On Uniformity (-1799)

"Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing, e.g., a quality. For whatever ratio is found to exist between intensity and intensity, in relating intensities of the same kind, a similar ratio is found to exist between line and line, and vice versa." (Nicole Oresme, "Tractatus de configurationibus qualitatum et motuum" ["A treatise on the uniformity and difformity of intensities"], 1352) [definition of a functional relationship between two variables]

"Reason may be employed in two ways to establish a point: first for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results [...]" (Saint Thomas Aquinas,"Summa Theologica", 1485)

"Those who devised the eccentrics seen thereby in large measure to have solved the problem of apparent motions with approximate calculations. But meanwhile they introduced a good many ideas which apparently contradict the first principles of uniform motion. Nor could they elicit or deduce from the eccentrics the principal consideration, that is, the structure of the universe and the true symmetry of its parts. " (Nicolaus Copernicus, "De revolutionibus orbium coelestium", 1543)

"To measure motion, space is as necessary to be considered as time....[They] are made use of to denote the position of finite: real beings, in respect one to another, in those infinite uniform oceans of duration and space." (John Locke, "An Essay Concerning Human Understanding", 1689)

"For chemistry is no science form’d à priori; ’tis no production of the human mind, framed by reasoning and deduction: it took its rise from a number of experiments casually made, without any expectation of what follow’d; and was only reduced into an art or system, by collecting and comparing the effects of such unpremeditated experiments, and observing the uniform tendency thereof. So far, then, as a number of experimenters agree to establish any undoubted truth; so far they may be consider'd as constituting the theory of chemistry." (Hermann Boerhaave, "A New Method of Chemistry", 1727)

"The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

On Uniformity (1800-1849)

"Is it not truly wonderful, that, in the constitution of the universe, time and space should every where be so happily combined, that notwithstanding the infinity of wheels and springs which mutually depend on each other, and which are all necessary to the play of the machine, the visible order of nature should, nevertheless, every where preserve the same air of simplicity aud uniformity." (Johann H Lambert, "The System of the World", 1800)

"Space and time: these man proposes to measure. The one circumscribes his momentary existence, the other accompanies his successive stages in life. These two dimensions are tied together through a necessary relationship, namely, motion. When motion is constant and uniform, space is known by time and time is measured by space. Man has nothing within him that is constant and uniform; continually modified every instant, he is changing, irregular. and hardly durable enough to be a measure of duration." (Sophie Germain, cca. 1809)

"In science, reason is the guide; in poetry, taste. The object of the one is truth, which is uniform and indivisible; the object of the other is beauty, which is multiform and varied." (Charles C Colton,"Lacon", 1820)

"Problems relative to the uniform propagation, or to the varied movements of heat in the interior of solids, are reduced […] to problems of pure analysis, and the progress of this part of physics will depend in consequence upon the advance which may be made in the art of analysis. The differential equations […] contain the chief results of the theory; they express, in the most general and concise manner, the necessary relations of numerical analysis to a very extensive class of phenomena; and they connect forever with mathematical science one of the most important branches of natural philosophy." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"Heat can evidently be a cause of motion only by virtue of the changes of volume or of form which it produces in bodies. These changes are not caused by uniform temperature but rather by alternations of heat and cold." (Nicolas L S Carnot, "Reflections on the Motive Power of Heat and on Machines Fitted to Develop Power", 1824)

"Instead of seeking to attain consistency and uniformity of system, as some modern writers have attempted, by banishing this thought of time from the higher Algebra, I seek to attain the same object, by systematically introducing it into the lower or earlier parts of the science." (William R Hamilton, "Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time", Transactions of the Royal Irish Academy, 1837)

On Uniformity (1850-1899)

"Isolated facts and experiments have in themselves no value, however great their number may be. They only become valuable in a theoretical or practical point of view when they make us acquainted with the law of a series of uniformly recurring phenomena, or, it may be, only give a negative result showing an incompleteness in our knowledge of such a law, till then held to be perfect." (Hermann von Helmholtz, "The Aim and Progress of Physical Science", 1869)

"Causation is defined by some modern philosophers as unconditional uniformity of succession, e.g., existence of fire follows from putting a lighted match to the fuel." (William K Clifford, Energy and Force", 1873)

