On Limits II (Systems)

"Clearly, if the state of the system is coupled to parameters of an environment and the state of the environment is made to modify parameters of the system, a learning process will occur. Such an arrangement will be called a Finite Learning Machine, since it has a definite capacity. It is, of course, an active learning mechanism which trades with its surroundings. Indeed it is the limit case of a self-organizing system which will appear in the network if the currency supply is generalized." (Gordon Pask, "The Natural History of Networks", 1960)

"Taking no action to solve these problems is equivalent of taking strong action. Every day of continued exponential growth brings the world system closer to the ultimate limits of that growth. A decision to do nothing is a decision to increase the risk of collapse." (Donella Meadows et al, "The Limits to Growth", 1972)

"Every day of continued exponential growth brings the world system closer to the ultimate limits of that growth." (Mihajlo D Mesarovic, "Mankind at the Turning Point", 1974)

"In a loosely coupled system there is more room available for self-determination by the actors. If it is argued that a sense of efficacy is crucial for human beings. when a sense of efficacy might be greater in a loosely coupled system with autonomous units than it would be in a tightly coupled system where discretion is limited." (Karl E Weick, "Educational organizations as loosely coupled systems", 1976)

"Prediction of the future is possible only in systems that have stable parameters like celestial mechanics. The only reason why prediction is so successful in celestial mechanics is that the evolution of the solar system has ground to a halt in what is essentially a dynamic equilibrium with stable parameters. Evolutionary systems, however, by their very nature have unstable parameters. They are disequilibrium systems and in such systems our power of prediction, though not zero, is very limited because of the unpredictability of the parameters themselves. If, of course, it were possible to predict the change in the parameters, then there would be other parameters which were unchanged, but the search for ultimately stable parameters in evolutionary systems is futile, for they probably do not exist… Social systems have Heisenberg principles all over the place, for we cannot predict the future without changing it." (Kenneth E Boulding, Evolutionary Economics, 1981)

"The phenomenon of self-organization is not limited to living matter but occurs also in certain chemical systems […] [Ilya] Prigogine has called these systems 'dissipative structures' to express the fact that they maintain and develop structure by breaking down other structures in the process of metabolism, thus creating entropy­ disorder - which is subsequently dissipated in the form of degraded waste products. Dissipative chemical structures display the dynamics of self-organization in its simplest form, exhibiting most of the phenomena characteristic of life self-renewal, adaptation, evolution, and even primitive forms of 'mental' processes." (Fritjof Capra, "The Turning Point: Science, Society, and the Turning Culture", 1982)

"Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"Computational reducibility may well be the exception rather than the rule: Most physical questions may be answerable only through irreducible amounts of computation. Those that concern idealized limits of infinite time, volume, or numerical precision can require arbitrarily long computations, and so be formally undecidable." (Stephen Wolfram, Undecidability and intractability in theoretical physics", Physical Review Letters 54 (8), 1985)

"Regarding stability, the state trajectories of a system tend to equilibrium. In the simplest case they converge to one point (or different points from different initial states), more commonly to one (or several, according to initial state) fixed point or limit cycle(s) or even torus(es) of characteristic equilibrial behaviour. All this is, in a rigorous sense, contingent upon describing a potential, as a special summation of the multitude of forces acting upon the state in question, and finding the fixed points, cycles, etc., to be minima of the potential function. It is often more convenient to use the equivalent jargon of 'attractors' so that the state of a system is 'attracted' to an equilibrial behaviour. In any case, once in equilibrial conditions, the system returns to its limit, equilibrial behaviour after small, arbitrary, and random perturbations." (Gordon Pask, "Different Kinds of Cybernetics", 1992)

"Systems, acting dynamically, produce (and incidentally, reproduce) their own boundaries, as structures which are complementary (necessarily so) to their motion and dynamics. They are liable, for all that, to instabilities chaos, as commonly interpreted of chaotic form, where nowadays, is remote from the random. Chaos is a peculiar situation in which the trajectories of a system, taken in the traditional sense, fail to converge as they approach their limit cycles or 'attractors' or 'equilibria'. Instead, they diverge, due to an increase, of indefinite magnitude, in amplification or gain." (Gordon Pask, "Different Kinds of Cybernetics", 1992)

"In spite of the insurmountable computational limits, we continue to pursue the many problems that possess the characteristics of organized complexity. These problems are too important for our well being to give up on them. The main challenge in pursuing these problems narrows down fundamentally to one question: how to deal with systems and associated problems whose complexities are beyond our information processing limits? That is, how can we deal with these problems if no computational power alone is sufficient?"  (George Klir, "Fuzzy sets and fuzzy logic", 1995)

"The dimensionality and nonlinearity requirements of chaos do not guarantee its appearance. At best, these conditions allow it to occur, and even then under limited conditions relating to particular parameter values. But this does not imply that chaos is rare in the real world. Indeed, discoveries are being made constantly of either the clearly identifiable or arguably persuasive appearance of chaos. Most of these discoveries are being made with regard to physical systems, but the lack of similar discoveries involving human behavior is almost certainly due to the still developing nature of nonlinear analyses in the social sciences rather than the absence of chaos in the human setting."  (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

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

"A population that grows logistically, initially increases exponentially; then the growth lows down and eventually approaches an upper bound or limit. The most well-known form of the model is the logistic differential equation." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)

"The methodology of feedback design is borrowed from cybernetics (control theory). It is based upon methods of controlled system model’s building, methods of system states and parameters estimation (identification), and methods of feedback synthesis. The models of controlled system used in cybernetics differ from conventional models of physics and mechanics in that they have explicitly specified inputs and outputs. Unlike conventional physics results, often formulated as conservation laws, the results of cybernetical physics are formulated in the form of transformation laws, establishing the possibilities and limits of changing properties of a physical system by means of control." (Alexander L Fradkov, "Cybernetical Physics: From Control of Chaos to Quantum Control", 2007)

"A characteristic of such chaotic dynamics is an extreme sensitivity to initial conditions (exponential separation of neighboring trajectories), which puts severe limitations on any forecast of the future fate of a particular trajectory. This sensitivity is known as the ‘butterfly effect’: the state of the system at time t can be entirely different even if the initial conditions are only slightly changed, i.e., by a butterfly flapping its wings." (Hans J Korsch et al, "Chaos: A Program Collection for the PC", 2008)

"A quantity growing exponentially toward a limit reaches that limit in a surprisingly short time." (Donella Meadows, "Thinking in systems: A Primer", 2008)

"A typical complex system consists of a vast number of identical copies of several generic processes, which are operating and interacting only locally or with a limited number of not necessary close neighbours. There is no global leader or controller associated to such systems and the resulting behaviour is usually very complex." (Jirí Kroc & Peter M A Sloot, "Complex Systems Modeling by Cellular Automata", Encyclopedia of Artificial Intelligence, 2009)

"Strange attractors, unlike regular ones, are geometrically very complicated, as revealed by the evolution of a small phase-space volume. For instance, if the attractor is a limit cycle, a small two-dimensional volume does not change too much its shape: in a direction it maintains its size, while in the other it shrinks till becoming a 'very thin strand' with an almost constant length. In chaotic systems, instead, the dynamics continuously stretches and folds an initial small volume transforming it into a thinner and thinner 'ribbon' with an exponentially increasing length." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010)

"We are beginning to see the entire universe as a holographically interlinked network of energy and information, organically whole and self-referential at all scales of its existence. We, and all things in the universe, are non-locally connected with each other and with all other things in ways that are unfettered by the hitherto known limitations of space and time." (Ervin László, "Cosmos: A Co-creator's Guide to the Whole-World", 2010)

"It should also be noted that the novel information generated by interactions in complex systems limits their predictability. Without randomness, complexity implies a particular non-determinism characterized by computational irreducibility. In other words, complex phenomena cannot be known a priori." (Carlos Gershenson, "Complexity", 2011)

