On Differential Equations IV

"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)

"It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations." (Felix Klein, "Lectures on Mathematics", 1911)

"Men have fallen in love with statues and pictures. I find it easier to imagine a man falling in love with a differential equation, and I am inclined to think that some mathematicians have done so. Even in a nonmathematician like myself, some differential equations evoke fairly violent physical sensations to those described by Sappho and Catallus when viewing their mistresses. Personally, I obtain an even greater 'kick' from finite difference equations, which are perhaps more like those which an up-to-date materialist would use to describe human behavior." (John B S Haldane, "The Inequality of Man and Other Essays", 1932)

"The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration of semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form." (Leonid V Kantorovich, "On one class of functional equations", 1936)

"The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 1944)

"The study of changes in the qualitative structure of the flow of a differential equation as parameters are varied is called bifurcation theory. At a given parameter value, a differential equation is said to have stable orbit structure if the qualitative structure of the flow does not change for sufficiently small variations of the parameter. A parameter value for which the flow does not have stable orbit structure is called a bifurcation value, and the equation is said to be at a bifurcation point." (Jack K Hale & Hüseyin Kocak, "Dynamics and Bifurcations", 1991)

"Dynamical systems that vary in discrete steps […] are technically known as mappings. The mathematical tool for handling a mapping is the difference equation. A system of difference equations amounts to a set of formulas that together express the values of all of the variables at the next step in terms of the values at the current step. […] For mappings, the difference equations directly express future states in terms of present ones, and obtaining chronological sequences of points poses no problems. For flows, the differential equations must first be solved. General solutions of equations whose particular solutions are chaotic cannot ordinarily be found, and approximations to the latter are usually determined by numerical methods." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Faced with the overwhelming complexity of the real world, time pressure, and limited cognitive capabilities, we are forced to fall back on rote procedures, habits, rules of thumb, and simple mental models to make decisions. Though we sometimes strive to make the best decisions we can, bounded rationality means we often systematically fall short, limiting our ability to learn from experience." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Following the traditional classification in the field of control systems, a system that describes the input-output behavior in a way similar to a mathematical mapping without involving a differential operator or equation is called a static system. In contrast, a system described by a differential operator or equation is called a dynamic system." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

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On Differentiability I

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

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

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

"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)

"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] differentiation is performed by focusing on the behavior of a function near one point. A quantity obtained in this manner is essentially a local quantity. Is it possible that such local quantities can show us something very basic about global properties such as smoothness? Does there exist a place in mathematics which would enable us to study the relationship between local and global quantities?" (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"To consider differentiable functions, we must introduce a coordinate system on the plane and thereby to concentrate on the world of numbers.[...] a continuous function defined on a plane can be differentiable or nondifferentiable depending on the choice of coordinates. [...] the choice of coordinates on the plane determines which functions among the continuous functions should be selected as differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future)." (Benoît Mandelbrot, "The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward", 2004)?

"Roughly speaking, a function defined on an open set of Euclidean space is differentiable at a point if we can approximate it in a neighborhood of this point by a linear map, which is called its differential (or total derivative). This differential can be of course expressed by partial derivatives, but it is the differential and not the partial derivatives that plays the central role." (Jacques Lafontaine, "An Introduction to Differential Manifolds", 2010)

"I turn away with fright and horror from the lamentable evil of functions which do not have derivatives." (Charles Hermite, [letter to Thomas J Stieltjes])

On Facts (1800-1824)

"Isolated facts, those that can only be obtained by rough estimate and that require development, can only be presented in memoires; but those that can be presented in a body, with details, and on whose accuracy one can rely, may be expounded in tables." (Emmanuel Duvillard, "Memoire sur le travail du Bureau de statistique", 1806)

"The foundations of chemical philosophy are observation, experiment, and analogy. By observation, facts are distinctly and minutely impressed on the mind. By analogy, similar facts are connected. By experiment, new facts are discovered; and, in the progression of knowledge, observation, guided by analogy, lends to experiment, and analogy confirmed by experiment, becomes scientific truth." (Sir Humphry Davy, "Elements of Chemical Philosophy" Vol. 4, 1812)

"The theory of which we have just given an overview may be considered from a point of view apt to set aside the obscure in what it presents, and which seems to be the primary aim, namely: to establish new notions on imaginary quantities. Indeed, putting to one side the question of whether these notions are true or false, we may restrict ourselves to viewing this theory as a means of research, to adopt the lines in direction only as signs of the real or imaginary quantities, and to see, in the usage to which we have put them, only the simple employment of a particular notation. For that, it suffices to start by demonstrating, through the first theorems of trigonometry, the rules of multiplication and addition given above; the applications will follow, and all that will remain is to examine the question of didactics. And if the employment of this notation were to be advantageous? And if it were to open up shorter and easier paths to demonstrate certain truths? That is what fact alone can decide." (Jean-Robert Argand, "Essai sur une manière de représenter les quantités imaginaires, dans les constructions géométriques", Annales Tome IV, 1813)

"Induction, analogy, hypotheses founded upon facts and rectified continually by new observations, a happy tact given by nature and strengthened by numerous comparisons of its indications with experience, such are the principal means for arriving at truth." (Pierre-Simon Laplace, "A Philosophical Essay on Probabilities", 1814)

"The substitution of analogy for fact is the bane of chemical philosophy; the legitimate use of analogy is to connect facts together and to guide to new experiments." (Sir Humphry Davy, "Journal of Science and the Arts", 1816)

"Force is not a fact at all, but an idea embodying what is approximately the fact." (William K Clifford, "The Common Sense of the Exact Sciences", 1823)

On Approximation (2000-2024)

"Optimization by individual agents, often used to derive competitive equilibria, are unnecessary for an actual economy to approximately attain such equilibria. From the failure of humans to optimize in complex tasks, one need not conclude that the equilibria derived from the competitive model are descriptively irrelevant. We show that even in complex economic systems, such equilibria can be attained under a range of surprisingly weak assumptions about agent behavior." (Antoni Bosch-Domènech & Shyam Sunder, "Tracking the Invisible Hand", 2000)

"Conventional wisdom, fooled by our misleading 'physical intuition', is that the real world is continuous, and that discrete models are necessary evils for approximating the 'real' world, due to the innate discreteness of the digital computer." (Doron Zeilberger, "'Real' Analysis is a Degenerate Case of Discrete Analysis", 2001)

"Most physical systems, particularly those complex ones, are extremely difficult to model by an accurate and precise mathematical formula or equation due to the complexity of the system structure, nonlinearity, uncertainty, randomness, etc. Therefore, approximate modeling is often necessary and practical in real-world applications. Intuitively, approximate modeling is always possible. However, the key questions are what kind of approximation is good, where the sense of 'goodness' has to be first defined, of course, and how to formulate such a good approximation in modeling a system such that it is mathematically rigorous and can produce satisfactory results in both theory and applications." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"But linearity is often an approximation to a more complicated reality. Most systems behave linearly only when they are close to equilibrium, and only when we don't push them too hard." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Mathematical modeling is as much ‘art’ as ‘science’: it requires the practitioner to (i) identify a so-called ‘real world’ problem (whatever the context may be); (ii) formulate it in mathematical terms (the ‘word problem’ so beloved of undergraduates); (iii) solve the problem thus formulated (if possible; perhaps approximate solutions will suffice, especially if the complete problem is intractable); and (iv) interpret the solution in the context of the original problem." (John A Adam, "Mathematics in Nature", 2003)

