George Sarton - Collected Quotes

"The more science enters into our lives, the more it must be 'humanized', and there is no better way to humanize it than to study its history." (George Sarton, "An Institute for the History of Science and Civilization", Science Vol. 40 (1100), 1917)

"From the point of view of the history of science, transmission is as essential as discovery." (George Sarton, "Introduction to the History of Science" Vol. 2, 1927)

"Mysteries which we have driven outside of the boundaries of our knowledge and which we have located and encompassed, such mysteries will not harm us; on the contrary they will stimulate and inspire us in many ways; the dangerous mysteries are those which are hopelessly mingled with our knowledge, and of which we are perhaps unaware." (George Sarton, "The History of Science and the New Humanism", 1928)

"Science is neither philosophy, nor religion, nor art; it is the totality of positive knowledge, as closely knit as possible; it is as different from its practical applications on the one hand, as it is from idle theorizing and blind faith on the other. It behooves us to make no extravagant claims for it, and to be as humble as we can." (George Sarton, "The History of Science and the New Humanism", 1928)

"Science tends to destroy the darkness where evil and injustice breed, but there is also some element of beauty and poetry in that darkness." (George Sarton, "The History of Science and the New Humanism", 1928)

"Science, like art and religion - neither more nor less - is a form of man's reaction against nature. It is an attempt to explain nature in its own terms, that is, to evidence its unity, wholeness, and congruency." (George Sarton, "The History of Science and the New Humanism", 1928)

"The study of history, and especially of the history of science, may thus be regarded, not only as a source of wisdom and humanism, but also as a regulator for our consciences: it helps us not to be complacent, arrogant, too sanguine of success, and yet remain grateful and hopeful, and never to cease working quietly for the accomplishment of our own task." (George Sarton, "The History of Science and the New Humanism", 1928)

"Mathematicians and other scientists, however great they may be, do not know the future. Their genius may enable them to project their purpose ahead of them; it is as if they had a special lamp, unavailable to lesser men, illuminating their path; but even in the most favorable cases the lamp sends only a very small cone of light into the infinite darkness." (George Sarton, "The Study of the History of Mathematics", 1936)

"Mathematics gives to science its innermost unity and cohesion, which can never be entirely replaced with props and buttresses or with roundabout connections, no matter how many of these may be introduced." (George Sarton, "The Study of the History of Mathematics", 1936)

"The concatenations of mathematical ideas are not divorced from life, far from it, but they are less influenced than other scientific ideas by accidents, and it is perhaps more possible, and more permissible, for a mathematician than for any other man to secrete himself in a tower of ivory." (George Sarton, "The Study of the History of Mathematics", 1936)

"The main source of mathematical invention seems to be within man rather than outside of him: his own inveterate and insatiable curiosity, his constant itching for intellectual adventure; and likewise the main obstacles to mathematical progress seem to be also within himself; his scandalous inertia and laziness, his fear of adventure, his need of conformity to old standards, and his obsession by mathematical ghosts." (George Sarton, "The Study of the History of Mathematics", 1936)

"The history of science is the only history which can illustrate the progress of mankind. In fact, progress has no definite and unquestionable meaning in fields other than the fields of science." (George Sarton,"The Study of The History of Science", 1936)

"The great intellectual division of mankind is not along geographical or racial lines, but between those who understand and practice the experimental method and those who do not understand and do not practice it." (George Sarton, "A History of Science", 1948)

"A deed happens in a definite place at a definite time, but if it be sufficiently great and pregnant, its virtue radiates everywhere in time and space." (George Sarton, "A History of Science" Vol. 2, 1959)

"Men of science have made abundant mistakes of every kind; their knowledge has improved only because of their gradual abandonment of ancient errors, poor approximations, and premature conclusions." (George Sarton, "A History of Science" Vol. 2, 1959)

"The main duty of the historian of mathematics, as well as his fondest privilege, is to explain the humanity of mathematics, to illustrate its greatness, beauty and dignity, and to describe how the incessant efforts and accumulated genius of many generations have built up that magnificent monument, the object of our most legitimate pride as men, and of our wonder, humility, and thankfulness, as individuals. The study of the history of mathematics will not make better mathematicians but gentler ones, it will enrich their minds, mellow their hearts, and bring out their finer qualities." (George Sarton, The American Mathematical Monthly, Vol. 102, No. 4, 1995)

Arthur Koestler - Collected Quotes

"The ultimate truth is penultimately always a falsehood. He who will be proved right in the end appears to be wrong and harmful before it." (Arthur Koestler, "Darkness at Noon", 1940)

"Artists treat facts as stimuli for the imagination, while scientists use their imagination to coordinate facts." (Arthur Koestler, "Insight and Outlook: An Inquiry into the Common Foundations of Science, Art and Social Ethics", 1949)

"Men cannot be treated as units in operations of political arithmetic because they behave like the symbols for zero and the infinite, which dislocate all mathematical operations." (Arthur Koestler, "Crossman", 1949)

"If time is treated in modern physics as a dimension on a par with the dimensions of space, why should we a priori exclude the possibility that we are pulled as well as pushed along its axis? The future has, after all, as much or as little reality as the past, and there is nothing logically inconceivable in introducing, as a working hypothesis, an element of finality, supplementary to the element of causality, into our equations. It betrays a great lack of imagination to believe that the concept of “purpose” must necessarily be associated with some anthropomorphic deity." (Arthur Koestler, "The Sleepwalkers: A History of Man’s Changing Vision of the Universe", 1959)

"Myths grow like crystals, according to their own, recurrent pattern; but there must be a suitable core to start their growth." (Arthur Koestler, "The Sleepwalkers: A History of Man’s Changing Vision of the Universe", 1959)

"We can add to our knowledge, but we cannot subtract from it." (Arthur Koestler, "The Sleepwalkers: A History of Man’s Changing Vision of the Universe", 1959) 

"It has been said that discovery consists in seeing an analogy which nobody had seen before." (Arthur Koestler, "The Act of Creation", 1964) 

"The moment of truth, the sudden emergence of new insight, is an act of intuition. Such intuitions give the appearance of miraculous flashes, or short circuits of reasoning. In fact they may be likened to an immersed chain, of which only the beginning and the end are visible above the surface of consciousness. The diver vanishes at one end of the chain and comes up at the other end, guided by invisible links." (Arthur Koestler, "The Act of Creation", 1964)

"The principle mark of genius is not perfection, but originality, the opening of new frontiers; once this is done, the conquered territory becomes common property." (Arthur Koestler, "The Act of Creation", 1964)

"True creativity often starts where language ends." (Arthur Koestler, "The Act of Creation", 1964)

"Without the hard little bits of marble which are called 'facts' or 'data' one cannot compose a mosaic; what matters, however, are not so much the individual bits, but the successive patterns into which you arrange them, then break them up and rearrange them." (Arthur Koestler, "The Act of Creation", 1964)

"Creative activity could be described as a type of learning process where teacher and pupil are located in the same individual." (Arthur Koestler, "Drinkers of Infinity: Essays 1955-1967", 1967)

"The progress of science is strewn, like an ancient desert trail, with the bleached skeletons of discarded theories which once seemed to possess eternal life." (Arthur Koestler, "The Ghost in the Machine", 1967)

