"[…] 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)
"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."
"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."
"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)
"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)
"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."
"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."
"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."
"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."
"[…] 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)
"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)
"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)
"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."
"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."
"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."
"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."
"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."
"[…] 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."
"[…] 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."
"[…] 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."
"Due to the problem of predicting outlier events, they are
not usually factored into the design of systems."
"[…] 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."
"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)
"Reality is a wave function traveling both backward and forward in time." (John L Casti)
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