"Fuzzy models can provide good numerical approximation of functions as well as linguistic information over the behavior of the functions.  […] Fuzzy models with embedded linguistic interpretability are useful to extract knowledge from data. This knowledge is represented as a set of IF–THEN rules where the antecedents and the consequences are semantically meaningful." (Jairo Espinosa et al, "Fuzzy Logic, Identification and Predictive Control", 2005)

"Logic is the study of methods and principles of reasoning, where reasoning means obtaining new propositions from existing propositions. In classical logic, propositions are required to be either true or false; that is, the truth value of a proposition is either 0 or 1. Fuzzy logic generalizes classical two-value logic by allowing the truth values of a proposition to be any numbers in [0, 1]. This generalization allows us to perform fuzzy reasoning, also called approximate reasoning; that is, deducing imprecise conclusions (fuzzy propositions) from a collection of imprecise premises (fuzzy propositions). In this section, we first introduce some basic concepts and principles in classical logic and then study their generalizations to fuzzy logic." (Huaguang Zhang & Derong Liu, "Fuzzy Modeling and Fuzzy Control", 2006)

"All models are approximations. Essentially, all models are wrong, but some are useful. However, the approximate nature of the model must always be borne in mind." (George E P Box & Norman R Draper, "Response Surfaces, Mixtures, and Ridge Analyses", 2007)

"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)

"Approximate symmetry is a softening of the hard dichotomy between symmetry and asymmetry. The extent of deviation from exact symmetry that can still be considered approximate symmetry will depend on the context and the application and could very well be a matter of personal taste." (Joe Rosen, "Symmetry Rules: How Science and Nature Are Founded on Symmetry", 2008)

"From the historical point of view, since inequalities are associated with order, they arose as soon as people started using numbers, making measurements, and later, finding approximations and bounds. Thus inequalities have a long and distinguished role in the evolution of mathematics." (Claudi Alsina & Roger B Nelsen, "When Less is More: Visualizing Basic Inequalities", 2009)

"Fuzzy logic is an application area of fuzzy set theory dealing with uncertainty in reasoning. It utilizes concepts, principles, and methods developed within fuzzy set theory for formulating various forms of sound approximate reasoning. Fuzzy logic allows for set membership values to range (inclusively) between 0 and 1, and in its linguistic form, imprecise concepts like 'slightly', 'quite' and 'very'. Specifically, it allows partial membership in a set." (Larbi Esmahi et al,  Adaptive Neuro-Fuzzy Systems, 2009)

"Stated loosely, models are simplified, idealized and approximate representations of the structure, mechanism and behavior of real-world systems. From the standpoint of set-theoretic model theory, a mathematical model of a target system is specified by a nonempty set - called the model’s domain, endowed with some operations and relations, delineated by suitable axioms and intended empirical interpretation." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"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)

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On Approximation (1975-1999)

"In the absolute universe all events can be regarded as absolutely deterministic, and if we can’t perceive the greater structures, it’s because our vision is faulty. If we had a real grasp of causality down to the molecular level, we wouldn’t need to rely on mathematical approximations, on statistics and probabilities, in making predictions. If our perceptions of cause and effect were only good enough, we’d be able to attain absolute knowledge of what is to come. We would make ourselves all-seeing." (Robert Silverberg, "The Stochastic Man", 1975)

"Whenever the Eastern mystics express their knowledge in words - be it with the help of myths, symbols, poetic images or paradoxical statements-they are well aware of the limitations imposed by language and 'linear' thinking. Modern physics has come to take exactly the same attitude with regard to its verbal models and theories. They, too, are only approximate and necessarily inaccurate. They are the counterparts of the Eastern myths, symbols and poetic images, and it is at this level that I shall draw the parallels. The same idea about matter is conveyed, for example, to the Hindu by the cosmic dance of the god Shiva as to the physicist by certain aspects of quantum field theory. Both the dancing god and the physical theory are creations of the mind: models to describe their authors' intuition of reality." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 1975)

"It seems that truth
Is progressive approximation
In which the relative fraction
Of our spontaneously tolerated residual error
Constantly diminishes." (R Buckminster Fuller, "And It Came to Pass - Not to Stay", 1976)

"Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate." (Gregory Bateson, "Number is Different from Quantity", CoEvolution Quarterly, 1978)

"Until now, physical theories have been regarded as merely models with approximately describe the reality of nature. As the models improve, so the fit between theory and reality gets closer. Some physicists are now claiming that supergravity is the reality, that the model and the real world are in mathematically perfect accord." (Paul C W Davies, "Superforce", 1984)

"The fact that [the model] is an approximation does not necessarily detract from its usefulness because models are approximations. All models are wrong, but some are useful." (George Box, 1987)

"The task of us all as thinking human beings is to discover the truth. The truth is absolute and objective, but we do not have it in the bag. We are constantly seeking it and often find it only with difficulty; and we keep trying to improve our approximation to the truth. If truth were not absolute and objective, we should not be able to go wrong. Or our mistakes would be as good as our truth." (Karl R Popper, "The Epistemological Position of Evolutionary Epistemology", [in "All Life is Problem Solving", 1999] 1987)

"When it comes to very highly organized systems, such as a living cell, the task of modeling by approximation to simple, continuous and smoothly varying quantities is hopeless. It is for this reason that attempts by sociologists and economists to imitate physicists and describe their subject matter by simple mathematical equations is rarely convincing." (Paul C W Davies, "The Cosmic Blueprint: New Discoveries in Nature’s Creative Ability to Order the Universe", 1987)

"A distinctive feature of mathematics, that feature in virtue of which it stands as a paradigmatically rational discipline, is that assertions are not accepted without proof. […] By proof is meant a deductively valid, rationally compelling argument which shows why this must be so, given what it is to be a triangle. But arguments always have premises so that if there are to be any proofs there must also be starting points, premises which are agreed to be necessarily true, self-evident, neither capable of, nor standing in need of, further justification. The conception of mathematics as a discipline in which proofs are required must therefore also be a conception of a discipline in which a systematic and hierarchical order is imposed on its various branches. Some propositions appear as first principles, accepted without proof, and others are ordered on the basis of how directly they can be proved from these first principle. Basic theorems, once proved, are then used to prove further results, and so on. Thus there is a sense in which, so long as mathematicians demand and provide proofs, they must necessarily organize their discipline along lines approximating to the pattern to be found in Euclid's Elements." (Mary Tiles, "Mathematics and the Image of Reason" , 1991)

"A model is something one tries to construct when one has to describe a complicated situation. A model is therefore an approximate description of reality and invariably involves many simplifying assumptions. […] models are convenient idealisations." (Ganeschan Venkataraman, "Chandrasekhar and His Limit", 1992)

"Symmetry is basically a geometrical concept. Mathematically it can be defined as the invariance of geometrical patterns under certain operations. But when abstracted, the concept applies to all sorts of situations. It is one of the ways by which the human mind recognizes order in nature. In this sense symmetry need not be perfect to be meaningful. Even an approximate symmetry attracts one's attention, and makes one wonder if there is some deep reason behind it." (Eguchi Tohru & K Nishijima , "Broken Symmetry: Selected Papers Of Y Nambu", 1995)

