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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On Knowledge (1850-1874)

"In every branch of knowledge the progress is proportional to the amount of facts on which to build, and therefore to the facility of obtaining data." (James C Maxwell, [Letter to Lewis Campbell] 1851) 

"Remember that accumulated knowledge, like accumulated capital, increases at compound interest: but it differs from the accumulation of capital in this; that the increase of knowledge produces a more rapid rate of progress, whilst the accumulation of capital leads to a lower rate of interest. Capital thus checks its own accumulation: knowledge thus accelerates its own advance. Each generation, therefore, to deserve comparison with its predecessor, is bound to add much more largely to the common stock than that which it immediately succeeds." (Charles Babbage, "The Exposition of 1851: Or the Views of Industry, Science and Government of England", 1851)

"All knowledge is profitable; profitable in its ennobling effect on the character, in the pleasure it imparts in its acquisition, as well as in the power it gives over the operations of mind and of matter. All knowledge is useful; every part of this complex system of nature is connected with every other. Nothing is isolated. The discovery of to-day, which appears unconnected with any useful process, may, in the course of a few years, become the fruitful source of a thousand inventions." (Joseph Henry, "Report of the Secretary" [Sixth Annual Report of the Board of Regents of the Smithsonian Institution for 1851], 1852)

"SYSTEM (σύστημα, σύν ἵστημιavu, to place together) - is a full and connected view of all the truths of some department of knowledge. An organized body of truth, or truths arranged under one and the same idea, which idea is as the life or soul which assimilates all those truths. No truth is altogether isolated. Every truth has relation to some other. And we should try to unite the facts of our knowledge so as to see them in their several bearings. This we do when we frame them into a system. To do so legitimately we must begin by analysis and end with synthesis. But system applies not only to our knowledge, but to the objects of our knowledge. Thus we speak of the planetary system, the muscular system, the nervous system. We believe that the order to which we would reduce our ideas has a foundation in the nature of things. And it is this belief that encourages us to reduce our knowledge of things into systematic order. The doing so is attended with many advantages. At the same time a spirit of systematizing may be carried too far. It is only in so far as it is in accordance with the order of nature that it can be useful or sound." (William Fleming, "Vocabulary of philosophy, mental, moral, and metaphysical; with quotations and references; for the use of students", 1857)

"Science asks no questions about the ontological pedigree or a priori character of a theory, but is content to judge it by its performance; and it is thus that a knowledge of nature, having all the certainty which the senses are competent to inspire, has been attained - a knowledge which maintains a strict neutrality toward all philosophical systems and concerns itself not with the genesis or a priori grounds of ideas." (Chauncey Wright, "The Philosophy of Herbert Spencer", North American Review, 1865)

"One of the greatest obstacles to the free and universal movement of human knowledge is the tendency that leads different kinds of knowledge to separate into systems." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

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

"As systematic unity is what first raises ordinary knowledge to the rank of science, that is, makes a system out of a mere aggregate of knowledge, architectonic is the doctrine of the scientific in our knowledge, and therefore necessarily forms part of the doctrine of method." (Immanuel Kant, "Critique of Pure Reason", 1871)

"Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress." (William T Kelvin, "Elements of Natural Philosophy", 1873)

"Numerical facts, like other facts, are but the raw materials of knowledge, upon which our reasoning faculties must be exerted in order to draw forth the principles of nature. [...] Numerical precision is the soul of science [...]" (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

On Continuity XIV (Progress)

"Nothing is accomplished all at once, and it is one of my great maxims, and one of the most completely verified, that Nature makes no leaps: a maxim which I have called the law of continuity." (Gottfried W Leibniz, "New Essays Concerning Human Understanding with an Appealing", 1704)

"Harmonious order governing eternally continuous progress - the web and woof of matter and force interweaving by slow degrees, without a broken thread, that veil which lies between us and the Infinite - that universe which alone we know or can know; such is the picture which science draws of the world, and in proportion as any part of that picture is in unison with the rest, so may we feel sure that it is rightly painted." (Thomas H Huxley, "Darwiniana", 1893–94)

"Mathematics is not a world empire where one man can exert a dominating influence over work along all the lines, but it is continually splitting up into self-determining republics." (Geroge A Miller, "Felix Klein and the History of Modern Mathematics", 1927)

"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)

"For it is true, generally speaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers." (David Hilbert, "Anschauliche Geometrie", 1932)

"Science does not mean an idle resting upon a body of certain knowledge; it means unresting endeavor and continually progressing development toward an end which the poetic intuition may apprehend, but which the intellect can never fully grasp." (Max Planck, "The Philosophy of Physics", 1936)

"And yet, on looking into the history of science, one is overwhelmed by evidence that all too often there is no regular procedure, no logical system of discovery, no simple, continuous development. The process of discovery has been as varied as the temperament of the scientist." (Gerald Holton, "Thematic Origins of Scientific Thought: Kepler to Einstein", 1973)

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 Technology III

"Technology means the systematic application of scientific or other organized knowledge to practical tasks." (John K Galbraith, "The New Industrial State", 1967)

"Networks constitute the new social morphology of our societies, and the diffusion of networking logic substantially modifies the operation and outcomes in processes of production, experience, power, and culture. While the networking form of social organization has existed in other times and spaces, the new information technology paradigm provides the material basis for its pervasive expansion throughout the entire social structure." (Manuel Castells, "The Rise of the Network Society", 1996)

"Beauty is more important in computing than anywhere else in technology because software is so complicated. Beauty is the ultimate defense against complexity." (David Gelernter, "Machine Beauty: Elegance And The Heart Of Technolog", 1998)

"Modelling techniques on powerful computers allow us to simulate the behaviour of complex systems without having to understand them.  We can do with technology what we cannot do with science.  […] The rise of powerful technology is not an unconditional blessing.  We have  to deal with what we do not understand, and that demands new  ways of thinking." (Paul Cilliers,"Complexity and Postmodernism: Understanding Complex Systems", 1998)

"A primary reason that evolution - of life-forms or technology - speeds up is that it builds on its own increasing order." (Ray Kurzweil, "The Age of Spiritual Machines: When Computers Exceed Human Intelligence", 1999) 

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

"We do not learn much from looking at a model - we learn more from building the model and manipulating it. Just as one needs to use or observe the use of a hammer in order to really understand its function, similarly, models have to be used before they will give up their secrets. In this sense, they have the quality of a technology - the power of the model only becomes apparent in the context of its use." (Margaret Morrison & Mary S Morgan, "Models as mediating instruments", 1999)

"Periods of rapid change and high exponential growth do not, typically, last long. A new equilibrium with a new dominant technology and/or competitor is likely to be established before long. Periods of punctuation are therefore exciting and exhibit unusual uncertainty. The payoff from establishing a dominant position in this short time is therefore extraordinarily high. Dominance is more likely to come from skill in marketing and positioning than from superior technology itself." (Richar Koch, "The Power Laws", 2000)