"It is the prerogative of Intellect to discover what is uniform and unchanging in the phenomena around us." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"The reason and the immediate purpose for the introduction of complex quantities into mathematics lie in the theory of uniform relations between variable quantities which are expressed by simple mathematical formulas. Using these relations in an extended sense, by giving complex values to the variable quantities involved, we discover in them a hidden harmony and regularity that would otherwise remain hidden." (Bernhard Riemann,"Gesammelte Mathematische Werke", 1876)

"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, [Report of the Forty-eighth Meeting of the British Association for the, Advancement of Science] 1878)

"The aim of scientific thought, then, is to apply past experience to new circumstances; the instrument is an observed uniformity in the course of events. By the use of this instrument it gives us information transcending our experience, it enables us to infer things that we have not seen from things that we have seen; and the evidence for the truth of that information depends on our supposing that the uniformity holds good beyond our experience." (William K Clifford, "Lectures and Essays", 1879)

"It is difficult to give an idea of the vast extent of modern mathematics. The word 'extent' is not the right one: I mean extent crowded with beautiful details - not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood and flower [...]" (Arthur Cayley, [address before the meeting of the British Association at Southport] 1883)

"We may conceive our space to have everywhere a nearly uniform curvature, but that slight variations of the curvature may occur from point to point, and themselves vary with the time. These variations of the curvature with the time may produce effects which we not unnaturally attribute to physical causes independent of the geometry of our space. We might even go so far as to assign to this variation of the curvature of space 'what really happens in that phenomenon which we term the motion of matter'." (William K Clifford, Richard C Rowe & Karl Pearson,"The Common Sense of the Exact Sciences", 1885)

"If we start with the assumption, grounded on experience, that there is uniformity in this average, and so long as this is secured to us, we can afford to be perfectly indifferent to the fate, as regards causation, of the individuals which compose the average." (John Venn, The Logic of Chance: An Essay on the Foundation and Province of the - Theory of Probability, Chance, Causation, and Design", 1887)

""[…] 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)

"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 Entropy of a system is the mechanical work it can perform without communication of heat, or alteration of its total volume, all transference of heat being performed by reversible engines. When the pressure and temperature of the system have become uniform the entropy is exhausted. The original energy of the system is equal to the sum of the entropy and the energy remaining in the state of uniform pressure and temperature. The entropy of a system consisting of several component systems is the same in whatever order the entropy of the parts is exhausted. It is therefore equal to the sum of the entropy of each component system, together with the entropy of the system consisting of the component systems, each with its own entropy exhausted." (James C Maxwell, "Theory of Heat", 1899)

On Uniformity (1900-1924)

"Mathematics is a type of thought which seems ingrained in the human mind, which manifests itself to some extent with even the primitive races, and which is developed to a high degree with the growth of civilization. […] A type of thought, a body of results, so essentially characteristic of the human mind, so little influenced by environment, so uniformly present in every civilization, is one of which no well-informed mind today can be ignorant." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"The equations of Newton's mechanics exhibit a two-fold invariance. Their form remains unaltered, firstly, if we subject the underlying system of spatial coordinates to any arbitrary change of position ; secondly, if we change its state of motion, namely, by imparting to it any uniform translatory motion ; furthermore, the zero point of time is given no part to play. We are accustomed to look upon the axioms of geometry as finished with, when we feel ripe for the axioms of mechanics, and for that reason the two invariances are probably rarely mentioned in the same breath. Each of them by itself signifies, for the differential equations of mechanics, a certain group of transformations. The existence of the first group is looked upon as a fundamental characteristic of space. The second group is preferably treated with disdain, so that we with un-troubled minds may overcome the difficulty of never being able to decide, from physical phenomena, whether space, which is supposed to be stationary, may not be after all in a state of uniform translation. Thus the two groups, side by side, lead their lives entirely apart. Their utterly heterogeneous character may have discouraged any attempt to compound them. But it is precisely when they are compounded that the complete group, as a whole, gives us to think." (Hermann Minkowski, "Space and Time", [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"An experiment is an observation that can be repeated, isolated and varied. The more frequently you can repeat an observation, the more likely are you to see clearly what is there and to describe accurately what you have seen. The more strictly you can isolate an observation, the easier does your task of observation become, and the less danger is there of your being led astray by irrelevant circumstances, or of placing emphasis on the wrong point. The more widely you can vary an observation, the more clearly will be the uniformity of experience stand out, and the better is your chance of discovering laws." (Edward B Titchener, "A Text-Book of Psychology", 1909)