"To select an appropriate fuzzy implication for approximate reasoning under each particular situation is a difficult problem. Although some theoretically supported guidelines are now available for some situations, we are still far from a general solution to this problem." (George Klir, "Fuzzy sets and fuzzy logic", 1995)

"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)

"The role of science, like that of art, is to blend proximate imagery with more distant meaning, the parts we already understand with those given as new into larger patterns that are coherent enough to be acceptable as truth. Biologists know this relation by intuition during the course of fieldwork, as they struggle to make order out of the infinitely varying patterns of nature." (Edward O Wilson, "In Search of Nature", 1996)

"Perhaps the most common complaint about the weakness of the random-walk theory is based on a distrust of mathematics and a misconception of what the theory means. 'The market isn't random', the complaint goes, 'and no mathematician is going to convince me it is'. [...] But, even if markets were dominated during certain periods by irrational crowd behavior, the stock market might still well be approximated by a random walk. The original illustrative analogy of a random walk concerned a drunken man staggering around an empty field. He is not rational, but he's not predictable either." (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)

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On Approximation (1950-1974)

"We have here no esoteric theory of the ultimate nature of concepts, nor a philosophical championing of the primacy of the 'operation'. We have merely a pragmatic matter, namely that we have observed after much experience that if we want to do certain kinds of things with our concepts, our concepts had better be constructed in certain ways. In fact one can see that the situation here is no different from what we always find when we push our analysis to the limit; operations are not ultimately sharp or irreducible any more than any other sort of creature. We always run into a haze eventually, and all our concepts are describable only in spiralling approximation." (Percy W Bridgman, "Reflections of a Physicist", 1950)

"Men of science have made abundant mistakes of every kind; their knowledge has improved only because of their gradual abandonment of ancient errors, poor approximations, and premature conclusions." (George Sarton, "A History of Science" Vol 2, 1952)

"The result of the struggle between the thought and the ability to express it, between dream and reality, is seldom more than a compromise or an approximation. Thus there is little chance that we will succeed in getting through to a large audience, and on the whole we are quite satisfied if we are understood and appreciated by a small number of sensitive, receptive people." (Maurits C Escher, "On Being a Graphic Artist", 1953)

"By the laws of statistics we could probably approximate just how unlikely it is that it would happen. But people forget - especially those who ought to know better, such as yourself - that while the laws of statistics tell you how unlikely a particular coincidence is, they state just as firmly that coincidences do happen." (Robert A Heinlein, "The Door Into Summer", 1957)

"The first consequence of the principle of bounded rationality is that the intended rationality of an actor requires him to construct a simplified model of the real situation in order to deal with it. He behaves rationally with respect to this model, and such behavior is not even approximately optimal with respect to the real world. To predict his behavior we must understand the way in which this simplified model is constructed, and its construction will certainly be related to his psychological properties as a perceiving, thinking, and learning animal." (Herbert A Simon, "Models of Man", 1957)

"Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. That is, mathematics determines what conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science is frequently of great service in suggesting new problems and points of view." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"A scientific theory is usually felt to be better than its predecessors not only in the sense that it is a better instrument for discovering and solving puzzles but also because it is somehow a better representation of what nature is really like. One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is 'really there'." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962)

"Probabilities must be regarded as analogous to the measurement of physical magnitudes; that is to say, they can never be known exactly, but only within certain approximation." (Félix E Borel, "Probabilities and Life", 1962)

"The most important maxim for data analysis to heed, and one which many statisticians seem to have shunned is this: ‘Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.’ Data analysis must progress by approximate answers, at best, since its knowledge of what the problem really is will at best be approximate." (John W Tukey, "The Future of Data Analysis", Annals of Mathematical Statistics, Vol. 33, No. 1, 1962)

"Far better an approximate answer to the right question, which is often vague, than the exact answer to the wrong question, which can always be made precise." (John Tukey, "The Future of Data Analysis", Annals of Mathematical Statistics, Vol. 33, No. 1, 1962)

"Because engineering is science in action - the practice of decision making at the earliest moment - it has been defined as the art of skillful approximation. No situation in engineering is simple enough to be solved precisely, and none worth evaluating is solved exactly. Never are there sufficient facts, sufficient time, or sufficient money for an exact solution, for if by chance there were, the answer would be of academic and not economic interest to society. These are the circumstances that make engineering so vital and so creative." (Ronald B Smith, "Engineering Is…", Mechanical Engineering Vol. 86 (5), 1964)

"Each piece, or part, of the whole of nature is always merely an approximation to the complete truth, or the complete truth so far as we know it. In fact, everything we know is only some kind of approximation, because we know that we do not know all the laws as yet. Therefore, things must be learned only to be unlearned again or, more likely, to be corrected." (Richard Feynman, "The Feynman Lectures on Physics" Vol. 1,1964)

"Engineering is the art of skillful approximation; the practice of gamesmanship in the highest form. In the end it is a method broad enough to tame the unknown, a means of combing disciplined judgment with intuition, courage with responsibility, and scientific competence within the practical aspects of time, of cost, and of talent." (Ronald B Smith, "Professional Responsibility of Engineering", Mechanical Engineering Vol. 86 (1), 1964)

"Measurement, we have seen, always has an element of error in it. The most exact description or prediction that a scientist can make is still only approximate." (Abraham Kaplan, "The Conduct of Inquiry: Methodology for Behavioral Science", 1964)

"A model is a qualitative or quantitative representation of a process or endeavor that shows the effects of those factors which are significant for the purposes being considered. A model may be pictorial, descriptive, qualitative, or generally approximate in nature; or it may be mathematical and quantitative in nature and reasonably precise. It is important that effective means for modeling be understood such as analog, stochastic, procedural, scheduling, flow chart, schematic, and block diagrams." (Harold Chestnut, "Systems Engineering Tools", 1965)

"One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently, generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is ‘really there’." (Thomas S Kuhn, "The Structure of Scientific Revolutions", 1970)

"One grievous error in interpreting approximations is to allow only good approximations." (Preston C Hammer, "Mind Pollution", Cybernetics, Vol. 14, 1971)

"A conceptual model is neither idle nor faithful: it is, or rather it is supposed to be and so taken until further notice, an approximate representation of a real thing." (Mario Bunge, "Philosophy of Physics", 1972)

"The idea of approximation to the truth is, in my view, one of the most important ideas in the theory of science. [...] The idea of approximation to the truth - like the idea of truth as are gulative principle - presupposes a realistic view of the world. It does not presuppose that reality is as our scientific theories describe it; but it does presuppose that there is a reality and that we and our theories - which are ideas we have ourselves created and are therefore always idealizations - can draw closer and closer to an adequate description of reality, if we employ the four-stage method of trial and error." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"The realistic view of the world, together with the idea of approximation to the truth, seem to me indispensable for an understanding of the perpetually idealizing character of science." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

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On Inequalities I

"The worst form of inequality is to try to make unequal things equal." (Aristotle)

"Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect. And those which are said to be opposite to each other, the superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too little." (Nicomachus of Gerasa,"Introductio Arithmetica", cca. 100 AD)

"Inequality is the cause of all local movements. There is no rest without equality." (Leonardo da Vinci, Codex Atlanticus, 1478)

"It is from this absolute indifference and tranquility of the mind, that mathematical speculations derive some of their most considerable advantages; because there is nothing to interest the imagination; because the judgment sits free and unbiased to examine the point. All proportions, every arrangement of quantity, is alike to the understanding, because the same truths result to it from all; from greater from lesser, from equality and inequality. (Edmund Burke, "On the Sublime and Beautiful", 1757)