"Complexity scientists concluded that there are just too many factors - both concordant and contrarian - to understand. And with so many potential gaps in information, almost nobody can see the whole picture. Complex systems have severe limits, not only to predictability but also to measurability. Some complexity theorists argue that modelling, while useful for thinking and for studying the complexities of the world, is a particularly poor tool for predicting what will happen." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

"A limit cycle is an isolated closed trajectory. Isolated means that neighboring trajectories are not closed; they spiral either toward or away from the limit cycle. If all neighboring trajectories approach the limit cycle, we say the limit cycle is stable or attracting. Otherwise the limit cycle is unstable, or in exceptional cases, half-stable. Stable limit cycles are very important scientifically - they model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)

On Puzzles (Unsourced)

"It is an outcome of faith that nature - as she is perceptible to our five senses - takes the character of such a well formulated puzzle." (Albert Einstein)

"Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man." (Maria Goeppert-Mayer)

"Science is a game - but a game with reality, a game with sharpened knives [..] If a man cuts a picture carefully into 1000 pieces, you solve the puzzle when you reassemble the pieces into a picture; in the success or failure, both your intelligences compete. In the presentation of a scientific problem, the other player is the good Lord. He has not only set the problem but also has devised the rules of the game - but they are not completely known, half of them are left for you to discover or to deduce. The experiment is the tempered blade which you wield with success against the spirits of darkness - or which defeats you shamefully. The uncertainty is how many of the rules God himself has permanently ordained, and how many apparently are caused by your own mental inertia, while the solution generally becomes possible only through freedom from its limitations." (Erwin Schrödinger)

"The art of simplicity is a puzzle of complexity." (Douglas Horton)

"Throughout science there is a constant alternation between periods when a particular subject is in a state of order, with all known data falling neatly into their places, and a state of puzzlement and confusion, when new observations throw all neatly arranged ideas into disarray." (Sir Hermann Bondi)

"While the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what anyone man will be up to, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant. So says the statistician." (Sir Arthur C Doyle)

Ben Bolker - Collected Quotes

"Frequentist statistics assumes that there is a 'true' state of the world (e.g. the difference between species in predation probability) which gives rise to a distribution of possible experimental outcomes. The Bayesian framework says instead that the experimental outcome - what we actually saw happen - is the truth, while the parameter values or hypotheses have probability distributions. The Bayesian framework solves many of the conceptual problems of frequentist statistics: answers depend on what we actually saw and not on a range of hypothetical outcomes, and we can legitimately make statements about the probability of different hypotheses or parameter values." (Ben Bolker, "Ecological Models and Data in R", 2007)

"Most modern statistics uses an approach called maximum likelihood estimation, or approximations to it. For a particular statistical model, maximum likelihood finds the set of parameters (e.g. seed removal rates) that makes the observed data (e.g. the particular outcomes of predation trials) most likely to have occurred. Based on a model for both the deterministic and stochastic aspects of the data, we can compute the likelihood (the probability of the observed outcome) given a particular choice of parameters. We then find the set of parameters that makes the likelihood as large as possible, and take the resulting maximum likelihood estimates (MLEs) as our best guess at the parameters." (Ben Bolker, "Ecological Models and Data in R", 2007)

"Normally distributed variables are everywhere, and most classical statistical methods use this distribution. The explanation for the normal distribution’s ubiquity is the Central Limit Theorem, which says that if you add a large number of independent samples from the same distribution the distribution of the sum will be approximately normal." (Ben Bolker, "Ecological Models and Data in R", 2007)

"Phenomenological models concentrate on observed patterns in the data, using functions and distributions that are the right shape and/or sufficiently flexible to match them; mechanistic models are more concerned with the underlying processes, using functions and distributions based on theoretical expectations. As usual, there are shades of gray; the same function could be classified as either phenomenological or mechanistic depending on why it was chosen." (Ben Bolker, "Ecological Models and Data in R", 2007)

"The dichotomy of mathematical vs. statistical modeling says more about the culture of modeling and how different disciplines go about thinking about models than about how we should actually model ecological systems. A mathematician is more likely to produce a deterministic, dynamic process model without thinking very much about noise and uncertainty (e.g. the ordinary differential equations that make up the Lotka-Volterra predator prey model). A statistician, on the other hand, is more likely to produce a stochastic but static model, that treats noise and uncertainty carefully but focuses more on static patterns than on the dynamic processes that produce them (e.g. linear regression)." (Ben Bolker, "Ecological Models and Data in R", 2007)

Joseph J Thomson - Collected Quotes

"From the point of view of the physicist, a theory of matter is a policy rather than a creed; its object is to connect or co-ordinate apparently diverse phenomena, and above all to suggest, stimulate and direct experiment. It ought to furnish a compass which, if followed, will lead to observer further and further into previously unexplored regions." (Sir Joseph J Thomson, "The Corpuscular Theory of Matter", 1907)

"It [a theory] ought to furnish a compass which, if followed, will lead the observer further and further into previously unexplored regions. Whether these regions will be barren or fertile experience alone will decide; but, at any rate, one who is guided in this way will travel onward in a definite direction, and will not wander aimlessly to and fro." (Sir Joseph J Thomson, "The Corpuscular Theory of Matter", 1907)

"Nature is far more wonderful and unconventional than anything we can evolve from our inner consciousness. The most far-reaching generalizations which may influence philosophy as well as revolutionize physics, may be suggested, nay, forced on the mind by the discovery of some trivial phenomenon." (Joseph J Thomson, "The Atomic Theory", 1914)

"It [relativity] was not a discovery of an outlying island, but of a whole continent of new scientific ideas of the greatest importance to some of the most fundamental questions connected with physics." (Joseph J Thomson, "Eclipse Showed Gravity Variation: Hailed as Epochmaking", The New York Times, 1919)

"If the modern conception of the atom is correct the barrier which separated physics from chemistry has been removed." (Sir Joseph J Thomson, "The Electron in Chemistry", 1923)

"A great discovery is not a terminus, but an avenue leading to regions hitherto unknown. We climb to the top of the peak and find that it reveals to us another higher than any we have yet seen, and so it goes on. The additions to our knowledge of physics made in a generation do not get smaller or less fundamental or less revolutionary, as one generation succeeds another. The sum of our knowledge is not like what mathematicians call a convergent series […] where the study of a few terms may give the general properties of the whole. Physics corresponds rather to the other type of series called divergent, where the terms which are added one after another do not get smaller and smaller, and where the conclusions we draw from the few terms we know, cannot be trusted to be those we should draw if further knowledge were at our disposal." (Sir Joseph J Thomson)

On Convergence III (Trivia)

"It is the principle of necessity towards which, as to their ultimate centre, all the ideas advanced in this essay immediately converge. In abstract theory the limits of this necessity are determined solely by considerations of man’s proper nature as a human being; but in the application we have to regard, in addition, the individuality of man as he actually exists. This principle of necessity should, I think, prescribe the grand fundamental rule to which every effort to act on human beings and their manifold relations should be invariably conformed. For it is the only thing which conducts to certain and unquestionable results. The consideration of the useful, which might be opposed to it, does not admit of any true and unswerving decision." (Wilhelm Von Humboldt, "The Limits of State Action", 1792)

"The more man inquires into the laws which regulate the material universe, the more he is convinced that all its varied forms arise from the action of a few simple principles. These principles themselves converge, with accelerating force, towards some still more comprehensive law to which all matter seems to be submitted. Simple as that law may possibly be, it must be remembered that it is only one amongst an infinite number of simple laws: that each of these laws has consequences at least as extensive as the existing one, and therefore that the Creator who selected the present law must have foreseen the consequences of all other laws." (Charles Babbage, "Passages From the Life of a Philosopher", 1864)

"The more progress physical sciences makes, the more they tend to enter the domain of mathematics, which is a kind of center to which they all converge. We may even judge of the degree of perfection to which a science has arrived by the facility with which it may be submitted to calculation." (Adolphe Quetelet, "Annual Report of the Board of Regents of the Smithsonian Institution", 1874)