"The term dispersion is used to indicate the facts that within a given group, the items differ from one another in size or in other words, there is lack of uniformity in their sizes." (Willford I King, "The Elements of Statistical Method", 1912)

"Statistical tables are essentially specific in their meaning, and they require data that are uniformly specific in the same kind and degree." (W. B Bailey & John Cummings, "Statistics", 1917)

On Uniformity (1925-1949)

"Physics has progressed because, in the first place, she accepted the uniformity of nature; because, in the next place, she early discovered the value of exact measurements; because, in the third place, she concentrated her attention on the regularities that underlie the complexities of phenomena as they appear to us; and lastly, and not the least significant, because she emphasized the importance of the experimental method of research. An ideal or crucial experiment is a study of an event, controlled so as to give a definite and measurable answer to a question - an answer in terms of specific theoretical ideas, or better still an answer in terms of better understood relations." (Thomas H Morgan, "The Relation of Biology to Physics", Science Vol. LXV" (1679), 1927)

"The rational concept of probability, which is the only basis of probability calculus, applies only to problems in which either the same event repeats itself again and again, or a great number of uniform elements are involved at the same time. Using the language of physics, we may say that in order to apply the theory of probability we must have a practically unlimited sequence of uniform observations." (Richard von Mises,Probability, Statistics and Truth", 1928)

"A Weltanschauung [worldview] is an intellectual construction which solves all the problems of our existence uniformly on the basis of one overriding hypothesis, which, accordingly, leaves no question unanswered and in which everything that interests us finds its fixed place [...] the worldview of science already departs noticeably from our definition. It is true that it too assumes the uniformity of the explanation of the universe; but it does so only as a programme, the fulfillment of which is relegated to the future. Sigmund Freud, "New introductory lectures on psycho-analysis", 1932)

"A Weltanschauung [worldview] is an intellectual construction which solves all the problems of our existence uniformly on the basis of one overriding hypothesis, which, accordingly, leaves no question unanswered and in which everything that interests us finds its fixed place [...] the worldview of science already departs noticeably from our definition. It is true that it too assumes the uniformity of the explanation of the universe; but it does so only as a programme, the fulfillment of which is relegated to the future. Sigmund Freud,"New introductory lectures on psycho-analysis", 1932)

"To the grand primary impression of the world power, the immensities, the pervading order, and the universal flux, with which the man of feeling has been nurtured from the old, modern science has added thrilling impressions of manifoldedness, intricacy, uniformity, inter-relatedness, and evolution. Science widens and clears the emotional window. There are great vistas to which science alone can lead, and they make for elevation of mind." (J Arthur Thomson, "The Outline of Science" Vol. 4, 1937)

"[…] in the world of immediate experience, the world of things is there. Trees grow, day follows night, and death supervenes upon life. One may not say that relations here are external or even internal. They are not relations at all. They are lost in the indescriptibility of things and events, which are what they are. The world which is the test of all observations and all scientific hypothetical reconstruction has in itself no system that can be isolated as a structure of laws, or uniformities, though all laws and formulations of uniformities must be brought to its court for its imprimatur." (Donald C May & George H Mead, "The Philosophy of the Act", 1938)

"For the state centralisation is the appropriate form of organisation, since it aims at the greatest possible uniformity in social life for the maintenance of political and social equilibrium. But for a movement whose very existence depends on prompt action at any favourable moment and on the independent thought and action of its supporters, centralism could but be a curse by weakening its power of decision and systematically repressing all immediate action. [...] Organisation is, after all, only a means to an end. When it becomes an end in itself, it kills the spirit and the vital initiative of its members and sets up that domination by mediocrity which is the characteristic of all bureaucracies." (Rudolf Rocker, "Anarcho-Syndicalism", 1938)

"Science [...] involves active, purposeful search; it discovers, accumulates, sifts, orders, and tests data; it is a slow, painstaking, laborious activity; it is a search after bodies of knowledge sufficiently comprehensive to lead to the discovery of uniformities, sequential orders or so-called 'laws'; it may be carried on by an individual, but it gains relevance only as it produces data which can be added to and tested by the findings of others." (Constantine Panunzio, "Major Social Institutions", 1939)