"Mythology and science both extend the scope of human beings. Like science and technology, mythology, as we shall see, is not about opting out of this world, but about enabling us to live more intensely within it." (Karen Armstrong, "A Short History Of Myth", 2004)

"In an age when technology is integrating us more tightly together and delivering tremendous flows of innovation, knowledge, connectivity and commerce, the future belongs to those who build webs not walls, who can integrate not separate, to get the most out of these flows." (Thomas L Friedman, The New York Times, 2016)

On Technology I

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

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

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

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

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

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

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

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

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

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

Thomas H Huxley - Collected Quotes

"Living things have no inertia, and tend to no equilibrium." (Thomas H Huxley, "On the Educational Value of the Natural History Sciences", 1854)

"Unity of plan everywhere lies hidden under the mask of diversity of structure - the complex is everywhere evolved out of the simple." (Thomas H Huxley, "A Lobster; or, the Study of Zoology", 1861)

"Science has fulfilled her function when she has ascertained and enunciated truth."  (Thomas H Huxley, "Man's Place in Nature.", 1863)

"The method of scientific investigation is nothing but the expression of the necessary mode of working of the human mind. (Thomas H Huxley, "Our Knowledge of the Causes of the Phenomena of Organic Nature", 1863)

"Mathematical training is almost purely deductive. The mathematician starts with a few simple propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them." (Thomas H Huxley, "Scientific Education: Notes of an After Dinner Speech", Macmillan’s Magazine Vol. XX, 1869)

"The mathematician starts with a few propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the same general nature: authority and tradition furnish the data, and the mental operations are deductive." (Thomas H Huxley, 1869)

"[...] there can be little doubt that the further science advances, the more extensively and consistently will all the phenomena of Nature be represented by materialistic formulae and symbols." (Thomas H Huxley, "On the Physical Basis of Life", 1869)

"[Mathematics] is that [subject] which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1870)

"The great tragedy of Science - the slaying of a beautiful hypothesis by an ugly fact." (Thomas H Huxley, "Biogenesis and abiogenesis", [address] 1870)

"The Mathematician deals with two properties of objects only, number and extension, and all the inductions he wants have been formed and finished ages ago. He is now occupied with nothing but deductions and verification." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1870)

"Therefore, the great business of the scientific teacher is, to imprint the fundamental, irrefragable facts of his science, not only by words upon the mind, but by sensible impressions upon the eye, and ear, and touch of the student, in so complete a manner, that every term used, or law enunciated, should afterwards call up vivid images of the particular structural, or other, facts which furnished the demonstration of the law, or the illustration of the term." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1870) 

"You may read any quantity of books, and you may be almost as ignorant as you were at starting, if you don’t have, at the back of your minds, the change for words in definite images which can only be acquired through the operation of your observing faculties on the phenomena of nature." (Thomas H Huxley, "Science and Education", 1877)

"It sounds paradoxical to say the attainment of scientific truth has been effected, to a great extent, by the help of scientific errors." (Thomas H Huxley, "The Progress of Science", 1887)

"All artificial education ought to be an anticipation of natural education."  (Thomas H Huxley, "Science and Education", 1891)

"All truth, in the long run, is only common sense clarified." (Thomas H Huxley, "Science and Education", 1891)

"Education is the instruction of the intellect in the laws of Nature, under which name I include not merely things and their forces, but men and their ways; and the fashioning of the affections and of the will into an earnest and loving desire to move in harmony with those laws." (Thomas H Huxley, "Science and Education", 1891)

"It is not a question whether one order of study or another should predominate. It is a question of what topics of education you shall elect which will combine all the needful elements in such due proportion as to give the greatest amount of food, support, and encouragement to those faculties which enable us to appreciate truth, and to profit by those sources of innocent happiness which are open to us, and, at the same time, to avoid that which is bad, and coarse, and ugly, and keep clear of the multitude of pitfalls and dangers which beset those who break through the natural or moral laws." (Thomas H Huxley, "Science and Education", 1891)

"Many of the faults and mistakes of the ancient philosophers are traceable to the fact that they knew no language but their own, and were often led into confusing the symbol with the thought which it embodied." (Thomas H Huxley, "Science and Education", 1891)

"There is but one right, and the possibilities of wrong are infinite." (Thomas H Huxley, "Science and Education", 1891)

"Accuracy is the foundation of everything else." (Thomas H Huxley, "Method and Results", 1893)

"Anyone who is practically acquainted with scientific work is aware that those who refuse to go beyond fact, rarely get as far as fact; and anyone who has studied the history of science knows that almost every great step therein has been made by the 'anticipation of Nature.'" (Thomas H Huxley, "Method and Results", 1893)

"The 'Law of Nature' is not a command to do, or to refrain from doing, anything. It contains, in reality, nothing but a statement of that which a given being tends to do under the circumstances of its existence; and which, in the case of a living and sensitive being, it is necessitated to do, if it is to escape certain kinds of disability, pain, and ultimate dissolution." (Thomas H Huxley, "Method and Results", 1893)

"The man of science has learned to believe in justification, not by faith, but by verification." (Thomas H Huxley, "Method and Results", 1893)

"The man of science, who, forgetting the limits of philosophical inquiry, slides from these formulæ and symbols into what is commonly understood by materialism, seems to me to place himself on a level with the mathematician, who should mistake the x's and y's with which he works his problems for real entities - and with this further disadvantage, as compared with the mathematician, that the blunders of the latter are of no practical consequence, while the errors of systematic materialism may paralyse the energies and destroy the beauty of a life." (Thomas H Huxley, "Method and Results", 1893)

"The scientific imagination always restrains itself within the limits of probability." (Thomas H Huxley, "Science and Christian Tradition", 1893)

"Thought is existence. More than that, so far as we are concerned, existence is thought, all our conceptions of existence being some kind or other of thought." (Thomas H Huxley, "Method and Results", 1893)

"Harmonious order governing eternally continuous progress - the web and woof of matter and force interweaving by slow degrees, without a broken thread, that veil which lies between us and the Infinite - that universe which alone we know or can know; such is the picture which science draws of the world, and in proportion as any part of that picture is in unison with the rest, so may we feel sure that it is rightly painted." (Thomas H Huxley, "Darwiniana", 1893–94)

"The method of scientific investigation is nothing but the expression of the necessary mode of working of the human mind. It is simply the mode in which all phenomena are reasoned about, rendered precise and exact." (Thomas H Huxley, "Method and Results", 1893)

