On Art X: Artists

"The one thing that marks the true artist is a clear perception and a firm, bold hand, in distinction from that imperfect mental vision and uncertain truth which give up the feeble pictures and the lumpy statues of the mere artisans on canvas or in stone." (Oliver W Holmes, "The Professor at the Breakfast Table Ticknor and Fields", 1860)

"A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. [...] we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it." (Henri Poincaré, "Notice sur Halphen", Journal de l'École Polytechnique, 1890)

"The true mathematician is always a great deal of an artist, an architect, yes, of a poet. Beyond the real world, though perceptibly connected with it, mathematicians have created an ideal world which they attempt to develop into the most perfect of all worlds, and which is being explored in every direction. None has the faintest conception of this world except him who knows it; only presumptuous ignorance can assert that the mathematician moves in a narrow circle. The truth which he seeks is, to be sure, broadly considered, neither more nor less than consistency; but does not his mastership show, indeed, in this very limitation? To solve questions of this kind he passes unenviously over others." (Alfred Pringsheim, Jaresberichte der Deutschen Mathematiker Vereinigung Vol 13, 1904)

"Imagine any sort of model and a copy of it done by an awkward artist: the proportions are altered, lines drawn by a trembling hand are subject to excessive deviation and go off in unexpected directions. From the point of view of metric or even projective geometry these figures are not equivalent, but they appear as such from the point of view of geometry of position [that is, topology]." (Henri Poincaré, "Dernières pensées", 1920)

"There is a science of simple things, an art of complicated ones. Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to it. The artist works out his own formulas; the interest of science lies in the art of making science." (Paul Valéry, "Moralités", 1932)

"Today the function of the artist is to bring imagination to science and science to imagination, where they meet, in the myth." (Cyril Connolly, The Unquiet Grave, 1945)

"Nature does not seem full of circles and triangles to the ungeometrical; rather, mastery of the theory of triangles and circles, and later of conic sections, has taught the theorist, the experimenter, the carpenter, and even the artist to find them everywhere, from the heavenly motions to the pose of a Venus." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"For some years now the activity of the artist in our society has been trending more toward the function of the ecologist: one who deals with environmental relationships. Ecology is defined as the totality or pattern of relations between organisms and their environment. Thus the act of creation for the new artist is not so much the invention of new objects as the revelation of previously unrecognized relation- ships between existing phenomena, both physical and metaphysical. So we find that ecology is art in the most fundamental and pragmatic sense, expanding our apprehension of reality." (Gene Youngblood, "Expanded Cinema", 1970)

"Though we can say that mathematics is not art, some mathematicians think of themselves as artists of pure form. It seems clear, however, that their elegant and near aesthetic forms fail as art, because they are secondary visual ideas, the product of an intellectual set of restraints, rather than the cause of a felt insight realized in and through visual form." (Robert E Mueller, "Idols of Computer Art", 1972) 

"How often people speak of art and science as though they were two entirely different things, with no interconnection. An artist is emotional, they think, and uses only his intuition; he sees all at once and has no need of reason. A scientist is cold, they think, and uses only his reason; he argues carefully step by step, and needs no imagination. That is all wrong. The true artist is quite rational as well as imaginative and knows what he is doing; if he does not, his art suffers. The true scientist is quite imaginative as well as rational, and sometimes leaps to solutions where reason can follow only slowly; if he does not, his science suffers." (Isaac Asimov, "The Roving Mind", 1983)

"Literature is as much a product of the technological and scientific milieu as it is of the artistic one. Some of the large ideas, call them theories or metaphors - that humans are machines, that the observer affects the phenomenon observed, that information can be quantified - alter the way work is done in art. Metaphors invented by artists imply new ways of seeing, demolish mere logic, provoke alternatives, and lead to new theories in science." (David Porush, "The Soft Machine", 1985)

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

"[...] in science there are collectors, classifiers, compulsory tidiers-up and permanent contesters, detectives, some artists and many artisans, there are poet-scientists and philosophers and even a few mystics." (Rolf M Zinkernagel, [Nobel lecture] 1996)

"'Doing mathematics' is thus working on the construction of some mathematical object and resembles other creative enterprises of the mind in a scientific or artistic domain. But while the mental exercise of creating mathematics is somehow related to that of creating art, it should remain clear that mathematical objects are very different from the artistic objects that occur in literature, music, or the visual arts." (David Ruelle, "The Mathematician's Brain", 2007)

"It is ironic but true: the one reality science cannot reduce is the only reality we will ever know. This is why we need art. By expressing our actual experience, the artist reminds us that our science is incomplete, that no map of matter will ever explain the immateriality of our consciousness." (Jonah Lehrer, "Proust Was a Neuroscientist", 2011)

On Art IX: Metaphysics

"Metaphysics may be, after all, only the art of being sure of something that is not so and logic only the art of going wrong with confidence." (Joseph W Krutch, "The Modern Temper", 1929)

"As to the role of emotions in art and the subconscious mechanism that serves as the integrating factor both in artistic creation and in man's response to art, they involve a psychological phenomenon which we call a sense of life. A sense of life is a pre-conceptual equivalent of metaphysics, an emotional, subconsciously integrated appraisal of man and of existence." (Ayn Rand, "The Romantic Manifesto: A Philosophy of Literature", 1969)

