On Continuity (1970-1989)

"The belief in algebra pervades the whole of eighteenth-century analysis. One sees it in the whole confusion over functions themselves. The distinctions between different kinds of function which matter mathematically are those of our diagram 1.1 : the classes of differentiable, continuous and discontinuous functions hinge on the existence, and equality or non- equality, of the left- and right-hand limiting values of a function and its derivative at a given point." (Ivor Grattan-Guinness, "The Development of the Foundations of Mathematical Analysis from Euler to Riemann", 1970)

"Related is the idea of structural stability and certain variations. This kind of is a property of a dynamical system itself (not of a or orbit) and asserts that nearby dynamical systems have the same structure. The 'same structure' can be defined in several interesting ways, but the basic idea is that two dynamical systems have 'the same structure' if they have the same gross behavior, or the same qualitative behavior. For example, the original definition of 'same structure' of two dynamical systems was that there was an orbit preserving continuous transformation between them. This yields the definition of structural stability proper. It is a recent theorem that every compact manifold admits structurally stable systems, and almost all gradient dynamical systems are structurally stable. But while there exists a rich set of structurally stable systems, there are also important examples which are not stable, and have good but weaker stability properties." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)

"[…] under plane transformations, like those encountered in the arbitrary stretching of a rubber sheet, certain properties of the figures involved are preserved. The mathematician has a name for them. They are called continuous transformations. This means that very close lying points pass into close lying points and a line is translated into a line under these transformations. Quite obviously, then, two intersecting lines will continue to intersect under a continuous transformation, and nonintersecting lines will not intersect; also, a figure with a hole cannot translate into a figure without a hole or into one with two holes, for that would require some kind of tearing or gluing - a disruption of the continuity." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"[…] topology, a science that studies the properties of geometric figures that do not change under continuous transformations." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"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)

"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)

"This notion of each point in a space having a collection of 'neighbourhoods', the neighbourhoods leading in turn to a good definition of continuous function, is the crucial one. Notice that in defining neighbourhoods in a euclidean space we used very strongly the euclidean distance between points. In constructing an abstract space we would like to retain the concept of neighbourhood but rid ourselves of any dependence on a distance function. (A topological equivalence does not preserve distances.)" (Mark A Armstrong, "Basic Topology", 1979)

"Topology has to do with those properties of a space which are left unchanged by the kind of transformation that we have called a topological equivalence or homeomorphism. But what sort of spaces interest us and what exactly do we mean by a 'space? The idea of a homeomorphism involves very strongly the notion of continuity [...]"  (Mark A Armstrong, "Basic Topology", 1979)

"However, it turns out that a one-to-one mapping of the points in a square into the points on a line cannot be continuous. As we move smoothly along a curve through the square, the points on the line which represent the successive points on the square necessarily jump around erratically, not only for the mapping described above but for any one-to-one mapping whatever. Any one-to-one mapping of the square onto the line is discontinuous." (John R Pierce, "An Introduction to Information Theory: Symbols, Signals & Noise" 2nd Ed., 1980)

"We intuitively know space to be a smooth continuum, an arena in which the fundamental particles move and interact. This assumption underpins our physical theories, and no experimental evidence has ever contradicted it. However, the possibility that space may not be smooth cannot be excluded." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The view of science is that all processes ultimately run down, but entropy is maximized only in some far, far away future. The idea of entropy makes an assumption that the laws of the space-time continuum are infinitely and linearly extendable into the future. In the spiral time scheme of the timewave this assumption is not made. Rather, final time means passing out of one set of laws that are conditioning existence and into another radically different set of laws. The universe is seen as a series of compartmentalized eras or epochs whose laws are quite different from one another, with transitions from one epoch to another occurring with unexpected suddenness." (Terence McKenna, "True Hallucinations", 1989)