"The scientific spirit is of more value than its products, and irrationally held truths may be more harmful than reasoned errors." (Thomas H Huxley, "Darwiniana", 1893–94)

"Whatever happens, science may bide her time in patience and in confidence." (Thomas H Huxley, "Science and Christian Tradition", 1893)

"Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascods, so pages of formulæ will not get a definite result out of loose data." (Thomas H Huxley, "Discourses, Biological and Geological", 1894)

"In the world of letters, learning and knowledge are one, and books are the source of both; whereas in science, as in life, learning and knowledge are distinct, and the study of things, and not of books, is the source of the latter." (Thomas H Huxley, "Discourses, Biological and Geological", 1894)

"Perhaps the most valuable result of all education is the ability to make yourself do the thing you have to do, when it ought to be done, whether you like it or not; it is the first lesson that ought to be learned; and however early a man's training begins, it is probably the last lesson that he learns thoroughly." (Thomas H Huxley)

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)

George Santayana - Collected Quotes

"Symmetry is evidently a kind of unity in variety, where a whole is determined by the rhythmic repetition of similar." (George Santayana, "The Sense of Beauty: Being the Outlines of Aesthetic Theory", 1896)

"The scientific value of truth is not, however, ultimate or absolute. It rests partly on practical, partly on aesthetic interests. As our ideas are gradually brought into conformity with the facts by the painful process of selection, - for intuition runs equally into truth and into error, and can settle nothing if not controlled by experience, - we gain vastly in our command over our environment. This is the fundamental value of natural science." (George Santayana, "The Sense of Beauty: Being the Outlines of Aesthetic Theory", 1896)

"No system would have ever been framed if people had been simply interested in knowing what is true, whatever it may be. What produces systems is the interest in maintaining against all comers that some favourite or inherited idea of ours is sufficient and right. A system may contain an account of many things which, in detail, are true enough; but as a system, covering infinite possibilities that neither our experience nor our logic can prejudge, it must be a work of imagination and a piece of human soliloquy: It may be expressive of human experience, it may be poetical; but how should anyone who really coveted truth suppose that it was true?" (George Santayana, "The Genteel Tradition in American Philosophy", 1911)

"If all the arts aspire to the condition of music, all the sciences aspire to the condition of mathematics." (George Santayana, Some Turns of Thought in Modern Philosophy: Five Essays, 1933)

"[…] mathematics is like music, freely exploring the possibilities of form. And yet, notoriously, mathematics holds true of things; hugs and permeates them far more closely than does confused and inconstant human perception; so that the dream of many exasperated critics of human error has been to assimilate all science to mathematics, so as to make knowledge safe by making it, as Locke wished, direct perception of the relations between ideas […]" (George Santayana, "The Realm of Truth: Book Third of Realms of Being", 1937)

"Science, then, is the attentive consideration of common experience; it is common knowledge extended and refined. Its validity is of the same order as that of ordinary perception; memory, and understanding. Its test is found, like theirs, in actual intuition, which sometimes consists in perception and sometimes in intent." (George Santayana, "The Life of Reason, or the Phases of Human Progress", 1954)

 "Theory helps us to bear our ignorance of facts." (George Santayana)

Hermann von Helmholtz - Collected Quotes

"There is a kind, I might almost say, of artistic satisfaction, when we are able to survey the enormous wealth of Nature as a regularly ordered whole - a kosmos, an image of the logical thought of our own mind." (Hermann von Helmholtz. "On the Conservation of Force", 1862)

"When I have before my eyes a pair of stereoscopic drawings which are hard to combine, it is difficult to bring the lines and points that correspond, to cover each other, and with every little motion of the eyes they glide apart. But if I chance to gain a lively mental image (Anschauungsbild) of the represented solid form (a thing that often occurs by lucky chance), I then move my two eyes with perfect certainty over the figure without the picture separating again." (Hermann von Helmholtz, "Tonempfindungen" ["Sensations of Tone"], 1863)

"Thus representations of the external world are images of the lawlike temporal succession of natural events, and if they are correctly formed in accordance with the laws of our thinking, and we are able correctly to translate them back again into actuality through our actions, then the representations that we have are also the uniquely true [ones] for our faculty of thought; all others would be false." (Hermann von Helmholtz, "Handbuch der physiologischen Optik" Vol. 3, 1867)

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

"A law of nature, however, is not a mere logical conception that we have adopted as a kind of memoria technical to enable us to more readily remember facts. We of the present day have already sufficient insight to know that the laws of nature are not things which we can evolve by any speculative method. On the contrary, we have to discover them in the facts; we have to test them by repeated observation or experiment, in constantly new cases, under ever-varying circumstances; and in proportion only as they hold good under a constantly increasing change of conditions, in a constantly increasing number of cases with greater delicacy in the means of observation, does our confidence in their trustworthiness rise." (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1873)

"It [geometry] escapes the tedious and troublesome task of collecting experimental facts, which is the province of the natural sciences in the strict sense of the word; the sole form of its scientific method is deduction." (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1873)

"A metaphysical conclusion is either a false conclusion or a concealed experimental conclusion." (Hermann von Helmholtz, "On Thought in Medicine", 1877)

"Mathematics and music, the most sharply contrasted fields of scientific activity which can be found, and yet related, supporting each other, as if to show forth the secret connection which ties together all the activities of our mind, and which leads us to surmise that the manifestations of the artist's genius are but the unconscious expressions of a mysteriously acting rationality." (Hermann von Helmholtz, "Vorträge und Reden", Bd. 1, 1884)

"I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain. [...] But the pride I might have held in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had almost always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors. I am fain to compare myself with a wanderer on the mountains who, not knowing the path, climbs slowly and painfully upwards and often has to retrace his steps because he can go no further—then, whether by taking thought or from luck, discovers a new track that leads him on a little till at length when he reaches the summit he finds to his shame that there is a royal road by which he might have ascended, had he only the wits to find the right approach to it. In my works, I naturally said nothing about my mistake to the reader, but only described the made track by which he may now reach the same heights without difficulty." (Hermann von Helmholtz, 1891)

"[In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity." (Hermann Helmholtz, "Vorträge und Reden", 1896)

"In proportion as the range of science extends, its system and organization must be improved, and it must inevitably come about that individual students will find themselves compelled to go through a stricter course of training than grammar is in a position to supply." (Hermann von Helmholtz, "On the Relation of Natural Science to Science in general", Popular Lectures on Scientific Subjects, 1900)