"For some years now the activity of the artist in our society has been trending more toward the function of the ecologist: one who deals with environmental relationships. Ecology is defined as the totality or pattern of relations between organisms and their environment. Thus the act of creation for the new artist is not so much the invention of new objects as the revelation of previously unrecognized relation- ships between existing phenomena, both physical and metaphysical. So we find that ecology is art in the most fundamental and pragmatic sense, expanding our apprehension of reality." (Gene Youngblood, "Expanded Cinema", 1970)

"The mystery of sound is mysticism; the harmony of life is religion. The knowledge of vibrations is metaphysics, the analysis of atoms is science, and their harmonious grouping is art. The rhythm of form is poetry, and the rhythm of sound is music. This shows that music is the art of arts and the science of all sciences; and it contains the fountain of all knowledge within itself." (Inayat Khan, "The Mysticism of Sound and Music", 1996)

 "Science, as usually taught to liberal arts students, emphasizes results rather than method, and tries to teach technique rather than to give insight into and understanding of the scientific habit of thought. What is needed, however, is not a dose of metaphysics but a truly humanistic teaching of science." (Harry D Gideonse)

On Art VIII: Physics

"It is impossible to follow the march of one of the greatest theories of physics, to see it unroll majestically its regular deductions starting from initial hypotheses, to see its consequences represent a multitude of experimental laws down to the smallest detail, without being charmed by the beauty of such a construction, without feeling keenly that such a creation of the human mind is truly a work of art." (Pierre-Maurice-Marie DuhemDuhem, "The Aim and Structure of Physical Theory", 1908)

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

"Mathematicians who build new spaces and physicists who find them in the universe can profit from the study of pictorial and architectural spaces conceived and built by men of art." (György Kepes, "The New Landscape In Art and Science", 1956)

"A more problematic example is the parallel between the increasingly abstract and insubstantial picture of the physical universe which modern physics has given us and the popularity of abstract and non-representational forms of art and poetry. In each case the representation of reality is increasingly removed from the picture which is immediately presented to us by our senses." (Harvey Brooks, "Scientific Concepts and Cultural Change", 1965)

"Part of the art and skill of the engineer and of the experimental physicist is to create conditions in which certain events are sure to occur." (Eugene P Wigner, "Symmetries and Reflections", 1979)

"There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depends heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood." (Paul Lockhart, "A Mathematician's Lament", 2009)

On Art VII: Knowledge

"The invention of what we may call primary or fundamental notation has been but little indebted to analogy, evidently owing to the small extent of ideas in which comparison can be made useful. But at the same time analogy should be attended to, even if for no other reason than that, by making the invention of notation an art, the exertion of individual caprice ceases to be allowable. Nothing is more easy than the invention of notation, and nothing of worse example and consequence than the confusion of mathematical expressions by unknown symbols. If new notation be advisable, permanently or temporarily, it should carry with it some mark of distinction from that which is already in use, unless it be a demonstrable extension of the latter." (Augustus De Morgan, "Calculus of Functions" Encyclopaedia of Pure Mathematics, 1847)

"A metaphor holds a truth and an untruth, felt as inextricably bound up with each other. If one takes it as it is and gives it some sensual form, in the shape of reality, one gets dreams and art; but between these two and real, full-scale life there is a glass partition. If one analyzes it for its rational content and separates the unverifiable from the verifiable, one gets truth and knowledge but kills the feeling." (Robert Musil, "Man Without Qualities", 1930)

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

"There is beauty in discovery. There is mathematics in music, a kinship of science and poetry in the description of nature, and exquisite form in a molecule. Attempts to place different disciplines in different camps are revealed as artificial in the face of the unity of knowledge. All illiterate men are sustained by the philosopher, the historian, the political analyst, the economist, the scientist, the poet, the artisan, and the musician." (Glenn T Seaborg, 1958)

"An analogy is a relationship between two entities, processes, or what you will, which allows inferences to be made about one of the things, usually that about which we know least, on the basis of what we know about the other. […] The art of using analogy is to balance up what we know of the likenesses against the unlikenesses between two things, and then on the basis of this balance make an inference as to what is called the neutral analogy, that about which we do not know." (Rom Harré," The Philosophies of Science", 1972)

"There is one great difficulty with a good hypothesis. When it is completed and rounded, the corners smooth and the content cohesive and coherent, it is likely to become a thing in itself, a work of art. It is then like a finished sonnet or a painting completed. One hates to disturb it. Even if subsequent information should shoot a hole in it, one hates to tear it down because it once was beautiful and whole."  (John Steinbeck, "Sea of Cortez", 1982)

"The role of science, like that of art, is to blend proximate imagery with more distant meaning, the parts we already understand with those given as new into larger patterns that are coherent enough to be acceptable as truth. Biologists know this relation by intuition during the course of fieldwork, as they struggle to make order out of the infinitely varying patterns of nature." (Edward O Wilson, "In Search of Nature", 1996)

