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 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 Engineering I

"In fact 'engineering' now often signifies a new system of thought, a fresh method of attack upon the world’s problems the antithesis of traditionalism, with its precedents and dogmas. (Alfred D Flinn, "Leadership in Economic Progress", Civil Engineering Vol. 2 (4), 1932)

"There may be said to be two kinds of engineering, that which is essentially creative, and that which is practiced in pursuit of known methods." (William L Emmet, "The Autobiography of an Engineer", 1940)

"Science acquires knowledge but has no interest in its practical applications. The applications are the work of engineers." (Edwin P Hubble, "The Nature of Science and Other Lectures", 1954)

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

"Science aims at the discovery, verification, and organization of fact and information [...] engineering is fundamentally committed to the translation of scientific facts and information to concrete machines, structures, materials, processes, and the like that can be used by men." (Eric A Walker, "Engineers and/or Scientists", Journal of Engineering Education Vol. 51, 1961)

"What, then, is science according to common opinion? Science is what scientists do. Science is knowledge, a body of information about the external world. Science is the ability to predict. Science is power, it is engineering. Science explains, or gives causes and reasons." (John Bremer "What Is Science?" [in "Notes on the Nature of Science"], 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)

"Engineering is a method and a philosophy for coping with that which is uncertain at the earliest possible moment and to the ultimate service to mankind. It is not a science struggling for a place in the sun. Engineering is extrapolation from existing knowledge rather than interpolation between known points. Because engineering is science in action - the practice of decision making at the earliest moment - it has been defined as the art of skillful approximation. No situation in engineering is simple enough to be solved precisely, and none worth evaluating is solved exactly. Never are there sufficient facts, sufficient time, or sufficient money for an exact solution, for if by chance there were, the answer would be of academic and not economic interest to society. These are the circumstances that make engineering so vital and so creative." (Ronald B Smith, "Engineering Is…", Mechanical Engineering Vol. 86 (5), 1964)

"Engineering is the conscious application of science to the problems of economic production." (Halbert P Gillette)

"Engineering is the professional and systematic application of science to the efficient utilization of natural resources to produce wealth." (T J Hoover & J C L Fish)

On Artificial Intelligence I

"There is no security against the ultimate development of mechanical consciousness, in the fact of machines possessing little consciousness now. A mollusc has not much consciousness. Reflect upon the extraordinary advance which machines have made during the last few hundred years, and note how slowly the animal and vegetable kingdoms are advancing. The more highly organized machines are creatures not so much of yesterday, as of the last five minutes, so to speak, in comparison with past time." (Samuel Butler, "Erewhon: Or, Over the Range", 1872)

"A computer would deserve to be called intelligent if it could deceive a human into believing that it was human." (Alan Turing, "Computing Machinery and Intelligence", 1950)

"The following are some aspects of the artificial intelligence problem: […] If a machine can do a job, then an automatic calculator can be programmed to simulate the machine. […] It may be speculated that a large part of human thought consists of manipulating words according to rules of reasoning and rules of conjecture. From this point of view, forming a generalization consists of admitting a new word and some rules whereby sentences containing it imply and are implied by others. This idea has never been very precisely formulated nor have examples been worked out. […] How can a set of (hypothetical) neurons be arranged so as to form concepts. […] to get a measure of the efficiency of a calculation it is necessary to have on hand a method of measuring the complexity of calculating devices which in turn can be done. […] Probably a truly intelligent machine will carry out activities which may best be described as self-improvement. […] A number of types of 'abstraction' can be distinctly defined and several others less distinctly. […] the difference between creative thinking and unimaginative competent thinking lies in the injection of a some randomness. The randomness must be guided by intuition to be efficient." (John McCarthy et al, "A Proposal for the Dartmouth Summer Research Project on Artificial Intelligence", 1955)

"We shall therefore say that a program has common sense if it automatically deduces for itself a sufficient wide class of immediate consequences of anything it is told and what it already knows. [...] Our ultimate objective is to make programs that learn from their experience as effectively as humans do." (John McCarthy, "Programs with Common Sense", 1958)

"When intelligent machines are constructed, we should not be surprised to find them as confused and as stubborn as men in their convictions about mind-matter, consciousness, free will, and the like." (Marvin Minsky, "Matter, Mind, and Models", Proceedings of the International Federation of Information Processing Congress Vol. 1 (49), 1965)