On Paradox II

"Nevertheless, there are three distinct types of paradoxes which do arise in mathematics. There are contradictory and absurd propositions, which arise from fallacious reasoning. There are theorems which seem strange and incredible, but which, because they are logically unassailable, must be accepted even though they transcend intuition and imagination. The third and most important class consists of those logical paradoxes which arise in connection with the theory of aggregates, and which have resulted in a re-examination of the foundations of mathematics." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"The most pervasive paradox of the human condition which we see is that the processes which allow us to survive, grow, change, and experience joy are the same processes which allow us to maintain an impoverished model of the world - our ability to manipulate symbols, that is, to create models. So the processes which allow us to accomplish the most extraordinary and unique human activities are the same processes which block our further growth if we commit the error of mistaking the model of the world for reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"The paradox of reality is that no image is as compelling as the one which exists only in the mind's eye." (Shana Alexander, "Talking Woman", 1976)

 "The world of science lives fairly comfortably with paradox. We know that light is a wave and also that light is a particle. The discoveries made in the infinitely small world of particle physics indicate randomness and chance, and I do not find it any more difficult to live with the paradox of a universe of randomness and chance and a universe of pattern and purpose than I do with light as a wave and light as a particle. Living with contradiction is nothing new to the human being." (Madeline L'Engle, "Two-Part Invention: The Story of a Marriage", 1988)

"Chaos demonstrates that deterministic causes can have random effects […] There's a similar surprise regarding symmetry: symmetric causes can have asymmetric effects. […] This paradox, that symmetry can get lost between cause and effect, is called symmetry-breaking. […] From the smallest scales to the largest, many of nature's patterns are a result of broken symmetry; […]" (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"The best reaction to a paradox is to invent a genuinely new and deep idea." (Ian Hacking, "An Introduction to Probability and Inductive Logic", 2001)

"Chaos theory, for example, uses the metaphor of the ‘butterfly effect’. At critical times in the formation of Earth’s weather, even the fluttering of the wings of a butterfly sends ripples that can tip the balance of forces and set off a powerful storm. Even the smallest inanimate objects sent back into the past will inevitably change the past in unpredictable ways, resulting in a time paradox." (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos", 2004)

"Nature does weird things. It lives on the edge. But it is careful to bob and weave from the fatal punch of logical paradox." (Brian Greene, "The Fabric of the Cosmos: Space, Time, and the Texture of Reality", 2004)

"Paradoxes often arise because theory routinely refuses to be subordinate to reality." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

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On Invention (1975-1999)

"Mathematicians do not agree among themselves whether mathematics is invented or discovered, whether such a thing as mathematical reality exists or is illusory." (Albert L Hammond, "Mathematics - Our invisible culture", 1978)

"For the great majority of mathematicians, mathematics is […] a whole world of invention and discovery - an art. The construction of a new theorem, the intuition of some new principle, or the creation of a new branch of mathematics is the triumph of the creative imagination of the mathematician, which can be compared to that of a poet, the painter and the sculptor." (George F J Temple, "100 Years of Mathematics: a Personal Viewpoint", 1981)

"The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticize and modify as we go along, so that it winds by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […]"(Michael Guillen, "Bridges to Infinity: The Human Side of Mathematics", 1983)

"History is a constant race between invention and catastrophe." (Frank Herbert, "God Emperor of Dune", 1984)

"I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar. Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission." (Carl Eckart, "Our Modern Idol: Mathematical Science", 1984) 

"Theoretical scientists, inching away from the safe and known, skirting the point of no return, confront nature with a free invention of the intellect. They strip the discovery down and wire it into place in the form of mathematical models or other abstractions that define the perceived relation exactly. The now-naked idea is scrutinized with as much coldness and outward lack of pity as the naturally warm human heart can muster. They try to put it to use, devising experiments or field observations to test its claims. By the rules of scientific procedure it is then either discarded or temporarily sustained. Either way, the central theory encompassing it grows. If the abstractions survive they generate new knowledge from which further exploratory trips of the mind can be planned. Through the repeated alternation between flights of the imagination and the accretion of hard data, a mutual agreement on the workings of the world is written, in the form of natural law." (Edward O Wilson, "Biophilia", 1984)

"One cannot ‘invent’ the structure of an object. The most we can do is to patiently bring it to the light of day, with humility - in making it known, it is ‘discovered’. If there is some sort of inventiveness in this work, and if it happens that we find ourselves the maker or indefatigable builder, we are in no sense ‘making’ or ’building’ these ‘structures’. They have not waited for us to find them in order to exist, exactly as they are! But it is in order to express, as faithfully as possible, the things that we have been detecting or discovering, the reticent structure which we are trying to grasp at, perhaps with a language no better than babbling. Thereby are we constantly driven to ‘invent’ the language most appropriate to express, with increasing refinement, the intimate structure of the mathematical object, and to ‘construct’ with the help of this language, bit by bit, those ‘theories’ which claim to give a fair account of what has been apprehended and seen. There is a continual coming and going, uninterrupted, between the apprehension of things, and the means of expressing them by a language in constant state improvement [...]. The sole thing that constitutes the true inventiveness and imagination of the researcher is the quality of his attention as he listens to the voices of things." (Alexander Grothendieck, "Récoltes et semailles –Rélexions et témoignage sur un passé de mathématicien", 1985)

"Because mathematical proofs are long, they are also difficult to invent. One has to construct, without making any mistakes, long chains of assertions, and see what one is doing, see where one is going. To see means to be able to guess what is true and what is false, what is useful and what is not. To see means to have a feeling for which definitions one should introduce, and what the key assertions are that will allow one to develop a theory in a natural manner." (David Ruelle, "Chance and Chaos", 1991)

"There is one qualitative aspect of reality that sticks out from all others in both profundity and mystery. It is the consistent success of mathematics as a description of the workings of reality and the ability of the human mind to discover and invent mathematical truths." (John D Barrow, "Theories of Everything", 1991)

"Ultimately, discovery and invention are both problems of classification, and classification is fundamentally a problem of finding sameness. When we classify, we seek to group things that have a common structure or exhibit a common behavior." (Grady Booch, "Object-oriented design: With Applications", 1991)

"As a result, surprisingly enough, scientific advance rarely comes solely through the accumulation of new facts. It comes most often through the construction of new theoretical frameworks. [..] To understand scientific development, it is not enough merely to chronicle new discoveries and inventions. We must also trace the succession of worldviews" (Nancy R Pearcey & Charles B Thaxton, "The Soul of Science: Christian Faith and Natural Philosophy", 1994)

"Mathematicians tells us that it is easy to invent mathematical theorems which are true, but that it is hard to find interesting ones. In analyzing music or writing its history, we meet the same difficulty, and it is compounded by another." (Charles Rosen, "The Frontiers of Meaning: Three Informal Lectures on Music", 1994)

"The controversy between those who think mathematics is discovered and those who think it is invented may run and run, like many perennial problems of philosophy. Controversies such as those between idealists and realists, and between dogmatists and sceptics, have already lasted more than two and a half thousand years. I do not expect to be able to convert those committed to the discovery view of mathematics to the inventionist view." (Paul Ernst, "Is Mathematics Discovered or Invented", 1996)