"It is ironic but true: the one reality science cannot reduce is the only reality we will ever know. This is why we need art. By expressing our actual experience, the artist reminds us that our science is incomplete, that no map of matter will ever explain the immateriality of our consciousness." (Jonah Lehrer, "Proust Was a Neuroscientist", 2011)

On Art VI: Algebra

"We think only through the medium of words. Languages are true analytical methods. Algebra, which is adapted to its purpose in every species of expression, in the most simple, most exact, and best manner possible, is at the same time a language and an analytical method. The art of reasoning is nothing more than a language well arranged. (Abbé de Condillac, "System of Logic", cca. 1781)

"Algebra, as an art, can be of no use to any one in the business of life; certainly not as taught in the schools. I appeal to every man who has been through the school routine whether this be not the case. Taught as an art it is of little use in the higher mathematics, as those are made to feel who attempt to study the differential calculus without knowing more of the principles than is contained in books of rules." (Augustus de Morgan, "Elements of Algebra", 1837)

"The difficulties which so many have felt in the doctrine of Negative and Imaginary Quantities in Algebra forced themselves long ago on my attention […] And while agreeing with those who had contended that negatives and imaginaries were not properly quantities at all, I still felt dissatisfied with any view which should not give to them, from the outset, a clear interpretation and meaning [...] It early appeared to me that these ends might be attained by our consenting to regard Algebra as being no mere Art, nor Language, nor primarily a Science of Quantity; but rather as the Science of Order in Progression." (William R Hamilton, "Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method…", 1853)

"By the help of God and with His precious assistance I say that algebra is a scientific art. The objects with which it deals are absolute numbers and (geometrical) magnitudes which, though themselves unknown, are related to things which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. […] What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of algebra as stated above." (Omar Khayyam [quoted by Daoud Suleiman Kasir in "The Algebra of Omar Khayyam", 1931)

"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire)

On Art V: Geometry

"Rules necessary for definitions. Not to leave any terms at all obscure or ambiguous without definition; Not to employ in definitions any but terms perfectly known or already explained. […] A few rules include all that is necessary for the perfection of the definitions, the axioms, and the demonstrations, and consequently of the entire method of the geometrical proofs of the art of persuading." (Blaise Pascal, "Pensées", 1670)

"We study art because we receive pleasure from the great works of the masters, and probably we appreciate them the more because we have dabbled a little in pigments or in clay. We do not expect to be composers, or poets, or sculptors, but we wish to appreciate music and letters and the fine arts, and to derive pleasure from them and be uplifted by them. […] So it is with geometry. We study it because we derive pleasure from contact with a great and ancient body of learning that has occupied the attention of master minds during the thousands of years in which it has been perfected, and we are uplifted by it." (David E Smith, "The Teaching of Geometry", 1911)

"And here is what makes this analysis situs interesting to us; it is that geometric intuition really intervenes there. When, in a theorem of metric geometry, one appeals to this intuition, it is because it is impossible to study the metric properties of a figure as abstractions of its qualitative properties, that is, of those which are the proper business of analysis situs. It has often been said that geometry is the art of reasoning correctly from badly drawn figures. This is not a capricious statement; it is a truth that merits reflection. But what is a badly drawn figure? It is what might be executed by the unskilled draftsman spoken of earlier; he alters the properties more or less grossly; his straight lines have disquieting zigzags; his circles show awkward bumps. But this does not matter; this will by no means bother the geometer; this will not prevent him from reasoning." (Henri Poincaré, "Dernières pensées", 1913)

"Projective Geometry: a boundless domain of countless fields where reals and imaginaries, finites and infinites, enter on equal terms, where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint - an enchanted realm where thought is double and flows throughout in parallel streams." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"Imagine any sort of model and a copy of it done by an awkward artist: the proportions are altered, lines drawn by a trembling hand are subject to excessive deviation and go off in unexpected directions. From the point of view of metric or even projective geometry these figures are not equivalent, but they appear as such from the point of view of geometry of position [that is, topology]." (Henri Poincaré, "Dernières pensées", 1920)

"[…] no branch of mathematics competes with projective geometry in originality of ideas, coordination of intuition in discovery and rigor in proof, purity of thought, logical finish, elegance of proofs and comprehensiveness of concepts. The science born of art proved to be an art." (Morris Kline, "Projective Geometry", Scientific America Vol. 192 (1), 1955)

"Nature does not seem full of circles and triangles to the ungeometrical; rather, mastery of the theory of triangles and circles, and later of conic sections, has taught the theorist, the experimenter, the carpenter, and even the artist to find them everywhere, from the heavenly motions to the pose of a Venus." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

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

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

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

On Art IV: Systems Thinking

"For some years now the activity of the artist in our society has been trending more toward the function of the ecologist: one who deals with environmental relationships. Ecology is defined as the totality or pattern of relations between organisms and their environment. Thus the act of creation for the new artist is not so much the invention of new objects as the revelation of previously unrecognized relation- ships between existing phenomena, both physical and metaphysical. So we find that ecology is art in the most fundamental and pragmatic sense, expanding our apprehension of reality." (Gene Youngblood, "Expanded Cinema", 1970)

