"What I am offering, is not a scientific theory, but a method; the first step in the construction of a model is to describe the dynamical models compatible with an empirically given morphology, and this is also the first step in understanding the phenomena under consideration. [...] We may hope that theoreticians will develop a quantitative model [for specific processes described by catastrophe theory ...] But this is only a hope." (René Thom, "Structural Stability and Morphogenesis", 1972)
"The catastrophe model is at the same time much less and much more than a scientific theory; one should consider it as a language, a method, which permits classification and systematization of given empirical data [...] In fact, any phenomenon at all can be explained by a suitable model from catastrophe theory." (René F Thom, 1973)
"On the plane of philosophy properly speaking, of metaphysics, catastrophe theory cannot, to be sure, supply any answer to the great problems which torment mankind. But it favors a dialectical, Heraclitean view of the universe, of a world which is the continual theatre of the battle between 'logoi', between archetypes." (René F Thom, "Catastrophe Theory: Its Present State and Future Perspectives", 1975)
"[...] if the behavior points for the entire control surface are plotted and then connected, they form a smooth surface: the behavior surface. The surface has an overall slope from high values where rage predominates to low values in the region where fear is the prevailing state of mind, but the slope is not its most distinctive feature. Catastrophe theory reveals that in the middle of the surface there must be a smooth double fold, creating a pleat without creases, which grows narrower from the front of the surface to the back and eventually disappears in a singular point where the three sheets of the pleat come together. It is the pleat that gives the model its most interesting characteristics. All the points on the behavior surface represent the most probable behavior [...], with the exception of those on the middle sheet, which represent least probable behavior. Through catastrophe theory we can deduce the shape of the entire surface from the fact that the behavior is bimodal for some control points." (E Cristopher Zeeman, "Catastrophe Theory", Scientific American, 1976)
"Catastrophe Theory is - quite likely - the first coherent attempt (since Aristotelian logic) to give a theory on analogy. When narrow-minded scientists object to Catastrophe Theory that it gives no more than analogies, or metaphors, they do not realise that they are stating the proper aim of Catastrophe Theory, which is to classify all possible types of analogous situations." (René F Thom," La Théorie des catastrophes: État présent et perspective", 1977)
"First, nature's line patterns are not all of the same sort; the triple junctions generic in mud cracks cannot occur with caustics. Second, the geometrical optics of cylindrically symmetric artifacts such as telescopes, where departures from the ideal point focus are treated as 'aberrations', is very different from the geometrical optics of nature, where the generic forms of caustic surfaces are governed by the mathematics of catastrophe theory." Michael V Berry & John F Nye, "Fine Structure in Caustic Junctions", Nature Vol. 267 (3606), 1977)
"the claims made for the theory are greatly exaggerated and its accomplishments, at least in the biological and social sciences, are insignificant. [...] Catastrophe theory is one of many attempts that have been made to deduce the world by thought alone [...] an appealing dream for mathematicians, but a dream that cannot come true." (Héctor J Sussmann & Raphael S Zahler, Nature, 1977)
"A catastrophe, in the very broad sense [René] Thom gives to the word, is any discontinuous transition that occurs when a system can have more than one stable state, or can follow more than one stable pathway of change. The catastrophe is the 'jump' from one state or pathway to another." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"'Catastrophe theory' denotes both a purely mathematical discipline describing certain singularities of smooth maps, as well as the concerted effort to apply these theorems to a wide variety of problems in fields ranging from linguistics and psychology to embryology, evolution, physics, and engineering." (Héctor J Sussmann & Raphael S Zahler, "Catastrophe Theory as Applied to the Social and Biological Sciences: A Critique" Synthese Vol. 37 (2), 1978)
"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 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)
"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)
"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)
"Catastrophes are often stimulated by the failure to feel the emergence of a domain, and so what cannot be felt in the imagination is experienced as embodied sensation in the catastrophe. (William I Thompson, "Gaia, a Way of Knowing: Political Implications of the New Biology", 1987)
"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)