"Creativity in science could be described as the art of putting two and two together to make five. In other words, it consists in combining previously unrelated mental structures in such a way that you get more out of the emergent whole than you have put in." (Arthur Koestler, "Janus: A Summing Up", 1978)

"The more original a discovery, the more obvious it seems afterwards." (Arthur Koestler)

Beyond the History of Mathematics IV

"[…] we are far from having exhausted all the applications of analysis to geometry, and instead of believing that we have approached the end where these sciences must stop because they  have reached the limit of the forces of the human spirit, we ought to avow rather we are only at the first steps of an immense career. These new [practical] applications, independently of the utility which they may have in themselves, are necessary to the progress of analysis in general; they give birth to questions which one would not think to propose; they demand that one create new methods. Technical processes are the children of need; one can say the same for the methods of the most abstract sciences. But we owe the latter to the needs of a more noble kind, the need to discover the new truths or to know better the laws of nature." (Nicolas de Condorcet, 1781)

"It would be difficult and rash to analyze the chances which the future offers to the advancement of mathematics; in almost all its branches one is blocked by insurmountable difficulties; perfection of detail seems to be the only thing which remains to be done. All of these difficulties appear to announce that the power of our analysis is practically exhausted." (Jean B J Delambre, "Rapport historique sur le progres des sciences mathematiques depuis 1789 et leur etat actuel, 1808)

"The progress of mathematics has been most erratic, and [...] intuition has played a predominant role in it. [...] It was the function of intuition to create new forms; it was the acknowledged right of logic to accept or reject these new forms, in whose birth it had no part. [...] the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied." (Tobias Dantzig, "Number: The Language of Science", 1930)

"The constructions of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow-mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth. Everybody who looks at the spectacle of modern algebra will be struck by this complementarity of freedom and necessity." (Hermann Weyl, "A Half-Century of Mathematics", The American Mathematical Monthly, 1951)

“There is a real role here for the history of mathematics - and the history of number in particular - for history emphasizes the diversity of approaches and methods which are possible and frees us from the straightjacket of contemporary fashions in mathematics education. It is, at the same time, both interesting and stimulating in its own right.” (Graham Flegg, “Numbers: Their History and Meaning”, 1983)

"[…] calling upon the needs of rigor to explain the development of mathematics constitutes a circular argument. In actual fact, new standards of rigor are formed when the old criteria no longer permit an adequate response to questions that arise in mathematical practice or to problems that are in a certain sense external to mathematics. When these are treated mathematically, they compel changes in the theoretical framework of mathematics. It is thus not by chance that mathematical physics and applied mathematics have generally been formidable stimuli to the development of pure mathematics." (Umberto Bottazzini, "The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass", 1986)

"The main duty of the historian of mathematics, as well as his fondest privilege, is to explain the humanity of mathematics, to illustrate its greatness, beauty and dignity, and to describe how the incessant efforts and accumulated genius of many generations have built up that magnificent monument, the object of our most legitimate pride as men, and of our wonder, humility, and thankfulness, as individuals. The study of the history of mathematics will not make better mathematicians but gentler ones, it will enrich their minds, mellow their hearts, and bring out their finer qualities." (George Sarton, The American Mathematical Monthly, Vol. 102, No. 4, 1995)

"Experience has taught most mathematicians that much that looks solid and satisfactory to one mathematical generation stands a fair chance of dissolving into cobwebs under the steadier scrutiny of the next." (Eric T Bell)

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David Berlinski - Collected Quotes

"The body of mathematics to which the calculus gives rise embodies a certain swashbuckling style of thinking, at once bold and dramatic, given over to large intellectual gestures and indifferent, in large measure, to any very detailed description of the world. It is a style that has shaped the physical but not the biological sciences, and its success in Newtonian mechanics, general relativity and quantum mechanics is among the miracles of mankind. But the era in thought that the calculus made possible is coming to an end. Everyone feels this is so and everyone is right." (David Berlinski, "A Tour of the Calculus", 1995)

"Yet everything has a beginning, everything comes to an end, and if the universe actually began in some dense explosion, thus creating time and space, so time and space are themselves destined to disappear, the measure vanishing with the measured, until with another ripple running through the primordial quantum field, something new arises from nothingness once again." (David Berlinski, "A Tour of the Calculus", 1995)

"The motion of the mind is conveyed along a cloud of meaning. There is this paradox that we get to meaning only when we strip the meaning from symbols." (David Berlinski, "The Advent of the Algorithm: The Idea that Rules the World", 2000)

"A definition in mathematics is an exercise in uncovering the essence of things, one reason that good definitions are so hard to pull off, since a definition brings the essence to light, and the light brings the definition to life." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"A five-dimensional space is not a strange deformation of ordinary space, one that only mathematicians can see, but a place where numbers are collected in ordered sets. When string theorists talk of the eleven dimensions required by their latest theory, they are not encouraging one another to search for eight otherwise familiar spatial dimensions that have somehow become lost. They are saying only that for their purposes, eleven numbers are needed to specify points. Where they are is no one’s business." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005) 

"A group is a collection of objects, one that is alive in the sense that some underlying principle of productivity is at work engendering new members from old. […] Like many other highly structured objects, groups have parts, and in particular they may well have subgroups as parts, one group nested within a large group, kangarette to kangaroo." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"A proof in mathematics is an argument and so falls under the controlling power of logic itself. […] Within mathematics, a proof is an intellectual structure in which premises are conveyed to their conclusions by specific inferential steps. Assumptions in mathematics are called axioms, and conclusions theorems. This definition may be sharpened a little bit. A proof is a finite series of statements such that every statement is either an axiom or follows directly from an axiom by means of tight, narrowly defined rules. The mathematician’s business is to derive theorems from his axioms; if his system has been carefully constructed, a gross cascade of theorems will flow from a collection of carefully chosen axioms." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Beyond the theory of complex numbers, there is the much greater and grander theory of the functions of a complex variable, as when the complex plane is mapped to the complex plane, complex numbers linking themselves to other complex numbers. It is here that complex differentiation and integration are defined. Every mathematician in his education studies this theory and surrenders to it completely. The experience is like first love." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"But like every profound mathematical idea, the concept of a group reveals something about the nature of the world that lies beyond the mathematician’s symbols. […] There is […] a royal road between group theory and the most fundamental processes in nature. Some groups represent- they are reflections of - continuous rotations, things that whiz around and around smoothly." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"If the method of proof offers the mathematician the prospect of certainty, it is a form of certainty that is itself conditional. A proof, after all, conveys assumptions to conclusions, or axioms to theorems. If the hammer of certainty falls on the theorems, it cannot fall on the axioms with equal force." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Mathematics is insight and invention and the flash of something grasped at once, but it is also something salt-cleaned and stout as a Gothic cathedral." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