On Invention (1850-1899)

"Our inventions are wont to be pretty toys, which distract our attention from serious things. They are but improved means to an unimproved end." (Henry D Thoreau, "Walden ou la vie dans les bois", 1854) 

"They say that each generation inherits from those that have gone before; if this were so there would be no limit to man's improvements or to his power of reaching perfection. But he is very far from receiving intact that storehouse of knowledge which the centuries have piled up before him; he may perfect some inventions, but in others, he lags behind the originators, and a great many inventions have been lost entirely. What he gains on the one hand, he loses on the other." (Eugène Delacroix, 1854) 

"Nevertheless so profound is our ignorance, and so high our presumption, that we marvel when we hear of the extinction of an organic being; and as we do not see the cause, we invoke cataclysms to desolate the world, or invent laws on the duration of the forms of life!" (Charles Darwin, "On the Origin of Species by Means of Natural Selection", 1866)

"As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention." (James J Sylvester, "A plea for the mathematician", Nature Vol. 1, 1870) 

"The culture of the geometric imagination, tending to produce precision in remembrance and invention of visible forms will, therefore, tend directly to increase the appreciation of works of belles-letters." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32, 1875)

"Great inventions are never, and great discoveries are seldom, the work of any one mind. Every great invention is really an aggregation of minor inventions, or the final step of a progression. It is not usually a creation, but a growth, as truly so as is the growth of the trees in the forest." (Robert H Thurston, "The Growth of the Steam Engine", Popular Science, 1877) 

"A discoverer is a tester of scientific ideas; he must not only be able to imagine likely hypotheses, and to select suitable ones for investigation, but, as hypotheses may be true or untrue, he must also be competent to invent appropriate experiments for testing them, and to devise the requisite apparatus and arrangements." (George Gore, "The Art of Scientific Discovery", 1878) 

"Mathematics renders its best service through the immediate furthering of rigorous thought and the spirit of invention."  (Johann F Herbart, "Mathematischer Lehrplan fur Realschulen", 1890)

"The natural world has its laws, and no man must interfere with them in the way of presentment any more than in the way of use; but they themselves may suggest laws of other kinds, and man may, if he pleases, invent a little world of his own, with its own laws; for there is that in him which delights in calling up new forms - which is the nearest, perhaps, he can come to creation. When such forms are new embodiments of old truths, we call them products of the Imagination; when they are mere inventions, however lovely, I should call them the work of the Fancy: in either case, Law has been diligently at work." (George MacDonald, "The Fantastic Imagination", 1893)

"In scientific investigations, it is permitted to invent any hypothesis and, if it explains various large and independent classes of facts, it rises to the ranks of a well-grounded theory." (Charles Darwin, "The Variations of Animals and Plants Under Domestication" Vol. 1, 1896) 

From Parts to Wholes (2000-2009)

"Each part in itself constitutes the whole to which it belongs." (José Saramago, "The Cave The Cave", 2000)

"In the existing sciences much of the emphasis over the past century or so has been on breaking systems down to find their underlying parts, then trying to analyze these parts in as much detail as possible. [...] But just how these components act together to produce even some of the most obvious features of the overall behavior we see has in the past remained an almost complete mystery." (Stephen Wolfram, "A New Kind of Science", 2002)

"Systems thinking means the ability to see the synergy of the whole rather than just the separate elements of a system and to learn to reinforce or change whole system patterns. Many people have been trained to solve problems by breaking a complex system, such as an organization, into discrete parts and working to make each part perform as well as possible. However, the success of each piece does not add up to the success of the whole. to the success of the whole. In fact, sometimes changing one part to make it better actually makes the whole system function less effectively." (Richard L Daft, "The Leadership Experience", 2002)

"A system is an open set of complementary, interacting parts, with properties, capabilities and behaviours of the set emerging both from the parts and from their interactions to synthesize a unified whole." (Derek Hitchins, "Advanced Systems Thinking, Engineering, and Management", 2003)

"Emergence is the phenomenon of properties, capabilities and behaviours evident in the whole system that are not exclusively ascribable to any of its parts." (Derek Hitchins, "Advanced Systems Thinking, Engineering and Management", 2003)

"Emergence is not really mysterious, although it may be complex. Emergence is brought about by the interactions between the parts of a system. The galloping horse illusion depends upon the persistence of the human retina/brain combination, for instance. Elemental gases bond in combination by sharing outer electrons, thereby altering the appearance and behavior of the combination. In every case of emergence, the source is interaction between the parts - sometimes, as with the brain, very many parts - so that the phenomenon defies simple explanation." (Derek Hitchins, "Advanced Systems Thinking, Engineering and Management", 2003)

"There exists an alternative to reductionism for studying systems. This alternative is known as holism. Holism considers systems to be more than the sum of their parts. It is of course interested in the parts and particularly the networks of relationships between the parts, but primarily in terms of how they give rise to and sustain in existence the new entity that is the whole whether it be a river system, an automobile, a philosophical system or a quality system." (Mike Jackson, "Systems Thinking: Creative Holism for Managers", 2003)

"The traditional, scientific method for studying such systems is known as reductionism. Reductionism sees the parts as paramount and seeks to identify the parts, understand the parts and work up from an understanding of the parts to an understanding of the whole. The problem with this is that the whole often seems to take on a form that is not recognizable from the parts. The whole emerges from the interactions between the parts, which affect each other through complex networks of relationships. Once it has emerged, it is the whole that seems to give meaning to the parts and their interactions." (Mike Jackson, "Systems Thinking: Creative Holism for Managers", 2003)

"The progress of science requires the growth of understanding in both directions, downward from the whole to the parts and upward from the parts to the whole." (Freeman J Dyson, "The Scientist As Rebel", 2006)

"This reduction principle - the reduction of the behavior of a complex system to the behavior of its parts - is valid only if the level of complexity of the system is rather low." (Andrzej P Wierzbicki & Yoshiteru Nakamori, "Creative Space: Models of Creative Processes for the Knowledge Civilization Age", Studies in Computational Intelligence Vol.10, 2006)

"The fabric of our complex society is woven too tightly to permit any part of it to be damaged without damaging the whole." (Joe Biden, "Promises to Keep", 2008)

Hermann Hankel - Collected Quotes

"The purely formal sciences, logic and mathematics, deal with those relations which are, or can be, independent of the particular content or the substance of objects. To mathematics in particular fall those relations between objects which involve the concepts of magnitude, of measure and of number." (Hermann Hankel, "Theorie der Complexen Zahlensysteme", 1867)