"Historical investigation not only promotes the understanding of that which now is, but also brings new possibilities before us, by showing that which exists to be in great measure conventional and accidental. From the higher point of view at which different paths of thought converge we may look about us with freer vision and discover routes before unknown." (Ernst Mach, "The Science of Mechanics", 1883)

"This is the reason why mechanical explanations are better understood than stories, even though they are more difficult to reproduce. The exposition, even if it is faulty, excites analogous schemas already existing in the listener’s mind; so that what takes place is not genuine understanding, but a convergence of acquired schemas of thought. In the case of stories, this convergence is not possible, and the schemas brought into play are usually divergent." (Jean Piaget, "The Language and Thought of the Child", 1926)

"No science has ever been born on a specific day. Each science emerges out of a convergence of an increased interest in some class of problems and the development of scientific methods, techniques, and tools which are adequate to solve these problems." (C West Churchman, "Introduction to Operations Research", 1957)

"Science, philosophy and religion are bound to converge as they draw nearer to the whole." (Pierre T de Chardin, "The Phenomenon of Man", 1959)

"Relativity is inherently convergent, though convergent toward a plurality of centers of abstract truths. Degrees of accuracy are only degrees of refinement and magnitude in no way affects the fundamental reliability, which refers, as directional or angular sense, toward centralized truths. Truth is a relationship." (R Buckminster Fuller, "The Designers and the Politicians", 1962)

"Knowledge is not a series of self-consistent theories that converges toward an ideal view; it is rather an ever increasing ocean of mutually incompatible (and perhaps even incommensurable) alternatives, each single theory, each fairy tale, each myth that is part of the collection forcing the others into greater articulation and all of them contributing, via this process of competition, to the development of our consciousness." (Paul K Feyerabend, "Against Method: Outline of an Anarchistic Theory of Knowledge", 1975)

"Today the world we see outside and the world we see within are converging. This convergence of two worlds is perhaps one of the important cultural events of our age." (Ilya Prigogine, "The Philosophy of Instability", Futures 21 (4), 1989)

"Scientists have discovered many peculiar things, and many beautiful things. But perhaps the most beautiful and the most peculiar thing that they have discovered is the pattern of science itself. Our scientific discoveries are not independent isolated facts; one scientific generalization finds its explanation in another, which is itself explained by yet another. By tracing these arrows of explanation back toward their source we have discovered a striking convergent pattern - perhaps the deepest thing we have yet learned about the universe." (Steven Weinberg, "Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature", 1992)

"A great discovery is not a terminus, but an avenue leading to regions hitherto unknown. We climb to the top of the peak and find that it reveals to us another higher than any we have yet seen, and so it goes on. The additions to our knowledge of physics made in a generation do not get smaller or less fundamental or less revolutionary, as one generation succeeds another. The sum of our knowledge is not like what mathematicians call a convergent series […] where the study of a few terms may give the general properties of the whole. Physics corresponds rather to the other type of series called divergent, where the terms which are added one after another do not get smaller and smaller, and where the conclusions we draw from the few terms we know, cannot be trusted to be those we should draw if further knowledge were at our disposal." (Sir Joseph J Thomson) 

On Convergence II (Systems)

"The system becomes more coherent as it is further extended. The elements which we require for explaining a new class of facts are already contained in our system. Different members of the theory run together, and we have thus a constant convergence to unity. In false theories, the contrary is the case." (William Whewell, "Philosophy of the Inductive Sciences", 1840)

"The power and beauty of stochastic approximation theory is that it provides simple, easy to implement gain sequences which guarantee convergence without depending (explicitly) on knowledge of the function to be minimized or the noise properties. Unfortunately, convergence is usually extremely slow. This is to be expected, as 'good performance' cannot be expected if no (or very little) knowledge of the nature of the problem is built into the algorithm. In other words, the strength of stochastic approximation (simplicity, little a priori knowledge) is also its weakness." (Fred C Scweppe, "Uncertain dynamic systems", 1973)

"When loops are present, the network is no longer singly connected and local propagation schemes will invariably run into trouble. [...] If we ignore the existence of loops and permit the nodes to continue communicating with each other as if the network were singly connected, messages may circulate indefinitely around the loops and process may not converges to a stable equilibrium. […] Such oscillations do not normally occur in probabilistic networks […] which tend to bring all messages to some stable equilibrium as time goes on. However, this asymptotic equilibrium is not coherent, in the sense that it does not represent the posterior probabilities of all nodes of the network." (Judea Pearl, "Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference", 1988)

"Regarding stability, the state trajectories of a system tend to equilibrium. In the simplest case they converge to one point (or different points from different initial states), more commonly to one (or several, according to initial state) fixed point or limit cycle(s) or even torus(es) of characteristic equilibrial behaviour. All this is, in a rigorous sense, contingent upon describing a potential, as a special summation of the multitude of forces acting upon the state in question, and finding the fixed points, cycles, etc., to be minima of the potential function. It is often more convenient to use the equivalent jargon of 'attractors' so that the state of a system is 'attracted' to an equilibrial behaviour. In any case, once in equilibrial conditions, the system returns to its limit, equilibrial behaviour after small, arbitrary, and random perturbations." (Gordon Pask, "Different Kinds of Cybernetics", 1992)

"Systems, acting dynamically, produce (and incidentally, reproduce) their own boundaries, as structures which are complementary (necessarily so) to their motion and dynamics. They are liable, for all that, to instabilities chaos, as commonly interpreted of chaotic form, where nowadays, is remote from the random. Chaos is a peculiar situation in which the trajectories of a system, taken in the traditional sense, fail to converge as they approach their limit cycles or 'attractors' or 'equilibria'. Instead, they diverge, due to an increase, of indefinite magnitude, in amplification or gain." (Gordon Pask, "Different Kinds of Cybernetics", 1992)

"The description of the evolutionary trajectory of dynamical systems as irreversible, periodically chaotic, and strongly nonlinear fits certain features of the historical development of human societies. But the description of evolutionary processes, whether in nature or in history, has additional elements. These elements include such factors as the convergence of existing systems on progressively higher organizational levels, the increasingly efficient exploitation by systems of the sources of free energy in their environment, and the complexification of systems structure in states progressively further removed from thermodynamic equilibrium." (Ervin László et al, "The Evolution of Cognitive Maps: New Paradigms for the Twenty-first Century", 1993) 

On Convergence I

"To the thought of considering the infinitely great not merely in the form of what grows without limits - and in the closely related form of the convergent infinite series first introduced in the seventeenth century-, but also fixing it mathematically by numbers in the determinate form of the completed-infinite, I have been logically compelled in the course of scientific exertions and attempts which have lasted many years, almost against my will, for it contradicts traditions which had become precious to me; and therefore I believe that no arguments can be made good against it which I would not know how to meet." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"Between mathematicians and astronomers some misunderstanding exists with respect to the meaning of the term 'convergence'. Mathematicians [...] stipulate that a series is convergent if the sum of the terms tends to a predetermined limit even if the first terms decrease very slowly. Conversely, astronomers are in the habit of saying that a series converges whenever the first twenty terms, for example, decrease rapidly even if the following terms might increase indefinitely. [...] Both rules are legitimate; the first for theoretical research and the second for numerical applications. Both must prevail, but in two entirely separate domains of which the boundaries must be accurately defined. Astronomers do not always know these boundaries accurately but rarely exceed them; the approximation with which they are satisfied usually keeps them far on this side of the boundary. In addition, their instinct guides them and, if they are wrong, a check on the actual observation promptly reveals their error [...]" (Henri Poincaré, "New Methods in Celestial Mechanics" ["Les méthodes nouvelles de la mécanique céleste"], 1892)