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

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

"The calculus is a theory of continuous change - processes that move smoothly and that do not stop, jerk, interrupt themselves, or hurtle over gaps in space and time. The supreme example of a continuous process in nature is represented by the motion of the planets in the night sky as without pause they sweep around the sun in elliptical orbits; but human consciousness is also continuous, the division of experience into separate aspects always coordinated by some underlying form of unity, one that we can barely identify and that we can describe only by calling it continuous." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"There is weirdness in non-Euclidean geometry, but not because of anything that geometers might say about the ordinary fond familiar world in which space is flat, angles sharp, and only curves are curved. Non-Euclidean geometry is an instrument in the enlargement of the mathematician’s self-consciousness, and so comprises an episode in a long, difficult, and extended exercise in which the human mind attempts to catch sight of itself catching sight of itself, and so without end." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"What a wealth of insight Euler’s formula reveals and what delicacy and precision of reasoning it exhibits. It provides a definition of complex exponentiation: It is a definition of complex exponentiation, but the definition proceeds in the most natural way, like a trained singer’s breath. It closes the complex circle once again by guaranteeing that in taking complex numbers to complex powers the mathematician always returns with complex numbers. It justifies the method of infinite series and sums. And it exposes that profound and unsuspected connection between exponential and trigonometric functions; with Euler’s formula the very distinction between trigonometric and exponential functions acquires the shimmer of a desert illusion." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Mathematicians are like pilots who maneuver their great lumbering planes into the sky without ever asking how the damn things stay aloft. […] The computer has in turn changed the very nature of mathematical experience, suggesting for the first time that mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen. [...] The existence and nature of mathematics is a more compelling and far deeper problem than any of the problems raised by mathematics itself." (David Berlinski)

Mathematics through Students' Eyes III

"Finally, students must learn to realize that mathematics is a science with a long history behind it, and that no true insight into the mathematics of the present day can be obtained without some acquaintance with its historical background. In the first-place time gives an additional dimension to one's mental picture both of mathematics as a whole, and of each individual branch." (André Weil, "The Mathematical Curriculum", 1954)

"Mathematics is a model of exact reasoning, an absorbing challenge to the mind, an esthetic experience for creators and some students, a nightmarish experience to other students, and an outlet for the egotistic display of mental power." (Morris Kline, "Mathematics and the Physical World", 1959)

"Formerly, the beginner was taught to crawl through the underbrush, never lifting his eyes to the trees; today he is often made to focus on the curvature of the universe, missing even the earth." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

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

"Students enjoy […] and gain in their understanding of today's mathematics through analyzing older and alternative approaches." (Lucas N H Bunt et al, "The Historical Roots of Elementary Mathematics", 1976)

"Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true." (Wesley R Hamming, "Coding and Information Theory", 1980)

"Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it." (Morris Kline, "Mathematics and the Search for Knowledge", 1985)

"Mathematics is often thought to be difficult and dull. Many people avoid it as much as they can and as a result much of the population is mathematically illiterate. This is in part due to the relative lack of importance given to numeracy in our culture, and to the way that the subject has been presented to students." (Julian Havil , "Gamma: Exploring Euler's Constant", 2003)

"As students, we learned mathematics from textbooks. In textbooks, mathematics is presented in a rigorous and logical way: definition, theorem, proof, example. But it is not discovered that way. It took many years for a mathematical subject to be understood well enough that a cohesive textbook could be written. Mathematics is created through slow, incremental progress, large leaps, missteps, corrections, and connections." (Richard S Richeson, "Eulers Gem: The Polyhedron Formula and the birth of Topology", 2008)

"A mathematical entity is a concept, a shared thought. Once you have acquired it, you have it available, for inspection or manipulation. If you understand it correctly (as a student, or as a professional) your ‘mental model’ of that entity, your personal representative of it, matches those of others who understand it correctly. (As is verified by giving the same answers to test questions.) The concept, the cultural entity, is nothing other than the collection of the mutually congruent personal representatives, the ‘mental models’, possessed by those participating in the mathematical culture." (Reuben Hersh, "Experiencing Mathematics: What Do We Do, when We Do Mathematics?", 2014)

On Inequalities I

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

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

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

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

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

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

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

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

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

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

On Generalization (1920-1929)

"If we are not content with the dull accumulation of experimental facts, if we make any deductions or generalizations, if we seek for any theory to guide us, some degree of speculation cannot be avoided. Some will prefer to take the interpretation which seems to be most immediately indicated and at once adopted as an hypothesis; others will rather seek to explore and classify the widest possibilities which are not definitely inconsistent with the facts. Either choice has its dangers: the first may be too narrow a view and lead progress into a cul-de-sac; the second may be so broad that it is useless as a guide and diverge indefinitely from experimental knowledge." (Sir Arthur S Eddington, "The Internal Constitution of the Stars Observatory", Vol. 43, 1920)

"It is well to be explicit when a positive generalization is made from negative experimental evidence." (Arthur Eddington, "Space, Time and Gravitation: An Outline of the General Relativity", 1920)

"Generalization is the golden thread which binds many facts into one simple description." (Joseph W Mellor, "A Comprehensive Treatise on Inorganic and Theoretical Chemistry", 1922)

"[…] a history of mathematics is largely a history of discoveries which no longer exist as separate items, but are merged into some more modern generalization, these discoveries have not been forgotten or made valueless. They are not dead, but transmuted." (John W N Sullivan, "The History of Mathematics in Europe", 1925)

"Number knows no limitations, either from the side of the infinitely great or from the side of the infinitely small, and the facility it offers for generalization is too great for us not to be tempted by it." (Émile Borel, "Space and Time", 1926)

"[…] the statistical prediction of the future from the past cannot be generally valid, because whatever is future to any given past, is in tum past for some future. That is, whoever continually revises his judgment of the probability of a statistical generalization by its successively observed verifications and failures, cannot fail to make more successful predictions than if he should disregard the past in his anticipation of the future. This might be called the ‘Principle of statistical accumulation’." (Clarence I Lewis, "Mind and the World-Order: Outline of a Theory of Knowledge", 1929)

"The true method of discovery is like the flight of an aeroplane. It starts from the ground of particular observation; it makes a flight in the thin air of imaginative generalization; and it again lands for renewed observation rendered acute by rational interpretation." (Alfred N Whitehead, "Process and Reality", 1929)

"Without doubt, if we are to go back to that ultimate, integral experience, unwarped by the sophistications of theory, that experience whose elucidation is the final aim of philosophy, the flux of things is one ultimate generalization around which we must weave our philosophical system." (Alfred N Whitehead, "Process and Reality: An Essay in Cosmology", 1929)

On Generalization (1930-1949)

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. […] it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)

"It is time, therefore, to abandon the superstition that natural science cannot be regarded as logically respectable until philosophers have solved the problem of induction. The problem of induction is, roughly speaking, the problem of finding a way to prove that certain empirical generalizations which are derived from past experience will hold good also in the future." (Alfred J Ayer, "Language, Truth and Logic", 1936)

"The problem of induction is, roughly speaking, the problem of finding a way to prove that certain empirical generalizations which are derived from past experience will hold good also in the future. There are only two ways of approaching this problem on the assumption that it is a genuine problem, and it is easy to see that neither of them can lead to its solution." (Alfred J Ayer, "Language, Truth, and Logic", 1936)

"The ethos of science involves the functionally necessary demand that theories or generalizations be evaluated in [terms of] their logical consistency and consonance with facts." (Robert K Merton, "Science and the Social Order", Philosophy of Science Vol 5 (3), 1938)

