On Creativity (Mathematics I)

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

"There are, roughly speaking, two kinds of mathematical creativity. One, akin to conquering a mountain peak, consists of solving a problem which has remained unsolved for a long time and has commanded the attention of many mathematicians. The other is exploring new territory." (Mark Kac, "Enigmas Of Chance", 1985)

"Music and higher mathematics share some obvious kinship. The practice of both requires a lengthy apprenticeship, talent, and no small amount of grace. Both seem to spring from some mysterious workings of the mind. Logic and system are essential for both, and yet each can reach a height of creativity beyond the merely mechanical." (Frederick Pratter, "How Music and Math Seek Truth in Beauty", Christian Science Monitor, 1995)

"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)

"Math is the beautiful, rich, joyful, playful, surprising, frustrating, humbling and creative art that speaks to something transcendental. It is worthy of much exploration and examination because it is intrinsically beautiful, nothing more to say. Why play the violin? Because it is beautiful! Why engage in math? Because it too is beautiful!" (James Tanton, "Thinking Mathematics")

"Mathematics is the summit of human thinking. It has all the creativity and imagination that you can find in all kinds of art, but unlike art-charlatans and all kinds of quacks will not succeed there." (Meir Shalev)

"No discovery has been made in mathematics, or anywhere else for that matter, by an effort of deductive logic; it results from the work of creative imagination which builds what seems to be truth, guided sometimes by analogies, sometimes by an esthetic ideal, but which does not hold at all on solid logical bases. Once a discovery is made, logic intervenes to act as a control; it is logic that ultimately decides whether the discovery is really true or is illusory; its role therefore, though considerable, is only secondary." (Henri Lebesgue)

"The essential feature of mathematical creativity is the exploration, under the pressure of powerful implosive forces, of difficult problems for whose validity and importance the explorer is eventually held accountable by reality." (Alfred Adler)

On Facts (Unsourced)

"A fact is not novel if it has an analogue which could have some interest. A fact which does not fi t in with a series of known facts is a fact which deserves particular attention. If the mind had to retain all individual facts, it could not manage and science would not exist; but when these facts can be connected by general laws and by theories, when a large number of these facts can be represented by a single one, one can remember them more easily, one can generalise one’s ideas, one can compare one general fact with another general fact and discoveries can succeed each other. It is only when laws can be introduced into a science that it assumes the true character of science." (Joseph L Gay-Lussac)

"All the pictures which science draws of Nature, and which alone seem capable of according with observational facts, are mathematical pictures." (Sir James Jeans)

"By observation, facts are distinctly and minutely impressed in the mind; by analogy, similar facts are connected ; by experiment, new facts are discovered ; and, in the progression of knowledge, observation, guided by analogy, leads to experiment, and analogy, confirmed by experiment, becomes scientific truth." (Sir Humphry Davy)

"Cognitive psychology has shown that the mind best understands facts when they are woven into a conceptual fabric, such as a narrative, mental map, or intuitive theory. Disconnected facts in the mind are like unlinked pages on the Web: They might as well not exist." (Steven Pinker) 

"Education is not the piling on of learning, information, data, facts, skills, or abilities - that's training or instruction - but is rather making visible what is hidden as a seed." (Thomas W Moore)

"Facts and values are entangled in science. It's not because scientists are biased, not because they are partial or influenced by other kinds of interests, but because of a commitment to reason, consistency, coherence, plausibility and replicability. These are value commitments." (Alva Noë)

"Facts are stubborn things, but statistics are pliable." (Mark Twain)

"[…] facts by themselves are silent. Observation discovers nothing directly of the actions of causes, but only of sequences in time." (Alfred Marshall)

"First accumulate a mass of Facts: and then construct a Theory." (Lewis Carroll)

[Maier’s Law:] "If the facts do not conform to the theory, they must be disposed of." (Norman R F Maier)

"Imagination, as well as reason, is necessary to perfection in the philosophical mind. A rapidity of combination, a power of perceiving analogies, and of comparing them by facts, is the creative source of discovery." (Sir Humphry Davy)

"In physical science the discovery of new facts is open to every blockhead with patience, manual dexterity, and acute senses; it is less effectually promoted by genius than by co-operation, and more frequently the result of accident than of design." (John Thomson)

"In the study of Nature conjecture must be entirely put aside, and vague hypothesis carefully guarded against. The study of Nature begins with facts, ascends to laws, and raises itself, as far as the limits of man’s intellect will permit, to the knowledge of causes, by the threefold means of observation, experiment and logical deduction." (Jean Baptiste-Andre Dumas)

"No good model ever accounted for all the facts, since some data was bound to be misleading if not plain wrong." (James Dewey Watson)

"Nothing in education is so astonishing as the amount of ignorance it accumulates in the form of inert facts." (Henry B Adams)

"One might describe the mathematical quality in Nature by saying that the universe is so constituted that mathematics is a useful tool in its description. However, recent advances in physical science show that this statement of the case is too trivial. The connection between mathematics and the description of the universe goes far deeper than this, and one can get an appreciation of it only from a thorough examination of the various facts that make it up." (Paul A M Dirac)

"Science does more than collect facts; it makes sense of them. Great scientists are virtuosi of the art of discovering the meaning of what otherwise might seem barren observations." (Theodosius Dobzhansky)

"Science is not a technique or a body of knowledge, though it uses both. It is rather an attitude of inquiry, or observation and reasoning, with respect to the world. It can be developed, not by memorizing facts or juggling formulas to get an answer, but only by actual practice of scientific observation and reasoning." (Karl T Compton)

"Science is opposed to theological dogmas because science is founded on fact. To me, the universe is simply a great machine which never came into being and never will end. The human being is no exception to the natural order. Man, like the universe, is a machine." (Nikola Tesla)

"Some facts can be seen more clearly by example than by proof." (Leonard Euler)

"The aim of education is the knowledge not of facts but of values." (William R Inge)

"The arguments […] by which you support my theories, are most ingenious, but not founded on demonstrated facts; analogy is no proof." (Louis Pasteur)

"The art of observation and that of experimentation are very distinct. In the first case, the fact may either proceed from logical reasons or be mere good fortune; it is sufficient to have some penetration and a sense of truth in order to profit by it. But the art of experimentation leads from the first to the last link of the chain, without hesitation and without a blank, making successive use of Reason, which suggests an alternative, and of Experience, which decides on it, until, starting from a faint glimmer, the full blaze of light is reached." (Jean Baptiste-Andre Dumas)

"The disclosure of a new fact, the leap forward, the conquest over yesterday’s ignorance, is an act not of reason but of imagination, of intuition." (Charles Nicolle)

"The experiment serves two purposes, often independent one from the other: it allows the observation of new facts, hitherto either unsuspected, or not yet well defined; and it determines whether." (René J Dubos)