"Incidentally, naive intuition, which is in large part an inherited talent, emerges unconsciously from the in-depth study of this or that field of science. The word ‘Anschauung’ has not perhaps been suitably chosen. I would like to include here the motoric sensation with which an engineer assesses the distribution of forces in something he is designing, and even that vague feeling possessed by the experienced number cruncher about the convergence of infinite processes with which he is confronted. I am saying that, in its fields of application, mathematical intuition understood in this way rushes ahead of logical thinking and in each moment has a wider scope than the latter " (Felix Klein, "Über Arithmetisierung der Mathematik", Zeitschrift für mathematischen und naturwissen-schaftlichen Unterricht 27, 1896)

"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)

"Mathematics has, of course, given the solution of the difficulties in terms of the abstract concept of converging infinite series. In a certain metaphysical sense this notion of convergence does not answer Zeno’s argument, in that it does not tell how one is to picture an infinite number of magnitudes as together making up only a finite magnitude; that is, it does not give an intuitively clear and satisfying picture, in terms of sense experience, of the relation subsisting between the infinite series and the limit of this series." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"Central to the development of the calculus were the concepts of convergence and limit, and with these concepts at hand it became at last possible to resolve the ancient paradoxes of infinity which had so much intrigued Zeno." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)

"If we know when a sequence approaches a point or, as we say, converges to a point, we can define a continuous mapping from one metric space to another by using the property that a converging sequence is mapped to the corresponding converging sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] the only characteristic property that continuous functions have is that near objects are sent to corresponding near objects, that is, a convergent sequence is mapped to the corresponding convergent sequence. It is reasonable to say that we cannot expect to extract from that property neither numerical consequences, nor a method to extensively study continuity. On the contrary, analytic functions can be represented by equations (precisely speaking, by infinite series). Compared to analytic functions, continuous functions, in general, are difficult to represent explicitly, although they exist as a concept." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

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

"Theoretically, the normal distribution is most famous because many distributions converge to it, if you sample from them enough times and average the results. This applies to the binomial distribution, Poisson distribution and pretty much any other distribution you’re likely to encounter (technically, any one for which the mean and standard deviation are finite)." (Field Cady, "The Data Science Handbook", 2017)

"At the basis of the distance concept lies, for example, the concept of convergent point sequence and their defined limits, and one can, by choosing these ideas as those fundamental to point set theory, eliminate the notions of distance." (Felix Hausdorff)

David Acheson - Collected Quotes

"It is sometimes possible to infer a great deal about a dynamical system simply by finding its equilibrium states and determining which of these are stable to small disturbances and which are unstable. This can also help explain sudden or 'catastrophic' jumps from one state to another as some parameter is gradually varied." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)

"Real dynamical problems typically involve nonlinear differential equations of second order, but these often simplify greatly if we investigate small oscillations about a position of equilibrium. Coupled oscillators are particularly interesting, an early example being the double pendulum, first studied by Euler and Daniel Bernoulli in the 1730s." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)

"Systems which exhibit chaotic oscillations typically do so for some ranges of the relevant parameters but not for others, so one matter of obvious interest is how the chaos appears (or disappears) as one of the parameters is gradually varied." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)

"While calculus is the mathematical key to an understanding of Nature, its roots lie really in problems of geometry." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)

"Differential equations provide, then, some of the deepest links between mathematics and the physical world." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"[…] it is all too easy in mathematics to jump to the wrong conclusion. And it is particularly dangerous to jump to some general conclusion on the basis of a few special cases." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002) 

"So, when trying to solve a problem in mathematics we have to watch out for subtle mistakes, otherwise, we can easily get the wrong solution." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"[…] the branch of mathematics which is most concerned with change is calculus. The key idea of calculus is in fact not so much change itself, but rather the rate at which change occurs." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"This, then, is the essence of chaos: irregular, erratic motion which is extremely sensitive to the initial conditions. […] A hallmark of chaos: two almost imperceptibly different starting conditions lead to two completely different outcomes, within a relatively short space of time." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"This general kind of behaviour, where a gradual change in some parameter can lead to a sudden and unexpected large change in the system as a whole, is known as a catastrophe." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

 "This strange number e pops up – like π – in all sorts of different places in mathematics. And it arises, in particular, in connection with a fundamental question involving the rate at which things change." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"Today, the whole subject of geometry extends way beyond the world of right-angled triangles, circles and so on. There are even branches of the subject in which the ideas of length, angle and area don’t really feature at all. One of these is topology – a sort of rubber-sheet geometry – where a recurring question is whether some geometric object can be deformed ‘smoothly’ into another one." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"While any one of us is fully entitled – of course – to a quite different opinion, this amazing connection between e, i and π is viewed by many mathematicians as, quite simply, the most stunning result in the whole subject … so far." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

On Fractals III

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

"Fractal geometry appears to have created a new category of art, next to art for art’s sake and art for the sake of commerce: art for the sake of science (and of mathematics). [...] The source of fractal art resides in the recognition that very simple mathematical formulas that seem completely barren may in fact be pregnant, so to speak, with an enormous amount of graphic structure. The artist’s taste can only affect the selection of formulas to be rendered, the cropping and the rendering. Thus, fractal art seems to fall outside the usual categories of ‘invention’, ‘discovery’ and ‘creativity’." (Benoît B Mandelbrot, "Fractals and an Art for the Sake of Science", 1989)

"What were the needs that led me to single out a few of these monsters, calling them fractals, to add some of their close or distant kin, and then to build a geometric language around them? The original need happens to have been purely utilitarian. That links exist between usefulness and beauty is, of course, well known. What we call the beauty of a flower attracts the insects that will gather and spread its pollen. Thus the beauty of a flower is useful - even indispensable - to the survival of its species. Similarly, it was the attractiveness of the fractal images that first brought them to the attention of many colleagues and then of a wide world." (Benoît B Mandelbrot, "Fractals and an Art for the Sake of Science", Leonardo [Supplemental Issue], 1989)

"Some fractals come close to qualifying as chaos by being produced by uncomplicated rules while appearing highly intricate and not just unfamiliar in structure. There is, however, one very close liaison between fractality and chaos; strange attractors are fractals." (Edward N Lorenz, "The Essence of Chaos", 1993)

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

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

"Wherever we look in our world the complex systems of nature and time seem to preserve the look of details at finer and finer scales. Fractals show a holistic hidden order behind things, a harmony in which everything affects everything else, and, above all, an endless variety of interwoven patterns. Fractal geometry allows bounded curves of infinite length, as well as closed surfaces with infinite area. It even allows curves with positive volume and arbitrarily large groups of shapes with exactly the same boundary." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"Fractals' simultaneous chaos and order, self-similarity, fractal dimension and tendency to scalability distinguish them from any other mathematically drawable figures previously conceived." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)

"One of the most important artistic properties of fractals is the randomness  governing the process of making them. Each fractal is essentially generated by a basic formula and one or more gradients that identify the colors of the fractal. Sometimes, however, fractals are generated by tens of different formulas and gradients." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)

"The concept of infinity embedded in fractals' identity provides  an infinity of possibilities to explore in  a single image. The repetition of a formula is the key to becoming more familiar with it. When trying a completely new formula, all fractal artists are engaged in the same activity - a random playing  around." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)

Magic in Mathematics II

 "In mathematics, it’s the limitations of a reasoned argument with the tools you have available, and with magic it’s to use your tools and sleight of hand to bring about a certain effect without the audience knowing what you’re doing. [...]When you’re inventing a trick, it’s always possible to have an elephant walk on stage, and while the elephant is in front of you, sneak something under your coat, but that’s not a good trick. Similarly with mathematical proof, it is always possible to bring out the big guns, but then you lose elegance, or your conclusions aren’t very different from your hypotheses, and it’s not a very interesting theorem." (Persi Diaconis, 2008)

"A genuine experience of the unexpected, in maths as much as in magic, demands of its performer at once originality of insight and a lightness of touch. Even a single step too many in a method renders ugly and clumsy the theorem or the trick." (Daniel Tammet, "Thinking in Numbers", 2012)

