On Chance: Definitions

"Chance is necessity hidden behind a veil." (Marie von Ebner-Eschenbach, Aphorisms, 1880/1893)

"Chance is only the measure of our ignorance." (Henri Poincaré," The Foundations of Science", 1913)

"[...] the conception of chance enters in the very first steps of scientific activity in virtue of the fact that no observation is absolutely correct. I think chance is a more fundamental conception that causality; for whether in a concrete case, a cause-effect relation holds or not can only be judged by applying the laws of chance to the observation." (Max Born, 1949)

"Chance is just as real as causation; both are modes of becoming. The way to model a random process is to enrich the mathematical theory of probability with a model of a random mechanism. In the sciences, probabilities are never made up or 'elicited' by observing the choices people make, or the bets they are willing to place.  The reason is that, in science and technology, interpreted probability exactifies objective chance, not gut feeling or intuition. No randomness, no probability." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"Chance is as relentless as necessity." (Simon Blackburn, Think, 1999) 

"Whether we shuffle cards or roll dice, chance is only a result of our human lack of deftness: we don't have enough control to immobilize a die at will or to individually direct the cards in a deck. The comparison is an important one nonetheless, and highlights the limits of this method of creating chance - it doesn't matter who rolls the dice, but we wouldn't let just anyone shuffle the cards." (Ivar Ekeland, "The Broken Dice, and Other Mathematical Tales of Chance", 1993)

"That randomness gives rise to innovation and diversity in nature is echoed by the notion that chance is also the source of invention in the arts and everyday affairs in which naturally occurring processes are balanced between tight organization, where redundancy is paramount, and volatility, in which little order is possible. One can argue that there is a difference in kind between the unconscious, and sometimes conscious, choices made by a writer or artist in creating a string of words or musical notes and the accidental succession of events taking place in the natural world. However, it is the perception of ambiguity in a string that matters, and not the process that generated it, whether it be man-made or from nature at large." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

Out of Context: On Chance (Definitions)

"Chance is a world void of sense; nothing can exist without a cause." (Voltaire, A Philosophical Dictionary, 1764)

"Our conception of chance is one of law and order in large numbers; it is not that idea of chaotic incidence which vexed the mediaeval mind." (Karl Pearson, "The Chances of Death", 1895)

"Chance is only the measure of our ignorance." (Henri Poincaré, "The Foundations of Science", 1913)

"Can there be laws of chance? The answer, it would seem should be negative, since chance is in fact defined as the characteristic of the phenomena which follow no law, phenomena whose causes are too complex to permit prediction." (Félix E Borel, "Probabilities and Life", 1943)

“Can there be laws of chance? The answer, it would seem should be negative, since chance is in fact defined as the characteristic of the phenomena which follow no law, phenomena whose causes are too complex to permit prediction.” (Félix E Borel, “Probabilities and Life”, 1962)

"Chance is commonly viewed as a self-correcting process in which a deviation in one direction induces a deviation in the opposite direction to restore the equilibrium. In fact, deviations are not 'corrected' as a chance process unfolds, they are merely diluted." (Amos Tversky & Daniel Kahneman, "Judgment Under Uncertainty: Heuristics and Biases", Science Vol. 185 (4157), 1974)

"Quantum chance is absolute. […] Quantum chance is not a measure of ignorance but an inherent property. […] Chance in quantum theory is absolute and irreducible." (F David Peat, "From Certainty to Uncertainty", 2002)

On Chance: Gamblers III

"Behavioural research shows that we tend to use simplifying heuristics when making judgements about uncertain events. These are prone to biases and systematic errors, such as stereotyping, disregard of sample size, disregard for regression to the mean, deriving estimates based on the ease of retrieving instances of the event, anchoring to the initial frame, the gambler’s fallacy, and wishful thinking, which are all affected by our inability to consider more than a few aspects or dimensions of any phenomenon or situation at the same time." (Hans G Daellenbach & Donald C McNickle, "Management Science: Decision making through systems thinking", 2005)

"People sometimes appeal to the ‘law of averages’ to justify their faith in the gambler’s fallacy. They may reason that, since all outcomes are equally likely, in the long run they will come out roughly equal in frequency. However, the next throw is very much in the short run and the coin, die or roulette wheel has no memory of what went before." (Alan Graham, "Developing Thinking in Statistics", 2006)

"Another kind of error possibly related to the use of the representativeness heuristic is the gambler’s fallacy, otherwise known as the law of averages. If you are playing roulette and the last four spins of the wheel have led to the ball’s landing on black, you may think that the next ball is more likely than otherwise to land on red. This cannot be. The roulette wheel has no memory. The chance of black is just what it always is. The reason people tend to think otherwise may be that they expect the sequence of events to be representative of random sequences, and the typical random sequence at roulette does not have five blacks in a row." (Jonathan Baron, "Thinking and Deciding" 4th Ed, 2008)

"The theory of randomness is fundamentally a codification of common sense. But it is also a field of subtlety, a field in which great experts have been famously wrong and expert gamblers infamously correct. What it takes to understand randomness and overcome our misconceptions is both experience and a lot of careful thinking." (Leonard Mlodinow, "The Drunkard’s Walk: How Randomness Rules Our Lives", 2008)

"Finding patterns is easy in any kind of data-rich environment; that's what mediocre gamblers do. The key is in determining whether the patterns represent signal or noise." (Nate Silver, "The Signal and the Noise: Why So Many Predictions Fail-but Some Don't", 2012)

"[…] many gamblers believe in the fallacious law of averages because they are eager to find a profitable pattern in the chaos created by random chance." (Gary Smith, "Standard Deviations", 2014)

On Algorithms: Definitions

"Algorithms are a set of procedures to generate the answer to a problem." (Stuart Kauffman, "At Home in the Universe: The Search for Laws of Complexity", 1995)

"An algorithm is a simple rule, or elementary task, that is repeated over and over again. In this way algorithms can produce structures of astounding complexity." (F David Peat, "From Certainty to Uncertainty", 2002)

"An algorithm refers to a successive and finite procedure by which it is possible to solve a certain problem. Algorithms are the operational base for most computer programs. They consist of a series of instructions that, thanks to programmers’ prior knowledge about the essential characteristics of a problem that must be solved, allow a step-by-step path to the solution." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009) 

"[...] algorithms, which are abstract or idealized process descriptions that ignore details and practicalities. An algorithm is a precise and unambiguous recipe. It’s expressed in terms of a fixed set of basic operations whose meanings are completely known and specified. It spells out a sequence of steps using those operations, with all possible situations covered, and it’s guaranteed to stop eventually." (Brian W Kernighan, "Understanding the Digital World", 2017)

"An algorithm is the computer science version of a careful, precise, unambiguous recipe or tax form, a sequence of steps that is guaranteed to compute a result correctly." (Brian W Kernighan, "Understanding the Digital World", 2017)

"Algorithms describe the solution to a problem in terms of the data needed to represent the  problem instance and a set of steps necessary to produce the intended result." (Bradley N Miller et al, "Python Programming in Context", 2019)

"An algorithm, meanwhile, is a step-by-step recipe for performing a series of actions, and in most cases 'algorithm' means simply 'computer program'." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

On Chaos: Definitions

"Chaos is but unperceived order; it is a word indicating the limitations of the human mind and the paucity of observational facts. The words ‘chaos’, ‘accidental’, ‘chance’, ‘unpredictable’ are conveniences behind which we hide our ignorance." (Harlow Shapley, "Of Stars and Men", 1958) 

"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic (that is fixed) rules (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order (a pattern) within disorder (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)

"The term chaos is also used in a general sense to describe the body of chaos theory, the complete sequence of behaviours generated by feed-back rules, the properties of those rules and that behaviour." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)

