resilience is dramatically reduced to unknown - and possibly unknowable - shocks and/or changes in its operating environment. In other words, there is an inescapable price to be paid in efficiency in order to gain the benefits of adaptability and survivability in a highly uncertain environment. There is no escape clause!" (John L Casti, "X-Events: The Collapse of Everything", 2012)

"Sustainability is a delicate balancing act calling upon us to remain on the narrow path between organization and chaos, simplicity and complexity." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"System theorists know that it's easy to couple simple-to-understand systems into a ‘super system’ that's capable of displaying behavioral modes that cannot be seen in any of its constituent parts. This is the process called ‘emergence’." (John L Casti, [interview with Austin Allen], 2012)

"The first path of increasing complexity via innovation often faces limits as to how much complexity can be added or reduced in a given system. This is because if you change the complexity level in one place, a compensating change in the opposite direction generally occurs somewhere else." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"[…] the law of requisite complexity […] states that in order to fully regulate/control a system, the complexity of the controller has to be at least as great as the complexity of the system that’s being controlled. To put it in even simpler terms, only complexity can destroy complexity." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"What is and isn’t complex depends to a large degree not only on a target system, but also on the system(s) the target interacts with, together with the overall context in which the interacting systems are embedded." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"A good analogy is stretching a rubber band. You can stretch and stretch and even feel the tension increase in the muscles in your hands and arms as the gap from one end of the band to the other widens. But at some point you reach the limits of elasticity of the band and it snaps. The same thing happens with human systems." (John L Casti)

"Reality is a wave function traveling both backward and forward in time." (John L Casti)

No comments:

Post a Comment

Related Posts Plugin for WordPress, Blogger...

On Theorizing

"Observation is so wide awake, and facts are being so rapidly added to the sum of human experience, that it appears as if the theorizer...