"The former distrust of specialization has been supplanted by its opposite, a distrust of generalization. Not only has man become a specialist in practice, he is being taught that special facts represent the highest form of knowledge." (Richard Weaver, "Ideas have Consequences", 1948)

Georg W F Hegel - Collected Quotes

"Knowledge is only real and can only be set forth fully in the form of science, in the form of system." (Georg W F Hegel, "The Phenomenology of Mind", 1807)

"The evident character of this defective cognition of which mathematics is proud, and on which it plumes itself before philosophy, rests solely on the poverty of its purpose and the defectiveness of its stuff, and is therefore of a kind that philosophy must spurn." (Georg W F Hegel, "The Phenomenology of Spirit", 1807)

"Each of the parts of philosophy is a philosophical whole, a circle rounded and complete in itself. In each of these parts, however, the philosophical Idea is found in a particular specificality or medium. The single circle, because it is a real totality, bursts through the limits imposed by its special medium, and gives rise to a wider circle. The whole of philosophy in this way resembles a circle of circles. The Idea appears in each single circle, but, at the same time, the whole Idea is constituted by the system of these peculiar phases, and each is a necessary member of the organisation." (Georg W F Hegel, "Encyclopedia of the Philosophical Sciences", 1816)

"It is because the method of physics does not satisfy the comprehension that we have to go on further." (Georg W F Hegel, "Encyclopedia of the Philosophical Sciences", 1816)

"Generally speaking, symbol is some form of external existence immediately present to the senses, which, however, is not accepted for its own worth, as it lies before us in its immediacy, but for the wider and more general significance which it offers to our reflection. We may consequently distinguish between two points of view equally applicable to the term: first, the significance, and, second, the mode in which such a significance is expressed. The first is a conception of the mind, or an object which stands wholly indifferent to any particular content; the latter is a form of sensuous existence or a representation of some kind or other" (Georg W F Hegel, "Ästhetik" Vol. 2, 1817)

"An idea is always a generalization, and generalization is a property of thinking. To generalize means to think." (Georg W F Hegel, "The Philosophy of Right", 1820)

"History in general is therefore the development of Spirit in Time, as Nature is the development of the Idea is Space." (Georg W F Hegel, "Lectures on the Philosophy of History", 1837)

"Poetry is the universal art of the spirit which has become free in itself and which is not tied down for its realization to external sensuous material; instead, it launches out exclusively in the inner space and the inner time of ideas and feelings." (G W Friedrich Hegel, "Introduction to Aesthetics", 1842)

"Education is the art of making man ethical." (Georg W F Hegel)

"Motion is the process, the transition of Time into Space and of Space into Time: Matter, on the other hand, is the relation of Space and Time as a peaceful identity." (Georg W F Hegel)

"Music is architecture translated or transposed from space into time; for in music, besides the deepest feeling, there reigns also a rigorous mathematical intelligence." (Georg W F Hegel)

"People who are too fastidious towards the finite never reach actuality, but linger in abstraction, and their light dies away." (Georg W F Hegel)

"Philosophy is by its nature something esoteric, neither made for the mob nor capable of being prepared for the mob." (Georg W F Hegel)

"Truth in philosophy means that concept and external reality correspond." (Georg W F Hegel)

Ernst Mach - Collected Quotes

"The aim of natural science is to obtain connections among phenomena. Theories, however, are like withered leaves, which drop off after having enabled the organism of science to breathe for a time." (Ernst Mach, "Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit", 1871)

"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)

"[…] not only a knowledge of the ideas that have been accepted and cultivated by subsequent teachers is necessary for the historical understanding of a science, but also that the rejected and transient thoughts of the inquirers, nay even apparently erroneous notions, may be very important and very instructive. The historical investigation of the development of a science is most needful, lest the principles treasured up in it become a system of half-understood prescripts, or worse, a system of prejudices." (Ernst Mach, "The Science of Mechanics", 1883)

"Purely mechanical phenomena do not exist […] are abstractions, made, either intentionally or from necessity, for facilitating our comprehension of things. The science of mechanics does not comprise the foundations, no, nor even a part of the world, but only an aspect of it." (Ernst Mach, "The Science of Mechanics", 1883)

"A person who knew the world only through the theatre, if brought behind the scenes and permitted to view the mechanism of the stage’s action, might possibly believe that the real world also was in need of a machine-room, and that if this were once thoroughly explored, we should know all. Similarly, we, too, should beware lest the intellectual machinery, employed in the representation of the world on the stage of thought, be regarded as the basis of the real world." (Ernst Mach, "The Science of Mechanics; a Critical and Historical Account of Its Development", 1893) 

"In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. […] One condition, therefore, though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equilibrium, is thus applied by symmetry." (Ernst Mach, "The Science of Mechanics: A Critical and Historical Account of Its Development", 1893)

"Properly speaking the world is not composed of 'things' as its elements, but colors, tones, pressures, spaces, times, in short what we ordinarily call individual sensations." (Ernst Mach, "The Science of Mechanics", 1893)

"That branch of physics which is at once the oldest and the simplest and which is therefore treated as introductory to other departments of this science, is concerned with the motions and equilibrium of masses. It bears the name of mechanics." (Ernst Mach, "The Science of Mechanics: A Critical and Historical Account of Its Development", 1893)

"The atomic theory plays a part in physics similar to that of certain auxiliary concepts in mathematics: it is a mathematical model for facilitating the mental reproduction of facts. Although we represent vibrations by the harmonic formula, the phenomena of cooling by exponentials, falls by squares of time, etc, no one would fancy that vibrations in themselves have anything to do with circular functions, or the motion of falling bodies with squares." (Ernst Mach, "The Science of Mechanic", 1893)

"In algebra we perform, as far as possible, all numerical operations which are identical in form once for all, so that only a remnant of work is left for the individual case. The use of the signs of algebra and analysis, which are merely symbols of operations to be performed, is due to the observation that we can materially disburden the mind in this way and spare its powers for more important and more difficult duties, by imposing all mechanical operations upon the hand." (Ernst Mach, "The Economical Nature of Physical Enquiry", Popular Scientific Lectures, 1895)

"Strange as it may sound, the power of mathematics rests upon its evasion of all unnecessary thought and on its wonderful saving of mental operation. Even those arrangement-signs which we call numbers are a system of marvelous simplicity and economy. When we employ the multiplication-table in multiplying numbers of several places, and so use the results of old operations of counting instead of performing the whole of each operation anew; when we consult our table of logarithms, replacing and saving thus new calculations by old ones already performed; when we employ determinants instead of always beginning afresh the solution of a system of equations; when we resolve new integral expressions into familiar old integrals; we see in this simply a feeble reflexion of the intellectual activity of a Lagrange or a Cauchy, who, with the keen discernment of a great military commander, substituted new operations for whole hosts of old ones. No one will dispute me when I say that the most elementary as well as the highest mathematics are economically-ordered experiences of counting, put in forms ready for use." (Ernst Mach, "Popular Scientific Lectures", 1895)

"The aim of research is the discovery of the equations which subsist between the elements of phenomena." (Mach Ernst, 1898)