"Cybernetics is simultaneously the most important science of the age and the least recognized and understood. It is neither robotics nor freezing dead people. It is not limited to computer applications and it has as much to say about human interactions as it does about machine intelligence. Today’s cybernetics is at the root of major revolutions in biology, artificial intelligence, neural modeling, psychology, education, and mathematics. At last there is a unifying framework that suspends long-held differences between science and art, and between external reality and internal belief." (Paul Pangaro, "New Order From Old: The Rise of Second-Order Cybernetics and Its Implications for Machine Intelligence", 1988)

"Model building is the art of selecting those aspects of a process that are relevant to the question being asked. As with any art, this selection is guided by taste, elegance, and metaphor; it is a matter of induction, rather than deduction. High science depends on this art." (John H Holland, "Hidden Order: How Adaptation Builds Complexity", 1995)

"Thus my advice to model builders in the social sciences is to think in terms of social processes that might require algebraic structures that could yield catastrophe potential. Build models from an intimate knowledge of these processes while remaining aware of the algebraic requirements for catastrophes. The art of nonlinear model building is a delicate dance with two partners, algebraic forms that produce known effects and a substantive understanding of the complexities of social phenomena. Coordinating the two by mixing structure to match complexity is the job of the theorist, and it is the single greatest creative challenge of any researcher." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"The love of complexity without reductionism makes art; the love of complexity with reductionism makes science." (Edward O. Wilson, "Consilience: The Unity of Knowledge", 1998)

"Much of the art of system dynamics modeling is discovering and representing the feedback processes, which, along with stock and flow structures, time delays, and nonlinearities, determine the dynamics of a system. […] the most complex behaviors usually arise from the interactions (feedbacks) among the components of the system, not from the complexity of the components themselves." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Models need to be judged by what they eliminate as much as by what they include - like stone carving, the art is in removing what you do not need." (John H Miller & Scott E Page, "Complex Adaptive Systems: An Introduction to Computational Models of Social Life", 2007)

"[...] cybernetics is the art of creating equilibrium in a world of possibilities and constraints. This is not just a romantic description, it portrays the new way of thinking quite accurately. Cybernetics differs from the traditional scientific procedure, because it does not try to explain phenomena by searching for their causes, but rather by specifying the constraints that determine the direction of their development." (Ernst von Glasersfeld, "Partial Memories: Sketches from an Improbable Life", 2010)

"Holism [is] the art - in contrast with reductionism - of seeing a complex system as a whole. Holism knows the limits to its understanding; it acknowledges that the system has its wildness, its privacy, its own reasons, its defenses against invasive explanation." (David Fleming, "Lean Logic", 2016)

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

On Art III: Probability & Statistics

"It is impossible for a Die, with such determin’d force and direction, not to fall on such a determin’d side, only I don’t know the force and direction which makes it fall on such a determin’d side, and therefore I call that Chance, which is nothing but want of Art [...]" (John Arbuthnot, "Of the Laws of Chance", 1692)

"The art of drawing conclusions from experiments and observations consists in evaluating probabilities and in estimating whether they are sufficiently great or numerous enough to constitute proofs. This kind of calculation is more complicated and more difficult than it is commonly thought to be […]" (Antoine-Laurent de Lavoisier, cca. 1790)

"[Statistics] is both a science and an art. It is a science in that its methods are basically systematic and have general application; and an art in that their successful application depends to a considerable degree on the skill and special experience of the statistician, and on his knowledge of the field of application, e.g. economics." (Leonard H C Tippett, "Statistics", 1943)

"Sampling is the science and art of controlling and measuring the reliability of useful statistical information through the theory of probability." (William E Deming, "Some Theory of Sampling", 1950) 

"Statistics is the fundamental and most important part of inductive logic. It is both an art and a science, and it deals with the collection, the tabulation, the analysis and interpretation of quantitative and qualitative measurements. It is concerned with the classifying and determining of actual attributes as well as the making of estimates and the testing of various hypotheses by which probable, or expected, values are obtained. It is one of the means of carrying on scientific research in order to ascertain the laws of behavior of things - be they animate or inanimate. Statistics is the technique of the Scientific Method." (Bruce D Greenschields & Frank M Weida, "Statistics with Applications to Highway Traffic Analyses", 1952)

"Statistics is the name for that science and art which deals with uncertain inferences - which uses numbers to find out something about nature and experience." (Warren Weaver, 1952)

"The fact is that, despite its mathematical base, statistics is as much an art as it is a science. A great many manipulations and even distortions are possible within the bounds of propriety. Often the statistician must choose among methods, a subjective process, and find the one that he will use to represent the facts." (Darell Huff, "How to Lie with Statistics", 1954)

"Statistics are the art of stating in precise terms that which one does not know." (William Kruskal, "Statistics, Moliere, and Henry Adams", American Scientist Magazine, 1967)