"The barrier to an appreciation of mathematical beauty is not insurmountable, however. […] The beauty adored by mathematicians can be pursued through the everyday: through games, and music, and magic." (Daniel Tammet, "Thinking in Numbers" , 2012)

"Good statistics are not a trick, although they are a kind of magic. Good statistics are not smoke and mirrors; in fact, they help us see more clearly. Good statistics are like a telescope for an astronomer, a microscope for a bacteriologist, or an X-ray for a radiologist. If we are willing to let them, good statistics help us see things about the world around us and about ourselves - both large and small - that we would not be able to see in any other way." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

"Pure mathematics is the magician's real wand." (Friederich von Hardenberg [Novalis])

"[Arithmetic] is another of the great master-keys of life. With it the astronomer opens the depths of the heavens; the engineer, the gates of the mountains; the navigator, the pathways of the deep. The skillful arrangement, the rapid handling of figures, is a perfect magician's wand." (Edward Everett)

Magic in Science I

"Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite. " (Ezra Pound, "The Spirit of Romance", 1910)

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

"In any case there is an intense modern interest in machines that imitate life. The great difference between magic and the scientific imitation of life is that where the former is content to copy external appearance, the latter is concerned more with performance and behavior." (William G Walter," An imitation of life", 1950)

"Man masters nature not by force but by understanding. That is why science has succeeded where magic failed: because it has looked for no spell to cast on nature." (Jacob Bronowski, "Science and Human Values", 1956)

"Thus science must begin with myths, and with the criticism of myths; neither with the collection of observations, nor with the invention of experiments, but with the critical discussion of myths, and of magical techniques and practices." (Karl Popper, "Conjectures and Refutations: The Growth of Scientific Knowledge", 1963)

"The truth is more magical - in the best and most exciting sense of the word - than any myth or made-up mystery or miracle. Science has its own magic: the magic of reality." (Richard Dawkins, "The Magic of Reality: How We Know What's Really True", 2011)

Magic in Mathematics I

"Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skillful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures."(Johann F Herbart, 1890)

"The mystery that clings to numbers, the magic of numbers, may spring from this very fact, that the intellect, in the form of the number series, creates an infinite manifold of well distinguishable individuals. Even we enlightened scientists can still feel it e.g. in the impenetrable law of the distribution of prime numbers." (Hermann Weyl, "Philosophy of Mathematics and Natural Science", 1927)

"Mathematics is not a compendium or memorizable formula and magically manipulated figures." (Scott Buchanan, "Poetry and Mathematics", 1929)

"There seems to be striking similarities between the role of economic statistics in our society and some of the functions which magic and divination play in primitive society." (Ely Devons, "Essays in Economics", 1929)

"[…] the social scientist who lacks a mathematical mind and regards a mathematical formula as a magic recipe, rather than as the formulation of a supposition, does not hold forth much promise. A mathematical formula is never more than a precise statement. It must not be made into a Procrustean bed - and that is what one is driven to by the desire to quantify at any cost. It is utterly implausible that a mathematical formula should make the future known to us, and those who think it can, would once have believed in witchcraft. The chief merit of mathematicization is that it compels us to become conscious of what we are assuming." (Bertrand de Jouvenel, "The Art of Conjecture", 1967)

"Symbols, formulae and proofs have another hypnotic effect. Because they are not immediately understood, they, like certain jokes, are suspected of holding in some sort of magic embrace the secret of the universe, or at least some of its more hidden parts." (Scott Buchanan, "Poetry and Mathematics", 1975)

"Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God." (Malba Tahan & Patricia R Baquero, "The Man Who Counted", 1993)

"Number theory [...] is a field of almost pristine irrelevance to everything except the wondrous demonstration that pure numbers, no more substantial than Plato's shadows, conceal magical laws and orders that the human mind can discover after all." (Sharon Begley, "New Answer for an Old Question", Newsweek, 1993)

"In fact, mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that - by some mysterious agency - capture patterns of the universe around us?" (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

On Teaching (1975-1999)

"A central problem in teaching mathematics is to communicate a reasonable sense of taste - meaning often when to, or not to, generalize, abstract, or extend something you have just done." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"Experience without theory teaches nothing." (William E Deming, "Out of the Crisis", 1986)

"One of the lessons that the history of mathematics clearly teaches us is that the search for solutions to unsolved problems, whether solvable or unsolvable, invariably leads to important discoveries along the way. (Carl B Boyer & Uta C Merzbach, "A History of Mathematics", 1991)

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

"Whoever teaches learns in the act of teaching, and whoever learns teaches in the act of learning." (Paulo Freire, "Pedagogy of Freedom", 1996)

"A teacher who cannot explain any abstract subject to a child does not himself thoroughly understand his subject; if he does not attempt to break down his knowledge to fit the child's mind, he does not understand teaching." (Fulton J Sheen, "Life Is Worth Living", 1999)

On Teaching (1950-1974)

"It has been said, often enough and certainly with good reason, that teaching mathematics affords a unique opportunity to teach demonstrative reasoning. I wish to add that teaching mathematics also affords an excellent opportunity to teach plausible reasoning. A student of mathematics should learn, of course, demonstrative reasoning; it is his profession and the distinctive mark of his science. Yet he should also learn plausible reasoning; this is the kind of reasoning on which his creative work will mainly depend, The general student should get a taste of demonstrative reasoning; he may have little opportunity to use it directly, but he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life. He needs, however, in all his endeavors plausible reasoning. At any rate, an ambitious teacher of mathematics should teach both kinds of reasoning to both kinds of students." (George Pólya, "On Plausible Reasoning", Proceedings of the International Congress of Mathematics, 1950)

"At bottom, the society of scientists is more important than their discoveries. What science has to teach us here is not its techniques but its spirit: the irresistible need to explore." (Jacob Bronowski, "Science and Human Values", 1956)

"Mathematical examination problems are usually considered unfair if insoluble or improperly described: whereas the mathematical problems of real life are almost invariably insoluble and badly stated, at least in the first balance. In real life, the mathematician's main task is to formulate problems by building an abstract mathematical model consisting of equations, which will be simple enough to solve without being so crude that they fail to mirror reality. Solving equations is a minor technical matter compared with this fascinating and sophisticated craft of model-building, which calls for both clear, keen common-sense and the highest qualities of artistic and creative imagination." (John Hammersley & Mina Rees, "Mathematics in the Market Place", The American Mathematical Monthly 65, 1958) 

"The diagrams and circles aid the understanding by making it easy to visualize the elements of a given argument. They have considerable mnemonic value […] They have rhetorical value, not only arousing interest by their picturesque, cabalistic character, but also aiding in the demonstration of proofs and the teaching of doctrines. It is an investigative and inventive art. When ideas are combined in all possible ways, the new combinations start the mind thinking along novel channels and one is led to discover fresh truths and arguments, or to make new inventions. Finally, the Art possesses a kind of deductive power." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"The world of today demands more mathematical knowledge on the part of more people than the world of yesterday and the world of tomorrow will demand even more. It is therefore important that mathematics be taught in a vital and imaginative way which will make students aware that it is a living, growing subject which plays an increasingly important part in the contemporary world." (Edward G Begle, "The School Mathematics Study Group," The Mathematics Teacher 51, 1958)

"We believe that student will come to understand mathematics when his textbook and teacher use unambiguous language and when he is enabled to discover generalizations by himself." (Max Beberman, "An Emerging Program of Secondary School Mathematics", 1958)

"The first [principle], is that a mathematical theory can only he developed axiomatically in a fruitful way when the student has already acquired some familiarity with the corresponding material - a familiarity gained by working long enough with it on a kind of experimental, or semiexperimental basis, i.e. with constant appeal to intuition. The other principle [...]  is that when logical inference is introduced in some mathematical question, it should always he presented with absolute honesty - that is, without trying to hide gaps or flaws in the argument; any other way, in my opinion, is worse than giving no proof at all." (Jean Dieudonné, "Thinking in School Mathematics", 1961)