"What is chaos? Everyone has an impression of what the word means, but scientifically chaos is more than random behavior, lack of control, or complete disorder. [...] Scientifically, chaos is defined as extreme sensitivity to initial conditions. If a system is chaotic, when you change the initial state of the system by a tiny amount you change its future significantly." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"Chaos is a phenomenon encountered in science and mathematics wherein a deterministic (rule-based) system behaves unpredictably. That is, a system which is governed by fixed, precise rules, nevertheless behaves in a way which is, for all practical purposes, unpredictable in the long run. The mathematical use of the word 'chaos' does not align well with its more common usage to indicate lawlessness or the complete absence of order. On the contrary, mathematically chaotic systems are, in a sense, perfectly ordered, despite their apparent randomness. This seems like nonsense, but it is not." (David P Feldman, "Chaos and Fractals: An Elementary Introduction", 2012)

"Chaos provides order. Chaotic agitation and motion are needed to create overall, repetitive order. This ‘order through fluctuations’ keeps dynamic markets stable and evolutionary processes robust. In essence, chaos is a phase transition that gives spontaneous energy the means to achieve repetitive and structural order." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

On Continuity: Definitions

"The Law of Continuity, as we here deal with it, consists in the idea that [...] any quantity, in passing from one magnitude to another, must pass through all intermediate magnitudes of the same class. The same notion is also commonly expressed by saying that the passage is made by intermediate stages or steps; [...] the idea should be interpreted as follows: single states correspond to single instants of time, but increments or decrements only to small areas of continuous time." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium", 1758)

"An essential difference between continuity and differentiability is whether numbers are involved or not. The concept of continuity is characterized by the qualitative property that nearby objects are mapped to nearby objects. However, the concept of differentiation is obtained by using the ratio of infinitesimal increments. Therefore, we see that differentiability essentially involves numbers." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[…] continuity appears when we try to mathematically express continuously changing phenomena, and differentiability is the result of expressing smoothly changing phenomena." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Intuitively speaking, a visual representation associated with the concept of continuity is the property that a near object is sent to a corresponding near object, that is, a convergent sequence is sent to a corresponding convergent sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"A continuous function preserves closeness of points. A discontinuous function maps arbitrarily close points to points that are not close. The precise definition of continuity involves the relation of distance between pairs of points. […] continuity, a property of functions that allows stretching, shrinking, and folding, but preserves the closeness relation among points." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"Continuity is the rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable - say x - is associated with a value of a dependent variable - say y. Continuity of a function is sometimes expressed by saying that if the x-values are close together, then the y-values of the function will also be close. But if the question 'How close?' is asked, difficulties arise." (Erik Gregersen [Ed.], "Math Eplained: The Britannica Guide to Analysis and Calculus", 2011)

"Continuity is only a mathematical technique for approximating very finely grained things. The world is subtly discrete, not continuous." (Carlo Rovelli, "The Order of Time", 2018)

On Science: Definitions

"By Science is understood a Knowledge acquired by, or founded on clear and self evident Principles, whence it follows that the Mathematicks may truly be stiled such." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702)

"Science is nothing but the finding of analogy, identity, in the most remote parts." (Ralph W Emerson, 1837)

"Science is the systematic classification of experience." (George H Lewes, "The Physical Basis of Mind", 1877)

"Science is the observation of things possible, whether present or past; prescience is the knowledge of things which may come to pass, though but slowly." (Leonardo da Vinci, "The Notebooks of Leonardo da Vinci", 1883)

"Science is not the monopoly of the naturalist or the scholar, nor is it anything mysterious or esoteric. Science is the search for truth, and truth is the adequacy of a description of facts." (Paul Carus, "Philosophy as a Science", 1909)

"Abstract as it is, science is but an outgrowth of life. That is what the teacher must continually keep in mind. […] Let him explain […] science is not a dead system - the excretion of a monstrous pedantism - but really one of the most vigorous and exuberant phases of human life." (George A L Sarton, "The Teaching of the History of Science", The Scientific Monthly, 1918)

"Science is but a method. Whatever its material, an observation accurately made and free of compromise to bias and desire, and undeterred by consequence, is science." (Hans Zinsser, "Untheological Reflections", The Atlantic Monthly, 1929)

"Science is a system of statements based on direct experience, and controlled by experimental verification. Verification in science is not, however, of single statements but of the entire system or a sub-system of such statements." (Rudolf Carnap, "The Unity of Science", 1934)

"Science is the attempt to discover, by means of observation, and reasoning based upon it, first, particular facts about the world, and then laws connecting facts with one another and (in fortunate cases) making it possible to predict future occurrences." (Bertrand Russell, "Religion and Science, Grounds of Conflict", 1935)

"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)

"Science is the organised attempt of mankind to discover how things work as causal systems. The scientific attitude of mind is an interest in such questions. It can be contrasted with other attitudes, which have different interests; for instance the magical, which attempts to make things work not as material systems but as immaterial forces which can be controlled by spells; or the religious, which is interested in the world as revealing the nature of God." (Conrad H Waddington, "The Scientific Attitude", 1941)

"Science, in the broadest sense, is the entire body of the most accurately tested, critically established, systematized knowledge available about that part of the universe which has come under human observation. For the most part this knowledge concerns the forces impinging upon human beings in the serious business of living and thus affecting man’s adjustment to and of the physical and the social world. […] Pure science is more interested in understanding, and applied science is more interested in control […]" (Austin L Porterfield, "Creative Factors in Scientific Research", 1941)

"Science is an interconnected series of concepts and schemes that have developed as a result of experimentation and observation and are fruitful of further experimentation and observation."(James B Conant, "Science and Common Sense", 1951)

"Science is the creation of concepts and their exploration in the facts. It has no other test of the concept than its empirical truth to fact." (Jacob Bronowski, "Science and Human Values", 1956)

"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)

"What, then, is science according to common opinion? Science is what scientists do. Science is knowledge, a body of information about the external world. Science is the ability to predict. Science is power, it is engineering. Science explains, or gives causes and reasons." (John Bremer "What Is Science?" [in "Notes on the Nature of Science"], 1962)

"Science is a matter of disinterested observation, patient ratiocination within some system of logically correlated concepts. In real-life conflicts between reason and passion the issue is uncertain. Passion and prejudice are always able to mobilize their forces more rapidly and press the attack with greater fury; but in the long run (and often, of course, too late) enlightened self-interest may rouse itself, launch a counterattack and win the day for reason." (Aldous L Huxley, "Literature and Science", 1963)

"Science is a way to teach how something gets to be known, what is not known, to what extent things are known (for nothing is known absolutely), how to handle doubt and uncertainty, what the rules of evidence are, how to think about things so that judgments can be made, how to distinguish truth from fraud, and from show." (Richard P Feynman, "The Problem of Teaching Physics in Latin America", Engineering and Science, 1963)

"Science is a product of man, of his mind; and science creates the real world in its own image." (Frank E Egler, "The Way of Science", 1970)

"Science is systematic organisation of knowledge about the universe on the basis of explanatory hypotheses which are genuinely testable. Science advances by developing gradually more comprehensive theories; that is, by formulating theories of greater generality which can account for observational statements and hypotheses which appear as prima facie unrelated." (Francisco J Ayala, "Studies in the Philosophy of Biology: Reduction and Related Problems", 1974)

"Science is not a heartless pursuit of objective information. It is a creative human activity, its geniuses acting more as artists than information processors. Changes in theory are not simply the derivative results of the new discoveries but the work of creative imagination influenced by contemporary social and political forces." (Stephen J Gould, "Ever Since Darwin: Reflections in Natural History", 1977)

"Science is a process. It is a way of thinking, a manner of approaching and of possibly resolving problems, a route by which one can produce order and sense out of disorganized and chaotic observations. Through it we achieve useful conclusions and results that are compelling and upon which there is a tendency to agree." (Isaac Asimov, "‘X’ Stands for Unknown", 1984)