"Artificial intelligence is the science of making machines do things that would require intelligence if done by men." (Marvin Minsky, 1968)

"Artificial intelligence is based on the assumption that the mind can be described as some kind of formal system manipulating symbols that stand for things in the world. Thus it doesn't matter what the brain is made of, or what it uses for tokens in the great game of thinking. Using an equivalent set of tokens and rules, we can do thinking with a digital computer, just as we can play chess using cups, salt and pepper shakers, knives, forks, and spoons. Using the right software, one system (the mind) can be mapped onto the other (the computer)." (George Johnson, Machinery of the Mind: Inside the New Science of Artificial Intelligence, 1986)

"The deep paradox uncovered by AI research: the only way to deal efficiently with very complex problems is to move away from pure logic. [...] Most of the time, reaching the right decision requires little reasoning.[...] Expert systems are, thus, not about reasoning: they are about knowing. [...] Reasoning takes time, so we try to do it as seldom as possible. Instead we store the results of our reasoning for later reference." (Daniel Crevier, "The Tree of Knowledge", 1993)

On Intuition (1990-1999)

 “Every theoretical explanation is a reduction of intuition.” (Peter Høeg,  “Smilla's Sense of Snow”, 1992)

“Intuition is the art, peculiar to the human mind, of working out the correct answer from data that is, in itself, incomplete or even, perhaps, misleading.” (Isaac Asimov, “Forward the Foundation”, 1993)

“All this points to an appealing intuition: that a faculty for analogical reasoning is an innate part of human cognition.” (Dedre Gentner & Michael Jeziorski, “Western science”, 1993)

“People have amazing facilities for sensing something without knowing where it comes from (intuition); for sensing that some phenomenon or situation or object is like something else (association); and for building and testing connections and comparisons, holding two things in mind at the same time (metaphor). These facilities are quite important for mathematics. Personally, I put a lot of effort into ‘listening’ to my intuitions and associations, and building them into metaphors and connections. This involves a kind of simultaneous quieting and focusing of my mind. Words, logic, and detailed pictures rattling around can inhibit intuitions and associations.” (William P Thurston, “On proof and progress in mathematics”, Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

“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 Lan, “Reponses to …”,m Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

“Many pages have been expended on polemics in favor of rigor over intuition, or of intuition over rigor. Both extremes miss the point: the power of mathematics lies precisely in the combination of intuition and rigor.” (Ian Stewart, “Concepts of Modern Mathematics”, 1995)

 “Physics is not difficult; it’s just weird. […] Physics is weird because intuition is false. To understand what an electron’s world is like, you’ve got to be an electron, or jolly nearly. Intuition is forged in the hellish fires of the everyday world, which makes it so eminently useful in our daily struggle for survival. For anything else, it is hopeless.” (Vincent Icke, “The Force of Symmetry”, 1995)

“The entrepreneur's instinct is to exploit the natural world. The engineer's instinct is to change it. The scientist's instinct is to try to understand it - to work out what's really going on. The mathematician's instinct is to structure that process of understanding by seeking generalities that cut across the obvious subdivisions.” (Ian Stewart, “Nature's Numbers”, 1995)

“Scientists reach their  conclusions  for the damnedest of reasons: intuition, guesses, redirections after wild-goose chases, all combing with a dollop of rigorous observation and logical  reasoning to be sure […] This  messy and personal side of science should not be  disparaged, or covered up, by  scientists for two  major reasons. First, scientists should proudly show this  human face to  display their kinship with all other  modes of creative human thought […] Second, while biases and references often impede understanding, these  mental idiosyncrasies  may  also serve as powerful, if  quirky and personal, guides to solutions.” (Stephen J Gould, “Dinosaur in a  Haystack: Reflections in natural  history”, 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)

“Intuition isn't direct perception of something external. It's the effect in the mind/brain of manipulating concrete objects - at a later stage, of making marks on paper, and still later, manipulating mental images. This experience leaves a trace, an effect, in your mind/brain.” (Reuben Hersh, “What Is Mathematics, Really?”, 1998)

On Intuition (1925-1949)

“Intuition does not denote something contrary to reason, but something outside of the province of reason.” (Carl Gustav Jung, “Psychological types: or, The psychology of individuation”, 1926)

