"Because of its foundation in topology, catastrophe theory is qualitative, not quantitative. Just as geometry treated the properties of a triangle without regard to its size, so topology deals with properties that have no magnitude, for example, the property of a given point being inside or outside a closed curve or surface. This property is what topologists call 'invariant' -it does not change even when the curve is distorted. A topologist may work with seven-dimensional space, but he does not and cannot measure (in the ordinary sense) along any of those dimensions. The ability to classify and manipulate all types of form is achieved only by giving up concepts such as size, distance, and rate. So while catastrophe theory is well suited to describe and even to predict the shape of processes, its descriptions and predictions are not quantitative like those of theories built upon calculus. Instead, they are rather like maps without a scale: they tell us that there are mountains to the left, a river to the right, and a cliff somewhere ahead, but not how far away each is, or how large." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"But there is another kind of change, too, change that is less suited to mathematical analysis: the abrupt bursting of a bubble, the discontinuous transition from ice at its melting point to water at its freezing point, the qualitative shift in our minds when we 'get' a pun or a play on words. Catastrophe theory is a mathematical language created to describe and classify this second type of change. It challenges scientists to change the way they think about processes and events in many fields." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"Catastrophe theory (in
particular its essential concept of structural stability) is really a paradigm
rather than a theory. It has attracted so much attention and generated so much
argument because its scope and application appear to be virtually unlimited."
"Catastrophe theory is a controversial new way of thinking about change-change in a course of events, change in an object's shape, change in a system's behavior, change in ideas themselves. Its name suggests disaster, and indeed the theory can be applied to literal catastrophes such as the collapse of a bridge or the downfall of an empire. But it also deals with changes as quiet as the dancing of sunlight on the bottom of a pool of water and as subtle as the transition from waking to sleep." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"For one thing, they
say, the classification of the elementary catastrophes depends on what is
called 'local' analysis of topological properties-in other words, analysis
that describes only the immediate neighborhood of the singularity. But the classification
theorem does not prove that a system's total range, its 'global' behavior, is like its behavior in that neighborhood. […] Since the topological
approach provides no scale, it requires an act of faith to identify a
mathematical jump on the catastrophe surface with an observed discontinuity in
nature."
"In any system governed by a potential, and in which the system's behavior is determined by no more than four different factors, only seven qualitatively different types of discontinuity are possible. In other words, while there are an infinite number of ways for such a system to change continuously (staying at or near equilibrium), there are only seven structurally stable ways for it to change discontinuously (passing through non-equilibrium states)." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"In fact, there is a
close correspondence between catastrophe theory and bifurcation theory, and in
many cases their mathematics are equivalent or overlapping. Bifurcation theory,
though, is much more analytic in spirit."
"In living systems,
equilibrium is dynamic rather than static, because organisms and societies are
always taking in and transforming energy. They tend to establish cycles in
which no one state is stable, but the whole series of states resists
disturbance like a spinning gyroscope."
"Is catastrophe theory
correct? In its mathematics, yes; in the natural philosophy that inspired it
and the scientific applications that flow from it, the only possible answer is
that it's too soon to say. There is always a chance of error whenever we try to
capture any aspect of reality in mathematical symbols, and another chance of
error when (after working with the symbols) we use them to generate
descriptions or predictions of reality."
"Is the [catastrophe] theory useful? in the rigorous applications, yes; in the illustrations, sometimes; in the 'invocations', both yes and no. Yes, because catastrophe theory provides a common vocabulary for features of many different processes. Someday it may be as natural to speak of a 'cusp situation' or a 'butterfly compromise' as it is today to speak of the 'point of diminishing returns' or of a 'quantum jump'. No, because when the theory is invoked for the suggestiveness of its images, it cannot usually tell us anything we did not know before (although it can make explicit certain features that other models tend to neglect)." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"It is not enough to know the critical stress, that is, the quantitative breaking point of a complex design; one should also know as much as possible of the qualitative geometry of its failure modes, because what will happen beyond the critical stress level can be very different from one case to the next, depending on just which path the buckling takes. And here catastrophe theory, joined with bifurcation theory, can be very helpful by indicating how new failure modes appear." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"Mathematics has two
cutting edges: one in its formal abstractions, the pure manipulation of ideas,
and one in its applications to the real world."
"The unfoldings are called catastrophes because each of them has regions where a dynamic system can jump suddenly from one state to another, although the factors controlling the process change continuously. Each of the seven catastrophes represents a pattern of behavior determined only by the number of control factors, not by their nature or by the interior mechanisms that connect them to the system's behavior. Therefore, the elementary catastrophes can be models for a wide variety of processes, even those in which we know little about the quantitative laws involved." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"The most complex system imaginable is the mind - by definition, since the mind must be at least one degree more complex than whatever it imagines. Catastrophe theory proposes that qualitative stability is a necessary attribute of thought; without it, recognition and memory would be impossible." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"The most widely used
mathematical tools in the social sciences are statistical, and the prevalence
of statistical methods has given rise to theories so abstract and so hugely
complicated that they seem a discipline in themselves, divorced from the world
outside learned journals. Statistical theories usually assume that the behavior
of large numbers of people is a smooth, average 'summing-up' of
behavior over a long period of time. It is difficult for them to take into
account the sudden, critical points of important qualitative change. The
statistical approach leads to models that emphasize the quantitative conditions
needed for equilibrium-a balance of wages and prices, say, or of imports and
exports. These models are ill suited to describe qualitative change and social
discontinuity, and it is here that catastrophe theory may be especially helpful."
"The qualitative type
of any stable discontinuity does not depend on the specific nature of the
potential involved, merely on its existence. It does not depend on the specific
conditions regulating behavior, merely on their number. It does not depend on
the specific quantitative, cause-and-effect relationship between the conditions
and the resultant behavior, merely on the empirical fact that such a
relationship exists."
"The social or 'inexact' sciences have an uneasy relationship with mathematics. To some extent, they seek a Newtonian goal of quantification and prediction. Yet the human and environmental variables they must deal with are so many and varied, the possibility of meaningful experiment so limited, and the data (both current and historical) so questionable, that the greatest achievements of sociology and economics so far are chiefly descriptive rather than analytic." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"Two assumptions are needed to apply catastrophe theory as it now stands: first, that the system described be governed by a potential, and second, that its behavior depend on a limited number of control factors. Without these assumptions, the classification of the elementary catastrophes is impossible." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"Yet wherever the cracks appear, they show a tendency to extend towards each other, to form characteristic networks, to form specific types of junctions. The location, the magnitude, and the timing of the cracks (their quantitative aspects) are beyond calculation, but their patterns of growth and the topology of their joining (the qualitative aspects) recur again and again." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
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