"[…] scientific research is somewhat like unraveling complicated tangles of strings, in which luck is almost as vital as skill and accurate observation." (Ernst Mach, "Knowledge and Error: Sketches on the Psychology of Enquiry", 1905)

"A symbolical representation of a method of calculation has the same significance for a mathematician as a model or a visualisable working hypothesis has for a physicist. The symbol, the model, the hypothesis runs parallel with the thing to be represented. But the parallelism may extend farther, or be extended farther, than was originally intended on the adoption of the symbol. Since the thing represented and the device representing are after all different, what would be concealed in the one is apparent in the other." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)

"Geometry, accordingly, consists of the application of mathematics to experiences concerning space. Like mathematical physics, it can become an exact deductive science only on the condition of its representing the objects of experience by means of schematizing and idealizing concepts." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)

"Physiological, and particularly visual, space appears as a distortion of 'geometrical space when derived from the metrical data of geometrical space. But the properties of continuity and threefold manifoldness are preserved in such a transformation, and all the consequences of these properties may be derived without recourse to physical experience, by our representative powers solely." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)

"Physics shares with mathematics the advantages of succinct description and of brief, compendious definition, which precludes confusion, even in ideas where, with no apparent burdening of the brain, hosts of others are contained." (Ernst Mach)

"Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses." (Ernst Mach)

William C Dampier - Collected Quotes

"[…] we can only study Nature through our senses - that is […] we can only study the model of Nature that our senses enable our minds to construct; we cannot decide whether that model, consistent though it be, represents truly the real structure of Nature; whether, indeed, there be any Nature as an ultimate reality behind its phenomena." (William C Dampier, "The Recent Development of Physical Science", 1904)

"Confronted with the mystery of the Universe, we are driven to ask if the model our minds have framed at all corresponds with the reality; if, indeed, there be any reality behind the image." (Sir William C Dampier, "The Recent Development of Physical Science", 1904)

"The different sciences are not even parts of a whole; they are but different aspects of a whole, which essentially has nothing in it corresponding to the divisions we make; they are, so to speak, sections of our model of Nature in certain arbitrary planes, cut in directions to suit our convenience." (Sir William C Dampier, "The Recent Development of Physical Science", 1904)

"The mind of man, learning consciously and unconsciously lessons of experience, gradually constructs a mental image of its surroundings - as the mariner draws a chart of strange coasts to guide him in future voyages, and to enable those that follow after him to sail the same seas with ease and safety." (William C Dampier, "The Recent Development of Physical Science" , 1904)

"We can only study Nature through our senses – that is […] we can only study the model of Nature that our senses enable our minds to construct; we cannot decide whether that model, consistent though it be, represents truly the real structure of Nature; whether, indeed, there be any Nature as an ultimate reality behind its phenomena." (Sir William C Dampier, "The Recent Development of Physical Science", 1904)

"A false hypothesis, if it serve as a guide for further enquiry, may, at the right stage of science, be as useful as, or more useful than, a truer one for which acceptable evidence is not yet at hand." (William C Dampier, "Science and the Human Mind" , 1912)

"Sometimes the probability in favor of a generalization is enormous, but the infinite probability of certainty is never reached." (William C Dampier, "Science and the Human Mind" , 1912)

"Indeed the intellectual basis of all empirical knowledge may be said to be a matter of probability, expressible only in terms of a bet." (William C Dampier, "A History of Science and its Relations with Philosophy and Religion", 1936) 

"The fundamental concepts of physical science, it is now understood, are abstractions, framed by our mind, so as to bring order to an apparent chaos of phenomena." (Sir William C Dampier, "A History of Science and its Relations with Philosophy and Religion", 1929)

John W N Sullivan - Collected Quotes

"Knowledge for the sake of knowledge, as the history of science proves, is an aim with an irresistible fascination for mankind, and which needs no defense. The mere fact that science does, to a great extent, gratify our intellectual curiosity, is a sufficient reason for its existence." (John W N Sullivan, "The Limitations of Science", 1915)

"[…] science deals with but a partial aspect of reality, and there is no faintest reason for supposing that everything science ignores is less real than what it accepts. [...] Why is it that science forms a closed system? Why is it that the elements of reality it ignores never come in to disturb it? The reason is that all the terms of physics are defined in terms of one another. The abstractions with which physics begins are all it ever has to do with." (John W N Sullivan, "The Limitations of Science", 1915)

"Science deals with but a partial aspect of reality, and […] there is no faintest reason for supposing that everything science ignores is less real than what it accepts. […] Why is it that science forms a closed system? Why is is that the elements of reality it ignores never come in to disturb it? The reason is that all the terms of physics are defined in terms of one another. The abstractions with which physics begins are all it ever has to do with […]" (John W N Sullivan, "The Limitations of Science", 1915)

"Science, like everything else that man has created, exists, of course, to gratify certain human needs and desires. The fact that it has been steadily pursued for so many centuries, that it has attracted an ever-wider extent of attention, and that it is now the dominant intellectual interest of mankind, shows that it appeals to a very powerful and persistent group of appetites." (John W N Sullivan, "The Limitations of Science", 1915)

"Since the primary object of the scientific theory is to express the harmonies which are found to exist in nature, we see at once that these theories must have an aesthetic value. The measure of the success of a scientific theory is, in fact, a measure of its aesthetic value, since it is a measure of the extent to which it has introduced harmony in what was before chaos." (John W N Sullivan, "The Justification of the Scientific Method", The Athenaeum (4644), 1919)

"The measure in which science falls short of art is the measure in which it is incomplete as science." (John W N Sullivan, "The Justification of the Scientific Method", The Athenaeum, 1919)

"[…] a history of mathematics is largely a history of discoveries which no longer exist as separate items, but are merged into some more modern generalization, these discoveries have not been forgotten or made valueless. They are not dead, but transmuted." (John W N Sullivan, "The History of Mathematics in Europe", 1925)

"The electron is not, for example, an enduring something that can be tracked through time. Its mathematical description does not involve that degree of definiteness. Any picture we form of the atom errs, as it were, by excess of solidity. The mathematical symbols refer to entities more indefinite than our pictorial imagination, limited as it is by experience of 'gross matter', can construct." (John W N Sullivan, "The Bases of Modern Science", 1929)

"The present tendency of physics is toward describing the universe in terms of mathematical relations between unimaginable entities." (John W N Sullivan, "The Bases of Modern Science", 1929)

"Science, indeed, tells us a very great deal less about the universe than we have been accustomed to suppose, and there is no reason to believe that all we can ever know must be couched in terms of its thin and largely arbitrary abstractions."  (John W N Sullivan, "Art and Reality", 1964)

"The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God." (John W N Sullivan)

On Simplicity XIII (Complexity vs Simplicity V)

"Cultivate simplicity or rather should I say banish elaborateness, for simplicity springs spontaneous from the heart." (Charles Lamb, [Letter to Coleridge] 1790)