"Every field of knowledge has its subject matter and its methods, along with a style for handling them. The field of Probability has a great deal of the Art component in it-not only is the subject matter rather different from that of other fields, but at present the techniques are not well organized into systematic methods. As a result each problem has to be "looked at in the right way" to make it easy to solve. Thus in probability theory there is a great deal of art in setting up the model, in solving the problem, and in applying the results back to the real world actions that will follow." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"In the laws of probability theory, likelihood distributions are fixed properties of a hypothesis. In the art of rationality, to explain is to anticipate. To anticipate is to explain." (Eliezer S. Yudkowsky, "A Technical Explanation of Technical Explanation", 2005)

"Statistics is the art and science of designing studies and analyzing the data that those studies produce. Its ultimate goal is translating data into knowledge and understanding of the world around us. In short, statistics is the art and science of learning from data." (Alan Agresti & Christine Franklin, "Statistics: The Art and Science of Learning from Data" 3rd Ed., 2013)

"The art of drawing conclusions from experiments and observations consists in evaluating probabilities and in estimating whether they are sufficiently great or numerous enough to constitute proofs." (Antoine Lavoisier)

On Art II: Mathematics

"The arts which are useful, and absolutely necessary to the architect are painting and mathematics." (Leon Battista Alberti, "Treatise on Architecture", 1452)

"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)

"The invention of what we may call primary or fundamental notation has been but little indebted to analogy, evidently owing to the small extent of ideas in which comparison can be made useful. But at the same time analogy should be attended to, even if for no other reason than that, by making the invention of notation an art, the exertion of individual caprice ceases to be allowable. Nothing is more easy than the invention of notation, and nothing of worse example and consequence than the confusion of mathematical expressions by unknown symbols. If new notation be advisable, permanently or temporarily, it should carry with it some mark of distinction from that which is already in use, unless it be a demonstrable extension of the latter." (Augustus De Morgan, "Calculus of Functions" Encyclopaedia of Pure Mathematics, 1847)

"In mathematics the art of asking questions is more valuable than solving problems." (Georg Cantor, [thesis’ title] 1867)

"A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. [...] we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it." (Henri Poincaré, "Notice sur Halphen", Journal de l'École Polytechnique, 1890)

"The true mathematician is always a great deal of an artist, an architect, yes, of a poet. Beyond the real world, though perceptibly connected with it, mathematicians have created an ideal world which they attempt to develop into the most perfect of all worlds, and which is being explored in every direction. None has the faintest conception of this world except him who knows it; only presumptuous ignorance can assert that the mathematician moves in a narrow circle. The truth which he seeks is, to be sure, broadly considered, neither more nor less than consistency; but does not his mastership show, indeed, in this very limitation? To solve questions of this kind he passes unenviously over others." (Alfred Pringsheim, Jaresberichte der Deutschen Mathematiker Vereinigung Vol 13, 1904)

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

"Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks or hewing wood, no more than painting is the art of mixing colors on a palette, no more than the science of geology is the art of breaking rocks, or the science of anatomy the art of butchering." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"It is not surprising that the greatest mathematicians have again and again appealed to the arts in order to find some analogy to their own work. They have indeed found it in the most varied arts, in poetry, in painting, and in sculpture, although it would certainly seem that it is in music, the most abstract of all the arts, the art of number and of time, that we find the closest analogy." (Havelock Ellis, "The Dance of Life", 1923)

"[…] mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creation of the mind. It is a science of the most rigorous kind, like logic but more comprehensive and very much fuller; it is a true art, along with sculpture and music, as needing the guidance of inspiration and as developing under great conventions of form […]" (Oswald Spengler, "The Decline of the West" Vol. 1, 1926)

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

"[…] the process of scientific discovery may be regarded as a form of art. This is best seen in the theoretical aspects of Physical Science. The mathematical theorist builds up on certain assumptions and according to well understood logical rules, step by step, a stately edifice, while his imaginative power brings out clearly the hidden relations between its parts. A well-constructed theory is in some respects undoubtedly an artistic production." (Ernest Rutherford, 1932)

"As an Art, Mathematics has its own standard of beauty and elegance which can vie with the more decorative arts. In this it is diametrically opposed to a Baroque art which relies on a wealth of ornamental additions. Bereft of superfluous addenda, Mathematics may appear, on first acquaintance, austere and severe. But longer contemplation reveals the classic attributes of simplicity relative to its significance and depth of meaning." (Dudley E Littlewood,"The Skeleton Key of Mathematics", 1949)

"Mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers." (David Hilbert, "Geometry and the Imagination", 1952)

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

"Mathematicians who build new spaces and physicists who find them in the universe can profit from the study of pictorial and architectural spaces conceived and built by men of art." (György Kepes, "The New Landscape In Art and Science", 1956)

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

"The question ‘What is mathematics?’ cannot be answered meaningfully by philosophical generalities, semantic definitions or journalistic circumlocutions. Such characterizations also fail to do justice to music or painting. No one can form an appreciation of these arts without some experience with rhythm, harmony and structure, or with form, color and composition. For the appreciation of mathematics actual contact with its substance is even more necessary." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964)

"The study of symmetry was born out of art and mathematics; art as the comprehension of the beauty of nature and mathematics as the comprehension of the world's harmony. " (N F Ovchinnikov, "Principles of Preservation", 1966)