"Nature is unfair? So much the better, inequality is the only bearable thing, the monotony of equality can only lead us to boredom." (Francis Picabia, "Comoedia", 1922)

"The fundamental results of mathematics are often inequalities rather than equalities." Edwin Beckenbach & Richard Bellman, "An Introduction to Inequalities", 1961)

"There are three reasons for the study of inequalities: practical, theoretical and aesthetic. On the aesthetic aspects, as has been pointed out, beauty is in the eyes of the beholder. However, it is generally agreed that certain pieces of music, art, or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive." (Richard E Bellman, 1978)

"Linear programming is concerned with the maximization or minimization of a linear objective function in many variables subject to linear equality and inequality constraints."  (George B Dantzig & Mukund N Thapa, "Linear Programming" Vol I, 1997)

"From the historical point of view, since inequalities are associated with order, they arose as soon as people started using numbers, making measurements, and later, finding approximations and bounds. Thus inequalities have a long and distinguished role in the evolution of mathematics." (Claudi Alsina & Roger B Nelsen, "When Less is More: Visualizing Basic Inequalities", 2009)

"Inequalities permeate mathematics, from the Elements of Euclid to operations research and financial mathematics. Yet too often. especially in secondary and collegiate mathematics. the emphasis is on things equal to one another rather than unequal. While equalities and identities are without doubti mportant, they do not possess the richness and variety that one finds with inequalities." (Claudi Alsina & Roger B Nelsen, "When Less is More: Visualizing Basic Inequalities", 2009)

On Generalization (1970-1979)

"Accordingly there are two main types of science, exact science [...] and empirical science [...] seeking laws which are generalizations from particular experiences and are verifiable (or, more strictly, 'probabilities') only by observation and experiment." (Errol E Harris, "Hypothesis and Perception: The Roots of Scientific Method", 1970)

"One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently, generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is ‘really there’." (Thomas S Kuhn, "The Structure of Scientific Revolutions", 1970)

"Science uses the senses but does not enjoy them; finally buries them under theory, abstraction, mathematical generalization." (Theodore Roszak, "Where the Wasteland Ends", 1972)

"A single observation that is inconsistent with some generalization points to the falsehood of the generalization, and thereby 'points to itself'." (Ian Hacking, "The Emergence Of Probability", 1975)

"The sciences have started to swell. Their philosophical basis has never been very strong. Starting as modest probing operations to unravel the works of God in the world, to follow its traces in nature, they were driven gradually to ever more gigantic generalizations. Since the pieces of the giant puzzle never seemed to fit together perfectly, subsets of smaller, more homogeneous puzzles had to be constructed, in each of which the fit was better." (Erwin Chargaff, "Voices in the Labyrinth", 1975)

"The word generalization in literature usually means covering too much territory too thinly to be persuasive, let alone convincing. In science, however, a generalization means a principle that has been found to hold true in every special case. [...] The principle of leverage is a scientific generalization." (Buckminster Fuller, "Synergetics: Explorations in the Geometry of Thinking", 1975)

"And when such claims are extraordinary, that is, revolutionary in their implications for established scientific generalizations already accumulated and verified, we must demand extraordinary proof." (Marcello Truzzi, Zetetic Scholar, Vol. 1 (1), 1976)

"If it is to be effective as a tool of thought, a notation must allow convenient expression not only of notions arising directly from a problem, but also of those arising in subsequent analysis, generalization, and specialization." (Kenneth E Iverson, "Notation as a Tool of Thought", 1979)

"Prediction can never be absolutely valid and therefore science can never prove some generalization or even test a single descriptive statement and in that way arrive at final truth." (Gregory Bateson, "Mind and Nature, A necessary unity", 1979)

On Technology I

"Unlike art, science is genuinely progressive. Achievement in the fields of research and technology is cumulative; each generation begins at the point where its predecessor left off." (Aldous Huxley, "Science, Liberty and Peace", 1946)

"Doing engineering is practicing the art of the organized forcing of technological change." (George Spencer-Brown, Electronics, Vol. 32 (47),  1959)

"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)

"Engineering is the art of skillful approximation; the practice of gamesmanship in the highest form. In the end it is a method broad enough to tame the unknown, a means of combing disciplined judgment with intuition, courage with responsibility, and scientific competence within the practical aspects of time, of cost, and of talent. This is the exciting view of modern-day engineering that a vigorous profession can insist be the theme for education and training of its youth. It is an outlook that generates its strength and its grandeur not in the discovery of facts but in their application; not in receiving, but in giving. It is an outlook that requires many tools of science and the ability to manipulate them intelligently In the end, it is a welding of theory and practice to build an early, strong, and useful result. Except as a valuable discipline of the mind, a formal education in technology is sterile until it is applied." (Ronald B Smith, "Professional Responsibility of Engineering", Mechanical Engineering Vol. 86 (1), 1964)

"It is a commonplace of modern technology that there is a high measure of certainty that problems have solutions before there is knowledge of how they are to be solved." (John K Galbraith, "The New Industrial State", 1967)

"Technological invention and innovation are the business of engineering. They are embodied in engineering change." (Daniel V DeSimone & Hardy Cross, "Education for Innovation", 1968)

"The future masters of technology will have to be lighthearted and intelligent. The machine easily masters the grim and the dumb." (Marshall McLuhan, "Counterblast", 1969)

"It follows from this that man's most urgent and pre-emptive need is maximally to utilize cybernetic science and computer technology within a general systems framework, to build a meta-systemic reality which is now only dimly envisaged. Intelligent and purposeful application of rapidly developing telecommunications and teleprocessing technology should make possible a degree of worldwide value consensus heretofore unrealizable." (Richard F Ericson, "Visions of Cybernetic Organizations", 1972)

"The march of science and technology does not imply growing intellectual complexity in the lives of most people. It often means the opposite." (Thomas Sowell, "Knowledge And Decisions", 1980)

"A chipped pebble is almost part of the hand it never leaves. A thrown spear declares a sort of independence the moment it is released. [...] The whole trend in technology has been to devise machines that are less and less under direct control and more and more seem to have the beginning of a will of their own." (Isaac Asimov, "Past, Present, and Future", 1987)

Mathematical Models III

"Mathematical model making is an art. If the model is too small, a great deal of analysis and numerical solution can be done, but the results, in general, can be meaningless. If the model is too large, neither analysis nor numerical solution can be carried out, the interpretation of the results is in any case very difficult, and there is great difficulty in obtaining the numerical values of the parameters needed for numerical results." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"Symmetries abound in nature, in technology, and - especially - in the simplified mathematical models we study so assiduously. Symmetries complicate things and simplify them. They complicate them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They simplify them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They violate all the hypotheses of our favorite theorems, yet lead to natural generalizations of those theorems. It is now standard to study the 'generic' behavior of dynamical systems. Symmetry is not generic. The answer is to work within the world of symmetric systems and to examine a suitably restricted idea of genericity." (Ian Stewart, "Bifurcation with symmetry", 1988)

"Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from ‘ultimate’ or ‘fundamental’ truth." (Clifford Truesdell & Walter Noll, "The Non-Linear Field Theories of Mechanics" 2nd Ed., 1992)

