"Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is stretched" in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)
"[…] the simplicity of nature which we at present grasp is really the result of infinite complexity; and that below the uniformity there underlies a diversity whose depths we have not yet probed, and whose secret places are still beyond our reach." (William Spottiswoode, 1879)
"In various fields of knowledge the problem of the relationship between cause and condition is solved in different ways, depending mainly on the complexity of the relationships that are being studied, their uniformity or, on the contrary, the distinctness and comparative importance of separate factors." (Alexander Spirkin, "Dialectical Materialism", 1983)
"In the real world, none of these assumptions are uniformly valid. Often people want to know why mathematics and computers cannot be used to handle the meaningful problems of society, as opposed, let us say, to the moon boondoggle and high energy-high cost physics. The answer lies in the fact that we don't know how to describe the complex systems of society involving people, we don't understand cause and effect, which is to say the consequences of decisions, and we don't even know how to make our objectives reasonably precise. None of the requirements of classical science are met. Gradually, a new methodology for dealing with these 'fuzzy' problems is being developed, but the path is not easy." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)
"Things are similar: this makes science possible. Things are different: this makes science necessary. At various times in the history of science important advances have been made either by abstracting away differences to reveal similarity or by emphasizing the richness of variation within a seeming uniformity. But either choice by itself is ultimately misleading. The general does not completely contain the particular as cases, but the empiricist refusal to group, generalize, and abstract reduces science to collecting - if not specimens, then examples." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)
"There is no coherent knowledge, i.e. no uniform comprehensive account of the world and the events in it. There is no comprehensive truth that goes beyond an enumeration of details, but there are many pieces of information, obtained in different ways from different sources and collected for the benefit of the curious. The best way of presenting such knowledge is the list - and the oldest scientific works were indeed lists of facts, parts, coincidences, problems in several specialized domains." (Paul K Feyerabend, "Farewell to Reason", 1987)
"There is no coherent knowledge, i.e. no uniform comprehensive account of the world and the events in it. There is no comprehensive truth that goes beyond an enumeration of details, but there are many pieces of information, obtained in different ways from different sources and collected for the benefit of the curious. The best way of presenting such knowledge is the list - and the oldest scientific works were indeed lists of facts, parts, coincidences, problems in several specialized domains." (Paul K Feyerabend,"Farewell to Reason", 1987)
"There is a technical difference between a bar chart and a histogram in that the number represented is proportional to the length of bar in the former and the area in the latter. This matters if non-uniform binning is used. Bar charts can be used for qualitative or quantitative data, whereas histograms can only be used for quantitative data, as no meaning can be attached to the width of the bins if the data are qualitative." (Roger J Barlow, "Statistics: A guide to the use of statistical methods in the physical sciences", 1989)
"Where did those symmetries come from? From the even more extensive set of symmetries of the (idealised) uniform state in the infinite dish. The instability of that state caused certain symmetries to be eliminated, but others persist. For target patterns, some rotations and reflections persist. For spirals, what persists is the space-time symmetries 'let time pass and then rotate back'. In a very curious sense, the patterns that we see in the spirals are evidence of other patterns that might have been - the unstable uniform state with its enormous amount of (totally boring) symmetry. They are 'caused' by something that doesn't actually happen." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)
"Why don't the chemicals take up the fully symmetric uniform state? Because it is unstable. Any tiny lack of uniformity grows, and destroys the uniform pattern. And in the real world there are always tiny lacks of uniformity - dust motes, bubbles, even just a few molecules vibrating because of heat. (All molecules vibrate because of heat - or more accurately 'heat' is what you get when molecules vibrate - but it only takes a few of them to trigger instability.) The instability is not intuitively obvious, but it's what happens both in the real world and in mathematical models, and here we can take it as given." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)
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