"[…] the mathematician learns early to accept no fact, to believe no statement, however apparently reasonable or obvious or trivial, until it has been proved, rigorously and totally by a series of steps proceeding from universally accepted first principles." (Alfred Adler)

"The object of education is not only to produce a man who knows, but one who does; who makes his mark in the straggle of life and succeeds well in whatever he undertakes: who can solve the problems of nature and of humanity as they arise, and who, when he knows he is right, can boldly convince the world of the fact." (Henry A Rowland)

"The present system of education is all wrong. The mind is crammed with facts before it knows how to think. Control of the mind should be taught first. It takes people a long time to learn things because they can't concentrate their minds at will." (Swami Vivekananda)

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

"There is no drawing the line between physics and metaphysics. If you examine every day facts at all closely, you are a physicist; but if you press your physics at all home, you become a metaphysician; if you press your metaphysics at all home, you are in a fog." (Samuel Butler)

"There is nothing more deceptive than an obvious fact." (Sir Arthur C Doyle)

On Generalization (1920-1929)

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

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

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

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

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

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

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

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

On Imagination (1925-1949)

"The sciences bring into play the imagination, the building of images in which the reality, of the past is blended with the ideals for the future, and from the picture there springs the prescience of genius." (William J Mayo, "Contributions of Pure Science to Progressive Medicine", The Journal of the American Medical Association Vol. 84 (20), 1925)

"We do not know why the imagination has accepted that image before the reason can reject it; or why such correspondences seem really to correspond to something in the soul." (Gilbert K Chesterton, "The Everlasting Man", 1925)

"The world is not run by thought, nor by imagination, but by opinion." (Elizabeth A Drew, "The Modern Novel", 1926)

"In this way things, external objects, are assimilated to more or less ordered motor schemas, and in this continuous assimilation of objects the child's own activity is the starting point of play. Not only this, but when to pure movement are added language and imagination, the assimilation is strengthened, and wherever the mind feels no actual need for accommodating itself to reality, its natural tendency will be to distort the objects that surround it in accordance with its desires or its fantasy, in short to use them for its satisfaction. Such is the intellectual egocentrism that characterizes the earliest form of child thought." (Jean Piaget, "The Moral Judgment of the Child", 1932)

"What is the inner secret of mathematical power? Briefly stated, it is that mathematics discloses the skeletal outlines of all closely articulated relational systems. For this purpose mathematics uses the language of pure logic with its score or so of symbolic words, which, in its important forms of expression, enables the mind to comprehend systems of relations otherwise completely beyond its power. These forms are creative discoveries which, once made, remain permanently at our disposal. By means of them the scientific imagination is enabled to penetrate ever more deeply into the rationale of the universe about us." (George D Birkhoff, "Mathematics: Quantity and Order", 1934)

"The scientist explores the world of phenomena by successive approximations. He knows that his data are not precise and that his theories must always be tested. It is quite natural that he tends to develop healthy skepticism, suspended judgment, and disciplined imagination." (Edwin P Hubble, 1938)

"The great instrument of moral good is the imagination; and poetry administers to the effect by acting on the cause." (Percy B Shelley, "A Defence of Poetry", 1840) [written 1821]

"The artist must bow to the monster of his own imagination." (Richard Wright, "Twelve Million Black Voices", 1941)

"Yet a review of receipt physics has shown that all attempts at mechanical models or pictures have failed and must fail. For a mechanical model or picture must represent things as happening in space and time, while it has recently become clear that the ultimate processes of nature neither occur in, nor admit of representation in, space and time. Thus an understanding of the ultimate processes of nature is for ever beyond our reach: we shall never be able - even in imagination - to open the case of our watch and see how the wheels go round. The true object of scientific study can never be the realities of nature, but only our own observations on nature." (James H Jeans, "Physics and Philosophy", 1942)

"The straight line of the geometers does not exist in the material universe. It is a pure abstraction, an invention of the imagination or, if one prefers, an idea of the Eternal Mind." (Eric T Bell, "The Magic of Numbers", 1946)

"For, in mathematics or symbolic logic, reason can crank out the answer from the symboled equations -even a calculating machine can often do so - but it cannot alone set up the equations. Imagination resides in the words which define and connect the symbols - subtract them from the most aridly rigorous mathematical treatise and all meaning vanishes." (Ralph W Gerard, "The Biological Basis of Imagination", American Thought, 1947)

"Imagination and fiction make up more than three-quarters of our real life." (Simone Weil, "Gravity and Grace", 1947)

"[...] when the pioneer in science sends for the groping feelers of his thoughts, he must have a vivid intuitive imagination, for new ideas are not generated by deduction, but by an artistically creative imagination. Nevertheless, the worth of a new idea is invariably determined, not by the degree of its intuitiveness - which, incidentally, is to a major extent a matter of experience and habit - but by the scope and accuracy of the individual laws to the discovery of which it eventually leads. (Max Planck, The Meaning and Limits of Exact Science, Science Vol. 110 (2857), 1949)

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On Deduction (1800-1849)

"One very reprehensible mode of theory-making consists, after honest deductions from a few facts have been made, in torturing other facts to suit the end proposed, in omitting some, and in making use of any authority that may lend assistance to the object desired; while all those which militate against it are carefully put on one side or doubted." (Henry De la Beche, "Sections and Views, Illustrative of Geological Phaenomena", 1830)

"Facts [...] are not truths; they are not conclusions; they are not even premises, but in the nature and parts of premises. The truth depends on, and is only arrived at, by a legitimate deduction from all the facts which are truly material." (Samuel T Coleridge, "The Table Talk and Omniana of Samuel Taylor Coleridge", 1831)

"The deduction of effect from cause is often blocked by some insuperable extrinsic obstacle: the true causes may be quite unknown." (Carl von Clausewitz, "On War", 1832)

"Physical astronomy is the science which compares and identifies the laws of motion observed on earth with the motions that take place in the heavens; and which traces, by an uninterrupted chain of deduction from the great principle that governs the universe, the revolutions and rotations of the planets, and the oscillations of the fluids at their surfaces; and which estimates the changes the system has hitherto undergone, or may hereafter experience - changes which require millions of years for their accomplishment." (Mary Somerville, "The Connection of the Physical Sciences", 1834)

"Every stage of science has its train of practical applications and systematic inferences, arising both from the demands of convenience and curiosity, and from the pleasure which, as we have already said, ingenious and active-minded men feel in exercising the process of deduction." (William Whewell, "The Philosophy of the Inductive Sciences Founded Upon Their History", 1840)

"These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences", 1840)

On Deduction (2000-2024)

"The process of abstracting out the common features of things in this way is known as induction. Applying what we have learned - generalising from one or a few examples to a whole range of new examples - is known as deduction. Induction is the process of moving from the particular to the general, and deduction is the process of going from the general to the particular." (S Ian Robertson, "Problem Solving", 2001)