"If some great mathematicians have known how to give lyrical expression to their enthusiasm for the beauty of their science, nobody has suggested examining it as if it were the object of an art - mathematical art -  and consequently the subject of a theory of aesthetics, the aesthetics of mathematics (François Le Lionnais, "Great Currents of Mathematical Thought", 1971)

"Though we can say that mathematics is not art, some mathematicians think of themselves as artists of pure form. It seems clear, however, that their elegant and near aesthetic forms fail as art, because they are secondary visual ideas, the product of an intellectual set of restraints, rather than the cause of a felt insight realized in and through visual form." (Robert E Mueller, "Idols of Computer Art", 1972) 

"Mathematical physics represents the purest image that the view of nature may generate in the human mind; this image presents all the character of the product of art; it begets some unity, it is true and has the quality of sublimity; this image is to physical nature what music is to the thousand noises of which the air is full […]" (Théophile de Donder, 1977)

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

"[...] 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", Mathematics Today, 1978)

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

"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)

"Mathematicians do not see their art as a way of simply calculating or ordering reality. They understand that math articulates, manipulates, and discovers reality. In that sense, it’s both a language and a literature; a box of tools and the edifices constructed from them." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

"We all know what we like in music, painting or poetry, but it is much harder to explain why we like it. The same is true in mathematics, which is, in part, an art form. We can identify a long list of desirable qualities: beauty, elegance, importance, originality, usefulness, depth, breadth, brevity, simplicity, clarity. However, a single work can hardly embody them all; in fact, some are mutually incompatible. Just as different qualities are appropriate in sonatas, quartets or symphonies, so mathematical compositions of varying types require different treatment." (Michael Atiyah, "Mathematics: Art and Science" Bulletin of the AMS 43, 2006)

"'Doing mathematics' is thus working on the construction of some mathematical object and resembles other creative enterprises of the mind in a scientific or artistic domain. But while the mental exercise of creating mathematics is somehow related to that of creating art, it should remain clear that mathematical objects are very different from the artistic objects that occur in literature, music, or the visual arts." (David Ruelle, "The Mathematician's Brain", 2007)

"The ever-present rigorous proof is both a science and an art." (Edward B. Burger, Zero To Infinity: A History of Numbers", 2007)

"There are three reasons for the study of inequalities: practical, theoretical and aesthetic. In many practical investigations, it is necessary to bound one quantity by another. The classical inequalities are very useful for this purpose. From the theoretical point of view, very simple questions give rise to entire theories. […] Finally, let us turn to the aesthetic aspects. As has been pointed out, beauty is in the eye 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." (Claudi Alsina & Roger B Nelsen, "When Less is More: Visualizing Basic Inequalities", 2009)

"There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depends heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood. (Paul Lockhart, "A Mathematician's Lament", 2009)

"In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life." (Michael F Atiyah, "The Art of Mathematics", 2010)

"The solution to a math problem is not a number; it’s an argument, a proof. We’re trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying." (Paul Lockhart, "Measurement", 2012)

"The theory of fractality is of importance from two distinct but related points of view: its origins and its results. Fractals are the fruit of the breaking down of traditional thought and philosophy that had governed mathematics and the sciences for centuries. They in turn had a revolutionary effect on diverse  sciences, mathematics, thought and arts in a very short period  of time. They upended linear philosophical conceptions of true or false, high or low, ordered or disordered, beautiful or ugly." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)

"Yet there is a distinct difference between the writer’s art and the mathematician’s. Whereas the writer is at liberty to use symbols in ways that contradict experience in order to jolt emotions or to create states of mind with deep-rooted meanings from a personal life’s journey, the mathematician cannot compose contradictions, aside from the standard argument that establishes a proof by contradiction. Mathematical symbols have a definite initial purpose: to tidily package complex information in order to facilitate understanding." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"Mathematics is pure language - the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms. (In a spoken language, there exist words, like "happiness", that defy definition.) It is also an art - the most intellectual and classical of the arts." (Alfred Adler)

"Mathematics is, as it were, a sensuous logic, and relates to philosophy as do the arts, music, and plastic art to poetry." (Friedrich von Schlegel)

"Pure mathematics can be practically useful and applied mathematics can be artistically elegant." (Paul R Halmos)

"The mathematician's best work is art […] a high and perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch each other." (M Gustav Mittag-Leffler)

On Art I: Science

"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)

"From that time, the universe has steadily become more complex and less reducible to a central control. With as much obstinacy as though it were human, it has insisted on expanding its parts; with as much elusiveness as though it were feminine, it has evaded the attempt to impose on it a single will. Modern science, like modern art, tends, in practice, to drop the dogma of organic unity. Some of the mediaeval habit of mind survives, but even that is said to be yielding before the daily evidence of increasing and extending complexity. The fault, then, was not in man, if he no longer looked at science or art as an organic whole or as the expression of unity. Unity turned itself into complexity, multiplicity, variety, and even contradiction." (Henry Adams, "Mont Saint Michel and Chartres", 1904)