"Science is defined as a set of observations and theories about observations." (F Albert Matsen, "The Role of Theory in Chemistry", Journal of Chemical Education Vol. 62 (5), 1985)

"Science is human experience systematically extended (by intent, methodology and instrumentation) for the purpose of learning more about the natural world and for the critical empirical testing and possible falsification of all ideas about the natural world. Scientific hypotheses may incorporate only elements of the natural empirical world, and thus may contain no element of the supernatural." (Robert E Kofahl, Correctly Redefining Distorted Science: A Most Essential Task", Creation Research Society Quarterly Vol. 23, 1986)

"Science is not a given set of answers but a system for obtaining answers. The method by which the search is conducted is more important than the nature of the solution. Questions need not be answered at all, or answers may be provided and then changed. It does not matter how often or how profoundly our view of the universe alters, as long as these changes take place in a way appropriate to science. For the practice of science, like the game of baseball, is covered by definite rules." (Robert Shapiro, "Origins: A Skeptic’s Guide to the Creation of Life on Earth", 1986)

"Science doesn't purvey absolute truth. Science is a mechanism. It's a way of trying to improve your knowledge of nature. It's a system for testing your thoughts against the universe and seeing whether they match. And this works, not just for the ordinary aspects of science, but for all of life. I should think people would want to know that what they know is truly what the universe is like, or at least as close as they can get to it." (Isaac Asimov, [Interview by Bill Moyers] 1988)

"Science doesn’t purvey absolute truth. Science is a mechanism, a way of trying to improve your knowledge of nature. It’s a system for testing your thoughts against the universe, and seeing whether they match." (Isaac Asimov, [interview with Bill Moyers in The Humanist] 1989)

"Science is (or should be) a precise art. Precise, because data may be taken or theories formulated with a certain amount of accuracy; an art, because putting the information into the most useful form for investigation or for presentation requires a certain amount of creativity and insight." (Patricia H Reiff, "The Use and Misuse of Statistics in Space Physics", Journal of Geomagnetism and Geoelectricity 42, 1990)

"Science is not about control. It is about cultivating a perpetual condition of wonder in the face of something that forever grows one step richer and subtler than our latest theory about it. It is about  reverence, not mastery." (Richard Power, "Gold Bug Variations", 1993)

"Clearly, science is not simply a matter of observing facts. Every scientific theory also expresses a worldview. Philosophical preconceptions determine where facts are sought, how experiments are designed, and which conclusions are drawn from them." (Nancy R Pearcey & Charles B. Thaxton, "The Soul of Science: Christian Faith and Natural Philosophy", 1994)

"Science is more than a mere attempt to describe nature as accurately as possible. Frequently the real message is well hidden, and a law that gives a poor approximation to nature has more significance than one which works fairly well but is poisoned at the root." (Robert H March, "Physics for Poets", 1996)

"Mathematics is the study of analogies between analogies. All science is. Scientists want to show that things that don’t look alike are really the same. That is one of their innermost Freudian motivations. In fact, that is what we mean by understanding." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"Religion is the antithesis of science; science is competent to illuminate all the deep questions of existence, and does so in a manner that makes full use of, and respects the human intellect. I see neither need nor sign of any future reconciliation." (Peter W Atkins, "Religion - The Antithesis to Science", 1997)

"Science is the art of the appropriate approximation. While the flat earth model is usually spoken of with derision it is still widely used. Flat maps, either in atlases or road maps, use the flat earth model as an approximation to the more complicated shape." (Byron K. Jennings, "On the Nature of Science", Physics in Canada Vol. 63 (1), 2007)

"Science isn’t about being right. It is about convincing others of the correctness of an idea through a methodology all will accept using data everyone can trust. New ideas take time to be accepted because they compete with others that have already passed the test." (Tom Koch, "Commentary: Nobody loves a critic: Edmund A Parkes and John Snow’s cholera", International Journal of Epidemiology Vol. 42 (6), 2013)

"Science, at its core, is simply a method of practical logic that tests hypotheses against experience. Scientism, by contrast, is the worldview and value system that insists that the questions the scientific method can answer are the most important questions human beings can ask, and that the picture of the world yielded by science is a better approximation to reality than any other." (John M Greer, "After Progress: Reason and Religion at the End of the Industrial Age", 2015)

On Models: Definitions

"A model is essentially a calculating engine designed to produce some output for a given input." (Richard C Lewontin, "Models, Mathematics and Metaphors", Synthese, Vol. 15, No. 2, 1963)

"A model is an abstract description of the real world. It is a simple representation of more complex forms, processes and functions of physical phenomena and ideas." (Moshe F Rubinstein & Iris R Firstenberg, "Patterns of Problem Solving", 1975)

"A model is an attempt to represent some segment of reality and explain, in a simplified manner, the way the segment operates." (E Frank Harrison, "The managerial decision-making process", 1975)

"A mathematical model is any complete and consistent set of mathematical equations which are designed to correspond to some other entity, its prototype. The prototype may be a physical, biological, social, psychological or conceptual entity, perhaps even another mathematical model." (Rutherford Aris, "Mathematical Modelling", 1978)

"A model is a representation containing the essential structure of some object or event in the real world." (David W Stockburger, "Introductory Statistics", 1996)

"A model is a deliberately simplified representation of a much more complicated situation. […] The idea is to focus on one or two causal or conditioning factors, exclude everything else, and hope to understand how just these aspects of reality work and interact." (Robert M Solow, "How Did Economics Get That Way and What Way Did It Get?", Daedalus, Vol. 126, No. 1, 1997)

"Models are synthetic sets of rules, pictures, and algorithms providing us with useful representations of the world of our perceptions and of their patterns." (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)

“A model is an imitation of reality and a mathematical model is a particular form of representation. We should never forget this and get so distracted by the model that we forget the real application which is driving the modelling. In the process of model building we are translating our real world problem into an equivalent mathematical problem which we solve and then attempt to interpret. We do this to gain insight into the original real world situation or to use the model for control, optimization or possibly safety studies." (Ian T Cameron & Katalin Hangos, “Process Modelling and Model Analysis”, 2001)

"[…] a conceptual model is a diagram connecting variables and constructs based on theory and logic that displays the hypotheses to be tested." (Mary Wolfinbarger Celsi et al, "Essentials of Business Research Methods", 2011)

"A model is a simplified representation of a system. It can be conceptual, verbal, diagrammatic, physical, or formal (mathematical)." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

“A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. [...] A mathematical model is never a completely accurate representation of a physical situation - it is an idealization." (James Stewart, “Calculus: Early Transcedentals” 8th Ed., 2016)

On Probability: Definitions

"Probability is the very guide of life." (Marcus Tullius Cicero, "De Natura Deorum" ["On the Nature of the Gods"], 45 BC)

"Probability is a degree of possibility." (Gottfried W Leibniz, "On estimating the uncertain", 1676)

"Probability is a degree of certainty and it differs from certainty as a part from a whole." (Jacob Bernoulli, "Ars Conjectandi" ["The Art of Conjecturing"], 1713)

"Probable evidence, in its very nature, affords but an imperfect kind of information, and is to be considered as relative only to beings of limited capacities. For nothing which is the possible object of knowledge, whether past, present, or future, can be probable to an infinite Intelligence; since it cannot but be discerned absolutely as it is in itself, certainly true, or certainly false. To us, probability is the very guide of life." (Joseph Butler, "The Analogy of Religion, Natural and Revealed, to the Constitution and Course of Nature", 1736)