"As the objects of abstract geometry cannot be totally grasped by space intuition, a rigorous proof in abstract geometry can never be based only on intuition, but it must be founded on logical deduction from valid and precise axioms. Nevertheless intuition maintains, also in precision geometry, its irreplaceable value that cannot be substituted by logical considerations. Intuition helps us to construct a proof and to gain an overview, it is, moreover, a source of inventions and new mental connections." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

“Scientific hypotheses are intuitive leaps in the dark.” (Alexander Goldenweiser, “Robots or Gods: An Essay on Craft and Mind”, American Journal of Sociology 37 (3), 1931)

“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)

“Men of science belong to two different types - the logical and the intuitive. Science owes its progress to both forms of minds. Mathematics, although a purely logical structure, nevertheless makes use of intuition. “ (Alexis Carrel, “Man the Unknown”, 1935)

“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)

“When a scientist is ahead of his times, it is often through misunderstanding of current, rather than intuition of future truth. In science there is never any error so gross that it won't one day, from some perspective, appear prophetic.” (Jean Rostand, “Pensées d'un Biologiste”,1939)

 “Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings.” (Alan Turing, "Systems of Logic Based on Ordinals”, Proceedings of the London Mathematical Society Vol 45 (2), 1939)

“The intellect is at home in that which is fixed only because it is done and over with, for intellect is itself just as much a deposit of past life as is the matter to which it is congenial. Intuition alone articulates in the forward thrust of life and alone lays hold of reality.” (John Dewey, “Time and Individuality”, 1940)

“Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science." (Richard Courant & Herbert Robbins, “What Is Mathematics?”, 1941)

“It is his intuition, his mystical insight into the nature of things, rather than his reasoning which makes a great scientist.” (Karl R Popper, “The Open Society and Its Enemies”, 1945)

"But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perception will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in future." (Kurt Gödel, "What is Cantor’s Continuum problem?", American Mathematical Monthly 54, 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)

Isn't it Obvious? - Part II

“It [science] involves an intelligent and persistent endeavor to revise current beliefs so as to weed out what is erroneous, to add to their accuracy, and, above all, to give them such shape that the dependencies of the various facts upon one another may be as obvious as possible.” (John Dewey, “Democracy and Education”, 1916)

“The reason the obvious in Nature goes unrecognized so long is because artful man misdoubts her plain message and in a spirit of subtlety reads in a farrago of irrelevancies between the lines.”  (George H Lepper, “From Nebula to Nebula”, 1917)

”It requires a very unusual mind to undertake the analysis of the obvious.” (Alfred N Whitehead, “Science in the Modern World”, 1925)

“’Obvious’ is the most dangerous word in mathematics.” (Eric T Bell, “Mathematics: Queen and Servant of Science”, 1951)

”The obvious is that which is never seen until someone expresses it simply.” (Kahlil Gibran, “Sand and Foam: A Book of Aphorisms”, 1959)

“Significant advances in science often have a peculiar quality: they contradict obvious, commonsense opinions.” (Salvador E Luria, “A Slot Machine, a Broken Test Tube”, 1984)

“[…] creativity is the ability to see the obvious over the long term, and not to be restrained by short-term conventional wisdom.” (Arthur J Birch, “To See the Obvious”, 1995)

On Theorems (1970-1979)

“In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.” (Martin Gardner, “Mathematical Games”, Scientific American, 1973)

“The world is anxious to admire that apex and culmination of modern mathematics: a theorem so perfectly general that no particular application of it is feasible.” (George Pólya, “A Story With a Moral”, Mathematical Gazette 57 (400), 1973) 

 „For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.“ (Michael Frayn, „Constructions“, 1974)

“Mathematics does not grow through a monotonous increase of the number of indubitably established theorems, but through the incessant improvement of guesses by speculation and criticism.” (Imre Lakatos, "Proofs and Refutations", 1976)

“The esthetic side of mathematics has been of overwhelming importance throughout its growth. It is not so much whether a theorem is useful that matters, but how elegant it is.” (Stanislaw Ulam “Adventures of a Mathematician”, 1976)

“At the heart of mathematics is a constant search for simpler and simpler ways to prove theorems  and solve problems.” (Martin Gardner, “Aha! Insight”, 1978)