"Science is the attempt to make the chaotic diversity of our sense experience correspond to a logically uniform system of thought." (Albert Einstein, "Considerations Concerning the Fundaments of Theoretical Physics", Science Vol. 91" (2369), 1940)

"An isolated system or a system in a uniform environment" (which for the present consideration we do best to include as a part of the system we contemplate) increases its entropy and more or less rapidly approaches the inert state of maximum entropy. We now recognize this fundamental law of physics to be just the natural tendency of things to approach the chaotic state" (the same tendency that the books of a library or the piles of papers and manuscripts on a writing desk display) unless we obviate it." (The analogue of irregular heat motion, in this case, is our handling those objects now and again without troubling to put them back in their proper places.)" (Erwin Schrödinger, "What is Life?", 1944)

"In a metamathematical sense our theory provides general concepts applicable to all branches of abstract mathematics, and so contributes to the current trend towards uniform treatment of different mathematical disciplines. In particular, it provides opportunities for the comparison of constructions and of the isomorphisms occurring in different branches of mathematics; in this way it may occasionally suggest new results by analogy." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth. […] No force of gravitation is […] needed to impress curvature on world-lines; the curvature is inherent in the space […]" (James H Jeans," The Growth of Physical Science", 1947)

"To some people, statistics is ‘quartered pies, cute little battleships and tapering rows of sturdy soldiers in diversified uniforms’. To others, it is columns and columns of numerical facts. Many regard it as a branch of economics. The beginning student of the subject considers it to be largely mathematics." (The Editors, "Statistics, The Physical Sciences and Engineering", The American Statistician, Vol. 2, No. 4, 1948)

On Uniformity (1950-1974)

"Science is the attempt to make the chaotic diversity of our sense-experience correspond to a logically uniform system of thought. " (Albert Einstein, "Out of My Later Years", 1950)

"The technical analysis of any large collection of data is a task for a highly trained and expensive man who knows the mathematical theory of statistics inside and out. Otherwise the outcome is likely to be a collection of drawings - quartered pies, cute little battleships, and tapering rows of sturdy soldiers in diversified uniforms - interesting enough in the colored Sunday supplement, but hardly the sort of thing from which to draw reliable inferences." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)

"When, for instance, I see a symmetrical object, I feel its pleasurable quality, but do not need to assert explicitly to myself, ‘How symmetrical!’. This characteristic feature may be explained as follows. In the course of individual experience it is found generally that symmetrical objects possess exceptional and desirable qualities. Thus our own bodies are not regarded as perfectly formed unless they are symmetrical. Furthermore, the visual and tactual technique by which we perceive the symmetry of various objects is uniform, highly developed, and almost instantaneously applied. It is this technique which forms the associative 'pointer.' In consequence of it, the perception of any symmetrical object is accompanied by an intuitive aesthetic feeling of positive tone." (George D Birkhoff, "Mathematics of Aesthetics", 1956)

"Exact truth of a null hypothesis is very unlikely except in a genuine uniformity trial." (David R Cox, "Some problems connected with statistical inference", Annals of Mathematical Statistics 29, 1958)

"[A] sequence is random if it has every property that is shared by all infinite sequences of independent samples of random variables from the uniform distribution." (Joel N Franklin, 1962)

"Science is the reduction of the bewildering diversity of unique events to manageable uniformity within one of a number of symbol systems, and technology is the art of using these symbol systems so as to control and organize unique events. Scientific observation is always a viewing of things through the refracting medium of a symbol system, and technological praxis is always handling of things in ways that some symbol system has dictated. Education in science and technology is essentially education on the symbol level." (Aldous L Huxley, "Essay", Daedalus, 1962)

"Statistics is the description in numerical terms of experiences concerning phenomena not subject to regular uniformity. […] Statistic is therefore a specific method of history." (Ludwig von Mises, "The Ultimate Foundation of Economic Science: An Essay on Method", 1962)

"Theories are usually introduced when previous study of a class of phenomena has revealed a system of uniformities. […] Theories then seek to explain those regularities and, generally, to afford a deeper and more accurate understanding of the phenomena in question. To this end, a theory construes those phenomena as manifestations of entities and processes that lie behind or beneath them, as it were." (Carl G Hempel, "Philosophy of Natural Science", 1966)