"In the original discovery of a proposition of practical utility, by deduction from general principles and from experimental data, a complex algebraical investigation is often not merely useful, but indispensable; but in expounding such a proposition as a part of practical science, and applying it to practical purposes, simplicity is of the importance: - and […] the more thoroughly a scientific man has studied higher mathematics, the more fully does he become aware of this truth – and […] the better qualified does he become to free the exposition and application of principles from mathematical intricacy." (William J M Rankine, "On the Harmony of Theory and Practice in Mechanics", 1856) 

"The first obligation of Simplicity is that of using the simplest means to secure the fullest effect. But although the mind instinctlvely rejects all needless complexity, we shall greatly err if we fail to recognise the fact, that what the mind recoils from is not the complexity, but the needlessness." (George H Lewes, "The Principles of Success in Literature", 1865)

"Expansion means complexity, and complexity decay." (C Northcote Parkinson, "In-laws and Outlaws", 1962)

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

"All propaganda or popularization involves a putting of the complex into the simple, but such a move is instantly deconstructive. For if the complex can be put into the simple, then it cannot be as complex as it seemed in the first place; and if the simple can be an adequate medium of such complexity, then it cannot after all be as simple as all that." (Terry Eagleton, Against The Grain, 1986) 

"There is no over-arching theory of complexity that allows us to ignore the contingent aspects of complex systems. If something really is complex, it cannot by adequately described by means of a simple theory. Engaging with complexity entails engaging with specific complex systems. Despite this we can, at a very basic level, make general remarks concerning the conditions for complex behaviour and the dynamics of complex systems. Furthermore, I suggest that complex systems can be modelled." (Paul Cilliers," Complexity and Postmodernism", 1998) 

"People who pride themselves on their 'complexity' and deride others for being 'simplistic' should realize that the truth is often not very complicated. What gets complex is evading the truth." (Thomas Sowell, "Barbarians inside the Gates and Other Controversial Essays", 1999) 

"History, as well as life itself, is complicated; neither life nor history is an enterprise for those who seek simplicity and consistency." (Jared Diamond, "Collapse: How Societies Choose to Fail or Succeed", 2005)

"Simplicity in a system tends to increase that system’s efficiency. Because less can go wrong with fewer parts, less will. Complexity in a system tends to increase that system’s inefficiency; the greater the number of variables, the greater the probability of those variables clashing, and in turn, the greater the potential for conflict and disarray. Because more can go wrong, more will. That is why centralized systems are inclined to break down quickly and become enmeshed in greater unintended consequences." (Lawrence K Samuels,"Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013) 

From Parts to Wholes (1-999 AD)

"No species remains constant: that great renovator of matter Nature, endlessly fashions new forms from old: there’s nothing in the whole universe that perishes, believe me; rather it renews and varies its substance. What we describe as birth is no more than incipient change from a prior state, while dying is merely to quit it. Though the parts may be transported hither and thither, the sum of all matter is constant." (Publius Ovidius Naso [Ovid], "Metamorphoses", 8 AD)

"A far greater glory is it to the wise to die for freedom, the love of which stands in very truth implanted in the soul like nothing else, not as a casual adjunct but an essential part of its unity, and cannot be amputated without the whole system being destroyed as a result." (Philo of Alexandria, "Every Good Man is Free",  cca. 15 - 45 AD)

"We can get some idea of a whole from a part, but never knowledge or exact opinion. Special histories therefore contribute very little to the knowledge of the whole and conviction of its truth. It is only indeed by study of the interconnexion of all the particulars, their resemblances and differences, that we are enabled at least to make a general survey, and thus derive both benefit and pleasure from history." (Polybius, "The Histories", cca. 150 BC)

"Either all things proceed from one intelligent source and come together as in one body, and the part ought not to find fault with what is done for the benefit of the whole; or there are only atoms, and nothing else than a mixture and dispersion. Why, then, art thou disturbed? Say to this ruling faculty, Art thou dead, art thou corrupted, art thou playing the hypocrite, art thou become a beast, dost thou herd and feed with the rest?" (Marcus Aurelius, "Meditations". cca. 121–180 AD)

"The other reason is that what happens to the individual is a cause of well-being in what directs the world - of its well-being, its fulfillment, or its very existence, even. Because the whole is damaged if you cut away anything - anything at all - from its continuity and its coherence. Not only its parts, but its purposes. And that's what you're doing when you complain: hacking and destroying." *Marcus Aurelius, "Meditations", cca. 121–180 AD)

"When, therefore, as will be clear to those who read, the passage as a connected whole is literally impossible, whereas the outstanding part of it is not impossible but even true, the reader must endeavor to grasp the entire meaning, connecting by an intellectual process the account of what is literally impossible with the parts that are not impossible but historically true, these being interpreted allegorically in common with the part which, so far as the letter goes, did not happen at all. For our contention with regard to the whole of divine scripture is that it all has a spiritual meaning, but not all a bodily meaning; for the bodily meaning is often proved to be an impossibility." (Origen Adamantius, "On First Principles", cca. 220-230)

From Parts to Wholes (BC)

"Further, the state is by nature clearly prior to the family and to the individual, since the whole is of necessity prior to the part; for example, if the whole body be destroyed, there will be no foot or hand, except in an equivocal sense, as we might speak of a stone hand; for when destroyed the hand will be no better than that. But things are defined by their working and power; and we ought not to say that they are the same when they no longer have their proper quality, but only that they have the same name." (Aristotle, Politics, 4th century BC)

"The infinite […] happens to subsist in a way contrary to what is asserted by others: for the infinite is not that beyond which there is nothing, but it is that of which there is always something beyond. […] But that pertaining to which there is nothing beyond is perfect and whole. […] that of which nothing is absent pertaining to the parts […] the whole is that pertaining to which there is nothing beyond. But that pertaining to which something external is absent, that is not all […] But nothing is perfect which has not an end; and the end is a bound. On this account […] Parmenides spoke better than Melissus: for the latter says that the infinite is a whole; but the former, that the whole is finite, and equally balanced from the middle: for to conjoin the infinite with the universe and the whole, is not to connect line with line." (Aristotle, Physics, cca. 4th century BC)

"The proof that the state is a creation of nature and prior to the individual is that the individual, when isolated, is not self-sufficing; and therefore he is like a part in relation to the whole." (Aristotle, Politics, 4th century BC)

"The totality is not, as it were, a mere heap, but the whole is something besides the parts." (Aristotle, "Metaphysics", cca. 335-323 BC)
 
"The whole is more than the sum of its parts." (Aristotle, "Metaphysics", cca. 335-323 BC)
 
"And the whole [is] greater than the part." (Euclid, "Elements", cca. 300 BC)

"We can get some idea of a whole from a part, but never knowledge or exact opinion. Special histories therefore contribute very little to the knowledge of the whole and conviction of its truth. It is only indeed by study of the interconnexion of all the particulars, their resemblances and differences, that we are enabled at least to make a general survey, and thus derive both benefit and pleasure from history." (Polybius, "The Histories", cca. 150 BC)
 
"I say, then, that the universe and all its parts both received their first order from divine providence, and are at all times administered by it." (Marcus T Cicero, "De Natura Deorum" ["On the Nature of the Gods"], 45 BC)
 
