"It is possible to pass continuously from any non-singular curve to any other such curve by interposition of other curves in such a way that during this procedure one will meet no other occurrence of singularities, apart from a finite number of times a curve with an ordinary double point, no matter whether the curve has at that point real or imaginary branches." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)
"A singularity is a place where the classical concepts of space and time break down as do all the known laws of physics." (Stephen W Hawking, "Breakdown of Predictability in Gravitational Collapse", Physical Review D, 1976)
"'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)
"Topology has been called 'rubber-sheet geometry', and it is helpful to think of the surfaces as made of rubber. They can be deformed in any way at an, as long as they are not torn and as long as no new qualitative features appear on them. That is, the actual behavior surface for a process with two control factors may be any distortion of the canonical cusp surface, but it can have no local singularity more complex than a pleat." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"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, 1970s)
"A theory that involves singularities and involves them unavoidably, moreover, carries within itself the seeds of its own destruction." (Peter Bergmann, 1980)
"In fact, all our theories of science are formulated on the assumption that space-time is smooth and nearly flat, so they break down at the big bang singularity, where the curvature of space-time is infinite." (Stephen W Hawking, "A Brief History of Time", 1988)
"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)
"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)
"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 problems associated with the initial singularity of the universe bring us to what is called the theory of everything. It is an all-encompassing theory that would completely explain me origin of the universe and everything in it. It would bring together general relativity and quantum mechanics, and explain everything there is to know about the elementary particles of the universe, and the four basic forces of nature (gravitational, electromagnetic, weak, and strong nuclear forces). Furthermore, it would explain the basic laws of nature and the fundamental constants of nature such as the speed of light and Planck's constant." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
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