On Smoothness I

"Since one could directly derive the expansion in series of algebraic functions according to the powers of an increment, the derivatives, and the integral, one not only held that it was possible to assume the existence of such a series, derivative, and integral for all functions in general, but one never even had the idea that herein lay an assertion, whether it now be an axiom or a theorem - so self-evident did the transfer of the properties of algebraic functions to transcendental ones seem in the light of the geometrical view of curves representing functions. And examples in which purely analytic functions displayed singularities that were clearly different from those of algebraic functions remained entirely unnoticed." (Hermann Hankel, 1870)

"It is not theory, but actual statistical experience, which forces us to the conclusion that, however little we know of what will happen in the individual instance, yet the frequency of a large number of instances is distributed round the mode in a manner more and more smooth and uniform the greater the number of individual instances. When this distribution round the mode does not take place [...] then we assert that some cause other than chance is at work." (Karl Pearson "The Chances of Death", 1895)

"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"Two Riemann surfaces which can be mapped conformally onto each other are (conformally) equivalent and are to be regarded as different representations of one and the same ideal Riemann surface. The intrinsic properties of a Riemann surface will include only those properties which are invariant under conformal maps; that is, those properties which, if possessed by one Riemann surface are possessed by every equivalent surface. Obviously all topological properties are intrinsic properties of a Riemann surface; similarly with those properties belonging to the surface by virtue of its smoothness." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"One of the operations of algebra that is important for a mechanical brain is approximation, the problem of getting close to the right value of a number. [...] Another important operation of algebra is interpolation, the problem of putting values smoothly in between other values."  (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"[...] 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' 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)

"Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in straight line. [...] Nature exhibits not simply a higher degree but an altogether different level of complexity." (Benoît B Mandelbrot, 1984)