"Generality of points of view and of methods, precision and elegance in presentation, have become, since Lagrange, the common property of all who would lay claim to the rank of scientific mathematicians. And, even if this generality leads at times to abstruseness at the expense of intuition and applicability, so that general theorems are formulated which fail to apply to a single special case, if furthermore precision at times degenerates into a studied brevity which makes it more difficult to read an article than it was to write it; if, finally, elegance of form has well-nigh become in our day the criterion of the worth or worthlessness of a proposition, - yet are these conditions of the highest importance to a wholesome  development, in that they keep the scientific material within the limits which are necessary both intrinsically and extrinsically if mathematics is not to spend itself in trivialities or smother in profusion." (Hermann Hankel, "Theorie der Complexen Zahlensysteme", 1867)

"If one would see how a science can be constructed and developed to its minutest details from a very small number of intuitively perceived axioms, postulates, and plain definitions, by means of rigorous, one would almost say chaste, syllogism, which nowhere makes use of surreptitious or foreign aids, if one would see how a science may thus be constructed one must turn to the elements of Euclid." (Hermann Hankel, "Theorie der Complexen Zahlensysteme", 1867)

"If two forms expressed in the general symbols of universal arithmetic are equal to each other, then they will also remain equal when the symbols cease to represent simple magnitudes, and the operations also consequently have another meaning of any kind." (Hermann Hankel, "Theorie der Complexen Zahlensysteme", 1867)

"If we compare a mathematical problem with an immense rock, whose interior we wish to penetrate, then the work of the Greek mathematicians appears to us like that of a robust stonecutter, who, with indefatigable perseverance, attempts to demolish the rock gradually from the outside by means of  hammer and chisel; but the modern mathematician resembles an expert miner, who first constructs a few passages through the rock and then explodes it with a single blast, bringing to light its inner treasures." (Hermann Hankel, "Die Entwickelung der Mathematik in den letzten Jahrhunderten", 1869)

"Isolated, so-called ‘pretty theorems’ have even less value in the eyes of a modern mathematician than the discovery of a new ‘pretty flower’ has to the scientific botanist, though the layman finds in these the chief charm of the respective Sciences." (Hermann Hankel, "Die Entwickelung der Mathematik in den letzten Jahrhunderten", 1869)

"It is, so to speak, a scientific tact, which must guide mathematicians in their investigations, and guard them from spending their forces on scientifically worthless problems and abstruse realms, a tact which is closely related to esthetic tact and which is the only thing in our science which cannot be taught or acquired, and is yet the indispensable endowment of every mathematician." (Hermann Hankel, "Die Entwickelung der Mathematik in den letzten Jahrhunderten", 1869)

"Mathematics pursues its own course unrestrained, not indeed with an unbridled licence which submits to no laws, but rather with the freedom which is determined by its own nature and in conformity with its own being." (Hermann Hankel, "Die Entwickelung der Mathematik in den letzten Jahrhunderten", 1869)

"Mathematics will not be properly esteemed in wider circles until more than the a b c of it is taught in the schools, and until the unfortunate impression is gotten rid of that mathematics serves no other purpose in instruction than the formal training of the mind. The aim of mathematics is its content, its form is a secondary consideration and need not necessarily be that historic form which is due to the circumstance that mathematics took permanent shape under the influence of Greek logic." (Hermann Hankel, "Die Entwickelung der Mathematik in den letzten Jahrhunderten", 1869)

"Since one could directly derive the expansion in series of algebraic functions according to the powers of an increment, the derivatives, and the integral, one not only held that it was possible to assume the existence of such a series, derivative, and integral for all functions in general, but one never even had the idea that herein lay an assertion, whether it now be an axiom or a theorem - so self-evident did the transfer of the properties of algebraic functions to transcendental ones seem in the light of the geometrical view of curves representing functions. And examples in which purely analytic functions displayed singularities that were clearly different from those of algebraic functions remained entirely unnoticed." (Hermann Hankel, 1870)

"In most sciences, one generation tears down what another has built and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure" (Hermann Hankel, "Die Entwicklung der Mathematik in den letzten Jahrhunderten, 1884)

On Analogy (1925-1949)

"Analogies prove nothing, that is quite true, but they can make one feel more at home." (Sigmund Freud, Neue Folge der Vorlesungen zur Einführung in die Psychoanalyse" ["New Introductory Lectures on Psychoanalysis"], 1933)

"If the scientist has, during the whole of his life, observed carefully, trained himself to be on the look-out for analogy and possessed himself of relevant knowledge, then the ‘instrument of feeling’ […] will become a powerful divining rod leading the scientist to discover order in the midst of chaos by providing him with a clue, a hint, or an hypothesis upon which to base his experiments.' (Rosamund E M Harding, "An Anatomy of Inspiration", 1940)

"The question of the origin of the hypothesis belongs to a domain in which no very general rules can be given; experiment, analogy and constructive intuition play their part here. But once the correct hypothesis is formulated, the principle of mathematical induction is often sufficient to provide the proof." (Richard Courant & Herbert Robbins, "What Is Mathematics?: An Elementary Approach to Ideas and Methods" , 1941) 

"Analogies are useful for analysis in unexplored fields. By means of analogies an unfamiliar system may be compared with one that is better known. The relations and actions are more easily visualized, the mathematics more readily applied, and the analytical solutions more readily obtained in the familiar system." (Harry F Olson, "Dynamical Analogies", 1943)

"Of course we have still to face the question why these analogies between different mechanisms - these similarities of relation-structure - should exist. To see common principles and simple rules running through such complexity is at first perplexing though intriguing. When, however, we find that the apparently complex objects around us are combinations of a few almost indestructible units, such as electrons, it becomes less perplexing." (Kenneth Craik, "The Nature of Explanation", 1943)

"This, however, is very speculative; the point of interest for our present enquiry is that physical reality is built up, apparently, from a few fundamental types of units whose properties determine many of the properties of the most complicated phenomena, and this seems to afford a sufficient explanation of the emergence of analogies between mechanisms and similarities of relation-structure among these combinations without the necessity of any theory of objective universals." (Kenneth Craik, "The Nature of Explanation", 1943)

"Analogy is a sort of similarity. Similar objects agree with each other in some respect, analogous objects agree in certain relations of their respective parts." (George Pólya, "How to solve it", 1945)

"[…] analogy [is] an important source of conjectures. In mathematics, as in the natural and physical sciences, discovery often starts from observation, analogy, and induction. These means, tastefully used in framing a plausible heuristic argument, appeal particularly to the physicist and the engineer." (George Pólya, "How to solve it", 1945) 

"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements." (George Pólya, "How to Solve It", 1945)

"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements. Analogy is used on very different levels. People often use vague, ambiguous, incomplete, or incompletely clarified analogies, but analogy may reach the level of mathematical precision. All sorts of analogy may play a role in the discovery of the solution and so we should not neglect any sort." (George Pólya, "How to solve it", 1945)