"Probability is a mathematical discipline with aims akin to those, for example, of geometry or analytical mechanics. In each field we must carefully distinguish three aspects of the theory: (a) the formal logical content, (b) the intuitive background, (c) the applications. The character, and the charm, of the whole structure cannot be appreciated without considering all three aspects in their proper relation." (William Feller, "An Introduction to Probability Theory and Its Applications", 1950)

"Probability is a mathematical discipline with aims akin to those, for example, of geometry or analytical mechanics. In each field we must carefully distinguish three aspects of the theory: (a) the formal logical content, (b) the intuitive background, (c) the applications. The character, and the charm, of the whole structure cannot be appreciated without considering all three aspects in their proper relation." (William Feller, "An Introduction to Probability Theory and Its Applications", 1957)

"Many modern philosophers claim that probability is relation between an hypothesis and the evidence for it." (Ian Hacking, "The Emergence of Probability", 1975)

"Probability is the mathematics of uncertainty. Not only do we constantly face situations in which there is neither adequate data nor an adequate theory, but many modem theories have uncertainty built into their foundations. Thus learning to think in terms of probability is essential. Statistics is the reverse of probability (glibly speaking). In probability you go from the model of the situation to what you expect to see; in statistics you have the observations and you wish to estimate features of the underlying model." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985) 

"Probabilities are summaries of knowledge that is left behind when information is transferred to a higher level of abstraction." (Judea Pearl, "Probabilistic Reasoning in Intelligent Systems: Network of Plausible, Inference", 1988)

"Phenomena having uncertain individual outcomes but a regular pattern of outcomes in many repetitions are called random. 'Random' is not a synonym for 'haphazard' but a description of a kind of order different from the deterministic one that is popularly associated with science and mathematics. Probability is the branch of mathematics that describes randomness." (David S Moore, "Uncertainty", 1990)

"Mathematics is not just a collection of results, often called theorems; it is a style of thinking. Computing is also basically a style of thinking. Similarly, probability is a style of thinking." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"Probability is not about the odds, but about the belief in the existence of an alternative outcome, cause, or motive." (Nassim N Taleb, "Fooled by Randomness", 2001)

"Probability is a mathematical language for quantifying uncertainty." (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)

"Although some people use them interchangeably, probability and odds are not the same and people often misuse the terms. Probability is the likelihood that an outcome will occur. The odds of something happening, statistically speaking, is the ratio of favorable outcomes to unfavorable outcomes." (John H Johnson & Mike Gluck, "Everydata: The misinformation hidden in the little data you consume every day", 2016)

See also: Out of Context: on Probability

On Maps: Definitions

"Two important characteristics of maps should be noticed. A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1958)

"For our purposes, a simple way to understand paradigms is to see them as maps. We all know that ‘the map is not the territory’. A map is simply an explanation of certain aspects of the territory. That’s exactly what a paradigm is. It is a theory, an explanation, or model of something else." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)

 "A map is a graphical representation of geographical or astronomical features, but this may range from a sketch of a subway system, to an interactive, zoomable, or animated map on a computer which constantly changes in front of the eyes."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"A map is designed to help one get around in the landscape it depicts. [...]"  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"The representational nature of maps, however, is often ignored - what we see when looking at a map is not the word, but an abstract representation that we find convenient to use in place of the world. When we build these abstract representations we are not revealing knowledge as much as are creating it." (Alan MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"Maps are models, and every model represents some aspect of reality or an idea that is of interest. A model is a simplification. It is an interpretation of reality that abstracts the aspects relevant to solving the problem at hand and ignores extraneous detail." (Eric Evans, "Domain-Driven Design: Tackling complexity in the heart of software", 2003)

"A map does not just chart, it unlocks and formulates meaning; it forms bridges between here and there, between disparate ideas that we did not know were previously connected." (Reif Larsen, "The Selected Works of T S Spivet", 2009)

Jacques Ozanam - Collected Quotes

"Although the Mathematicks according to its Etymology, signifies only Discipline, yet it merits the Name of Science better than any other, because its Principles are self-evident, and independent on any sensible Experience, and its Propositions demonstrated beyond all possible Doubt or Opposition. Youth were anciently instructed herein before Philosophy, on which Account Aristotle called it the Science of Children. This was taught them not only to raise and excite their Genius, but also as a fit preparative to the Study of Nature; and it was upon this Account that the Divine Plato inscribed on his School... that none wholly ignorant of Geometry should be admitted there." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"By Science is understood a Knowledge acquired by, or founded on clear and self evident Principles, whence it follows that the Mathematicks may truly be stiled such." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"Mathematicks therefore is a Science which teaches or contemplates whatever is capable of Measure or Number as such. When it relates to Number, it is called Arithmetick; but when to measure, as Length, Breadth, Depth, Degrees of Velocity in Motion, Intenseness or Remissness of Sounds, Augmentation or Diminution of Quality, etc. it is called Geometry." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"The Essential Parts of the Simple or Pure Mathematicks are Arithmetick and Geometry, which mutually assist one another, and are independent on any other Sciences, except perhaps on Artificial Logick: But doubtless Natural Logick may be sufficient to a Man of Sense. The other parts are chiefly Physical Subjects explained by the Principles of Arithmetics or Geometry." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702)

"The Usefulness of the Mathematicks in General, and of some Parts of them in Particular, in the common Affairs of Humane Life, has rendered some competent Knowledge of them very necessary to a great Part of Mankind, and very convenient to all the Rest that are any way conversant beyond the Limits of their own particular callings." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"There are two general Methods made use of in the Mathematicks, viz. Synthesis and Analysis, which we shall explain, after having acquainted the Reader, that the Method we make use of to resolve a Mathematical Problem, is called Zetetick; and that that Method which determines when, and by what way, and how many different ways a Problem may be resolved, is called Poristick. But in treating of Methods, we will first premise, that in general, a Method is the Art of disposing a Train of Arguments or Consequences in a right Order, either to discover the Truth of a Theorem, which we would find out, or to demonstrate it to others, when found." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"To be perfectly ignorant in all the Terms of them is only tolerable in those, who think their Tongues of as little Use to them, as generally their Understandings are. Those whom Necessity has obliged to get their Bread by Manual Industry, where some Degree of Art is required to go along with it, and who have had some Insight into these Studies, have very often found Advantages from them sufficient to reward the Pains they were at in acquiring them. And whatever may have been imputed (how justly I'm not now to determine) to some other Studies, under the Notion of Insignificancy and Loss of Time ; yet these, I believe, never caused Repentance in any, except it was for their Remissness in the Prosecution of them." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"Arithmetic and geometry, according to Plato, are the two wings of the mathematician. The object indeed of all mathematical questions, is to determine the ratios of numbers, or of magnitudes ; and it may even be said, to continue the comparison of the ancient philosopher, that arithmetic is the mathematician's right wing; for it is an incontestable truth, that geometrical determinations would, for the most part, present nothing satisfactory to the mind, if the ratios thus determined could not be reduced to numerical ratios. This justifies the common practice, which we shall here follow, of beginning with arithmetic." (Jacques Ozanam et al, "Recreations in Mathematics and Natural Philosophy", 1803)

On Chance: Gamblers II

"Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere - Archimedes did it in a bathtub." (Richard J Trudeau, "Dots and Lines", 1976)

"Probability does pervade the universe, and in this sense, the old chestnut about baseball imitating life really has validity. The statistics of streaks and slumps, properly understood, do teach an important lesson about epistemology, and life in general. The history of a species, or any natural phenomenon, that requires unbroken continuity in a world of trouble, works like a batting streak. All are games of a gambler playing with a limited stake against a house with infinite resources. The gambler must eventually go bust. His aim can only be to stick around as long as possible, to have some fun while he's at it, and, if he happens to be a moral agent as well, to worry about staying the course with honor!" (Stephen J Gould, 1991)