"Order gives due measure to the members of a work considered separately, and symmetrical agreement to the proportions of the whole. It is an adjustment according to quantity. By this I mean the selection of modules from the members of the work itself and, starting from these individual parts of members, constructing the whole work to correspond." (Marcus Vitruvius Pollio, "De architectura" ["On Architecture"], cca. 15 BC)

On Equilibrium (1990-1999)

"Living systems are never in equilibrium. They are inherently unstable. They may seem stable, but they're not. Everything is moving and changing. In a sense, everything is on the edge of collapse. Michael Crichton, "Jurassic Park", 1990)

"Everywhere […] in the Universe, we discern that closed physical systems evolve in the same sense from ordered states towards a state of complete disorder called thermal equilibrium. This cannot be a consequence of known laws of change, since […] these laws are time symmetric- they permit […] time-reverse. […] The initial conditions play a decisive role in endowing the world with its sense of temporal direction. […] some prescription for initial conditions is crucial if we are to understand […]" (John D Barrow, "Theories of Everything: The Quest for Ultimate Explanation", 1991)

"History has so far shown us only two roads to international stability: domination and equilibrium." (Henry Kissinger, [Times] 1991)

"Three laws governing black hole changes were thus found, but it was soon noticed that something unusual was going on. If one merely replaced the words 'surface area' by 'entropy' and 'gravitational field' by 'temperature', then the laws of black hole changes became merely statements of the laws of thermodynamics. The rule that the horizon surface areas can never decrease in physical processes becomes the second law of thermodynamics that the entropy can never decrease; the constancy of the gravitational field around the horizon is the so-called zeroth law of thermodynamics that the temperature must be the same everywhere in a state of thermal equilibrium. The rule linking allowed changes in the defining quantities of the black hole just becomes the first law of thermodynamics, which is more commonly known as the conservation of energy." (John D Barrow, "Theories of Everything: The Quest for Ultimate Explanation", 1991)

"[...] it's essentially meaningless to talk about a complex adaptive system being in equilibrium: the system can never get there. It is always unfolding, always in transition. In fact, if the system ever does reach equilibrium, it isn't just stable. It's dead." (M Mitchell Waldrop, "Complexity: The Emerging Science at the Edge of Order and Chaos", 1992)

"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 key to making discontinuity emerge from smoothness is the observation that the overall behavior of both static and dynamical systems is governed by what's happening near the critical points. These are the points at which the gradient of the function vanishes. Away from the critical points, the Implicit Function Theorem tells us that the behavior is boring and predictable, linear, in fact. So it's only at the critical points that the system has the possibility of breaking out of this mold to enter a new mode of operation. It's at the critical points that we have the opportunity to effect dramatic shifts in the system's behavior by 'nudging' lightly the system dynamics, one type of nudge leading to a limit cycle, another to a stable equilibrium, and yet a third type resulting in the system's moving into the domain of a 'strange attractor'. It's by these nudges in the equations of motion that the germ of the idea of discontinuity from smoothness blossoms forth into the modern theory of singularities, catastrophes and bifurcations, wherein we see how to make discontinuous outputs emerge from smooth inputs." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"The new information technologies can be seen to drive societies toward increasingly dynamic high-energy regions further and further from thermodynamical equilibrium, characterized by decreasing specific entropy and increasingly dense free-energy flows, accessed and processed by more and more complex social, economic, and political structures." (Ervin László, "Information Technology and Social Change: An Evolutionary Systems Analysis", Behavioral Science 37, 1992)

"An equilibrium is defined to be stable if all sufficiently small disturbances away from it damp out in time. Thus stable equilibria are represented geometrically by stable fixed points. Conversely, unstable equilibria, in which disturbances grow in time, are represented by unstable fixed points." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)

"[…] chaos and fractals are part of an even grander subject known as dynamics. This is the subject that deals with change, with systems that evolve in time. Whether the system in question settles down to equilibrium, keeps repeating in cycles, or does something more complicated, it is dynamics that we use to analyze the behavior." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)

"Democracy is the only system capable of reflecting the humanist premise of equilibrium or balance. The key to its secret is the involvement of the citizen." (John R Saul, "The Doubter's Companion", 1994)

"In many parts of the economy, stabilizing forces appear not to operate. Instead, positive feedback magnifies the effects of small economic shifts; the economic models that describe such effects differ vastly from the conventional ones. Diminishing returns imply a single equilibrium point for the economy, but positive feedback – increasing returns – makes for many possible equilibrium points. There is no guarantee that the particular economic outcome selected from among the many alternatives will be the ‘best’ one."  (W Brian Arthur, "Returns and Path Dependence in the Economy", 1994)

"Objects in nature have provided and do provide models for stimulating mathematical discoveries. Nature has a way of achieving an equilibrium and an exquisite balance in its creations. The key to understanding the workings of nature is with mathematics and the sciences. [.] Mathematical tools provide a means by which we try to understand, explain, and copy natural phenomena. One discovery leads to the next." (Theoni Pappas, "The Magic of Mathematics: Discovering the spell of mathematics", 1994)

"The model of competitive equilibrium which has been discussed so far is set in a timeless environment. People and companies all operate in a world in which there is no future and hence no uncertainty." (Paul Ormerod, "The Death of Economics", 1994)

"We need to abandon the economist's notion of the economy as a machine, with its attendant concept of equilibrium. A more helpful way of thinking about the economy is to imagine it as a living organism." (Paul Ormerod, "The Death of Economics", 1994)

"Initially, it may seem that such systems constitute a very special class of processes. And, in fact, that is indeed the case. However, nature has providentially worked things out so that a lot of processes of practical concern just happen to belong to this class - including many of the systems of classical physics like passive electrical circuits, damped vibrating springs, and bending beams. Moreover, when we observe these kinds of processes in real life, what we usually see is the system when it is at or very near to equilibrium. For these reasons catastrophe theory can be of great value in helping us understand how these kinds of systems can shift abruptly from one equilibrium state to another as various parameters, like spring constants or unemployment rates, are varied just a little bit." (John L Casti, "Five Golden Rules", 1995)

"Self-organization refers to the spontaneous formation of patterns and pattern change in open, nonequilibrium systems. […] Self-organization provides a paradigm for behavior and cognition, as well as the structure and function of the nervous system. In contrast to a computer, which requires particular programs to produce particular results, the tendency for self-organization is intrinsic to natural systems under certain conditions." (J A Scott Kelso, "Dynamic Patterns : The Self-organization of Brain and Behavior", 1995)

"The reason catastrophe theory can tell us about such abrupt changes in a system's behavior is that we usually observe a dynamical system when it's at or near its steady-state, or equilibrium, position. And under various assumptions about the nature of the system's dynamical law of motion, the set of all possible equilibrium states is simply the set of critical points of a smooth function closely related to the system dynamics. When these critical points are nondegenerate, Morse's Theorem applies. But it is exactly when they become degenerate that the system can move sharply from one equilibrium position to another. The Thorn Classification Theorem tells when such shifts will occur and what direction they will take." (John L Casti, "Five Golden Rules", 1995)

"Contrary to what happens at equilibrium, or near equilibrium, systems far from equilibrium do not conform to any minimum principle that is valid for functions of free energy or entropy production." (Ilya Prigogine, "The End of Certainty: Time, Chaos, and the New Laws of Nature", 1996) 