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

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

"[…] the process of scientific discovery may be regarded as a form of art. This is best seen in the theoretical aspects of Physical Science. The mathematical theorist builds up on certain assumptions and according to well understood logical rules, step by step, a stately edifice, while his imaginative power brings out clearly the hidden relations between its parts. A well-constructed theory is in some respects undoubtedly an artistic production." (Ernest Rutherford, 1932)

"There is a science of simple things, an art of complicated ones. Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to it. The artist works out his own formulas; the interest of science lies in the art of making science." (Paul Valéry, "Moralités", 1932)

"It is impossible to make a clear cut between science, religion, and art. The whole is never equal simply to the sum of its various parts." (Max Planck, "The Philosophy of Physics", 1936)

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

"Today the function of the artist is to bring imagination to science and science to imagination, where they meet, in the myth." (Cyril Connolly, The Unquiet Grave, 1945)

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

"All art originates in an act of intuition or vision." (Herbert Read, "Form in Modern Poetry", 1948)

"Whether our work is art or science or the daily work of society, it is only the form in which we explore our experience which is different." (Jacob Bronowski, "Science and Human Values", 1956)

"Both science and art form in the course of the centuries a human language by which we can speak about the more remote parts of reality, and the coherent sets of concepts as well as the different styles of art are different words or groups of words in this language." (Werner K Heisenberg, "Physics and Philosophy", 1958)

"The important distinction between science and those other systematizations [i.e., art, philosophy, and theology] is that science is self-testing and self-correcting. Here the essential point of science is respect for objective fact. What is correctly observed must be believed [...] the competent scientist does quite the opposite of the popular stereotype of setting out to prove a theory; he seeks to disprove it. (George G Simpson, "Notes on the Nature of Science", 1962)

"All discoveries in art and science result from an accumulation of errors." (Marshall McLuhan, "Culture Is Our Business", 1970)

"If some great mathematicians have known how to give lyrical expression to their enthusiasm for the beauty of their science, nobody has suggested examining it as if it were the object of an art - mathematical art -  and consequently the subject of a theory of aesthetics, the aesthetics of mathematics (François Le Lionnais, "Great Currents of Mathematical Thought", 1971)

"There is no such thing as a Scientific Mind. Scientists are people of very dissimilar temperaments doing different things in very different ways. Among scientists are collectors, classifiers and compulsive tidiers-up; many are detectives by temperament and many are explorers; some are artists and others artisans. There are poet-scientists and philosopher-scientists and even a few mystics. What sort of mind or temperament can all these people be supposed to have in common? Obligative scientists must be very rare, and most people who are in fact scientists could easily have been something else instead." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"How often people speak of art and science as though they were two entirely different things, with no interconnection. An artist is emotional, they think, and uses only his intuition; he sees all at once and has no need of reason. A scientist is cold, they think, and uses only his reason; he argues carefully step by step, and needs no imagination. That is all wrong. The true artist is quite rational as well as imaginative and knows what he is doing; if he does not, his art suffers. The true scientist is quite imaginative as well as rational, and sometimes leaps to solutions where reason can follow only slowly; if he does not, his science suffers." (Isaac Asimov, "The Roving Mind", 1983)

"Literature is as much a product of the technological and scientific milieu as it is of the artistic one. Some of the large ideas, call them theories or metaphors - that humans are machines, that the observer affects the phenomenon observed, that information can be quantified - alter the way work is done in art. Metaphors invented by artists imply new ways of seeing, demolish mere logic, provoke alternatives, and lead to new theories in science." (David Porush, "The Soft Machine", 1985)

"There is an art to science, and science in art; the two are not enemies, but different aspects of the whole." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Science is (or should be) a precise art. Precise, because data may be taken or theories formulated with a certain amount of accuracy; an art, because putting the information into the most useful form for investigation or for presentation requires a certain amount of creativity and insight." (Patricia H Reiff, "The Use and Misuse of Statistics in Space Physics", Journal of Geomagnetism and Geoelectricity 42, 1990)

"What is done for science must also be done for art: accepting undesirable side effects for the sake of the main goal, and moreover diminishing their importance by making this main goal more magnificent. For one should reform forward, not backward: social illnesses, revolutions, are evolutions inhibited by a conserving stupidity. (Robert Musil", "Precision and Soul: Essays and Addresses", 1990)

"Science with all its faults has brought education and the arts to more people - a larger percentage - than has ever existed before science. In that respect it is science that is the great humanizer. And, if we are going to solve the problems that science has brought us, it will be done by science and in no other way." (Isaac Asimov, "Essay 400: A Way of Thinking, "The Magazine of Fantasy and Science Fiction", 1994)

"[...] in science there are collectors, classifiers, compulsory tidiers-up and permanent contesters, detectives, some artists and many artisans, there are poet-scientists and philosophers and even a few mystics." (Rolf M Zinkernagel, [Nobel lecture] 1996)

"Science is what we understand well enough to explain to a computer. Art is everything else we do." (Donald E Knuth, [foreword to the book "A=B" by Marko Petkovsek et al] 1996)

 "The art of science is knowing which observations to ignore and which are the key to the puzzle." (Edward W Kolb, "Blind Watchers of the Sky", 1996)