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

"Inference by analogy appears to be the most common kind of conclusion, and it is possibly the most essential kind. It yields more or less plausible conjectures which may or may not be confirmed by experience and stricter reasoning." (George Pólya, "How to Solve It", 1945)

"[…] the number of available analogies is a determining factor in the growth and progress of science." (Morris R Cohen, "The Meaning of Human History", 1947)

"A man desiring to understand the world looks about for a clue to its comprehension. He pitches upon some area of commonsense fact and tries to understand other areas in terms of this one. The original area becomes his basic analogy or root metaphor." (Stephen Pepper, "World Hypotheses: A Study in Evidence", 1948)

"The solution of the difficulty is that the two mental pictures which experiment lead us to form - the one of the particles, the other of the waves - are both incomplete and have only the validity of analogies which are accurate only in limiting cases." (Werner Heisenberg, "The Physical Principles of the Quantum Theory", 1949)

On Analogy (1900-1924)

"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments." (David Hilbert," Mathematical Problems", Bulletin American Mathematical Society Vol. 8, 1901-1902)

"Analogies are figures intended to serve as fatal weapons if they succeed, and as innocent toys if they fail." (Henry Adams, "Mont Saint Michel and Chartres", 1904)

"So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality." (Henri Poincaré, "The Value of Science", 1905)

"Without this language [mathematics] most of the intimate analogies of things would have remained forever unknown to us; and we should forever have been ignorant of the internal harmony of the world, which is the only true objective reality." (Henri Poincaré," The Value of Science", Popular Science Monthly, 1906) 

"The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features." (Eliakim H Moore, "Introduction to a Form of General Analysis", 1910)

"The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another." (Henri Poincaré, 1913)

“It must be gently but firmly pointed out that analogy is the very corner-stone of scientific method. A root-and-branch condemnation would invalidate any attempt to explain the unknown in terms of the known, and thus prune away every hypothesis.” (Archie E Heath, “On Analogy”, The Cambridge Magazine, 1918) 

"[…] analogies are not ‘aids’ to the establishment of theories; they are an utterly essential part of theories, without which theories would be completely valueless and unworthy of the name. It is often suggested that the analogy leads to the formulation of the theory, but that once the theory is formulated the analogy has served its purpose and may be removed or forgotten. Such a suggestion is absolutely false and perniciously misleading." (Norman R Campbell, "Physics: The Elements", 1920)

"To regard analogy as an aid to the invention of theories is as absurd as to regard melody as an aid to the composition of sonatas." (Norman R Campbell, "Physics: The Elements", 1920)

Nicolas de Condorcet - Collected Quotes

"[…] 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)

"[…] determine the probability of a future or unknown event not on the basis of the number of possible combinations resulting in this event or in its complementary event, but only on the basis of the knowledge of order of familiar previous events of this kind" (Nicolas de Condorcet, "Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix", 1785)

"We must therefore establish a form of decision-making in which voters need only ever pronounce on simple propositions, expressing their opinions only with a yes or a no. […] Clearly, if anyone’s vote was self-contradictory (intransitive), it would have to be discounted, and we should therefore establish a form of voting which makes such absurdities impossible." (Nicolas de Condorcet, "On the form of decisions made by plurality vote", 1788)

"It has never yet been supposed, that all the facts of nature, and all the means of acquiring precision in the computation and analysis of those facts, and all the connections of objects with each other, and all the possible combinations of ideas, can be exhausted by the human mind." (Nicolas de Condorcet, "Outlines Of An Historical View Of The Progress Of The Human Mind", 1795)

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

"As the mind learns to understand more complicated combinations of ideas, simpler formulae soon reduce their complexity; so truths that were discovered only by great effort, that could at first only be understood by men capable of profound thought, are soon developed and proved by methods that are not beyond the reach of common intelligence. The strength and the limits of man." (Nicolas de Condorcet) 

"Mathematics is the science that yields the best opportunity to observe the working of the mind. Its study is the best training of our abilities as it develops both the power and the precision of our thinking. Mathematics is valuable on account of the number and variety of its applications. And it is equally valuable in another respect: By cultivating it, we acquire the habit of a method of reasoning which can be applied afterwards to the study of any subject and can guide us in life's great and little problems." (Nicolas de Condorcet)

"This adventure of the physical sciences […] could not be observed without enlightened men seeking to follow it up in the other sciences; at each step it held out to them the model to be followed." (Nicolas de Condorcet)

"Those sciences, created almost in our own days, the object of which is man himself, the direct goal of which is the happiness of man, will enjoy a progress no less sure than that of the physical sciences; and this sweet idea - that our nephews will surpass us in wisdom as in enlightenment - is no longer an illusion." (Nicolas de Condorcet)

On Certainty (1950-1959)

"To say that observations of the past are certain, whereas predictions are merely probable, is not the ultimate answer to the question of induction; it is only a sort of intermediate answer, which is incomplete unless a theory of probability is developed that explains what we should mean by ‘probable’ and on what ground we can assert probabilities." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"We cannot define truth in science until we move from fact to law. And within the body of laws in turn, what impresses us as truth is the orderly coherence of the pieces. They fit together like the characters of a great novel, or like the words of a poem. Indeed, we should keep that last analogy by us always, for science is a language, and like a language it defines its parts by the way they make up a meaning. Every word in a sentence has some uncertainty of definition, and yet the sentence defines its own meaning and that of its words conclusively. It is the internal unity and coherence of science which gives it truth, and which makes it a better system of prediction than any less orderly language." (Jacob Bronowski, "The Common Sense of Science", 1953)

"[…] it is imperative in science to doubt; it is absolutely necessary, for progress of science, to have uncertainty as a fundamental part of your inner nature." (Richard P Feynman, Engineering and Science Vol. 19, 1956)

"Science does not mean an idle resting upon a body of certain knowledge; it means unresting endeavor and continually progressing development toward an end which the poetic intuition may apprehend, but which the intellect can never fully grasp." (Max Planck, "The New Science", 1959)

"With the idol of certainty (including that of degrees of imperfect certainty or probability) there falls one of the defences of obscurantism which bar the way of scientific advance. For the worship of this idol hampers not only the boldness of our questions, but also the rigor and the integrity of our tests. The wrong view of science betrays itself in the craving to be right; for it is not his possession of knowledge, of irrefutable truth, that makes the man of science, but his persistent and recklessly critical quest for truth." (Karl R Popper, "The Logic of Scientific Discovery", 1959)