"[…] interval mathematics and fuzzy logic together can provide a promising alternative to mathematical modeling for many physical systems that are too vague or too complicated to be described by simple and crisp mathematical formulas or equations. When interval mathematics and fuzzy logic are employed, the interval of confidence and the fuzzy membership functions are used as approximation measures, leading to the so-called fuzzy systems modeling." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"Modeling, in a general sense, refers to the establishment of a description of a system (a plant, a process, etc.) in mathematical terms, which characterizes the input-output behavior of the underlying system. To describe a physical system […] we have to use a mathematical formula or equation that can represent the system both qualitatively and quantitatively. Such a formulation is a mathematical representation, called a mathematical model, of the physical system." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"An important aspect of the global theory of dynamical systems is the stability of the orbit structure as a whole. The motivation for the corresponding theory comes from applied mathematics. Mathematical models always contain simplifying assumptions. Dominant features are modeled; supposed small disturbing forces are ignored. Thus, it is natural to ask if the qualitative structure of the set of solutions - the phase portrait - of a model would remain the same if small perturbations were included in the model. The corresponding mathematical theory is called structural stability." (Carmen Chicone, "Stability Theory of Ordinary Differential Equations" [Mathematics of Complexity and Dynamical Systems, 2012])

"Models do not and need not match reality in all of its aspects and details to be adequate. A mathematical model is usually developed for a specific class of target systems, and its validity is determined relative to its intended applications. A model is considered valid within its intended domain of applicability provided that its predictions in that domain fall within an acceptable range of error, specified prior to the model’s development or identification." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"Simplified description of a real world system in mathematical terms, e. g., by means of differential equations or other suitable mathematical structures." (Benedetto Piccoli, Andrea Tosin, "Vehicular Traffic: A Review of Continuum Mathematical Models" [Mathematics of Complexity and Dynamical Systems, 2012])

"Stated loosely, models are simplified, idealized and approximate representations of the structure, mechanism and behavior of real-world systems. From the standpoint of set-theoretic model theory, a mathematical model of a target system is specified by a nonempty set - called the model’s domain, endowed with some operations and relations, delineated by suitable axioms and intended empirical interpretation." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

On Coherence (-1949)

"Principles taken upon trust, consequences lamely deduced from them, want of coherence in the parts, and of evidence in the whole, these are every where to be met with in the systems of the most eminent philosophers, and seem to have drawn disgrace upon philosophy itself." (David Hume, "A Treatise of Human Nature", 1739-40)

"The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics." (Cassius J Keyser, Science, New Series, Vol. 35 (904), 1912)

"A system is said to be coherent if every fact in the system is related every other fact in the system by relations that are not merely conjunctive. A deductive system affords a good example of a coherent system." (Lizzie S Stebbing, "A modern introduction to logic", 1930)

"Even these humble objects reveal that our reality is not a mere collocation of elemental facts, but consists of units in which no part exists by itself, where each part points beyond itself and implies a larger whole. Facts and significance cease to be two concepts belonging to different realms, since a fact is always a fact in an intrinsically coherent whole. We could solve no problem of organization by solving it for each point separately, one after the other; the solution had to come for the whole. Thus we see how the problem of significance is closely bound up with the problem of the relation between the whole and its parts. It has been said: The whole is more than the sum of its parts. It is more correct to say that the whole is something else than the sum of its parts, because summing is a meaningless procedure, whereas the whole-part relationship is meaningful." (Kurt Koffka, "Principles of Gestalt Psychology", 1935)

"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)

"[…] reality is a system, completely ordered and fully intelligible, with which thought in its advance is more and more identifying itself. We may look at the growth of knowledge […] as an attempt by our mind to return to union with things as they are in their ordered wholeness. […] and if we take this view, our notion of truth is marked out for us. Truth is the approximation of thought to reality […] Its measure is the distance thought has travelled […] toward that intelligible system […] The degree of truth of a particular proposition is to be judged in the first instance by its coherence with experience as a whole, ultimately by its coherence with that further whole, all comprehensive and fully articulated, in which thought can come to rest." (Brand Blanshard, "The Nature of Thought" Vol. II, 1939)

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"[…] there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"Induction is the process of discovering general laws by the observation and combination of particular instances. […] Induction tries to find regularity and coherence behind the observations. Its most conspicuous instruments are generalization, specialization, analogy. Tentative generalization starts from an effort to understand the observed facts; it is based on analogy, and tested by further special cases." (George Pólya, "How to solve it", 1945)

On Structure: Structure in Mathematics (2000-2009)

"Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!" (Gregory J Chaitin, "A century of controversy over the foundations of mathematics", 2000)

"Most physical systems, particularly those complex ones, are extremely difficult to model by an accurate and precise mathematical formula or equation due to the complexity of the system structure, nonlinearity, uncertainty, randomness, etc. Therefore, approximate modeling is often necessary and practical in real-world applications. Intuitively, approximate modeling is always possible. However, the key questions are what kind of approximation is good, where the sense of 'goodness' has to be first defined, of course, and how to formulate such a good approximation in modeling a system such that it is mathematically rigorous and can produce satisfactory results in both theory and applications." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"Our world resonates with patterns. The waxing and waning of the moon. The changing of the seasons. The microscopic cell structure of all living things have patterns. Perhaps that explains our fascination with prime numbers which are uniquely without pattern. Prime numbers are among the most mysterious phenomena in mathematics." (Manindra Agrawal, 2003)

"[…] mathematicians are much more concerned for example with the structure behind something or with the whole edifice. Mathematicians are not really puzzlers. Those who really solve mathematical puzzles are the physicists. If you like to solve mathematical puzzles, you should not study mathematics but physics!" (Carlo Beenakker, [interview] 2006)

"A good problem solver must also be a conceptual mathematician, with a good intuitive grasp of structures. But structures remain tools for the problem solver, instead of the main object of study." (David Ruelle, "The Mathematician's Brain", 2007)

"A proof in mathematics is a psychological device for convincing some person, or some audience, that a certain mathematical assertion is true. The structure, and the language used, in formulating that proof will be a product of the person creating it; but it also must be tailored to the audience that will be receiving it and evaluating it. Thus there is no ‘unique’ or ‘right’ or ‘best’ proof of any given result. A proof is part of a situational ethic." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"It thus stands to reason that mathematical structures have a dual origin: in part human, in part purely logical. Human mathematics requires short formulations (because of our poor memory, etc.). But mathematical logic dictates that theorems with a short formulation may have very long proofs, as shown by Gödel. Clearly you don't want to go through the same long proof again and again. You will try instead to use repeatedly the short theorem that you have obtained. And an important tool to obtain short formulations is to give short names to mathematical objects that occur repeatedly. These short names describe new concepts. So we see how concept creation arises in the practice of mathematics as a consequence of the inherent logic of the sub ject and of the nature of human mathematicians." (David Ruelle, "The Mathematician's Brain", 2007)

"Logic moves in one direction, the direction of clarity, coherence, and structure. Ambiguity moves in the other direction, that of fluidity, openness, and release. Mathematics moves back and forth between these two poles. Mathematics is not a fixed, static entity that can be structured definitively. It is dynamic, alive: its dynamism a function of the relationship between the two poles that have been described above. It is the interactions between these different aspects that give mathematics its power." (William Byers, "How Mathematicians Think", 2007)

"Mathematics as done by mathematicians is not just heaping up statements logically deduced from the axioms. Most such statements are rubbish, even if perfectly correct. A good mathematician will look for interesting results. These interesting results, or theorems, organize themselves into meaningful and natural structures, and one may say that the object of mathematics is to find and study these structures." (David Ruelle, "The Mathematician's Brain", 2007)