"But both managed to understand mathematics and to make a 'fair' number of contributions to the subject. Rigorous proof is not nearly so important as proving the worth of what we are teaching; and most teachers, instead of being concerned about their failure to be sufficiently rigorous, should really be concerned about their failure to provide a truly intuitive approach.. The general principle, then, is that the rigor should be suited to the mathematical age of the student and not to the age of mathematics." (Morris Kline, "Mathematics: A Cultural Approach", 1962) 

"Creativity is the heart and soul of mathematics at all levels. The collection of special skills and techniques is only the raw material out of which the subject itself grows. To look at mathematics without the creative side of it, is to look at a black-and-white photograph of a Cezanne; outlines may be there, but everything that matters is missing." (Robert C Buck "Teaching Machines and Mathematics Programs",  The American Mathematical Monthly 69, 1962) 

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

"Teaching is more difficult than learning because what teaching calls for is this: to let learn. The real teacher, in fact, let nothing else be learned than learning. His conduct, therefore, often produces the impression that we properly learn nothing from him, if by ‘learning’ we now suddenly understand merely the procurement of useful information." (Martin Heidegger, "What is called thinking?", 1968)

"Many teachers and textbook writers have never recognized the power of sheer intellectual curiosity as a motive for the highest type of work in mathematics, and as a consequence they have failed to organize and present the work in a manner designed to stimulate the student’s interest through a challenge to his curiosity." (Charles H Butler & F Lynwood Wren, "The Teaching of Secondary Mathematics" 5th Ed., 1970)

Out of Context: Aim of Mathematics

"[…] mathematics is not, never was, and never will be, anything more than a particular kind of language, a sort of shorthand of thought and reasoning. The purpose of it is to cut across the complicated meanderings of long trains of reasoning with a bold rapidity that is unknown to the mediaeval slowness of the syllogisms expressed in our words." (Charles Nordmann, "Einstein and the Universe", 1922)

"Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means." (Hermann Weyl, "The Open World: Three Lectures In the Metaphysical Implications of Science", 1932)

"Just as mathematics aims to study such entities as numbers, functions, spaces, etc., the subject matter of metamathematics is mathematics itself." (Frank C DeSua, "Mathematics: A Non-Technical Exposition", American Scientist, 1954)

"There are at least four fundamental purposes that the study of mathematics should attain. First, it should serve as a functional tool in solving our individual everyday problems. [...] In the second place, mathematics serves as a handmaiden for the explanation of the quantitative situations in other subjects, such as economics, physics, navigation, finance, biology, and even the arts. [...] In the third place, mathematics, when properly conceived, becomes a model for thinking, for developing scientific structure, for drawing conclusions, and for solving problems. [...] In the fourth place, mathematics is the best describer of the universe about us." (Howard F Fehr,  "Reorientation in Mathematics Education", Teachers Record 54, 1953) 

"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)

On Teaching (1925-1949)

"Science is a magnificent force, but it is not a teacher of morals. It can perfect machinery, but it adds no moral restraints to protect society from the misuse of the machine. It can also build gigantic intellectual ships, but it constructs no moral rudders for the control of storm tossed human vessel. It not only fails to supply the spiritual element needed but some of its unproven hypotheses rob the ship of its compass and thus endangers its cargo." (William J Bryan, "Undelivered Trial Summation Scopes Trial", 1925)

"The primary purposes of the teaching of mathematics should be to develop those powers of understanding und analyzing relations of quantity and of space which are necessary to an insight into and a control over our environment and to an appreciation of the progress of civilization its various aspects, and to develop those habits of thought and of action which will make those powers in effective in the life of the individual." (J W Young [Ed] The Reorganization of Mathematics in Secondary Education, 1927)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"Before teachers can properly correlate mathematics with other fields, they ought to learn how to correlate the various parts of mathematics. They should first learn how and where arithmetic and informal geometry can be correlated, how and where algebra can be best correlated with arithmetic and informational geometry, and so on. Unless we can do this, there is small chance that we can successfully correlate mathematics with science, music, the arts, and other applied fields." (W. D. Reeve "Mathematics and the Integrated Program in Secondary Schools", Teachers College Record 36, 1935) 

"Many teachers and textbook writers have never recognized the power of sheer intellectual curiosity as a motive for the highest type of work in mathematics. and as a consequence they have failed to organize and present the work in a manner designed to stimulate the student's interest through a challenge to his curiosity." (Charles H Butler & E Lynwood Wren, "The Teaching of Secondary Mathematics, 1941)

"One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles." (George Pólya, "How to Solve It", 1945)

"The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach." (George Pólya, "How to solve it", 1945) 

"When a student makes really silly blunders or is exasperatingly slow, the trouble is almost always the same; he has no desire at all to solve the problem, even no desire to understand it properly, and so he has not understood it. Therefore, a teacher wishing seriously to help the student should. first of all, stir up his curiosity, give him some desire to solve the problem. The teacher should also allow some time to the student to make up his mind to settle down to his task. Teaching to solve problems is education of the will. Solving problems which are not too easy for him, the student learns to persevere through success, to appreciate small advance, to wait for the essential idea, to concentrate with all his might when it appears, If the student had no opportunity in school to familiarize himself with the varying emotions of the struggle for the solution his mathematical education failed in the most vital point." (George Pólya, "How to Solve It", 1945) 

"Unfortunately, the mechanical way in which calculus sometimes is taught fails to present the subject as the outcome of a dramatic intellectual struggle which has lasted for twenty-five hundred years or more, which is deeply rooted in many phases of human endeavors and which will continue as long as man strives to understand himself as well as nature. Teachers, students, and scholars who really want to comprehend the forces and appearances of science must have some understanding of the present aspect of knowledge as a result of historical evolution." (Richard Curand [forward to Carl B Boyer’s "The History of the Calculus and Its Conceptual Development", 1949])

On Teaching (1900-1924)

"The chief end of mathematical instruction is to develop certain powers of the mind, and among these the intuition is not the least precious. By it the mathematical world comes in contact with the real world, and even if pure mathematics could do without it, it would always be necessary to turn to it to bridge the gulf between symbol and reality. The practician will always need it, and for one mathematician there are a hundred practicians. However, for the mathematician himself the power is necessary, for while we demonstrate by logic, we create by intuition; and we have more to do than to criticize others’ theorems, we must invent new ones, this art, intuition teaches us." (Henri Poincaré, "The Value of Science", 1905)

"The aims in teaching geometry should he, according to my views: (1)That the pupil should acquire an accurate thorough knowledge of geometrical truths. 2. That he should develop the power o! original, logical, geometrical reasoning. 3. That he should acquire a habit of thought which will give him a practical sagacity; which will develop his judgment, increase his resourcefulness, and fit him to cope more successfully with the many and varied problems of his after life; which will teach him to rake a many-sided view of things, that if the avenue of attack is blocked, he should able to promptly, cheerfully and successfully attack from another." (W E Bond, "The Aims in Teaching Geometry and HOW to Attain Them", The Mathematics Teacher, 1908) [source]

"There is no science which teaches the harmonies of nature more clearly than mathematics […]." , (William Andrews, "Magic Squares and Cubes", 1908)

"It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction." (Felix Klein, "Lectures on Mathematics", 1911)

"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used. In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)

"To humanize the teaching of mathematics means so to present the subject, so to interpret its ideas and doctrines, that they shall appeal, not merely to the computatory faculty or to the logical faculty but to all the great powers and interests of the human mind." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"To come very near to a true theory, and to grasp its precise application, are two very different things, as the history of a science teaches us. Everything of importance has been said before by somebody who did not discover it." (Alfred N Whitehead, "The Organization of Thought", 1917)