"Chaos and catastrophe theories are among the most interesting recent developments in nonlinear modeling, and both have captured the interests of scientists in many disciplines. It is only natural that social scientists should be concerned with these theories. Linear statistical models have proven very useful in a great deal of social scientific empirical analyses, as is evidenced by how widely these models have been used for a number of decades. However, there is no apparent reason, intuitive or otherwise, as to why human behavior should be more linear than the behavior of other things, living and nonliving. Thus an intellectual movement toward nonlinear models is an appropriate evolutionary movement in social scientific thinking, if for no other reason than to expand our paradigmatic boundaries by encouraging greater flexibility in our algebraic specifications of all aspects of human life." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"[...] chaos and catastrophe theories per se address behavioral phenomena that are consequences of two general types of nonlinear dynamic behavior. In the most elementary of behavioral terms, chaotic phenomena are a class of deterministic processes that seem to mimic random or stochastic dynamics. Catastrophe phenomena, on the other hand, are a class of dynamic processes that exhibit a sudden and large scale change in at least one variable in correspondence with relatively small changes in other variables or, in some cases, parameters." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"Chaos and catastrophe theories directly address the social scientists' need to understand classes of nonlinear complexities that are certain to appear in social phenomena. The probabilistic properties of many chaos and catastrophe models are simply not known, and there is little likelihood that general procedures will be developed soon to alleviate the difficulties inherent with probabilistic approaches in such complicated settings." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"Fundamental to catastrophe theory is the idea of a bifurcation. A bifurcation is an event that occurs in the evolution of a dynamic system in which the characteristic behavior of the system is transformed. This occurs when an attractor in the system changes in response to change in the value of a parameter. A catastrophe is one type of bifurcation. The broader framework within which catastrophes are located is called dynamical bifurcation theory." (Courtney Brown, "Chaos and Catastrophe Theories", 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)
"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 reason catastrophe theory can tell us about such abrupt changes in a system's behavior is that we usually observe a dynamical system when it's at or near its steady-state, or equilibrium, position. And under various assumptions about the nature of the system's dynamical law of motion, the set of all possible equilibrium states is simply the set of critical points of a smooth function closely related to the system dynamics. When these critical points are nondegenerate, Morse's Theorem applies. But it is exactly when they become degenerate that the system can move sharply from one equilibrium position to another. The Thorn Classification Theorem tells when such shifts will occur and what direction they will take." (John L Casti, "Five Golden Rules", 1995)
"A catastrophe is a universal unfolding of a singular function germ. The singular function germs are called organization centers of the catastrophes. [...] Catastrophe theory is concerned with the mathematical modeling of sudden changes - so called 'catastrophes' - in the behavior of natural systems, which can appear as a consequence of continuous changes of the system parameters. While in common speech the word catastrophe has a negative connotation, in mathematics it is neutral." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])
"Catastrophe theory can be thought of as a link between classical analysis, dynamical systems, differential topology (including singularity theory),modern bifurcation theory and the theory of complex systems. [...] The name ‘catastrophe theory’ is used for a combination of singularity theory and its applications. [...] From the didactical point of view, there are two main positions for courses in catastrophe theory at university level: Trying to teach the theory as a perfect axiomatic system consisting of exact definitions, theorems and proofs or trying to teach mathematics as it can be developed from historical or from natural problems." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])
"Classification is only one of the mathematical aspects of catastrophe theory. Another is stability. The stable states of natural systems are the ones that we can observe over a longer period of time. But the stable states of a system, which can be described by potential functions and their singularities, can become unstable if the potentials are changed by perturbations. So stability problems in nature lead to mathematical questions concerning the stability of the potential functions." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])
"It is more a philosophy than mathematics, and even as a philosophy it doesn't explain the real world [...] as mathematics, it brings together two of the most basic ideas in modern math: the study of dynamic systems and the study of the singularities of maps. Together, they cover a very wide area - but catastrophe theory brings them together in an arbitrary and constrained way." (Steven Smale)
"While it must be granted that a number of immoderate claims in the form of 'catastrophe theory can do everything' have been made in the literature, on the basis of too little experience, it doesn't seem that the proper response is an equally immoderate claim that 'catastrophe theory can do nothing' on the basis of that same body of experience." (Robert Rosen)
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