"The panoply of technical tools of mathematics reflects the inside structure of mathematics and is basically all we know about this inside structure, so that building a new theory may change the way we understand the structural relations of different parts of mathematics." (David Ruelle, "The Mathematician's Brain", 2007)

"We must admit, however, that our knowledge of the logical structure of mathematics and of the workings of the human mind remain quite limited, so that we have only partial answers to some questions, while others remain quite open." (David Ruelle, "The Mathematician's Brain", 2007)

"In mathematics, beauty is a very important ingredient. Beauty exists in mathematics as in architecture and other things. It is a difficult thing to define but it is something you recognise when you see it. It certainly has to have elegance, simplicity, structure and form. All sorts of things make up real beauty. There are many different kinds of beauty and the same is true of mathematical theorems. Beauty is an important criterion in mathematics because basically there is a lot of choice in what you can do in mathematics and science. It determines what you regard as important and what is not." (Michael Atiyah, 2009)

"Philosophers have sometimes made a distinction between analytic and synthetic truths. Analytic truths are not verified by observation; true analytic statements are tautologies and are true by virtue of the definitions of their terms and their logical structure. Synthetic truths relate to the material world; the truth of synthetic statements depends on their correspondence to how physical reality works. Mathematics, according to this distinction, deals exclusively with analytic truths. Its statements are all tautologies and are (analytically) true by virtue of their adherence to formal rules of construction." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

On Nonlinearity III

"Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. (Edward N Lorenz, "Deterministic Nonperiodic Flow", Journal of the Atmospheric Science 20, 1963)

"We've seen that even in the simplest situations nonlinearities can interfere with a linear approach to aggregates. That point holds in general: nonlinear interactions almost always make the behavior of the aggregate more complicated than would be predicted by summing or averaging." (Lewis Mumford, "The Myth of the Machine" Vol 1, 1967)

"The structure of a complex system is not a simple feedback loop where one system state dominates the behavior. The complex system has a multiplicity of interacting feedback loops. Its internal rates of flow are controlled by non‐linear relationships. The complex system is of high order, meaning that there are many system states (or levels). It usually contains positive‐feedback loops describing growth processes as well as negative, goal‐seeking loops." (Jay F Forrester, "Urban Dynamics", 1969)

"I would therefore urge that people be introduced to [the logistic equation] early in their mathematical education. This equation can be studied phenomenologically by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student’s intuition about nonlinear systems. Not only in research but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties." (Robert M May, "Simple Mathematical Models with Very Complicated Dynamics", Nature Vol. 261 (5560), 1976)

"Most physical systems, particularly those complex ones, are extremely difficult to model by an accurate and precise mathematical formula or equation due to the complexity of the system structure, nonlinearity, uncertainty, randomness, etc. Therefore, approximate modeling is often necessary and practical in real-world applications. Intuitively, approximate modeling is always possible. However, the key questions are what kind of approximation is good, where the sense of 'goodness' has to be first defined, of course, and how to formulate such a good approximation in modeling a system such that it is mathematically rigorous and can produce satisfactory results in both theory and applications." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001) 

"Thus, nonlinearity can be understood as the effect of a causal loop, where effects or outputs are fed back into the causes or inputs of the process. Complex systems are characterized by networks of such causal loops. In a complex, the interdependencies are such that a component A will affect a component B, but B will in general also affect A, directly or indirectly.  A single feedback loop can be positive or negative. A positive feedback will amplify any variation in A, making it grow exponentially. The result is that the tiniest, microscopic difference between initial states can grow into macroscopically observable distinctions." (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"Where simplifications fail, causing the most damage, is when something nonlinear is simplified with the linear as a substitute. That is the most common Procrustean bed." (Nassim N Taleb, "Antifragile: Things that Gain from Disorder", 2012)

"Complex systems defy intuitive solutions. Even a third-order, linear differential equation is unsolvable by inspection. Yet, important situations in management, economics, medicine, and social behavior usually lose reality if simplified to less than fifth-order nonlinear dynamic systems. Attempts to deal with nonlinear dynamic systems using ordinary processes of description and debate lead to internal inconsistencies. Underlying assumptions may have been left unclear and contradictory, and mental models are often logically incomplete. Resulting behavior is likely to be contrary to that implied by the assumptions being made about' underlying system structure and governing policies." (Jay W Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"There is no linear additive process that, if all the parts are taken together, can be understood to create the total system that occurs at the moment of self-organization; it is not a quantity that comes into being. It is not predictable in its shape or subsequent behavior or its subsequent qualities. There is a nonlinear quality that comes into being at the moment of synchronicity." (Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)

"Exponentially growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems. How exponential growth emerges in nonlinear systems is mathematically unclear. […] The emergence of exponential growth from a multivariable nonlinear network is not mathematically intuitive. This indicates that the network structure and the flux functions of the modeled system must be subjected to constraints to result in long-term exponential dynamics." (Wei-Hsiang Lin et al, "Origin of exponential growth in nonlinear reaction networks", PNAS 117 (45), 2020)

Complexity vs Mathematics I

"But ignorance of the different causes involved in the production of events, as well as their complexity, taken together with the imperfection of analysis, prevents our reaching the same certainty about the vast majority of phenomena. Thus there are things that are uncertain for us, things more or less probable, and we seek to compensate for the impossibility of knowing them by determining their different degrees of likelihood. So it was that we owe to the weakness of the human mind one of the most delicate and ingenious of mathematical theories, the science of chance or probability." (Pierre-Simon Laplace, "Recherches, 1º, sur l'Intégration des Équations Différentielles aux Différences Finies, et sur leur Usage dans la Théorie des Hasards", 1773)

"So completely is nature mathematical that some of the more exact natural sciences, in particular astronomy and physics, are in their theoretic phases largely mathematical in character, while other sciences which have hitherto been compelled by the complexity of their phenomena and the inexactitude of their data to remain descriptive and empirical, are developing towards the mathematical ideal, proceeding upon the fundamental assumption that mathematical relations exist between the forces and the phenomena, and that nothing short, of the discovery and formulations of these relations would constitute definitive knowledge of the subject. Progress is measured by the closeness of the approximation to this ideal formulation." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analyzing, to greater and greater abstractness and logical simplicity." (Bertrand Russell, "Introduction to Mathematical Philosophy", 1919)

"We love to discover in the cosmos the geometrical forms that exist in the depths of our consciousness. The exactitude of the proportions of our monuments and the precision of our machines express a fundamental character of our mind. Geometry does not exist in the earthly world. It has originated in ourselves. The methods of nature are never so precise as those of man. We do not find in the universe the clearness and accuracy of our thought. We attempt, therefore, to abstract from the complexity of phenomena some simple systems whose components bear to one another certain relations susceptible of being described mathematically." (Alexis Carrel, "Man the Unknown", 1935)

"Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize." (George Pólya, "Mathematical Methods in Science", 1977)

"Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or ‘mathematics envy’." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"Somehow the breathless world that we witness seems far removed from the timeless laws of Nature which govern the elementary particles and forces of Nature. The reason is clear. We do not observe the laws of Nature: we observe their outcomes. Since these laws find their most efficient representation as mathematical equations, we might say that we see only the solutions of those equations not the equations themselves. This is the secret which reconciles the complexity observed in Nature with the advertised simplicity of her laws." (John D Barrow, "New Theories of Everything", 1991)