 “A good theorem will almost always have a wide-ranging influence on later mathematics, simply by virtue of the fact that it is true. Since it is true, it must be true for some reason; and if that reason lies deep, then the uncovering of it will usually require a deeper understanding of neighboring facts and principles.” (Ian Richards, “Number theory”, 1978)

„[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility.“ (Lynn A Steen, „Mathematics Today: Twelve Informal Essays“, 1978)

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

On Theorems (1950-1969)

“On the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem-proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory.” (Kurt Gödel, 1951)

"Mathematicians do not know what they are talking about because pure mathematics is not concerned with physical meaning. Mathematicians never know whether what they are saying is true because, as pure mathematicians, they make no effort to ascertain whether their theorems are true assertions about the physical world." (Morris Kline, “Mathematics in Western Culture”, 1953)

"The construction of hypotheses is a creative act of inspiration, intuition, invention; its essence is the vision of something new in familiar material. The process must be discussed in psychological, not logical, categories; studied in autobiographies and biographies, not treatises on scientific method; and promoted by maxim and example, not syllogism or theorem." (Milton Friedman, "Essays in Positive Economics", 1953)

“You have to guess the mathematical theorem before you prove it: you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies: you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing” (George Polya, “Mathematics and plausible reasoning” Vol. 1, 1954)

“Mathematics, springing from the soil of basic human experience with numbers and data and space and motion, builds up a far-flung architectural structure composed of theorems which reveal insights into the reasons behind appearances and of concepts which relate totally disparate concrete ideas.” (Saunders MacLane, “Of Course and Courses”, The American Mathematical Monthly, Vol. 61, No. 3, March, 1954)

”Mathematics is a creation of the mind. To begin with, there is a collection of things, which exist only in the mind, assumed to be distinguishable from one another; and there is a collection of statements about these things, which are taken for granted. Starting with the assumed statements concerning these invented or imagined things, the mathematician discovers other statements, called theorems, and proves them as necessary consequences. This, in brief, is the pattern of mathematics. The mathematician is an artist whose medium is the mind and whose creations are ideas.” (Hubert Stanley Wall, “Creative Mathematics”, 1963)

“So the first thing we have to accept is that even in mathematics you can start in different places. If all these various theorems are interconnected by reasoning there is no real way to say ‘These are the most fundamental axioms’, because if you were told something different instead you could also run the reasoning the other way. It is like a bridge with lots of members, and it is over-connected; if pieces have dropped out you can reconnect it another way.” (Richard Feynman, “The Character of Physical Law”, 1965)

"A mathematical proof, as usually written down, is a sequence of expressions in the state space. But we may also think of the proof as consisting of the sequence of justifications of consecutive proof steps - i.e., the references to axioms, previously-proved theorems, and rules of inference that legitimize the writing down of the proof steps. From this point of view, the proof is a sequence of actions (applications of rules of inference) that, operating initially on the axioms, transform them into the desired theorem." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

“A theorem is no more proved by logic and computation than a sonnet is written by grammar and rhetoric, or than a sonata is composed by harmony and counterpoint, or a picture painted by balance and perspective.” (George Spencer-Brown, “Laws of Form”, 1969)

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

On Beauty: Beauty and Mathematics (1975-1999)

"[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility." (Lynn A Steen, „Mathematics Today: Twelve Informal Essays", 1978)

"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) 

"The test of the intelligibility of any statement that overwhelms us with its air of profundity is its translatability into language that lacks the elevation and verve of the original statement but can pass muster as a simple and clear statement in ordinary, everyday speech. Most of what has been written about beauty will not survive this test. In the presence of many of the most eloquent statements about beauty, we are left speechless - speechless in the sense that we cannot find other words for expressing what we think or hope we understand." (Mortimer J Adler, Six Great Ideas, 1981)

"[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility." (Lynn A Steen," Mathematics Today: Twelve Informal Essays", 1978)

"Perhaps the best way to approach the question of what mathematics is, is to start at the beginning. In the far distant prehistoric past, where we must look for the beginnings of mathematics, there were already four major faces of mathematics. First, there was the ability to carry on the long chains of close reasoning that to this day characterize much of mathematics. Second, there was geometry, leading through the concept of continuity to topology and beyond. Third, there was number, leading to arithmetic, algebra, and beyond. Finally there was artistic taste, which plays so large a role in modern mathematics. There are, of course, many different kinds of beauty in mathematics. In number theory it seems to be mainly the beauty of the almost infinite detail; in abstract algebra the beauty is mainly in the generality. Various areas of mathematics thus have various standards of aesthetics." (Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly Vol. 87 (2), 1980)