"Both induction and deduction, reasoning from the particular and the general, and back again from the universal to the specific, form the essence to scientific thinking." (Hans Christian von Baeyer, "Information, The New Language of Science", 2003)

"Entropy is not about speeds or positions of particles, the way temperature and pressure and volume are, but about our lack of information." (Hans C von Baeyer," Information, The New Language of Science", 2003)

"Paradox is the sharpest scalpel in the satchel of science. Nothing concentrates the mind as effectively, regardless of whether it pits two competing theories against each other, or theory against observation, or a compelling mathematical deduction against ordinary common sense." (Hans Christian von Baeyer, "Information, The New Language of Science", 2003)

"The concept of proof perhaps marks the true beginning of mathematics as the art of deduction rather than just numerological observation, the point at which mathematical alchemy gave way to mathematical chemistry." (Marcus du Sautoy, "The Music of the Primes", 2004)

"It seems that scientists are often attracted to beautiful theories in the way that insects are attracted to flowers - not by logical deduction, but by something like a sense of smell." (Steven Weinberg, "Physics Today", 2005)

"Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception." (David Ruelle, "The Mathematician's Brain", 2007)

"Knowledge about things beyond our immediate environment may be acquired through deduction, if the initial premises are believed to be correct." (Nayef Al-Rodhan, "Sustainable History and the Dignity of Man", 2009)

On Deduction (1950-1974)

"Insight is not the same as scientific deduction, but even at that it may be more reliable than statistics." (Anthony Standen, "Science Is a Sacred Cow", 1950)

"[…] the chief reason in favor of any theory on the principles of mathematics must always be inductive, i.e., it must lie in the fact that the theory in question enables us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence the early deductions, until they reach this point, give reasons rather from them, than for believing the premises because true consequences follow from them, than for believing the consequences because they follow from the premises." (Alfred N Whitehead, "Principia Mathematica", 1950)

"All followers of the axiomatic method and most mathematicians think that there is some such thing as an absolute ‘mathematical rigor’ which has to be satisfied by any deduction if it is to be valid. The history of mathematics shows that this is not the case, that, on the contrary, every generation is surpassed in rigor again and again by its successors." (Richard von Mises, "Positivism: A Study in Human Understanding", 1951)

"We are driven to conclude that science, like mathematics, is a system of axioms, assumptions, and deductions; it may start from being, but later leaves it to itself, and ends in the formation of a hypothetical reality that has nothing to do with existence; or it is the discovery of an ideal being which is, of course, present in what we call actuality, and renders it an existence for us only by being present in it." (Poolla T Raju, "Idealistic Thought of India", 1953)

"[…] the grand aim of all science […] is to cover the greatest possible number of empirical facts by logical deductions from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)

"The theory of relativity is a fine example of the fundamental character of the modern development of theoretical science. The initial hypotheses become steadily more abstract and remote from experience. On the other hand, it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deduction from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)

"This method of deduction [...] is often called 'combinatory'. Its usefulness is not exhausted at this stage, but it does even at the outset lead to some valuable conclusions [...]" (John Chadwick, "The Decipherment of Linear B", 1958)

"The functional validity of a working hypothesis is not a priori certain, because often it is initially based on intuition. However, logical deductions from such a hypothesis provide expectations (so called prognoses) as to the circumstances under which certain phenomena will appear in nature. Such a postulate or working hypothesis can then be substantiated by additional observations or by experiments especially arranged to test details. The value of the hypothesis is strengthened if the observed facts fit the expectation within the limits of permissible error." (R Willem van Bemmelen, "The Scientific Character of Geology", The Journal of Geology Vol 69 (4), 1961)

"Intellect begins with the observation of nature, proceeds to memorize and classify the facts thus observed, and by logical deduction builds up that edifice of knowledge properly called science. But admittedly we also know by feeling, and we can combine the two faculties, and present knowledge in the guise of art." (Herbert Read, "Selected Writings: Poetry and Criticism", 1963)

"Perfect logic and faultless deduction make a pleasant theoretical structure, but it may be right or wrong; the experimenter is the only one to decide, and he is always right." (Léon Brillouin, "Scientific Uncertainty and Information", 1964)

"The interplay between generality and individuality, deduction and construction, logic and imagination - this is the profound essence of live mathematics. Anyone or another of these aspects of mathematics can be found at the center of a given achievement. In a far reaching development all of them will be involved. Generally speaking, such a development will start from the 'concrete', then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy: after this flight comes the crucial test for learning and reaching specific goals in the newly surveyed low plains of individual 'reality'. In brief, the flight into abstract generality must start from and return again to the concrete and specific." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964) 

"So in order to understand the physics one must always have a neat balance and contain in his head all of the various propositions and their interrelationships because the laws often extend beyond the range of their deductions. This will only have no importance when all the laws are known." (Richard Feynman, "The Character of Physical Law", 1965)

"[…] the human reason discovers new relations between things not by deduction, but by that unpredictable blend of speculation and insight […] induction, which - like other forms of imagination - cannot be formalized." (Jacob Bronowski, "The Reach of Imagination", 1967)

"Mathematics in itself, as I say, is independent of experience. It begins with the free choice of symbols, to which are freely assigned properties, and it then proceeds to deduce the necessary rational implications of those properties." (Herbert Dingle, "Science at the Crossroads", 1972)

"To give a causal explanation of an event means to deduce a statement which describes it, using as premises of the deduction one or more universal laws, together with certain singular statements, the initial conditions. [...] We have thus two different kinds of statement, both of which are necessary ingredients of a complete causal explanation." (Karl Popper, "The Philosophy of Karl Popper", 1974)

On Deduction (1850-1874)

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

"The principle of deduction is, that things which agree with the same thing agree with one another. The principle of induction is, that in the same circumstances and in the same substances, from the same causes the same effects will follow. The mathematical and metaphysical sciences are founded on deduction; the physical sciences rest on induction." (William Fleming, "A vocabulary of the philosophical sciences", 1857)

"This science, Geometry, is one of indispensable use and constant reference, for every student of the laws of nature; for the relations of space and number are the alphabet in which those laws are written. But besides the interest and importance of this kind which geometry possesses, it has a great and peculiar value for all who wish to understand the foundations of human knowledge, and the methods by which it is acquired. For the student of geometry acquires, with a degree of insight and clearness which the unmathematical reader can but feebly imagine, a conviction that there are necessary truths, many of them of a very complex and striking character; and that a few of the most simple and self-evident truths which it is possible for the mind of man to apprehend, may, by systematic deduction, lead to the most remote and unexpected results." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"If an idea presents itself to us, we must not reject it simply because it does not agree with the logical deductions of a reigning theory." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Observe this: the abstraction of the philosopher is meant to keep the object itself, with its perturbing suggestions, out of sight, allowing only one quality to fill the field of vision; whereas the abstraction of the poet is meant to bring the object itself into more vivid relief, to make it visible by means of the selected qualities. In other words, the one aims at abstract symbols, the other at picturesque effects. The one can carry on his deductions by the aid of colourless signs, X or Y. The other appeals to the emotions through the symbols which will most vividly express the real objects in their relations to our sensibilities." (George H Lewes, "The Principles of Success in Literature", 1865)