"To find the simple in the complex, the finite in the infinite - that is not a bad description of the aim and essence of mathematics. (Jacob T Schwartz, "Discrete Thoughts: Essays on Mathematics, Science, and Philosophy", 1992)

"Mathematical truth is not totally objective. If a mathematical statement is false, there will be no proofs, but if it is true, there will be an endless variety of proofs, not just one! Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, many proofs of that fact. [...] each proof will emphasize different aspects of the problem, each proof will lead in a different direction. Each one will have different corollaries, different generalizations. [...] the world of mathematical truth has infinite complexity […]" (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"[All phenomena] are equally susceptible of being calculated, and all that is necessary, to reduce the whole of nature to laws similar to those which Newton discovered with the aid of the calculus, is to have a sufficient number of observations and a mathematics that is complex enough." (Nicolas de Condorcet) 

Lotfi A Zadeh - Collected Quotes

"A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint." (Lotfi A Zadeh, "Fuzzy Sets", 1965)

"The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply denned criteria of class membership rather than the presence of random variables." (Lotfi A Zadeh, "Fuzzy Sets", 1965)

"In general, complexity and precision bear an inverse relation to one another in the sense that, as the complexity of a problem increases, the possibility of analysing it in precise terms diminishes. Thus 'fuzzy thinking' may not be deplorable, after all, if it makes possible the solution of problems which are much too complex for precise analysis." (Lotfi A Zadeh, "Fuzzy languages and their relation to human intelligence", 1972)

"As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics." (Lotfi A Zadeh, 1973)

"The closer one looks at a real-world problem, the fuzzier becomes its solution." (Lotfi A Zadeh, 1973)

"[Fuzzy logic is] a logic whose distinguishing features are (1) fuzzy truth-values expressed in linguistic terms, e. g., true, very true, more or less true, or somewhat true, false, nor very true and not very false, etc.; (2) imprecise truth tables; and (3) rules of inference whose validity is relative to a context rather than exact." (Lotfi A Zadeh, "Fuzzy logic and approximate reasoning", 1975)

"[...] much of the information on which human decisions are based is possibilistic rather than probabilistic in nature, and the intrinsic fuzziness of natural languages - which is a logical consequence of the necessity to express information in a summarized form - is, in the main, possibilistic in origin." (Lotfi A Zadeh, "Fuzzy Sets as the Basis for a Theory of Possibility", Fuzzy Sets and Systems, 1978) 

"Fuzziness, then, is a concomitant of complexity. This implies that as the complexity of a task, or of a system for performing that task, exceeds a certain threshold, the system must necessarily become fuzzy in nature. Thus, with the rapid increase in the complexity of the information processing tasks which the computers are called upon to perform, we are reaching a point where computers will have to be designed for processing of information in fuzzy form. In fact, it is the capability to manipulate fuzzy concepts that distinguishes human intelligence from the machine intelligence of current generation computers. Without such capability we cannot build machines that can summarize written text, translate well from one natural language to another, or perform many other tasks that humans can do with ease because of their ability to manipulate fuzzy concepts." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)

"In essence, a basic assumption in classical mathematics is that a concept must admit of precise definition which partitions the class of all objects into two classes: (i) those objects which are instances of the concept; and (ii) those which are not, with no borderline cases allowed. For example, a function is either continuous or discontinuous; it cannot be continuous to a degree. Similarly, a matrix is either symmetric or not symmetric; it cannot be somewhat symmetric or more or asymmetric or symmetric to a degree." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)

"It is important to observe that there is an intimate connection between fuzziness and complexity. Thus, a basic characteristic of the human brain, a characteristic shared in varying degrees with all information processing systems, is its limited capacity to handle classes of high cardinality, that is, classes having a large number of members. Consequently, when we are presented with a class of very high cardinality, we tend to group its elements together into subclasses in such a way as to reduce the complexity of the information processing task involved. When a point is reached where the cardinality of the class of subclasses exceeds the information handling capacity of the human brain, the boundaries of the subclasses are forced to become imprecise and fuzziness becomes a manifestation of this imprecision." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)

"As systems became more varied and more complex, we find that no single methodology suffices to deal with them. This is particularly true of what may be called information intelligent systems - systems which form the core of modern technology. To conceive, design, analyze and use such systems we frequently have to employ the totality of tools that are available. Among such tools are the techniques centered on fuzzy logic, neurocomputing, evolutionary computing, probabilistic computing and related methodologies. It is this conclusion that formed the genesis of the concept of soft computing." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)

"[...] fuzzy logic [FL] has many distinct facets - facets which overlap and have unsharp boundaries. Among these facets there are four that stand out in importance. They are (i) the logical facet; (ii) the set-theoretic facet: (iii) the relational facet, and (iv) the epistemic facet. […] The logical facet of FL, FL/L, is a logical system or, more accurately, a collection of logical systems which includes as a special case both two-valued and multiple-valued systems. […] The set-theoretic facet of FL, FL/S, is concerned with classes or sets whose boundaries are not sharply defined. […] The relational facet of FL, FL/R, is concerned in the main with representation and manipulation of imprecisely defined functions and relations. […] The epistemic facet of FL, FL/E, is linked to its logical facet and is centered on applications of FL to knowledge representation, information systems, fuzzy databases and the theories of possibility imprecise probabilities." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)

"In science, it is a long-standing tradition to deal with perceptions by converting them into measurements. But what is becoming increasingly evident is that, to a much greater extent than is generally recognized, conversion of perceptions into measurements is infeasible, unrealistic or counter-productive. With the vast computational power at our command, what is becoming feasible is a counter-traditional move from measurements to perceptions. […] To be able to compute with perceptions it is necessary to have a means of representing their meaning in a way that lends itself to computation." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)

"A concept which has a position of centrality in fuzzy logic is that of a fuzzy set. Informally, a fuzzy set is a class with a fuzzy boundary, implying a gradual transition from membership to nonmembership. A fuzzy set is precisiated through graduation, that is, through association with a scale of grades of membership. Thus, membership in a fuzzy set is a matter of degree. Importantly, in fuzzy logic everything is or is allowed to be graduated, that is, be a matter of degree. Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a clump of attribute-values drawn together by indistinguishability, equivalence, similarity, proximity or functionality. Graduation and granulation form the core of fuzzy logic. Graduated granulation is the basis for the concept of a linguistic variable – a variable whose values are words rather than numbers. The concept of a linguistic variable is employed in almost all applications of fuzzy logic." (Lofti A Zadeh, "Fuzzy Logic", 2009) 

"In essence, logic is concerned with formalization of reasoning. Correspondently, fuzzy logic is concerned with formalization of fuzzy reasoning, with the understanding that precise reasoning is a special case of fuzzy reasoning." (Lofti A Zadeh, "Fuzzy Logic", 2009

"Science deals not with reality but with models of reality. In large measure, scientific progress is driven by a quest for better models of reality. In the real world, imprecision, uncertainty and complexity have a pervasive presence. In this setting, construction of better models of reality requires a better understanding of how to deal effectively with imprecision, uncertainty and complexity. To a significant degree, development of fuzzy logic has been, and continues to be, motivated by this need." (Lofti A Zadeh, "Fuzzy Logic", 2009) 