"In lieu of the traditional confrontation between theory and experiment, superstring theorists pursue an inner harmony where elegance, uniqueness and beauty define truth. The theory depends for its existence upon magical coincidences, miraculous cancellations and relations among seemingly unrelated (and possibly undiscovered) fields of mathematics." (Sheldon L Glashow, "Desperately Seeking Superstrings?", Physics Today, 1986)

"The principle of mathematical beauty, like related aesthetic principles, is problematical. The main problem is that beauty is essentially subjective and hence cannot serve as a commonly defined tool for guiding or evaluating science. It is, to say the least, difficult to justify aesthetic judgment by rational arguments. Within literary and art criticism there is, indeed, a long tradition of analyzing the idea of beauty, including many attempts to give the concept an objective meaning. Objectivist and subjectivist theories of aesthetic judgment have been discussed for centuries without much progress, and today the problem seems as muddled as ever. Apart from the confused state of art in aesthetic theory, it is uncertain to what degree this discussion is relevant to the problem of scientific beauty. I, at any rate, can see no escape from the conclusion that aesthetic judgment in science is rooted in subjective and social factors. The sense of aesthetic standards is pan of the socialization that scientists acquire; but scientists, as well as scientific communities, may have widely different ideas of how to judge the aesthetic merit of a particular theory. No wonder that eminent physicists do not agree on which theories are beautiful and which are ugly." (Helge Kragh, 1990)

"Order wherever it reigns, brings beauty with it. Theory not only renders the group of physical laws it represents easier to handle, more convenient, and more useful, but also more beautiful." (Pierre Maurice Marie Duhem, "The Aim and Structure of Physical Theory", 1991) 

"The material world begins to seem so trivial, so arbitrary, so ephemeral when contrasted with the timeless beauty of mathematics." (William Dunham, "The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities", 1994)

"Mathematics-as-science naturally starts with mysterious phenomena to be explained, and leads (if you are successful) to powerful and harmonious patterns. Mathematics-as-a-game not only starts with simple objects and rules, but involves all the attractions of games like chess: neat tactics, deep strategy, beautiful combinations, elegant and surprising ideas. Mathematics-as-perception displays the beauty and mystery of art in parallel with the delight of illumination, and the satisfaction of feeling that now you understand." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"There is much beauty in nature's clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"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)

 "Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations." (James R Brown, “Philosophy of Mathematics”, 1999)

"The spirit of mathematics and the essence of its beauty is remarkably fragile, because mathematics is about ideas and about thought. Mathematics takes place in the mind, and no two minds are the same. After many years of study and work, a mathematician may stumble on a vast and beautiful vista that unifies and simplifies many things that once appeared disparate and complicated. Mathematicians can share a beautiful mathematical vista with one another, but there is no camera that can easily capture an image of such a vista to convey it in full to people who have not trudged along many of the same trails." (Silvio Levy, "The Eightfold Way: The Beauty of Klein’s Quartic Curve", 1999)

Poets and Mathematicians

"[…] it is impossible to be a mathematician without being a poet in soul […] imagination and invention are identical […] the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing." (Sophia Kovalevskaya)

“A mathematician who is not also something of a poet will never be a complete mathematician.” (Karl Weierstrass)

"A mathematician, like a painter or a poet, is a maker of patterns. [...]. The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." (Godfrey Harold Hardy, “A Mathematician's Apology”, 1941)

"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." (William James)

“The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.” (Gilbert Keith Chesterton)

"Imagination does not breed insanity. Exactly what does breed insanity is reason. Poets do not go mad […] mathematicians go mad.” (Gilbert Keith Chesterton)

"The imagination in a mathematician who creates makes no less difference than in a poet who invents […]." (Jean Le Rond D'Alembert, Discours Preliminaire de L'Encyclopedie, 1967)

"[…] mathematicians and poets are people who believe in the power of words, of concepts and giving names to concepts" (Cédric Villani)