"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)

"Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in Nature." (Augustus de Morgan, "A Budget of Paradoxes", 1872)

"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", 1872)

"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)

"Deduction is certain and infallible, in the sense that each step in deductive reasoning will lead us to some result, as certain as the law itself. But it does not follow that deduction will lead the reasoner to every result of a law or combination of laws." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Mathematics is a science of Observation, dealing with reals, precisely as all other sciences deal with reals. It would be easy to show that its Method is the same: that, like other sciences, having observed or discovered properties, which it classifies, generalises, co-ordinates and subordinates, it proceeds to extend discoveries by means of Hypothesis, Induction, Experiment and Deduction." (George H Lewes, "Problems of Life and Mind: The Method of Science and its Application", 1874)

"Whatever lies beyond the limits of experience, and claims another origin than that of induction and deduction from established data, is illegitimate." (George H Lewes, "The Foundations of a Creed", 1874-5)

On Deduction (1875-1899)

"Whatever lies beyond the limits of experience, and claims another origin than that of induction and deduction from established data, is illegitimate." (George H Lewes, "The Foundations of a Creed", 1875)

"I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction." (Gottlob Frege, "The Foundations of Arithmetic", 1884)

"[…] deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts." (Charles S Peirce, 1885)

"There is as great a distinction between mathematics and the mathematical sciences as there is between induction and the inductive sciences. Practically, few cases of induction do not involve, to a greater or less extent, deductions; so few mathematical processes do not involve some strictly logical procedure." (Charles C Everett, "The Science of Thought", 1890)

"By investigating the universe by an inductive method (endeavoring from the much which is observable to arrive at a little which may be verified and is indubitable) the new science refuses to recognise dogma as truth, but through reason, by a slow and laborious method of investigation, strives for and attains to true deductions." (Dmitri Mendeleev, "Principles of Chemistry", 1891)

"In deduction the mind is under the dominion of a habit or association by virtue of which a general idea suggests in each case a corresponding reaction. This is the way the hind legs of a frog separated from the rest of the body, reason, when you pinch them. It is the lowest form of psychical manifestation." (Charles S Peirce, "The Law of Mind", 1892)

"In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction." (Giuseppe Peano, "Notations de Logique Mathématique", 1894)

"All deduction rests ultimately upon the data derived from experience. This is the tortoise that supports our conception of the cosmos." (Percival Lowell, "Mars", 1895)

"Deduction is that mode of reasoning which examines the state of things asserted in the premises, forms a diagram of that state of things, perceives in the parts of the diagram relations not explicitly mentioned in the premises, satisfies itself by mental experiments upon the diagram that these relations would always subsist, or at least would do so in a certain proportion of cases, and concludes their necessary, or probable, truth." (Charles S Peirce, "Kinds of Reasoning", cca. 1896)

On Deduction (1900-1924)

"[...] it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning." (David Hilbert, "Mathematical Problems", 1900)

"Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? […] If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism." (Henri Poincaré, "Science and Hypothesis", 1901)

"The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle." (Henri Poincaré, "Science and Hypothesis", 1901)

"I may as well say at once that I do not distinguish between inference and deduction. What is called induction appears to me to be either disguised deduction or a mere method of making plausible guesses." (Bertrand Russell, "Principles of Mathematics", 1903)

"The mathematical formula is the point through which all the light gained by science passes in order to be of use to practice; it is also the point in which all knowledge gained by practice, experiment, and observation must be concentrated before it can be scientifically grasped. The more distant and marked the point, the more concentrated will be the light coming from it, the more unmistakable the insight conveyed. All scientific thought, from the simple gravitation formula of Newton, through the more complicated formulae of physics and chemistry, the vaguer so called laws of organic and animated nature, down to the uncertain statements of psychology and the data of our social and historical knowledge, alike partakes of this characteristic, that it is an attempt to gather up the scattered rays of light, the different parts of knowledge, in a focus, from whence it can be again spread out and analyzed, according to the abstract processes of the thinking mind. But only when this can be done with a mathematical precision and accuracy is the image sharp and well-defined, and the deductions clear and unmistakable. As we descend from the mechanical, through the physical, chemical, and biological, to the mental, moral, and social sciences, the process of focalization becomes less and less perfect, - the sharp point, the focus, is replaced by a larger or smaller circle, the contours of the image become less and less distinct, and with the possible light which we gain there is mingled much darkness, the sources of many mistakes and errors. But the tendency of all scientific thought is toward clearer and clearer definition; it lies in the direction of a more and more extended use of mathematical measurements, of mathematical formulae." (John T Merz, "History of European Thought in the 19th Century" Vol. 1, 1904)

"[…] mathematical verities flow from a small number of self-evident propositions by a chain of impeccable reasonings; they impose themselves not only on us, but on nature itself. They fetter, so to speak, the Creator and only permit him to choose between some relatively few solutions. A few experiments then will suffice to let us know what choice he has made. From each experiment a number of consequences will follow by a series of mathematical deductions, and in this way each of them will reveal to us a corner of the universe. This, to the minds of most people, and to students who are getting their first ideas of physics, is the origin of certainty in science." (Henri Poincaré, "The Foundations of Science", 1913)

"Mathematics is merely an apparatus for analyzing the deductions which can be drawn from any particular premises, supplied by common sense, or by more refined scientific observation, so far as these deductions depend on the forms of the propositions." (Alfred N Whitehead, "The Organization of Thought", Science N.S Vol. 44 1134, 1916)

"The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them."(Albert Einstein, "Principles of Research", 1918)

"Accidental truth of a conclusion is no compensation for erroneous deduction." (Arthur Eddington, "Space, Time and Gravitation: An Outline of the General Relativity", 1920)

"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; "Die logischen Grundlagen der Mathematik." Mathematische Annalen 88 (1), 1923)

"The question whether any branch of science can ever become purely deductive is easily answered. It cannot. If science deals with the external world, as we believe it does, and not merely with the relations of propositions then no branch of science can ever be purely deductive. Deductive reasoning by itself can never tell us about facts. The use of deduction in science is to serve as a calculus to make our observations go further, not to take the place of observation." (Arthur D Ritchie, "Scientific Method: An Inquiry into the Character and Validity of Natural Laws", 1923)

Complexity vs Mathematics II

"[Mathematics] guides our minds in an orderly way, and furnishes us simple and rational principles by means of which ambiguities are clarified, disorder is converted into order, and complexities are analyzed into their component parts." (Johann B Mencken, "The Charlatanry of the Learned", 1715)