"Gambling was the place where statistics and profound human consequences met most nakedly, after all, and cards, even more than dice or the numbers on a roulette wheel, seemed able to define and perhaps even dictate a player's... luck." (Tim Powers, "Last Call", 1992)

"Probability theory has a right and a left hand. On the right is the rigorous foundational work using the tools of measure theory. The left hand 'thinks probabilistically', reduces problems to gambling situations, coin-tossing, motions of a physical particle." (Leo Breiman, "Probability", 1992)

"Losing streaks and winning streaks occur frequently in games of chance, as they do in real life. Gamblers respond to these events in asymmetric fashion: they appeal to the law of averages to bring losing streaks to a speedy end. And they appeal to that same law of averages to suspend itself so that winning streaks will go on and on. The law of averages hears neither appeal. The last sequence of throws of the dice conveys absolutely no information about what the next throw will bring. Cards, coins, dice, and roulette wheels have no memory." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"Time is the dominant factor in gambling. Risk and time are opposite sides of the same coin, for if there were no tomorrow there would be no risk. Time transforms risk, and the nature of risk is shaped by the time horizon: the future is the playing field." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"A random walk is one in which future steps or directions cannot be predicted on the basis of past history. When the term is applied to the stock market, it means that short-run changes in stock prices are unpredictable. Investment advisory services, earnings forecasts, and chart patterns are useless. [...] What are often called 'persistent patterns' in the stock market occur no more frequently than the runs of luck in the fortunes of any gambler playing a game of chance. This is what economists mean when they say that stock prices behave very much like a random walk." (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)

"All this, though, is to miss the point of gambling, which is to accept the imbalance of chance in general yet deny it for the here and now. Individually we know, with absolute certainty, that 'the way things hap pen' and what actually happens to us are as different as sociology and poetry." (John Haigh," Taking Chances: Winning With Probability", 1999)

"The psychology of gambling includes both a conviction that the unusual must happen and a refusal to believe in it when it does. We are caught by the confusing nature of the long run; just as the imperturbable ocean seen from space will actually combine hurricanes and dead calms, so the same action, repeated over time, can show wide deviations from its normal expected results - deviations that do not themselves break the laws of probability. In fact, they have probabilities of their own." (John Haigh," Taking Chances: Winning With Probability", 1999)

"This notion of 'being due' - what is sometimes called the gambler’s fallacy - is a mistake we make because we cannot help it. The problem with life is that we have to live it from the beginning, but it makes sense only when seen from the end. As a result, our whole experience is one of coming to provisional conclusions based on insufficient evidence: read ing the signs, gauging the odds." (John Haigh," Taking Chances: Winning With Probability", 1999)

On Chance: Gamblers I

"The gambling reasoner is incorrigible; if he would but take to the squaring of the circle, what a load of misery would be saved." (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"In moderation, gambling possesses undeniable virtues. Yet it presents a curious spectacle replete with contradictions. While indulgence in its pleasures has always lain beyond the pale of fear of Hell’s fires, the great laboratories and respectable insurance palaces stand as monuments to a science originally born of the dice cup." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)

"A misunderstanding of Bernoulli’s theorem is responsible for one of the commonest fallacies in the estimation of probabilities, the fallacy of the maturity of chances. When a coin has come down heads twice in succession, gamblers sometimes say that it is more likely to come down tails next time because ‘by the law of averages’ (whatever that may mean) the proportion of tails must be brought right some time." (William Kneale, "Probability and Induction", 1949)

"The classical theory of probability was devoted mainly to a study of the gamble's gain, which is again a random variable; in fact, every random variable can be interpreted as the gain of a real or imaginary gambler in a suitable game." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950)

"The painful experience of many gamblers has taught us the lesson that no system of betting is successful in improving the gambler's chances. If the theory of probability is true to life, this experience must correspond to a provable statement." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950)

"The picture of scientific method drafted by modern philosophy is very different from traditional conceptions. Gone is the ideal of a universe whose course follows strict rules, a predetermined cosmos that unwinds itself like an unwinding clock. Gone is the ideal of the scientist who knows the absolute truth. The happenings of nature are like rolling dice rather than like revolving stars; they are controlled by probability laws, not by causality, and the scientist resembles a gambler more than a prophet. He can tell you only his best posits - he never knows beforehand whether they will come true. He is a better gambler, though, than the man at the green table, because his statistical methods are superior. And his goal is staked higher - the goal of foretelling the rolling dice of the cosmos. If he is asked why he follows his methods, with what title he makes his predictions, he cannot answer that he has an irrefutable knowledge of the future; he can only lay his best bets. But he can prove that they are best bets, that making them is the best he can do - and if a man does his best, what else can you ask of him?" (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"Control is an attribute of a system. This word is not used in the way in which either an office manager or a gambler might use it; it is used as a name for connectiveness. That is, anything that consists of parts connected together will be called a system." (Stafford Beer, "Cybernetics and Management", 1959)

"There always remains an orbit that to the limited knowledge of man appears as an orbit of pure chance and marks life as a gamble. Man and his works are always exposed to the impact of unforeseen and uncontrollable events." (Ludwig von Mises, "The Ultimate Foundation of Economic Science: An Essay on Method", 1962)

On Fractals: Definitions

 "A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales [...]" (Benoît Mandelbrot, "The Fractal Geometry of Nature", 1982)

"A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." (Benoît Mandelbrot, "The Fractal Geometry of Nature", 1982)

"In the mind's eye, a fractal is a way of seeing infinity." (James Gleick, "Chaos: Making a New Science, A Geometry of Nature", 1987)

"Fractals are geometric shapes that are equally complex in their details as in their overall form. That is, if a piece of a fractal is suitably magnified to become of the same size as the whole, it should look like the whole, either exactly, or perhaps after a slight limited deformation." (Benoît B Mandelbrot, "Fractals and an Art for the Sake of Science", 1989)

"Fractals are patterns which occur on many levels. This concept can be applied to any musical parameter. I make melodic fractals, where the pitches of a theme I dream up are used to determine a melodic shape on several levels, in space and time. I make rhythmic fractals, where a set of durations associated with a motive get stretched and compressed and maybe layered on top of each other. I make loudness fractals, where the characteristic loudness of a sound, its envelope shape, is found on several time scales. I even make fractals with the form of a piece, its instrumentation, density, range, and so on. Here I’ve separated the parameters of music, but in a real piece, all of these things are combined, so you might call it a fractal of fractals." (Györgi Ligeti, [interview] 1999)

"Mathematical fractals are generated by repeating the same simple steps at ever decreasing scales. In this way an apparently complex shape, containing endless detail, can be generated by the repeated application of a simple algorithm. In turn these fractals mimic some of the complex forms found in nature. After all, many organisms and colonies also grow though the repetition of elementary processes such as, for example, branching and division." (F David Peat, "From Certainty to Uncertainty", 2002)

"A fractal is a geometric object which is self-similar and characterized by an effective dimension which is not an integer." (Leonard M Sander, "Fractal Growth Processes", 2009) 

"A fractal is a structure which can be subdivided into parts, where the shape of each part is similar to that of the original structure." (Yakov M Strelniker, "Fractals and Percolation", 2009) 

"A fractal is an image that comprises two distinct attributes: infinite detail and self-similarity." (Daniel C Doolan et al, "Unlocking the Hidden Power of the Mobile", 2009)

"Fractals are complex mathematical objects that are invariant with respect to dilations (self-similarity) and therefore do not possess a characteristic length scale. Fractal objects display scale-invariance properties that can either fluctuate from point to point (multifractal) or be homogeneous (monofractal). Mathematically, these properties should hold over all scales. However, in the real world, there are necessarily lower and upper bounds over which self-similarity applies." (Alain Arneodo et al, "Fractals and Wavelets: What Can We Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences?", 2009) 