"As complexity rises, precise statements lose meaning and meaningful statements lose precision." (Lotfi A Zadeh) 

From Parts to Wholes (1960-1969)

"Every part of the system is so related to every other part that a change in a particular part causes a changes in all other parts and in the total system." (Arthur D Hall, "A methodology for systems engineering", 1962)

"Roughly, by a complex system I mean one made up of a large number of parts that interact in a nonsimple way. In such systems, the whole is more than the sum of the parts, not in an ultimate, metaphysical sense, but in the important pragmatic sense that, given the properties of the parts and the laws of their interaction, it is not a trivial matter to infer the properties of the whole." (Herbert A Simon, "The Architecture of Complexity", Proceedings of the American Philosophical Society, Vol. 106 (6), 1962)

"To say a system is 'self-organizing' leaves open two quite different meanings. There is a first meaning that is simple and unobjectionable. This refers to the system that starts with its parts separate (so that the behavior of each is independent of the others' states) and whose parts then act so that they change towards forming connections of some type. Such a system is 'self-organizing' in the sense that it changes from 'parts separated' to 'parts joined'. […] In general such systems can be more simply characterized as 'self-connecting', for the change from independence between the parts to conditionality can always be seen as some form of 'connection', even if it is as purely functional […]  'Organizing' […] may also mean 'changing from a bad organization to a good one' […] The system would be 'self-organizing' if a change were automatically made to the feedback, changing it from positive to negative; then the whole would have changed from a bad organization to a good." (W Ross Ashby, "Principles of the self-organizing system", 1962)

"Each piece, or part, of the whole of nature is always merely an approximation to the complete truth, or the complete truth so far as we know it. In fact, everything we know is only some kind of approximation, because we know that we do not know all the laws as yet. Therefore, things must be learned only to be unlearned again or, more likely, to be corrected.” (Richard Feynman, "The Feynman Lectures on Physics" Vol. 1, 1963)

"In a general way it may be said that to think in terms of systems seems the most appropriate conceptual response so far available when the phenomena under study - at any level and in any domain--display the character of being organized, and when understanding the nature of the interdependencies constitutes the research task. In the behavioral sciences, the first steps in building a systems theory were taken in connection with the analysis of internal processes in organisms, or organizations, when the parts had to be related to the whole." (Fred Emery, "The Causal Texture of Organizational Environments", 1963)

"Formulating and structuring a system provide methods for relating (1) what the system consists of in the mind of the persons or group desiring it; (2)what it means in terms of the persons or group designing and building it; and (3) in terms of the persons or groups operating, using and servicing it. They provide a set of 'reasonable"'parts and methods of relating them so that the many persons working on the system can understand the whole in sufficient detail for their purposes, and their particular parts in explicit detail so that they may contribute their best efforts to the extent required. A further purpose of system formulation is to recognize the magnitude of the job, including the possible pitfalls." (Harold Chestnut, "Systems Engineering Tools", 1965)

"Formulating consists of determining the system inputs, outputs, requirements, objectives, constraints. Structuring the system provides one or more methods of organizing the solution, the method of operation, the selection of parts, and the nature of their performance requirements. It is evident that the processes of formulating a system and structuring it are strongly related." (Harold Chestnut, "Systems Engineering Tools", 1965)

"The process of formulating and structuring a system are important and creative, since they provide and organize the information, which each system. "establishes the number of objectives and the balance between them which will be optimized". Furthermore, they help identify and define the system parts. Furthermore, they help identify and define the system parts which make up its 'diverse, specialized structures and subfunctions'." (Harold Chestnut, "Systems Engineering Tools", 1965)

"The Systems engineering method recognizes each system is an integrated whole even though composed of diverse, specialized structures and sub-functions. It further recognizes that any system has a number of objectives and that the balance between them may differ widely from system to system. The methods seek to optimize the overall system functions according to the weighted objectives and to achieve maximum compatibility of its parts." (Harold Chestnut, "Systems Engineering Tools", 1965)

"Just as no thing or organism exists on its own, it does not act on its own. Furthermore, every organism is a process: thus the organism is not other than its actions. To put it clumsily: it is what it does. More precisely, the organism, including its behavior, is a process which is to be understood only in relation to the larger and longer process of its environment. For what we mean by 'understanding' or 'comprehension' is seeing how parts fit into a whole, and then realizing that they don't compose the whole, as one assembles a jigsaw puzzle, but that the whole is a pattern, a complex wiggliness, which has no separate parts. Parts are fictions of language, of the calculus of looking at the world through a net which seems to chop it up into bits. Parts exist only for purposes of figuring and describing, and as we figure the world out we become confused if we do not remember this all the time." (Alan Watts, "The Book on the Taboo Against Knowing Who You Are", 1966)

"Nor does complexity deny the valid simplification which is part of the process of analysis, and even a method of achieving complex architecture itself." (Robert Venturi, "Complexity and Contradiction in Architecture Complexity and Contradiction in Architecture", 1966)

"A synthetic approach where piecemeal analysis is not possible due to the intricate interrelationships of parts that cannot be treated out of context of the whole;" (Walter F Buckley, "Sociology and modern systems theory", 1967)

"Now we are looking for another basic outlook on the world - the world as organization. Such a conception - if it can be substantiated - would indeed change the basic categories upon which scientific thought rests, and profoundly influence practical attitudes. This trend is marked by the emergence of a bundle of new disciplines such as cybernetics, information theory, general system theory, theories of games, of decisions, of queuing and others; in practical applications, systems analysis, systems engineering, operations research, etc. They are different in basic assumptions, mathematical techniques and aims, and they are often unsatisfactory and sometimes contradictory. They agree, however, in being concerned, in one way or another, with ‘systems’, ‘wholes’ or ‘organizations’; and in their totality, they herald a new approach." (Ludwig von Bertalanffy, "General System Theory", 1968)

"System' is the concept that refers both to a complex of interdependencies between parts, components, and processes, that involves discernible regularities of relationships, and to a similar type of interdependency between such a complex and its surrounding environment." (Talcott Parsons, "Systems Analysis: Social Systems", 1968)

"The system problem is essentially the problem of the limitation of analytical procedures in science. This used to be expressed by half-metaphysical statements, such as emergent evolution or ‘the whole is more than the sum of its parts,’ but has a clear operational meaning." (Ludwig von Bertalanffy, "General System Theory", 1968)

"You cannot sum up the behavior of the whole from the isolated parts, and you have to take into account the relations between the various subordinate systems which are super-ordinated to them in order to understand the behavior of the parts." (Ludwig von Bertalanffy, "General System Theory", 1968)

"In complex systems cause and effect are often not closely related in either time or space. The structure of a complex system is not a simple feedback loop where one system state dominates the behavior. The complex system has a multiplicity of interacting feedback loops. Its internal rates of flow are controlled by nonlinear relationships. The complex system is of high order, meaning that there are many system states (or levels). It usually contains positive-feedback loops describing growth processes as well as negative, goal-seeking loops. In the complex system the cause of a difficulty may lie far back in time from the symptoms, or in a completely different and remote part of the system. In fact, causes are usually found, not in prior events, but in the structure and policies of the system." (Jay Wright Forrester, "Urban dynamics", 1969)

"Synergy is the only word in our language that means behavior of whole systems unpredicted by the separately observed behaviors of any of the system's separate parts or any subassembly of the system's parts." (Buckminster Fuller, "Operating Manual for Spaceship Earth", 1969)