"These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences", 1840)

"The value of mathematical instruction as a preparation for those more difficult investigations, consists in the applicability not of its doctrines but of its methods. Mathematics will ever remain the past perfect type of the deductive method in general; and the applications of mathematics to the simpler branches of physics furnish the only school in which philosophers can effectually learn the most difficult and important of their art, the employment of the laws of simpler phenomena for explaining and predicting those of the more complex." (John S Mill, "System of Logic", 1843)

"It is certainly true that all physical phenomena are subject to strictly mathematical conditions, and mathematical processes are unassailable in themselves. The trouble arises from the data employed. Most phenomena are so highly complex that one can never be quite sure that he is dealing with all the factors until the experiment proves it. So that experiment is rather the criterion of mathematical conclusions and must lead the way." (Amos E Dolbear, "Matter, Ether, Motion", 1894)

"Mathematics, the science of the ideal, becomes the means of investigating, understanding and making known the world of the real. The complex is expressed in terms of the simple. From one point of view mathematics may be defined as the science of successive substitutions of simpler concepts for more complex [...]" (William F White, "A Scrap-book of Elementary Mathematics", 1908)

"A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to  applications to other departments of science." (Ernst W Hobson, Nature Vol. 84, [address] 1910)

"Elegance may produce the feeling of the unforeseen by the unexpected meeting of objects we are not accustomed to bring together; there again it is fruitful, since it thus unveils for us kinships before unrecognized. It is fruitful even when it results only from the contrast between the simplicity of the means and the complexity of the problem set; it makes us then think of the reason for this contrast and very often makes us see that chance is not the reason; that it is to be found in some unexpected law. In a word, the feeling of  mathematical elegance is only the satisfaction due to any adaptation of the solution to the needs of our mind, and it is because of this very adaptation that this solution can be for us an instrument. Consequently this esthetic satisfaction is bound up with the economy of thought." (Jules Henri Poincaré, "The Future of Mathematics", Monist Vol. 20, 1910)

"The mathematical laws presuppose a very complex elaboration. They are not known exclusively either a priori or a posteriori, but are a creation of the mind; and this creation is not an arbitrary one, but, owing to the mind’s resources, takes place with reference to experience and in view of it. Sometimes the mind starts with intuitions which it freely creates; sometimes, by a process of elimination, it gathers up the axioms it regards as most suitable for producing a harmonious development, one that is both simple and fertile. The mathematics is a voluntary and intelligent adaptation of thought to things, it represents the forms that will allow of qualitative diversity being surmounted, the moulds into which reality must enter in order to become as intelligible as possible." (Émile Boutroux, "Natural Law in Science and Philosophy", 1914)

"No equation, however impressive and complex, can arrive at the truth if the initial assumptions are incorrect." (Arthur C Clarke, "Profiles of the Future: An Inquiry into the Limits of the Possible", 1973)

"Economists are all too often preoccupied with petty mathematical problems of interest only to themselves. This obsession with mathematics is an easy way of acquiring the appearance of scientificity without having to answer the far more complex questions posed by the world we live in." (Thomas Piketty, Capital in the Twenty-First Century, 2013)

On Symbols (1910-1919)

"Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition." (Graham Fitch, "The Fourth Dimension simply Explained", 1910)

"Things and events explain themselves, and the business of thought is to brush aside the verbal and conceptual impediments which prevent them from doing so. Start with the notion that it is you who explain the Object, and not the Object that explains itself, and you are bound to end in explaining it away. It ceases to exist, its place being taken by a parcel of concepts, a string of symbols, a form of words, and you find yourself contemplating, not the thing, but your theory of the thing." (Lawrence P Jacks, "The Usurpation Of Language", 1910)

"A symbol which has not been properly defined is not a symbol at all. It is merely a blot of ink on paper which has an easily recognized shape. Nothing can be proved by a succession of blot, except the existence of a bad pen or a careless writer." (Alfred N Whitehead, "An Introduction to Mathematics", 1911)

"The symbols organized by knowledge, or concepts, themselves belong to social nature as its ideological elements. Therefore, by operating upon them, knowledge is able to expand its organizing function much more broadly than labour in its technological operation of real things; and as we have already seen that many things, which are not organized in practice, can be organized by knowledge, i.e. in symbols: where the ingression of things is absent, the ingression of their concepts is still possible." (Alexander A Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"This diagrammatic method has, however, serious inconveniences as a method for solving logical problems. It does not show how the data are exhibited by cancelling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i.e., 'throwing of the problem into an equation' and the transformation of the premises, nor with the subsequent steps, i.e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form." (Louis Couturat, "The Algebra of Logic", 1914)

"The rigor of mathematics is not absolute - absolute rigor is an ideal, to be, like other ideals, aspired unto, forever approached, but never quite attained, for such attainment would mean that every possibility of error or indetermination, however slight, had been eliminated from idea, from symbol, and from argumentation." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"In obedience to the feeling of reality, we shall insist that, in the analysis of propositions, nothing 'unreal' is to be admitted. But, after all, if there is nothing unreal, how, it may be asked, could we admit anything unreal? The reply is that, in dealing with propositions, we are dealing in the first instance with symbols, and if we attribute significance to groups of symbols which have no significance, we shall fall into the error of admitting unrealities, in the only sense in which this is possible, namely, as objects described." (Bertrand Russell, "Introduction to Mathematical Philosophy" , 1919)

On Method I (Mathematical Method)

"Mathematics is a science of Observation, dealing with reals, precisely as all other sciences deal with reals. It would be easy to show that its Method is the same: that, like other sciences, having observed or discovered properties, which it classifies, generalises, co-ordinates and subordinates, it proceeds to extend discoveries by means of Hypothesis, Induction, Experiment and Deduction." (George H Lewes, "Problems of Life and Mind: The Method of Science and its Application", 1874)

"The value of mathematical instruction as a preparation for those more difficult investigations, consists in the applicability not of its doctrines but of its methods. Mathematics will ever remain the past perfect type of the deductive method in general; and the applications of mathematics to the simpler branches of physics furnish the only school in which philosophers can effectually learn the most difficult and important of their art, the employment of the laws of simpler phenomena for explaining and predicting those of the more complex." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1843)

“Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols.” (Christian H Dillmann, "Die Mathematik die Fackelträgerin einer neuen Zeit", 1889)

"Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof." (David Hilbert, "Mathematical Problems", Bulletin of the American Mathematical Society Vol. 8, 1902)

“With the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and to cast aside some more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.” (David Hilbert, “Mathematical Problems”, Bulletin of the American Mathematical Society Vol. 8, 1902)

“Mathematics is merely a shorthand method of recording physical intuition and physical reasoning, but it should not be a formalism leading from nowhere to nowhere, as it is likely to be made by one who does not realize its purpose as a tool.” (Charles P Steinmetz, “Transactions of the American Institute of Electrical Engineers”, 1909)