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

"No student ought to complete a course in mathematics without the feeling that there must be something in it, without catching a glimpse, however fleeting, of its possibilities, without at least a few moments of pleasure in achievement and insight." (Helen A Merrill, 'Why Students Fail in Mathematics", The Mathematics Teacher, 1918) [source]

"Most teachers waste their time by asking questions which are intended to discover what a pupil does not know whereas the true art of questioning has for its purpose to discover what the pupil knows or is capable of knowing." (Albert Einstein, 1920)

"[…] teachers are simply your guides. You yourselves must do the travelling." (William J Gies, "Research in Destiny", 1921)

"Our work is great in the classroom it we feel the nobility of that work, if we love the human souls we work with more than the division of fractions, if we love our subject so much that we make our pupils love it, and if we remember that our duty to the world is to help fix in the minds of our pupils the facts of number that they must have in after life." (David E Smith, "The Progress of Arithmetic", 1923) 

"We have come to believe that a pupil in school should feel that he is living his own life naturally. with a minimum of restraint and without tasks that are unduly irksome; that he should find his way through arithmetic largely hoy his own spirit of curiosity; and that he should be directed in arithmetic as he would he directed in any other game, - not harshly driven, hardly even led, but proceeding with the feeling that he is being accompanied and that he is doing his share in finding the way." (David E Smith, "The Progress of Arithmetic", 1923)

Roger J Boscovich - Collected Quotes

"Especially when we investigate the general laws of Nature, induction has very great power; & there is scarcely any other method beside it for the discovery of these laws. By its assistance, even the ancient philosophers attributed to all bodies extension, figurability, mobility, & impenetrability; & to these properties, by the use of the same method of reasoning, most of the later philosophers add inertia & universal gravitation. Now, induction should take account of every single case that can possibly happen, before it can have the force of demonstration; such induction as this has no place in establishing the laws of Nature. But use is made of an induction of a less rigorous type ; in order that this kind of induction may be employed, it must be of such a nature that in all those cases particularly, which can be examined in a manner that is bound to lead to a definite conclusion as to whether or no the law in question is followed, in all of them the same result is arrived at; & that these cases are not merely a few. Moreover, in the other cases, if those which at first sight appeared to be contradictory, on further & more accurate investigation, can all of them be made to agree with the law; although, whether they can be made to agree in this way better than in any Other whatever, it is impossible to know directly anyhow. If such conditions obtain, then it must be considered that the induction is adapted to establishing the law." (Roger J Boscovich, "De Lege Continuitatis" ["On the law of continuity"], 1754)

"Any point has a real mode of existence, through which it is where it is; & another, due to which it exists at the time when it does exist. These real modes of existence are to me real time & space ; the possibility of these modes, hazily apprehended by us, is, to my mind, empty space & again empty time, so to speak ; in other words, space & imaginary time." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Further we believe that GOD Himself is present everywhere throughout the whole of the undoubtedly divisible space that all bodies occupy; & yet He is onefold in the highest degree & admits not of any composite nature whatever. Moreover, the same idea seems to depend on an analogy between space & time. For, just as rest is a conjunction with a continuous series of all the instants In the interval of time during which the rest endures; so also this virtual extension is a conjunction of one instant of time with a continuous series of all the points of space throughout which this one-fold entity extends virtually. Hence, just as rest is believed to exist in Nature, so also are we bound to admit virtual extension; & if this is admitted, then it will be possible for the primary elements of matter to be simple, & yet not absolutely non-extended." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Hence I acknowledge continuity in motion only, which is something successive and not co-existent ; & also in it alone, or because of it alone, in corporeal entities at any rate, lies my reason for admitting the Law of Continuity. From this it will be all the more clear that, as I remarked above, Nature accurately observes the Law of Continuity, or at least tries to do so. Nature observes it in motions & in distance, & tries to in many other cases, with which continuity, as we have defined it above, is in no wise in agreement; also in certain other cases, in which continuity cannot be completely obtained. This continuity does not present itself to us at first sight, unless we consider the subjects somewhat more deeply & study them closely." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Hence the whole of geometry is imaginary; but the hypothetical propositions that are deduced from it are true, if the conditions assumed by it exist, & also the conditional things deduced from them, in every case; & the relations between the imaginary distances of points, derived by the help of geometry from certain conditions, will always be real, & such as they are found to be by geometry, when those conditions exist for real distances of points." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"If matter is continuous, it may & must be subject to infinite divisibility; but actual division carried on indefinitely brings in its train difficulties that are truly inextricable; however, this infinite division is required by those who do not admit that there are any particles, no matter how small, in bodies that are perfectly free from, & incapable of, compression." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"In the same way, this should also happen with regard to time, namely, that between a preceding continuous time & the next following there should be a single instant, which is the indivisible boundary of either. There cannot be two instants, as we intimated above, contiguous to one another; but between one instant & another there must always intervene some interval of continuous time divisible indefinitely. In the same way, in any quantity which lasts for a continuous interval of time, there must be obtained a series of magnitudes of such a kind that to each instant of time there is its corresponding magnitude; & this magnitude connects the one that precedes with the one that follows it, & differs from the former by some definite magnitude. Nay even in that class of quantities, in which we cannot have two magnitudes at the same time, this very point can be deduced far more clearly, namely, that there cannot be any sudden change from one to another. For at that instant, when the sudden change should take place, & the series be broken by some momentary definite addition, two -magnitudes would necessarily be obtained, namely, the last of the first series & the first of the next. Now this very point is still more clearly seen in those states of things, in which on the one hand there must be at any instant some state so that at no time can the thing be without some state of the kind, whilst on the other hand it can never have two states of the kind simultaneously." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

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

"The theory of non-extension is also convenient for eliminating from Nature all idea of a coexistent continuum — to explain which philosophers have up till now laboured so very hard & generally in vain. Assuming non-extension, no division of a real entity can be carried on indefinitely ; we shall not be brought to a standstill when we seek to find out whether the number of parts that are actually distinct & separable is finite or infinite ; nor with it will there come in any of those other truly innumerable difficulties that, with the idea of continuous composition, have given so much trouble- to philosophers. For if the primary elements of matter are perfectly non-extended & indivisible points separated from one another by some definite interval, then the number of points in any given mass must bc finite ; because all the distances are finite." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"(1) There is absolutely no argument that can be brought forward to prove that matter has continuous extension, that it is not rather made up of perfectly indivisible points separated from one another by a definite interval ; nor is there any reason apart from prejudice in favour of continuous extension in preference to composition from points that are perfectly indivisible, non-extended, forming no extended continuum of any sort. (2) There are arguments, & fairly strong ones too, which will prove that this composition from indivisible points is preferable to continuous extension." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"There really must be, in the commencement of contact, in that indivisible instant of time which is an indivisible limit between the continuous time that preceded the contact & that subsequent to it (just in the same way as a point in geometry is an indivisible limit between two segments of a continuous line), a change of velocity taking place suddenly, without any passage through intermediate stages; & this violates the Law of Continuity, which absolutely denies the possibility of a passage from one magnitude to another without passing through intermediate stages." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

On Complex Numbers XIX

"At this point it may be useful to observe that a certain type of intellect is always worrying itself and others by discussion as to the applicability of technical terms. Are the incommensurable numbers properly called numbers? Are the positive and negative numbers really numbers? Are the imaginary numbers imaginary, and are they numbers?-are types of such futile questions. Now, it cannot be too clearly understood that, in science, technical terms are names arbitrarily assigned, like Christian names to children. There can be no question of the names being right or wrong. They may be judicious or injudicious; for they can sometimes be so arranged as to be easy to remember, or so as to suggest relevant and important ideas. But the essential principle involved was quite clearly enunciated in Wonderland to Alice by Humpty Dumpty, when he told her, apropos of his use of words, 'I pay them extra and make them mean what I like.' So we will not bother as to whether imaginary numbers are imaginary, or as to whether they are numbers, but will take the phrase as the arbitrary name of a certain mathematical idea, which we will now endeavour to make plain." (Alfred N Whitehead, "Introduction to Mathematics", 1911)