[fractal:] "A fragmented geometric shape that can be split up into secondary pieces, each of which is approximately a smaller replica of the whole, the phenomenon commonly known as self similarity." (Khondekar et al, "Soft Computing Based Statistical Time Series Analysis, Characterization of Chaos Theory, and Theory of Fractals", 2013)

John Haigh - Collected Quotes

"All this, though, is to miss the point of gambling, which is to accept the imbalance of chance in general yet deny it for the here and now. Individually we know, with absolute certainty, that 'the way things hap pen' and what actually happens to us are as different as sociology and poetry." (John Haigh," Taking Chances: Winning With Probability", 1999)

"As so often happens in mathematics, a convenient re-statement of a problem brings us suddenly up against the deepest questions of knowledge." (John Haigh," Taking Chances: Winning With Probability", 1999)

"But despite their frequently good intuition, many people go wrong in two places in particular. The first is in appreciating the real differences in magnitude that arise with rare events. If a chance is expressed as 'one in a thousand' or 'one in a million', the only message registered maybe that the chance is remote and yet one figure is a thousand times bigger than the other. Another area is in using partial information […]" (John Haigh," Taking Chances: Winning With Probability", 1999)

"It is the same with the numbers generated by roulette: the smoothness of probability in the long term allows any amount of local lumpiness on which to exercise our obsession with pattern. As the sequence of data lengthens, the relative proportions of odd or even, red or black, do indeed approach closer and closer to the 50-50 ratio predicted by probability, but the absolute discrepancy between one and the other will increase." (John Haigh," Taking Chances: Winning With Probability", 1999)

"Normal is safe; normal is central; normal is unexceptional. Yet it also means the pattern from which all others are drawn, the standard against which we measure the healthy specimen. In its simplest statistical form, normality is represented by the mean (often called the 'average') of a group of measurements." (John Haigh," Taking Chances: Winning With Probability", 1999)

"Probability therefore is a kind of corrective lens, allowing us, through an understanding of the nature of Chance, to refine our conclusions and approximate, if not achieve, the perfection of Design." (John Haigh," Taking Chances: Winning With Probability", 1999)

"The psychology of gambling includes both a conviction that the unusual must happen and a refusal to believe in it when it does. We are caught by the confusing nature of the long run; just as the imperturbable ocean seen from space will actually combine hurricanes and dead calms, so the same action, repeated over time, can show wide deviations from its normal expected results - deviations that do not themselves break the laws of probability. In fact, they have probabilities of their own." (John Haigh," Taking Chances: Winning With Probability", 1999)

"These so-called stochastic processes show up everywhere randomness is applied to the output of another random function. They provide, for instance, a method for describing the chance component of financial markets: not every value of the Dow is possible every day; the range of chance fluctuation centers on the opening price. Similarly, shuffling takes the output of the previous shuffle as its input. So, if you’re handed a deck in a given order, how much shuffling does it need to be truly random?" (John Haigh," Taking Chances: Winning With Probability", 1999)

"This notion of 'being due' - what is sometimes called the gambler’s fallacy - is a mistake we make because we cannot help it. The problem with life is that we have to live it from the beginning, but it makes sense only when seen from the end. As a result, our whole experience is one of coming to provisional conclusions based on insufficient evidence: read ing the signs, gauging the odds." (John Haigh," Taking Chances: Winning With Probability", 1999)

"We search for certainty and call what we find destiny. Everything is possible, yet only one thing happens - we live and die between these two poles, under the rule of probability. We prefer, though, to call it Chance: an old familiar embodied in gods and demons, harnessed in charms and rituals. We remind one another of fortune’s fickleness, each secretly believing himself exempt. I am master of my fate; you are dicing with danger; he is living in a fool’s paradise." (John Haigh," Taking Chances: Winning With Probability", 1999)

"Winning and losing is not simply a pastime; it is the model science uses to explore the universe. Flipping a coin or rolling a die is really asking a question: success or failure can be defined as getting a yes or no. So the distribution of probabilities in a game of chance is the same as that in any repeated test - even though the result of any one test is unpredictable." (John Haigh," Taking Chances: Winning With Probability", 1999)

On Geometry: Definitions

"Geometry is knowledge of the eternally existent."  (Pythagoras of Samos, cca. 6th century BC)

"Indeed, many geometric things can be discovered or elucidated by algebraic principles, and yet it does not follow that algebra is geometrical, or even that it is based on geometric principles (as some would seem to think). This close affinity of arithmetic and geometry comes about, rather, because geometry is, as it were, subordinate to arithmetic, and applies universal principles of arithmetic to its special objects." (John Wallis, "Mathesis Universalis", 1657)

"Mathematicks therefore is a Science which teaches or contemplates whatever is capable of Measure or Number as such. When it relates to Number, it is called Arithmetick; but when to measure, as Length, Breadth, Depth, Degrees of Velocity in Motion, Intenseness or Remissness of Sounds, Augmentation or Diminution of Quality, etc. it is called Geometry." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"It hath been an old remark, that Geometry is an excellent Logic. And it must be owned that when the definitions are clear; when the postulata cannot be refused, nor the axioms denied; when from the distinct contemplation and comparison of figures, their properties are derived, by a perpetual well-connected chain of consequences, the objects being still kept in view, and the attention ever fixed upon them; there is acquired a habit of reasoning, close and exact and methodical; which habit strengthens and sharpens the mind, and being transferred to other subjects is of general use in the inquiry after truth." (George Berkeley, "The Analyst; Or, A Discourse Addressed to an Infidel Mathematician", 1734)

"Geometry is a true natural science: - only more simple, and therefore more perfect than any other. We must not suppose that, because it admits the application of mathematical analysis, it is therefore a purely logical science, independent of observation. Everybody studied by geometers presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone." (Auguste Comte,"Course of Positive Philosophy", 1830)

"Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art are constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic." (William Chambers & Robert Chambers, "Chambers's Information for the People" Vol. 2, 1835)

"Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space." (James J Sylvester, "A Probationary Lecture on Geometry", 1844)

"Algebra is but written geometry and geometry is but figured algebra." (Sophie Germain, "Mémoire sur la surfaces élastiques", 1880)

"[…] geometry is the art of reasoning well from badly drawn figures; however, these figures, if they are not to deceive us, must satisfy certain conditions; the proportions may be grossly altered, but the relative positions of the different parts must not be upset." (Henri Poincaré, 1895)

"We see that experience plays an indispensable role in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science. If it were experimental it would be only approximative and provisional. And what rough approximation!" (Henri Poincaré, "Science and Hypothesis", 1901)

"And here is what makes this analysis situs interesting to us; it is that geometric intuition really intervenes there. When, in a theorem of metric geometry, one appeals to this intuition, it is because it is impossible to study the metric properties of a figure as abstractions of its qualitative properties, that is, of those which are the proper business of analysis situs. It has often been said that geometry is the art of reasoning correctly from badly drawn figures. This is not a capricious statement; it is a truth that merits reflection. But what is a badly drawn figure? It is what might be executed by the unskilled draftsman spoken of earlier; he alters the properties more or less grossly; his straight lines have disquieting zigzags; his circles show awkward bumps. But this does not matter; this will by no means bother the geometer; this will not prevent him from reasoning." (Henri Poincaré, "Dernières pensées", 1913)

"We must [...] maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"With a literature much vaster than those of algebra and arithmetic combined, and as least as extensive as that of analysis, geometry is a richer treasure house of more interesting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathematics." (Eric T Bell, "The Development of Mathematics", 1945)

“Geometry is the study of shapes. Although true, this definition is so broad that it is almost meaningless. The judge of a beauty contest is, in a sense, a geometrician because he is judging […] shapes, but this is not quite what we want the word to mean. It has been said that a curved line is the most beautiful distance between two points. Even though this statement is about curves, a proper element of geometry, the assertion seems more to be in the domain of aesthetics rather than mathematics.” (Martin Gardner, "Aha! Insight", 1978) 