"Mathematics is merely a shorthand method of recording physical intuition and physical reasoning, but it should not be a formalism leading from nowhere to nowhere, as it is likely to be made by one who does not realize its purpose as a tool." (Charles P Steinmetz, “Transactions of the American Institute of Electrical Engineers”, 1909)

"Mathematicians attach great importance to the elegance of their methods and their results. This is not pure dilettantism. What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details." (Henri Poincaré, "The Future of Mathematics", Monist Vol. 20, 1910)

“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)

"The mathematical method is the essence of mathematics. He who fully comprehends the method is a mathematician." (Friederich von Hardenberg [Novalis])

On Observation (1900-1919)

"To observe is not enough. We must use our observations, and to do that we must generalize." (Henri Poincaré, "Science and Hypothesis", 1902)

"An isolated sensation teaches us nothing, for it does not amount to an observation. Observation is a putting together of several results of sensation which are or are supposed to be connected with each other according to the law of causality, so that some represent causes and others their effects." (Thorvald N Thiele, "Theory of Observations", 1903)

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

"Unadulterated, unsweetened observations are what the real nature-lover craves. No man can invent incidents and traits as interesting as the reality." (John Burroughs, "Ways of Nature", 1905)

"Man's determination not to be deceived is precisely the origin of the problem of knowledge. The question is always and only this: to learn to know and to grasp reality in the midst of a thousand causes of error which tend to vitiate our observation." (Federigo Enriques, "Problems of Science", 1906)

"The motive for the study of mathematics is insight into the nature of the universe. Stars and strata, heat and electricity, the laws and processes of becoming and being, incorporate mathematical truths. If language imitates the voice of the Creator, revealing His heart, mathematics discloses His intellect, repeating the story of how things came into being. And the value of mathematics, appealing as it does to our energy and to our honor, to our desire to know the truth and thereby to live as of right in the household of God, is that it establishes us in larger and larger certainties. As literature develops emotion, understanding, and sympathy, so mathematics develops observation, imagination, and reason." (William E Chancellor, "A Theory of Motives, Ideals and Values in Education" 1907)

"In fact, we only attain laws by violating nature, by isolating more or less artificially a phenomenon from the whole, by checking those influences which would have falsified the observation. Thus the law cannot directly express reality. The phenomenon as it is envisaged by it, the ‘pure’ phenomenon, is rarely observed without our intervention, and even with this it remains imperfect, disturbed by accessory phenomena. […] Doubtless, if nature were not ordered, if it did not present us with similar objects, capable of furnishing generalized concepts, we could not formulate laws." (Emile Meyerson, "Identity and Reality", 1908)

"An experiment is an observation that can be repeated, isolated and varied. The more frequently you can repeat an observation, the more likely are you to see clearly what is there and to describe accurately what you have seen. The more strictly you can isolate an observation, the easier does your task of observation become, and the less danger is there of your being led astray by irrelevant circumstances, or of placing emphasis on the wrong point. The more widely you can vary an observation, the more clearly will be the uniformity of experience stand out, and the better is your chance of discovering laws." (Edward B Titchener, "A Text-Book of Psychology", 1909)

"It is experience which has given us our first real knowledge of Nature and her laws. It is experience, in the shape of observation and experiment, which has given us the raw material out of which hypothesis and inference have slowly elaborated that richer conception of the material world which constitutes perhaps the chief, and certainly the most characteristic, glory of the modern mind." (Arthur J Balfour, "The Foundations of Belief", 1912)

"Neither logic without observation, nor observation without logic, can move one step in the formation of science." (Alfred N Whitehead, "The Organization of Thought", 1916)

On Certainty (1900-1924)

"Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty." (Henri Poincaré, "Science and Hypothesis", 1902)

"The apodictic quality of mathematical thought, the certainty and correctness of its conclusions, are due, not to a special mode of ratiocination, but to the character of the concepts with which it deals. What is that distinctive characteristic? I answer: precision, sharpness, completeness of definition. But how comes your mathematician by such completeness? There is no mysterious trick involved; some ideas admit of such precision, others do not; and the mathematician is one who deals with those that do." (Cassius J Keyser, "The Universe and Beyond", Hibbert Journal Vol. 3, 1904–1905)

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

"No matter how solidly founded a prediction may appear to us, we are never absolutely sure that experiment will not contradict it, if we undertake to verify it . […] It is far better to foresee even without certainty than not to foresee at all." (Henri Poincaré, "The Foundations of Science", 1913)

"[…] mathematical verities flow from a small number of self-evident propositions by a chain of impeccable reasonings; they impose themselves not only on us, but on nature itself. They fetter, so to speak, the Creator and only permit him to choose between some relatively few solutions. A few experiments then will suffice to let us know what choice he has made. From each experiment a number of consequences will follow by a series of mathematical deductions, and in this way each of them will reveal to us a corner of the universe. This, to the minds of most people, and to students who are getting their first ideas of physics, is the origin of certainty in science." (Henri Poincaré, "The Foundations of Science", 1913)

"The very name calculus of probabilities is a paradox. Probability opposed to certainty is what we do not know, and how can we calculate what we do not know?" (Henri Poincaré, "The Foundations of Science", 1913)

"It would be a mistake to suppose that a science consists entirely of strictly proved theses, and it would be unjust to require this. […] Science has only a few apodeictic propositions in its catechism: the rest are assertions promoted by it to some particular degree of probability. It is actually a sign of a scientific mode of thought to find satisfaction in these approximations to certainty and to be able to pursue constructive work further in spite of the absence of final confirmation." (Sigmund Freud, "Introductory Lectures on Psycho-Analysis", 1916)

"Certitude is not the test of certainty. We have been cocksure of many things that were not so." (Oliver W Holmes Jr., "Natural Law", Harvard Law Review Vol. 32 (1), 1918)

On Discovery (1850-1899)

"To get to know, to discover, to publish - this is the destiny of a scientist." (François Arago, "De L’Utilité des Pensions", 1855)

"We learn wisdom from failure much more than from success. We often discover what will do, by finding out what will not do; and probably he who never made a mistake never made a discovery." (Samuel Smiles, “Facilities and Difficulties”, 1859)

"Nothing can be more puerile than the complaints sometimes made by certain cultivators of a science, that it is very difficult to make discoveries now that the soil has been exhausted, whereas they were so easily made when the ground was first broken. It is an error begotten by ignorance out of indolence. The first discovery did not drop upon the expectant idler who, with placid equanimity waited for the goods the gods might send, but was heavily obtained by patient, systematic, and intelligent labour; and, beyond all question, the same labour of the same mind which made the first discoveries in the new science, would now succeed in making many more, trampled though the field may be by the restless feet of those unmethodical inquirers who, running to and fro, anxiously exclaim, 'Who will show us any good thing?'" (George Gore, "Psychological Inquiries", Journal of Mental Science, 1862)