"It is a curious fact that the first introduction of the imaginaries occurred in the theory of cubic equations, in the case where it was clear that real solutions existed though in an unrecognisable form, and not in the theory of quadratic equations, where our present textbooks introduce them." (Dirk J Struik, A Concise History of Mathematics", 1948)

"There are many useful connections between these two disciplines [geometry and algebra]. Many applications of algebra to geometry and of geometry to algebra were known in antiquity; nearer to our time there appeared the important subject of analytical geometry, which led to algebraic geometry, a vast and rapidly developing science, concerned equally with algebra and geometry. Algebraic methods are now used in projective geometry, so that it is uncertain whether projective geometry should be called a branch of geometry or algebra. In the same way the study of complex numbers, which arises primarily within the bounds of algebra, proved to be very closely connected with geometry; this can be seen if only from the fact that geometers, perhaps, made a greater contribution to the development of the theory than algebraists." (Isaak M Yaglom, "Complex Numbers in Geometry", 1968)

"Besides being essential in modern physics, the complex-number field provides pure mathematics with a multitude of brain-boggling theorems. It is worth keeping in mind that complex numbers, although they include the reals.as a subset, differ from real numbers in startling ways. One cannot, for example, speak of a complex number as being either positive or negative: those properties apply only to the reals and the pure imaginaries. It is equally meaningless to say that one complex number is larger or smaller than another." (Martin Gardner, "Fractal Music, Hypercards and More... Mathematical Recreations from Scientific American Magazine", 1992)

"The seemingly preposterous assumption that there is a square root of -1 was justified on pragmatic grounds: it simplified certain calculations and so could be used as long as 'real' values were obtained at the end. The parallel with the rules for using negative numbers is striking. If you are trying to determine how many cows there are in a field (that is, if you are working in the domain of positive integers), you may find negative numbers useful in the calculation, but of course the final answer must be in terms of positive numbers because there is no such thing as a negative cow." (Martin Gardner, "Fractal Music, Hypercards and More... Mathematical Recreations from Scientific American Magazine", 1992)

"Likewise, complex functions are actually better behaved than real functions, and the subject of complex analysis is known for its regularity and order, while real analysis is known for wildness and pathology A smooth complex function is predictable, in the sense that the values of the function in an arbitrarily small region determine its values everywhere. A smooth real function can be completely unpredictable for example, it can be constantly zero for a long interval, then smoothly change to the value 1." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"The word 'complex' was introduced m a well-meaning attempt to dispel the mystery surrounding 'imaginary' or 'impossible' numbers, and (presumably) because two dimensions are more complex than one Today, 'complex' no longer seems such a good choice of word. It is usually interpreted as 'complicated', and hence is almost as prejudicial as its predecessors. Why frighten people unnecessarily? If you are not sure what 'analysis' is, you won't want to know about 'complex analysis' - but it is the best part of analysis." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Complex numbers do not fit readily into many people’s schema for ‘number’, and students often reject the concept when it is first presented. Modern mathematicians look at the situation with the aid of an enlarged schema in which the facts make sense." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"When we extend the system of natural numbers and counting to embrace infinite cardinals, the larger system need not have all of the properties of the smaller one. However, familiarity with the smaller system leads us to expect certain properties, and we can become confused when the pieces don’t seem to fit. Insecurity arose when the square of a complex number violated the real number principle that all squares are positive. This was resolved when we realised that the complex numbers cannot be ordered in the same way as their subset of reals." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Much of the final resistance to complex numbers faded as it became clear that their behavior posed no threat to the rules and operations of algebra. On the contrary, quite often the complex realm opened paths that made already existing results easier to prove." (David Perkins, "φ, π, e & i", 2017)

David Perkins - Collected Quotes

"A function acts like a set of rules for turning some numbers into others, a machine with parts that we can manipulate to accomplish anything we can imagine." (David Perkins, "Calculus and Its Origins", 2012)

"Despite its deductive nature, mathematics yields its truths much like any other intellectual pursuit: someone asks a question or poses a challenge, others react or propose solutions, and gradually the edges of the debate are framed and a vocabulary is built." (David Perkins, "Calculus and Its Origins", 2012)

"If we wish the word ‘continuous’ to prohibit jumps in a function, its definition must somehow control the vertical change of the function at a sort of microscopic level. That is, at any point on a ‘continuous’ function, the nearby points ought to be as ‘close’ as possible." (David Perkins, "Calculus and Its Origins", 2012)

"Mathematicians approach problems the way rock climbers do cliffs: the more difficult the pitch, the more exhilarating the ascent. After a climb has been solved, others look for new routes, or try equipment that no one else has used, simply for the joy of pioneering." (David Perkins, "Calculus and Its Origins", 2012)

"One trick to seeing beauty in mathematics is to nurture this sense of 'odd as it may seem' while at the same time understanding the subject well enough to know that oddities arise despite our attempts to set the subject on a simple, straightforward footing." (David Perkins, "Calculus and Its Origins", 2012)

"Ever since the discovery of irrational numbers fractured the Greek belief that all numbers were proportions, mathematicians have sorted numbers into categories and hunted for numbers that defied existing categories." (David Perkins, "φ, π, e & i", 2017)

"Imagine that each proof in this book is like a painting that one sees upon entering a gallery full of artwork, in which each work presents an artist’s unique vision of the same theme." (David Perkins, "φ, π, e & i", 2017) 

"Mathematicians linger on cherished topics, illuminating them from a variety of viewpoints, much like artists and poets try over and over to capture truths about the human condition. Re-proving something important in a new way brings joy to both the discoverer and the audience." (David Perkins, "φ, π, e & i", 2017)

"Much of the final resistance to complex numbers faded as it became clear that their behavior posed no threat to the rules and operations of algebra. On the contrary, quite often the complex realm opened paths that made already existing results easier to prove." (David Perkins, "φ, π, e & i", 2017)

Johann I Bernoulli - Collected Quotes

"A quantity diminished or enlarged by an infinitely smaller quantity is neither diminished nor enlarged." (Johann Bernoulli, cca 1691-92)

"Since the nature of differentials […] consists in their being infinitely small and infinitely changeable up to zero, in being only quantitates evanescentes, evanescentia divisibilia, they will be always smaller than any given quantity whatsoever. In fact, some difference which one can assign between two magnitudes which only differ by a differential, the continuous and imperceptible variability of that infinitely small differential, even at the very point of becoming zero, always allows one to find a quantity less than the proposed difference." (Johann Bernoulli, cca. 1692–1702)

"Here, we call function of a variable magnitude, a quantity formed in whatever manner with that variable magnitude and constants." (Johann I Bernoulli, 1718)

"If nature could pass from one extremity to another, for example, from rest to movement, from movement to rest, or from a movement in one direction to a movement in the opposite direction, without passing through all the imperceptible movements that lead from the one to the other; the first state must be destroyed, without nature knowing to which new state it must become; for in the end by what reason should one be chosen for preference, and of which one could not ask why this one and not that one? since having no necessary connection between these two states, no passage from movement to rest, from rest to movement, or from a movement [in one direction] to a movement in an opposite direction; no reason at all will determine producing one thing rather than any other." (Johann Bernoulli, "Discours sur les Loix de la Communication du Mouvement", 1727)

"In fact, a similar principle of hardness cannot exist; it is a chimera which offends that general law which nature constantly observes in all its operations; I speak of that immutable and perpetual order, established since the creation of the Universe, that can be called the LAW OF CONTINUITY, by virtue of which everything that takes place, takes place by infinitely small degrees. It seems that common sense dictates that no change can take place at a jump; natura non operatur per saltion; nothing can pass from one extreme to the other without passing through all the degrees in between." (Johann Bernoulli, "Discours sur les Loix de la Communication du Mouvement", 1727)

"But just as much as it is easy to find the differential of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not." (Johann I Bernoulli) [attributed to]