"Geometry is the study of form and shape. Our first encounter with it usually involves such figures as triangles, squares, and circles, or solids such as the cube, the cylinder, and the sphere. These objects all have finite dimensions of length, area, and volume - as do most of the objects around us. At first thought, then, the notion of infinity seems quite removed from ordinary geometry. That this is not so can already be seen from the simplest of all geometric figures - the straight line. A line stretches to infinity in both directions, and we may think of it as a means to go 'far out' in a one-dimensional world." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)

"It has been said that the art of geometry is to reason well from false diagrams." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"Since geometry is the mathematical idealization of space, a natural way to organize its study is by dimension. First we have points, objects of dimension O. Then come lines and curves, which are one-dimensional objects, followed by two-dimensional surfaces, and so on. A collection of such objects from a given dimension forms what mathematicians call a 'space'. And if there is some notion enabling us to say when two objects are 'nearby' in such a space, then it's called a topological space." (John L Casti, "Five Golden Rules", 1995)

"Traditional geometry is the study of the properties of spaces or objects that have integral dimensions. This can be generalized to allow effective fractional dimensions of objects, called fractals, that are embedded in an integral dimension space. […] Fractals are often defined as geometric objects whose spatial structure is self-similar. This means that by magnifying one part of the object, we find the same structure as of the original object. The object is characteristically formed out of a collection of elements: points, line segments, planar sections or volume elements. These elements exist in a space of the same or higher dimension to the elements themselves." (Yaneer Bar-Yamm, "Dynamics of Complexity", 1997)

"Practical geometry is an empirical undertaking, living and breathing and sweating in the real world where measurements are always approximate and things are fudged or smeared or jumbled up. Within Euclidean geometry points are concentrated, lines straightened, angles narrowed; idealizations are made, and some parts of experience discarded and other parts embraced. (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"In plain English, fractal geometry is the geometry of the irregular, the geometry of nature, and, in general, fractals are characterized by infinite detail, infinite length, and the absence of smoothness or derivative." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about physical world. The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry." (Harold S M Coxeter) 

"Geometry is the science which restores the situation that existed before the creation of the world and tries to fill the 'gap', relinquishing the help of matter." (Lucian Blaga)

On Statistics: Definitions

"By statistical is meant in Germany an inquiry for the purpose of ascertaining the political strength of a country, or questions concerning matters of state; whereas the idea I annexed to the term is an inquiry into the state of a country, for the purpose of ascertaining the quantum of happiness enjoyed by its inhabitants and the means of its future improvement." (Sir John Sinclair, "The Statistical Account of Scotland", 1791)

"[Statistics] are the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of man." (Sir Francis Galton, "Natural Inheritance", 1889)

"[…] statistics is the science of the measurement of the social organism, regarded as a whole, in all its manifestations." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Statistics may rightly be called the science of averages. […] Great numbers and the averages resulting from them, such as we always obtain in measuring social phenomena, have great inertia. […] It is this constancy of great numbers that makes statistical measurement possible. It is to great numbers that statistical measurement chiefly applies." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Statistics may, for instance, be called the science of counting. Counting appears at first sight to be a very simple operation, which any one can perform or which can be done automatically; but, as a matter of fact, when we come to large numbers, e.g., the population of the United Kingdom, counting is by no means easy, or within the power of an individual; limits of time and place alone prevent it being so carried out, and in no way can absolute accuracy be obtained when the numbers surpass certain limits." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Statistics may be regarded as (i) the study of populations, (ii) as the study of variation, and (iii) as the study of methods of the reduction of data." (Sir Ronald A Fisher, "Statistical Methods for Research Worker", 1925)

"Statistics is a scientific discipline concerned with collection, analysis, and interpretation of data obtained from observation or experiment. The subject has a coherent structure based on the theory of Probability and includes many different procedures which contribute to research and development throughout the whole of Science and Technology." (Egon Pearson, 1936)

"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)

"For the most part, Statistics is a method of investigation that is used when other methods are of no avail; it is often a last resort and a forlorn hope. A statistical analysis, properly conducted, is a delicate dissection of uncertainties, a surgery of suppositions. The surgeon must guard carefully against false incisions with his scalpel. Very often he has to sew up the patient as inoperable. The public knows too little about the statistician as a conscientious and skilled servant of true science." (Michael J Moroney, "Facts from Figures", 1951)

"Statistics is the name for that science and art which deals with uncertain inferences - which uses numbers to find out something about nature and experience." (Warren Weaver, 1952)

"Statistics is the fundamental and most important part of inductive logic. It is both an art and a science, and it deals with the collection, the tabulation, the analysis and interpretation of quantitative and qualitative measurements. It is concerned with the classifying and determining of actual attributes as well as the making of estimates and the testing of various hypotheses by which probable, or expected, values are obtained. It is one of the means of carrying on scientific research in order to ascertain the laws of behavior of things - be they animate or inanimate. Statistics is the technique of the Scientific Method." (Bruce D Greenschields & Frank M Weida, "Statistics with Applications to Highway Traffic Analyses", 1952)

"[Statistics] is concerned with things we can count. In so far as things, persons, are unique or ill-defi ned, statistics are meaningless and statisticians silenced; in so far as things are similar and definite - so many workers over 25, so many nuts and bolts made during December - they can be counted and new statistical facts are born." (Maurice S Bartlett, "Essays on Probability and Statistics", 1962)

"Statistics is the branch of scientific method which deals with the data obtained by counting or measuring the properties of populations of natural phenomena." (Sir Maurice G Kendall & Alan Stuart, "The Advanced Theory of Statistics", 1963)

"Statistics may be defined as the discipline concerned with the treatment of numerical data derived from groups of individuals." (Peter Armitage, "Statistical Methods in Medical Research", 1971)

"We provisionally define statistics as the study of how information should be employed to reflect on, and give guidance for action in, a practical situation involving uncertainty." (Vic Barnett, "Comparative Statistical Inference" 2nd Ed., 1982)

"Statistics is a tool. In experimental science you plan and carry out experiments, and then analyse and interpret the results. To do this you use statistical arguments and calculations. Like any other tool - an oscilloscope, for example, or a spectrometer, or even a humble spanner - you can use it delicately or clumsily, skillfully or ineptly. The more you know about it and understand how it works, the better you will be able to use it and the more useful it will be." (Roger Barlow, "Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences", 1989)

"The science of statistics may be described as exploring, analyzing and summarizing data; designing or choosing appropriate ways of collecting data and extracting information from them; and communicating that information. Statistics also involves constructing and testing models for describing chance phenomena. These models can be used as a basis for making inferences and drawing conclusions and, finally, perhaps for making decisions." (Fergus Daly et al, "Elements of Statistics", 1995)

"Statistics is a general intellectual method that applies wherever data, variation, and chance appear. It is a fundamental method because data, variation and chance are omnipresent in modern life. It is an independent discipline with its own core ideas rather than, for example, a branch of mathematics. […] Statistics offers general, fundamental, and independent ways of thinking." (David S Moore, "Statistics among the Liberal Arts", Journal of the American Statistical Association, 1998)

"Statistics is the branch of mathematics that uses observations and measurements called data to analyze, summarize, make inferences, and draw conclusions based on the data gathered." (Allan G Bluman, "Probability Demystified", 2005)

"Statistics is the science of collecting, organizing, analyzing, and interpreting data in order to make decisions." (Ron Larson & Betsy Farber, "Elementary Statistics: Picturing the World" 5th Ed., 2011)

"Statistics is the discipline of using data samples to support claims about populations." (Allen B Downey, "Think Stats: Probability and Statistics for Programmers", 2011)