"The process of discovery is very simple. An unwearied and systematic application of known laws to nature, causes the unknown to reveal themselves. Almost any mode of observation will be successful at last, for what is most wanted is method." (Henry Thoreau, "A Week on the Concord and Merrimack Rivers", 1862)

"It has often been said that, to make discoveries, one must be ignorant. This opinion, mistaken in itself, nevertheless conceals a truth. It means that it is better to know nothing than to keep in mind fixed ideas based on theories whose confirmation we constantly seek, neglecting meanwhile everything that fails to agree with them." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"The discoverer and the poet are inventors; and they are so because their mental vision detects the unapparent, unsuspected facts, almost as vividly as ocular vision rests on the apparent and familiar." (George H Lewes, "Principles of Success in Literature", 1865)

"Every process has laws, known or unknown, according to which it must take place. A consciousness of them is so far from being necessary to the process, that we cannot discover what they are, except by analyzing the results it has left us." (Lord William T Kelvin , "An Outline of the Necessary Laws of Thought", 1866)

"It is notorious that the same discovery is frequently made simultaneously and quite independently, by different persons. […] It would seem, that discoveries are usually made when the time is ripe for them - that is to say, when the ideas from which they naturally flow are fermenting in the minds of many men." (Sir Francis Galton, "Hereditary Genius", 1869)

"Accurate and minute measurement seems to the nonscientific imagination a less lofty and dignified work than looking for something new. But nearly all the grandest discoveries of science have been but the rewards of accurate measurement and patient long contained labor in the minute sifting of numerical results." (William T Kelvin, "Report of the British Association For the Advancement of Science" Vol. 41, 1871)

"Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in Nature." (Augustus de Morgan, "A Budget of Paradoxes", 1872)

"Science arises from the discovery of Identity amid Diversity." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"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) 

"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. […] The truest theories involve suppositions which are inconceivable, and no limit can really be placed to the freedom of hypotheses." (W Stanley Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 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)

"The philosopher believes that the value of his philosophy lies in the whole, in the building: posterity discovers it in the bricks with which he built and which are then often used again for better building: in the fact, that is to say, that building can be destroyed and nonetheless possess value as material." (Friedrich Nietzsche, "Human, all-too-human", 1878)

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

"In that pure enjoyment experienced on approaching to the ideal, in that eagerness to draw aside the veil from the hidden truth, and even in that discord which exists between the various workers, we ought to see the surest pledges of further scientific success. Science thus advances, discovering new truths, and at the same time obtaining practical results." (Dmitry I Mendeleev, "The Principles of Chemistry" Vol. 1, 1891)

"All great scientists have, in a certain sense, been great artists; the man with no imagination may collect facts, but he cannot make great discoveries." (Karl Pearson, "The Grammar of Science", 1892)

"There is no subject more captivating, more worthy of study, than nature. To understand this great mechanism, to discover the forces which are active, and the laws which govern them, is the highest aim of the intellect of man." (Nikola Tesla, "The Inventions, Researches and Writings of Nikola Tesla|, 1894)

"It is they who hold the secret of the mysterious property of the mind by which error ministers to truth, and truth slowly but irrevocably prevails. Theirs is the logic of discovery, the demonstration of the advance of knowledge and the development of ideas, which as the earthly wants and passions of men remain almost unchanged, are the charter of progress, and the vital spark in history." (Lord John Acton, "The Study of History", [lecture delivered at Cambridge] 1895)

"The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us." (Paul Valéry, "Introduction to the Method of Leonardo da Vinci", 1895)

On Truth (2010-2019)

“A proof in mathematics is a compelling argument that a proposition holds without exception; a disproof requires only the demonstration of an exception. A mathematical proof does not, in general, establish the empirical truth of whatever is proved. What it establishes is that whatever is proved - usually a theorem - follows logically from the givens, or axioms.” (Raymond S Nickerson, “Mathematical Reasoning”, 2010)

“What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?” (Raymond S Nickerson, “Mathematical Reasoning:  Patterns, Problems, Conjectures, and Proofs”, 2010)

“A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked.” (Sara Negri  & Jan von Plato, “Proof Analysis”, 2011)

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

“Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress. Hence those who are fluent in this new language will be on the cutting edge of progress.” (Edward Frenkel, “Love and Math”, 2014)

"In mathematics, we often depend on the proof of a statement to offer not only a justification of its truth, but also a way of understanding its implications, its connections to other established truths - a way, in short of explaining the statement. But sometimes even though a proof does its job of showing the truth of a result it still leaves us with the nagging question of why.’ It may be elusive - given a specific proof - to describe in useful terms the type of explanation the proof actually offers. It would be good to have an adequate vocabulary to help us think about the explanatory features of mathematics (and, more generally, of science)." (Barry Mazur, "On the word ‘because’ in mathematics, and elsewhere", 2017)

“Scientists generally agree that no theory is 100 percent correct. Thus, the real test of knowledge is not truth, but utility.” (Yuval N Harari, “Sapiens: A brief history of humankind”, 2017) 

On Proofs (1990-1999)

"[...] mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.“ (Ivars Peterson, „Islands of Truth: A Mathematical Mystery Cruise“, 1990)

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

"Notice also that scientists generally avoid the use of the word proof. Evidence can support a hypothesis or a theory, but it cannot prove a theory to be true. It is always possible that in the future a new idea will provide a better explanation of the evidence." (James E McLaren, “Heath Biology”, 1991)

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 1993)

"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic. Only such a chain of deductions can establish the validity of a mathematical law, a theorem. And unless this process has been satisfactorily carried out, no relation - regardless of how often it may have been confirmed by observation - is allowed to become a law. It may be given the status of a hypothesis or a conjecture, and all kinds of tentative results may be drawn from it, but no mathematician would ever base definitive conclusions on it. (Eli Maor, "e: The Story of a Number", 1994)

“Mathematicians apparently don’t generally rely on the formal rules of deduction as they are thinking. Rather, they hold a fair bit of logical structure of a proof in their heads, breaking proofs into intermediate results so that they don’t have to hold too much logic at once. In fact, it is common for excellent mathematicians not even to know the standard formal usage of quantifiers (for all and there exists), yet all mathematicians certainly perform the reasoning that they encode.” (William P Thurston, “On Proof and Progress in Mathematics”, 1994)

"Mathematics is about theorems: how to find them; how to prove them; how to generalize them; how to use them; how to understand them. […] But great theorems do not stand in isolation; they lead to great theories. […] And great theories in mathematics are like great poems, great paintings, or great literature: it takes time for them to mature and be recognized as being 'great'." (John L Casti, "Five Golden Rules", 1995)