"[A] sequence is random if it has every property that is shared by all infinite sequences of independent samples of random variables from the uniform distribution." (J. N. Franklin" (1962)"[…] random numbers should not be generated with a method chosen at random. Some theory should be used." (Donald E. Knuth, "The Art of Computer Programming" Vol. II, 1968)

"In the definition of a coordinate system we have required that the coordinate neighborhood and the range in Rd be open sets. This is contrary to popular usage, or at least more specific than the usage of curvilinear coordinates in advanced calculus. For example, spherical coordinates are used even along points of the z axis where they are not even 1-1. The reasons for the restriction to open sets are that it forces a uniformity in the local structure which simplifies analysis on a manifold" (there are no 'edge points') and, even if local uniformity were forced in some other way, it avoids the problem of. spelling out what we mean by differentiability at boundary points of the coordinate neighborhood; that is, one-sided derivatives need not be mentioned. On the other hand, in applications, boundary value problems frequently arise, the setting for which is a manifold with boundary. These spaces are more general than manifolds and the extra generality arises from allowing a boundary manifold of one dimension less. The points of the boundary manifold have a coordinate neighborhood in the boundary manifold which is attached to a coordinate neighborhood of the interior in much the same way as a face of a cube is attached to the interior. Just as the study of boundary value problems is more difficult than the study of spatial problems, the study of manifolds with boundary is more difficult than that of mere manifolds, so we shall limit ourselves to the latter." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"Scientific knowledge is not created solely by the piecemeal mining of discrete facts by uniformly accurate and reliable individual scientific investigations. The process of criticism and evaluation, of analysis and synthesis, are essential to the whole system. It is impossible for each one of us to be continually aware of all that is going on around us, so that we can immediately decide the significance of every new paper that is published. The job of making such judgments must therefore be delegated to the best and wisest among us, who speak, not with their own personal voices, but on behalf of the whole community of Science. […] It is impossible for the consensus - public knowledge - to be voiced at all, unless it is channeled through the minds of selected persons, and restated in their words for all to hear." (John M Ziman, "Public Knowledge: An Essay Concerning the Social Dimension of Science", 1968)

"Scientific knowledge is not created solely by the piecemeal mining of discrete facts by uniformly accurate and reliable individual scientific investigations." (John M Ziman, "Public Knowledge: An Essay Concerning the Social Dimension of Science", 1968)

"The machine rules. Human life is rigorously controlled by it, dominated by the terribly precise will of mechanisms. These creatures of man are exacting. They are now reacting on their creators, making them like themselves. They want well-trained humans; they are gradually wiping out the differences between men, fitting them into their own orderly functioning, into the uniformity of their own regimes. They are thus shaping humanity for their own use, almost in their own image." (Paul A Valéry, "Fairy Tales for Computers", 1969)

"The figures which excite in us the ideas of beauty seem to be those in which there is uniformity amidst variety. […] What we call beautiful in objects, to speak in the mathematical style, seems to be in compound ratio of uniformity and variety: so that where the uniformity of bodies is equal, the beauty is as the variety; and where the variety is equal, the beauty is as the uniformity." (Francis Hutcheson,"An Inquiry Concerning Beauty, Order, Harmony, Design", 1973)

"In order to describe meanders, then, we can invoke a model involving scour and centrifugal force, a model that describes the uniform expenditure of energy, or a model based on probability theory. All three models describe the same phenomenon. As far as meanders are concerned, all three models happen to be interrelated - but not out of any fundamental necessity. That is to say, the cross circulation induced by centrifugal force need not necessarily result in a uniform distribution of effort or produce a path that is especially probable. In a world of limited patterns, however, the meander answers several entirely different sets of specifications, so that scour, uniform effort, and probability produce the same design." (Peter B Stevens, "Patterns in Nature", 1974

"The spiral is beautifully uniform; it curves around on itself in a perfectly regular manner. It can fill all of two-dimensional space, being capable of infinite expansion, and it is also quite short. But [...], as measured by the mean of distances to its center, the spiral is extremely indirect." (Peter B Stevens, "Patterns in Nature", 1974)