"It seems to me that science has a much greater likelihood of being true in the main than any philosophy hitherto advanced (I do not, of course, except my own). In science there are many matters about which people are agreed; in philosophy there are none. Therefore, although each proposition in a science may be false, and it is practically certain that there are some that are false, yet we shall be wise to build our philosophy upon science, because the risk of error in philosophy is pretty sure to be greater than in science. If we could hope for certainty in philosophy, the matter would be otherwise, but so far as I can see such a hope would be a chimerical." (Bertrand Russel, "The Philosophy of Logical Atomism", 1959)

"There is a logic of language and a logic of mathematics. The former is supple and lifelike, it follows our experience. The latter is abstract and rigid, more ideal. The latter is perfectly necessary, perfectly reliable: the former is only sometimes reliable and hardly ever systematic. But the logic of mathematics achieves necessity at the expense of living truth, it is less real than the other, although more certain. It achieves certainty by a flight from the concrete into abstraction." (Thomas Merton, "The Secular Journal of Thomas Merton", 1959)

"Years ago a statistician might have claimed that statistics deals with the processing of data [...]today’s statistician will be more likely to say that statistics is concerned with decision making in the face of uncertainty." (Herman Chernoff & Lincoln E Moses, "Elementary Decision Theory", 1959)

On Discovery (1925-1949)

"In our recognition that order is universal, a fact confirmed by myriads of observations of patient, indefatigable, and devoted investigators, the old saying that 'an irreverent astronomer is mad' can apply with equal force to the physicist. Man learns something of his own minute and colossal stature, and he comes to feel that his own intelligence, which enables him to make such sublime discoveries, is the supreme achievement of evolution." (Harvey B Lemon, "Atomic Structure", 1927)

"Great scientific discoveries have been made by men seeking to verify quite erroneous theories about the nature of things." (Aldous L Huxley, "Life and Letters and the London Mercury" Vol. 1, 1928)

"Since the fundamental character of the living thing is its organization, the customary investigation of the single parts and processes cannot provide a complete explanation of the vital phenomena. This investigation gives us no information about the coordination of parts and processes. Thus, the chief task of biology must be to discover the laws of biological systems (at all levels of organization). We believe that the attempts to find a foundation for theoretical biology point at a fundamental change in the world picture. This view, considered as a method of investigation, we shall call ‘organismic biology’ and, as an attempt at an explanation, ‘the system theory of the organism’" (Ludwig von Bertalanffy, "Kritische Theorie der Formbildung", 1928)

"The art of discovery is confused with the logic of proof and an artificial simplification of the deeper movements of thought results. We forget that we invent by intuition though we prove by logic." (Sarvepalli Radhakrishnan, "An Idealist View of Life", 1929)

"To those who study her, Nature reveals herself as extraordinarily fertile and ingenious in devising means, but she has no ends which the human mind has been able to discover or comprehend." (Joseph W Krutch, "The Modern Temper", 1929)

"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, [letter to G P Thomson], 1930)

"The man who discovers a new scientific truth has previously had to smash to atoms almost everything he had learnt, and arrives at the new truth with hands blood stained from the slaughter of a thousand platitudes." (Jose Ortega y Gasset, "The Revolt of the Masses", 1930)

"It is this ideal of progress through cumulative effort rather than through genius - progress by organised effort, progress which does not wait for some brilliant stroke, some lucky discovery, or the advent of some superman, has been the chief gift of science to social philosophy." (William Wickenden, [Address to 48th annual summer convention of the American Institute of Electriccal Engineers, Cleveland] 1932)

"Scientific discovery and scientific knowledge have been achieved only by those who have gone in pursuit of them without any practical purpose whatsoever in view." (Max Planck, "Where is Science Going?", 1932)

"There is no such thing as a logical method of having new ideas or a logical reconstruction of this process […] very discovery contains an ‘irrational element’ or a ‘creative intuition’." (Karl Popper, "The logic of scientific discover", 1934)

"A scientist, whether theorist or experimenter, puts forward statements, or systems of statements, and tests them step by step. In the field of the empirical sciences, more particularly, he constructs hypotheses, or systems of theories, and tests them against experience by observation and experiment." (Karl Popper, "The Logic of Scientific Discovery", 1935)

"In experimental science facts of the greatest importance are rarely discovered accidentally: more frequently new ideas point the way towards them." (Erwin Schrödinger, "Science and the Human Temperament", 1935)

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

"A scientifically unimportant discovery is one which, however true and however interesting for other reasons, has no consequences for a system of theory with which scientists in that field are concerned." (Talcott Parsons, "The Structure of Social Action", 1937)

"Creating a new theory is not like destroying an old barn and erecting a skyscraper in its place. It is rather like climbing a mountain, gaining new and wider views, discovering unexpected connections between our starting point and its rich environment. But the point from which we started out still exists and can be seen, although it appears smaller and forms a tiny part of our broad view gained by the mastery of the obstacles on our adventurous way up." (Albert Einstein & Leopold Infeld, "The Evolution of Physics", 1938)

"The scientist takes off from the manifold observations of predecessors, and shows his intelligence, if any, by his ability to discriminate between the important and the negligible, by selecting here and there the significant stepping-stones that will lead across the difficulties to new understanding. The one who places the last stone and steps across the terra firma of accomplished discovery gets all the credit. Only the initiated know and honor those whose patient integrity and devotion to exact observation have made the last step possible." (Hans Zinsser, "As I Remember Him: The Biography of R.S.", 1940)

"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery." (George Polya, "How to solve it", 1945)

"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements. Analogy is used on very different levels. People often use vague, ambiguous, incomplete, or incompletely clarified analogies, but analogy may reach the level of mathematical precision. All sorts of analogy may play a role in the discovery of the solution and so we should not neglect any sort." (George Pólya, "How to solve it", 1945)

"In mathematics as in the physical sciences we may use observation and induction to discover general laws. But there is a difference. In the physical sciences, there is no higher authority than observation and induction but In mathematics there is such an authority: rigorous proof." (George Pólya, "How to solve it", 1945)

"The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea." (George Pólya, "How to solve it", 1945)

"It is always more easy to discover and proclaim general principles than it is to apply them." (Winston Churchill, "The Second World War: The gathering storm", 1948)

"Scientific discovery consists in the interpretation for our own convenience of a system of existence which has been made with no eye to our convenience at all." (Norbert Wiener, "The Human Use of Human Beings", 1949)

"Since the real world, in the absolute sense of the word, is independent of individual personalities, and in fact of all human intelligence, every discovery made by an individual acquires a completely universal significance. This gives the inquirer, wrestling with his problem in quiet seclusion, the assurance that every discovery will win the unhesitating recognition of all experts throughout the entire world, and in this feeling of the importance of his work lies his happiness. It compensates him fully for many a sacrifice which he must make in his daily life." (Max Planck, "The Meaning and Limits of Exact Science", 1949)