"The ingredient that knits this landscape together is proof. Proof determines the route from one fact to another. To professional mathematicians, no statement is considered valid unless it is proved beyond any possibility of logical error. But there are limits to what can be proved, and how it can be proved. A great deal of work in philosophy and the foundations of mathematics has established that you can't prove everything, because you have to start somewhere; and even when you've decided where to start, some statements may be neither provable nor disprovable." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

“What's so awful about using intuition or using inductive arguments? […] without them we would have virtually no mathematics at all; for, until the last few centuries, mathematics was advanced almost solely by intuition, inductive observation, and arguments designed to compel belief, not by laboured proofs, and certainly not through proofs of the ghastliness required by today's academic journals” (Jon MacKeman, “What's the point of proof?”, Mathematics Teaching 155, 1996)

"The lack of beauty in a piece of mathematics is of frequent occurrence, and it is a strong motivation for further mathematical research. Lack of beauty is associated with lack of definitiveness. A beautiful proof is more often than not the definitive proof (though a definitive proof need not be beautiful); a beautiful theorem is not likely to be improved upon or generalized." (Gian-Carlo Rota, "The phenomenology of mathematical proof", Synthese, 111(2), 1997)

"The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. […] A beautiful theorem may not be blessed with an equally beautiful proof; beautiful theorems with ugly proofs frequently occur. When a beautiful theorem is missing a beautiful proof, attempts are made by mathematicians to provide new proofs that will match the beauty of the theorem, with varying success. It is, however, impossible to find beautiful proofs of theorems that are not beautiful.” (Gian-Carlo Rota, “The Phenomenology of Mathematical Beauty”, 1997)

"The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof – which we know and can recognize, without the formal advice of the logicians. […] Intuition is glorious, but the heaven of mathematics requires much more. Physics has provided mathematics with many fine suggestions and new initiatives, but mathematics does not need to copy the style of experimental physics. Mathematics rests on proof - and proof is eternal." (Saunders Mac Lane, "Reponses to …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"In practice, proofs are simply whatever it takes to convince colleagues that a mathematical idea is true." (Claudia Henrion, "Women in Mathematics", 1997)

"Cleaning up old proofs is an important part of the mathematical enterprise that often yields new insights that can be used to solve new problems and build more beautiful and encompassing theories." (Bruce Schecter, "My Brain is Open", 1998)

See also:
Proofs I, II, III, IV, V, VI, VIII, IX
Theorems I, II, III, IV, V, VI, VII, VIII, IX, X

Mental Models XI

"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)

"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)

"For of an image one requires some kind of sameness with the pictured object, of a statue sameness of form, of a delineation sameness of perspective projection in the visual field, of a painting also sameness of color."  (Heinrich Hertz, "The Facts in Perception", 1878) 

"[...] the task of the theory consists in constructing a picture of the external world that exists purely internally and must be our guiding star in all thought and experiment; that is in completing, as it were, the thinking process and carrying out globally what on a small scale occurs within us whenever we form an idea.” (Ludwig E Boltzmann, “On the Significance of Theories”, 1890) 

“We form ourselves images or symbols of external objects; and the form which we give them is such that the necessary consequents of the images in thought are always the images of the necessary consequents in nature of the things pictured." (Heinrich Hertz," Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt", 1894)

"An icon is a representamen of what it represents and for the mind that interprets it as such, by  virtue of its being an immediate image, that is to say by virtue of characters which belong to it in itself as a sensible object, and which it would possess just the same were there no object in nature that it resembled, and though it never were interpreted as a sign. It is of the nature of an appearance, and as such, strictly speaking, exists only in consciousness, although for convenience in ordinary parlance and when extreme precision is not called for, we extend the term icon to the outward objects which excite in consciousness the image itself." (Charles S Peirce, “On Existential Graphs, Euler's Diagrams, and Logical”, 1903)

“Deduction is that mode of reasoning which examines the state of things asserted in the premises, forms a diagram of that state of things, perceives in the parts of the diagram relations not explicitly mentioned in the premises, satisfies itself by mental experiments upon the diagram that these relations would always subsist, or at least would do so in a certain proportion of cases, and concludes their necessary, or probable, truth.” (Charles S Peirce, “Kinds of Reasoning”, cca. 1896)

"Intellectual work is an act of creation. It is as if the mental image that is studied over a period of time were to sprout appendages like an ameba - outgrowths that extend in all directions while avoiding one obstacle after another - before interdigitating with related ideas." (Santiago Ramón y Cajal, "Reglas y Consejos sobre Investigacíon Cientifica: Los tónicos de la voluntad", 1897)

“We imagine cases, place mental diagrams before our mind's eye, and multiply these cases, until a habit is formed of expecting that always to turn out the case, which has been seen to be the result in all the diagrams. To appeal to such a habit is a very different thing from appealing to any immediate instinct of rationality. That the process of forming a habit of reasoning by the use of diagrams is often performed there is no room for doubt. It is perfectly open to consciousness.” (Charles S Peirce, “Fallibility of Reasoning and the Feeling of Rationality”, cca. 1902)

Mathematical Truth II

“The mathematician is perfect only in so far as he is a perfect man, in so far as he senses in himself the beauty of truth; only then will his work be thorough, transparent, prudent, pure, clear, graceful, indeed elegant.” (Plato)

"In Pure Mathematics, where all the various truths are necessarily connected with each other, (being all necessarily connected with those hypotheses which are the principles of the science), an arrangement is beautiful in proportion as the principles are few; and what we admire perhaps chiefly in the science, is the astonishing variety of consequences which may be demonstrably deduced from so small a number of premises.” (Dugald Stewart, “Elements of the Philosophy of the Human Mind" Vol. 3, 1827)

“The aim of proof is, in fact, not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of one truth upon another After we have convinced ourselves that a boulder is immovable, by trying unsuccessfully to move it, there remains the further question, what is it that supports it so securely." (Gottlob Frege, "The Foundations of Arithmetic", 1884)

“It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all.” (James J Sylvester, "On certain inequalities related to prime numbers",  Nature Vol. 38, 1888)

"The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connection of its parts, the infinite hierarchy and absolute evidence of the truths with which mathematical science is concerned, these, and such like, are the surest groimds of its title of human regard, and would remain unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.” (James J Sylvester, "A Plea for the Mathematician", Nature, 1908)

"Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted." (Henri Poincaré," Science and Method", 1908)

“[…] because mathematics contains truth, it extends its validity to the whole domain of art and the creatures of the constructive imagination.” (James B Shaw, “Lectures on the Philosophy of Mathematics”, 1918)

"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; “Die logischen Grundlagen der Mathematik.“ Mathematische Annalen 88 (1), 1923)

“Mathematics is the science of number and space. It starts from a group of self-evident truths and by infallible deduction arrives at incontestable conclusions […] the facts of mathematics are absolute, unalterable, and eternal truths.” (E Russell Stabler, “An Interpretation and Comparison of Three Schools of Thought in the Foundations of Mathematics”, The Mathematics Teacher Vol 26, 1935)