On Uniformity (1975-1999)

"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)

On Uniformity (2000-)

"There is no ‘scientific worldview’ just as there is no uniform enterprise ‘science’- except in the minds of metaphysicians, school masters, and scientists blinded by the achievements of their own particular niche." (Paul Feyerabend, "Conquest of Abundance", 2001)

"When natural frequencies are transformed into conditional probabilities, the base rate information is taken out (this is called normalization). The benefit of this normalization is that the resulting values fall within the uniform range of 0 and 1. The cost, however, is that when drawing inferences from probabilities (as opposed to natural frequencies), one has to put the base rates back in by multiplying the conditional probabilities by their respective base rates." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002

"Limiting factors in population dynamics play the role in ecology that friction does in physics. They stop exponential growth, not unlike the way in which friction stops uniform motion. Whether or not ecology is more like physics in a viscous liquid, when the growth-rate-based traditional view is sufficient, is an open question. We argue that this limit is an oversimplification, that populations do exhibit inertial properties that are noticeable. Note that the inclusion of inertia is a generalization - it does not exclude the regular rate-based, first-order theories. They may still be widely applicable under a strong immediate density dependence, acting like friction in physics." (Lev Ginzburg & Mark Colyvan, "Ecological Orbits: How Planets Move and Populations Grow", 2004)

"Statistical tests are just a way of working out the probability of obtaining the observed, or an even more extreme, difference among samples (or between an observed and expected value) if a specific hypothesis (usually the null of no difference) is true. Once the probability is known, the experimenter can make a decision about the difference, using criteria that are uniformly used and understood." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005) 

"One of the apparent paradoxes in probability is that, while the outcome of the next roll of a die or toss of a coin may be unpredictable, there are nevertheless underlying patterns in the outcomes overall. Specifically, when a fair die is rolled many times, there is a 'settling down' effect as the proportion of each outcome (1, 2, 3, …, 6) gradually approaches 1/6. In the limiting case, as the number of rolls reaches infinity, the shape of the probability distribution becomes uniform." (Alan Graham, "Developing Thinking in Statistics", 2006)

"Let's face it, the universe is messy. It is nonlinear, turbulent, and chaotic. It is dynamic. It spends its time in transient behavior on its way to somewhere else, not in mathematically neat equilibria. It self-organizes and evolves. It creates diversity, not uniformity. That's what makes the world interesting, that's what makes it beautiful, and that's what makes it work." (Donella H Meadow, "Thinking in Systems: A Primer", 2008)

"[…] topology is the study of those properties of geometric objects which remain unchanged under bi-uniform and bi-continuous transformations. Such transformations can be thought of as bending, stretching, twisting or compressing or any combination of these." (Lokenath Debnath, "The Legacy of Leonhard Euler - A Tricentennial Tribute", 2010)

"The invariance of physical laws with respect to position or orientation (i.e., the symmetry of space) gives rise to conservation laws for linear and angular momentum. Sometimes the implications of symmetry invariance are far more complicated or sophisticated than might at first be supposed; the invariance of the forces predicted by electromagnetic theory when measurements are made in observation frames moving uniformly at different speeds (inertial frames) was an important clue leading Einstein to the discovery of special relativity. With the advent of quantum mechanics, considerations of angular momentum and spin introduced new symmetry concepts into physics. These ideas have since catalyzed the modern development of particle theory." (George B Arfken et al, "Mathematical Methods for Physicists: A comprehensive guide", 2013)

"The No Free Lunch theorems prove that under a uniform distribution over induction problems" (search problems or learning problems), all induction algorithms perform equally.  […] the importance of the theorems arises by using them to analyze scenarios involving non-uniform distributions, and to compare different algorithms, without any assumption about the distribution over problems at all. In particular, the theorems prove that anti-cross-validation" (choosing among a set of candidate algorithms based on which has worst out-of-sample behavior) performs as well as cross-validation, unless one makes an assumption - which has never been formalized - about how the distribution over induction problems, on the one hand, is related to the set of algorithms one is choosing among using" (anti-)cross validation, on the other. In addition, they establish strong caveats concerning the significance of the many results in the literature which establish the strength of a particular algorithm without assuming a particular distribution." (David H Wolpert,What is important about the No Free Lunch theorems?", 2020)