"[…] self-organization is the spontaneous emergence of new structures and new forms of behavior in open systems far from equilibrium, characterized by internal feedback loops and described mathematically by nonlinear equations.” (Fritjof Capra, “The web of life: a new scientific understanding of living systems”, 1996)

"Complex systems operate under conditions far from equilibrium. Complex systems need a constant flow of energy to change, evolve and survive as complex entities. Equilibrium, symmetry and complete stability mean death. Just as the flow, of energy is necessary to fight entropy and maintain the complex structure of the system, society can only survive as a process. It is defined not by its origins or its goals, but by what it is doing." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"Financial markets are supposed to swing like a pendulum: They may fluctuate wildly in response to exogenous shocks, but eventually they are supposed to come to rest at an equilibrium point and that point is supposed to be the same irrespective of the interim fluctuations." (George Soros, "The Crisis of Global Capitalism", 1998)

"No one has yet succeeded in deriving the second law from any other law of nature. It stands on its own feet. It is the only law in our everyday world that gives a direction to time, which tells us that the universe is moving toward equilibrium and which gives us a criteria for that state, namely, the point of maximum entropy, of maximum probability. The second law involves no new forces. On the contrary, it says nothing about forces whatsoever." (Brian L Silver, "The Ascent of Science", 1998)

"There has to be a constant flow of energy to maintain the organization of the system and to ensure its survival. Equilibrium is another word for death." (Paul Cilliers, "Complexity and Postmodernism", 1998)

On Spacetime (1800-1849)

"Genius and science have burst the limits of space, and few observations, explained by just reasoning, have unveiled the mechanism of the universe. Would it not also be glorious for man to burst the limits of time, and, by a few observations, to ascertain the history of this world, and the series of events which preceded the birth of the human race?" (Georges Cuvier, "Essays on the Theory of the Earth", 1822)

"History in general is therefore the development of Spirit in Time, as Nature is the development of the Idea is Space." (Georg W F Hegel, "Lectures on the Philosophy of History", 1837)

"Yet time and space are but inverse measures of the force of the soul. The spirit sports with time." (Ralph W Emerson, "Essays", 1841)

"Great distance in either time or space has wonderful power to lull and render quiescent the human mind." (Abraham Lincoln, [An Address Delivered by Abraham Lincoln], 1842)

"Poetry is the universal art of the spirit which has become free in itself and which is not tied down for its realization to external sensuous material; instead, it launches out exclusively in the inner space and the inner time of ideas and feelings." (G W Friedrich Hegel, "Introduction to Aesthetics", 1842)

"Language has time as its element; all other media have space as their element." (Søren Kierkegaard, "Either/Or: A Fragment of Life", 1843)

Jean L R Agassiz - Collected Quotes

"I may say that here, as in most cases where the operations of nature interfere with the designs of man, it is not by a direct intervention on our part that we may remedy the difficulties, but rather by a precise knowledge of [nature’s] causes, which may enable us, if not to check, at least to avoid the evil consequences." (Jean L R Agassiz, "Annual Report of the Superintendent of the Coast Survey, Showing the Progress of that Work During the Year Ending November", 1851)

"As long as men inquire, they will find opportunities to know more upon these topics than those who have gone before them, so inexhaustibly rich is nature in the innermost diversity of her treasures of beauty, order, and intelligence." (Jean L R Agassiz, “Essay on Classification”, 1859)

"Are our systems the inventions of naturalists, or only their reading of the Book of Nature? and can that book have more than one reading? If these classifications are not mere inventions, if they are not an attempt to classify for our own convenience the objects we study, then they are thoughts which, whether we detect them or not, are expressed in Nature, - then Nature is the work of thought, the production of intelligence carried out according to plan, therefore premeditated, - and in our study of natural objects we are approaching the thoughts of the Creator, reading His conceptions, interpreting a system that is His and not ours." (Jean L R Agassiz, "Methods of Study in Natural History", 1863)

"[...] it must be for truth’s sake, and not even for the sake of its usefulness to humanity, that the scientific man studies Nature." (Jean L R Agassiz, "Methods of Study in Natural History", 1863)

"The education of a naturalist now consists chiefly in learning how to compare." (Jean L R Agassiz, "Methods of Study in Natural History", 1863)

"[...] the time has come when scientific truth must cease to be the property of the few, when it must be woven into the common life of the world; for we have reached the point where the results of science touch the very problem of existence, and all men listen for the solving of that mystery." (Jean L R Agassiz, "Methods of Study in Natural History", 1863)

"Philosophers and theologians have yet to learn that a physical fact is as sacred as a moral principle." (Jean L R Agassiz, "Evolution and Permanence of Type", The Atlantic Monthly, 1874)

"Facts are stupid things until brought into connection with some general law." (Jean L R Agassiz)

"Lay aside all conceit. Learn to read the book of nature for yourself. Those who have succeeded best have followed for years some slim thread which has once in a while broadened out and disclosed some treasure worth a life-long search." (Jean L R Agassiz)

"The only true scientific system must be one in which the thought, the intellectual structure, rises out of, and is based upon, facts." (Jean L R Agassiz)

"The study of nature is an intercourse with the highest mind. You should never trifle with nature. At the lowest her works are the works of the highest powers - the highest something, in whatever way we may look at it." (Jean L R Agassiz)

Carl B Boyer - Collected Quotes

"As the sensations of motion and discreteness led to the abstract notions of the calculus, so may sensory experience continue thus to suggest problem for the mathematician, and so may she in turn be free to reduce these to the basic formal logical relationships involved. Thus only may be fully appreciated the twofold aspect of mathematics: as the language of a descriptive interpretation of the relationships discovered in natural phenomena, and as a syllogistic elaboration of arbitrary premise." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"Mathematics is an aspect of culture as well as a collection of algorithms." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"Materialistic and idealistic philosophies have both failed to appreciate the nature of mathematics, as accepted at the present time. Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

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

"[...] mathematics is not free to develop as it will, but is bound by certain restrictions: by conceptions derived either a posteriori from natural science, or assumed to be imposed a priori by an absolutistic philosophy." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"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)

"The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"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", 1976)

Florian Cajori - Collected Quotes

"The best review of arithmetic consists in the study of algebra." (Florian Cajori, "Teaching and History of Mathematics in U. S.", 1896)

"The contemplation of the various steps by which mankind has come into possession of the vast stock of mathematical knowledge can hardly fail to interest the mathematician. He takes pride in the fact that his science, more than any other, is an exact science, and that hardly anything ever done in mathematics has proved to be useless." (Florian Cajori, "A History of Mathematics", 1893)

"The history of mathematics is important […] as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development." (Florian Cajori, "A History of Mathematics", 1893)

"The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store." (Florian Cajori, "A History of Mathematics", 1893)

"This history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of notational problems of the present time." (Florian Cajori, "A History of Mathematical Notations", 1928)

"One of the most baneful delusions by which the minds, not only of students, but even of many teachers of mathematics in our classical colleges, have been afflicted is that mathematics can be mastered by the favored few, but lies beyond the grasp and power of the ordinary mind." (Florian Cajori, "The Teaching and History of Mathematics in the United States", 1890)