"[… ] statistics is about understanding the role that variability plays in drawing conclusions based on data. […] Statistics is not about numbers; it is about data - numbers in context. It is the context that makes a problem meaningful and something worth considering." (Roxy Peck et al, "Introduction to Statistics and Data Analysis" 4th Ed., 2012)

"Statistics is the scientific discipline that provides methods to help us make sense of data. […] The field of statistics teaches us how to make intelligent judgments and informed decisions in the presence of uncertainty and variation." (Roxy Peck & Jay L Devore, "Statistics: The Exploration and Analysis of Data" 7th Ed, 2012)

"[…] statistics is a method of pursuing truth. At a minimum, statistics can tell you the likelihood that your hunch is true in this time and place and with these sorts of people. This type of pursuit of truth, especially in the form of an event’s future likelihood, is the essence of psychology, of science, and of human evolution." (Arthhur Aron et al, "Statistics for Phsychology" 6th Ed., 2012)

"Statistics is the scientific discipline that provides methods to help us make sense of data. Statistical methods, used intelligently, offer a set of powerful tools for gaining insight into the world around us." (Roxy Peck et al, "Introduction to Statistics and Data Analysis" 4th Ed., 2012)

"Statistics is the art and science of designing studies and analyzing the data that those studies produce. Its ultimate goal is translating data into knowledge and understanding of the world around us. In short, statistics is the art and science of learning from data." (Alan Agresti & Christine Franklin, "Statistics: The Art and Science of Learning from Data" 3rd Ed., 2013)

"Statistics is a science that helps us make decisions and draw conclusions in the presence of variability." (Douglas C Montgomery & George C Runger, "Applied Statistics and Probability for Engineers" 6th Ed., 2014)

"Statistics is an integral part of the quantitative approach to knowledge. The field of statistics is concerned with the scientific study of collecting, organizing, analyzing, and drawing conclusions from data." (Kandethody M Ramachandran & Chris P Tsokos, "Mathematical Statistics with Applications in R" 2nd Ed., 2015)

"Statistics can be defined as a collection of techniques used when planning a data collection, and when subsequently analyzing and presenting data." (Birger S Madsen, "Statistics for Non-Statisticians", 2016)

"Statistics is the science of collecting, organizing, and interpreting numerical facts, which we call data. […] Statistics is the science of learning from data." (Moore McCabe & Alwan Craig, "The Practice of Statistics for Business and Economics" 4th Ed., 2016)

"Statistics is the science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions. In addition, statistics is about providing a measure of confidence in any conclusions." (Michael Sullivan, "Statistics: Informed Decisions Using Data", 5th Ed., 2017)

On Chaos VIII

"The unifying concept underlying fractals, chaos, and power laws is self-similarity. Self-similarity, or invariance against changes in scale or size, is an attribute of many laws of nature and innumerable phenomena in the world around us. Self-similarity is, in fact, one of the decisive symmetries that shape our universe and our efforts to comprehend it." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"[…] the world is not complete chaos. Strange attractors often do have structure: like the Sierpinski gasket, they are self-similar or approximately so. And they have fractal dimensions that hold important clues for our attempts to understand chaotic systems such as the weather." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Mathematics is sometimes described as the science which generates eternal notions and concepts for the scientific method: derivatives‚ continuity‚ powers‚ logarithms are examples. The notions of chaos‚ fractals and strange attractors are not yet mathematical notions in that sense‚ because their final definitions are not yet agreed upon." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science", 2004)

"Chaos can leave statistical footprints that look like noise. This can arise from simple systems that are deterministic and not random. [...] The surprising mathematical fact is that most systems are chaotic. Change the starting value ever so slightly and soon the system wanders off on a new chaotic path no matter how close the starting point of the new path was to the starting point of the old path. Mathematicians call this sensitivity to initial conditions but many scientists just call it the butterfly effect. And what holds in math seems to hold in the real world - more and more systems appear to be chaotic." (Bart Kosko, "Noise", 2006)

"'Chaos' refers to systems that are very sensitive to small changes in their inputs. A minuscule change in a chaotic communication system can flip a 0 to a 1 or vice versa. This is the so-called butterfly effect: Small changes in the input of a chaotic system can produce large changes in the output. Suppose a butterfly flaps its wings in a slightly different way. can change its flight path. The change in flight path can in time change how a swarm of butterflies migrates." (Bart Kosko, "Noise", 2006)

"Entropy is the crisp scientific name for waste, chaos, and disorder. As far as we know, the sole law of physics with no known exceptions anywhere in the universe is this: All creation is headed to the basement. Everything in the universe is steadily sliding down the slope toward the supreme equality of wasted heat and maximum entropy." (Kevin Kelly, "What Technology Wants", 2010)

On Sets: Definitions

"[a set is] an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference." (Bernard Bolzano, 1847)

"By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law, and with this I believe to define something that is akin to the Platonic εἷδος [form] or ἷδεα [idea]." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit. If these or similar statements were set down as definitions, then it could be objected with good reason that they define idem per idemi or even obscurum per obscurius. However, we can consider them as expository, as references to a primitive concept, familiar to us all, whose resolution into more fundamental concepts would perhaps be neither competent nor necessary." (Felix Hausdorff, "Set Theory", 1962)

"Set theory is concerned with abstract objects and their relation to various collections which contain them. We do not define what a set is but accept it as a primitive notion. We gain an intuitive feeling for the meaning of sets and, consequently, an idea of their usage from merely listing some of the synonyms: class, collection, conglomeration, bunch, aggregate. Similarly, the notion of an object is primitive, with synonyms element and point. Finally, the relation between elements and sets, the idea of an element being in a set, is primitive." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"Unfortunately, we are not in a position to give a rigorous definition of the fundamental concept of the theory : the concept of set. Of course, we could say that a set is a collection, a union, an ensemble, a family, a system, a class, etc. But this would not be a mathematical definition, but rather a misuse of the multitude of words available in the English language." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"Set theory is unusual in that it deals with remarkably simple but apparently ineffable objects. A set is a collection, a class, an ensemble, a batch, a bunch, a lot, a troop, a tribe. To anyone incapable of grasping the concept of a set, these verbal digressions are apt to be of little help. […] A set may contain finitely many or infinitely many members. For that matter, a set such as {} may contain no members whatsoever, its parentheses vibrating around a mathematical black hole. To the empty set is reserved the symbol Ø, the figure now in use in daily life to signify access denied or don’t go, symbolic spillovers, I suppose, from its original suggestion of a canceled eye." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Unlike the classic set theory where a set is represented as an indicator function to specify if an object belongs or not to it, a fuzzy set is an extension of a classic set where a subset is represented as a membership function to characterize the degree that an object belongs to it. The indicator function of a classic set takes value of 1 or 0, whereas the membership function of a fuzzy set takes value between 1 and 0." (Jianchao Han & Nick Cerone, Principles and Perspectives of Granular Computing, 2009) 

"The invariance principle states that the result of counting a set does not depend on the order imposed on its elements during the counting process. Indeed, a mathematical set is just a collection without any implied ordering. A set is the collection of its elements - nothing more." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"A set is a unity of which its elements are the constituents. It is a fundamental property of the mind to comprehend multitudes into unities. Sets are multitudes which are also unities. A multitude is the opposite of a unity. How can anything be both a multitude and a unity? Yet a set is just that. It is a seemingly contradictory fact that sets exist. It is surprising that the fact that multitudes are also unities leads to no contradictions: this is the main fact of mathematics. Thinking a plurality together seems like a triviality: and this appears to explain why we have no contradiction. But 'many things for one' is far from trivial." (Kurt Gödel)

"A set is a Many that allows itself to be thought of as a One." (Georg Cantor)