"The mystery of sound is mysticism; the harmony of life is religion. The knowledge of vibrations is metaphysics, the analysis of atoms is science, and their harmonious grouping is art. The rhythm of form is poetry, and the rhythm of sound is music. This shows that music is the art of arts and the science of all sciences; and it contains the fountain of all knowledge within itself." (Inayat Khan, "The Mysticism of Sound and Music", 1996)

"The role of science, like that of art, is to blend proximate imagery with more distant meaning, the parts we already understand with those given as new into larger patterns that are coherent enough to be acceptable as truth. Biologists know this relation by intuition during the course of fieldwork, as they struggle to make order out of the infinitely varying patterns of nature." (Edward O Wilson, "In Search of Nature", 1996)

"The love of complexity without reductionism makes art; the love of complexity with reductionism makes science." (Edward O. Wilson, "Consilience: The Unity of Knowledge", 1998)

"Science and art, by their very nature, differ in that science can be learned in a systematic and logical way, whereas expertise in art has to be acquired by example, experience, and practice." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"It is ironic but true: the one reality science cannot reduce is the only reality we will ever know. This is why we need art. By expressing our actual experience, the artist reminds us that our science is incomplete, that no map of matter will ever explain the immateriality of our consciousness." (Jonah Lehrer, "Proust Was a Neuroscientist", 2011)

"I think that science may be styled the knowledge of universals, or abstract wisdom; and art is science reduced to practice - or science is reason, and art the mechanism of it - and may be called practical science. Science, in fine, is the theorem, and art the problem." (Laurence Sterne)

"Science is the art of creating suitable illusions which the fool believes or argues against, but the wise man enjoys for their beauty or their ingenuity, without being blind to the fact that they are human veils and curtains concealing the abysmal darkness of the unknowable." (Carl G Jung

"Science and art belong to the whole world, and the barriers of nationality vanish before them." (Johann Wolfgang von Goethe)

"The sciences and arts are not cast in a mold, but formed and shaped little by little, by repeated handling and polishing, as bears lick their cubs into shape at leisure." (Michel de Montaigne)

"The two forms of experiment, scientific and artistic, share an attitude of detached yet intense observation toward the events precipitated. Yet the two activities are fundamentally opposed. Scientific experiment, relying on empirical rigor in refining its methods and verifying its results, seeks to extend and consolidate our grasp of order in the universe. Artistic experiment sets out to breed disorder, thwart determinism, and open up a space for individual freedom and consciousness." (Roger Shattuck)

Artistic Experience I

"Music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics. Notwithstanding all the experience I may have acquired in music from being associated with it for so long, I must confess that only with the aid of mathematics did my ideas become clear and did light replace a certain obscurity of which I was unaware before." (Jean-Philippe Rameau, "Treatise on Harmony reduced to its natural principles", 1722)

"Music is like geometric figures and numbers, which are the universal forms of all possible objects of experience." (Friedrich Nietzsche, "Birth of Tragedy", 1872)

"A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. [...] we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it." (Henri Poincaré, "Notice sur Halphen", Journal de l'École Polytechnique, 1890)

"Art is a human activity consisting in this, that one consciously, by means of certain external symbols, conveys to others the feelings one has experienced, whereby people so infected by these feelings, also experience them." (Leo Tolstoy, "What is Art?", 1897)

"True artistic experience is never passive, for the spectator is obliged to participate, as it were, in the continuous or discontinuous variations of proportions, positions, lines and planes. Moreover, he must see clearly how this play of repeated or non-repeated changes may give rise to a new harmony of relations which will constitute the unity of the work. Every part becomes organized into a whole with the other parts. All the parts contribute to the unity of the composition, none of them assuming a dominant place in the whole." (Theo van Doesburg, 'Grundbegriffe der neuen Gestaltenden Kunst', 1921-23)

"Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)

"Music is the pleasure the human soul experiences from counting without being aware it is counting." (Gottfried W Leibniz)

Arthur Koestler - Collected Quotes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

On Art: Poetry and Mathematics V

"The true mathematician is always a great deal of an artist, an architect, yes, of a poet. Beyond the real world, though perceptibly connected with it, mathematicians have created an ideal world which they attempt to develop into the most perfect of all worlds, and which is being explored in every direction. None has the faintest conception of this world except him who knows it; only presumptuous ignorance can assert that the mathematician moves in a narrow circle. The truth which he seeks is, to be sure, broadly considered, neither more nor less than consistency; but does not his mastership show, indeed, in this very limitation? To solve questions of this kind he passes unenviously over others." (Alfred Pringsheim, Jaresberichte der Deutschen Mathematiker Vereinigung Vol 13, 1904)

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

"[...] mathematics and poetry move together between two extremes of mysticism, the mysticism of the commonplace where ideas illuminate and create facts, and the mysticism of the extraordinary where God, the Infinite, the Real, poses the riddles of desire and disappointment, sin and salvation, effort and failure, question and paradoxical answer [...]" (Scott Buchanan, "Poetry and Mathematics", 1929)

"[…] the major mathematical research acquires an organization and orientation similar to the poetical function which, adjusting by means of metaphor disjunctive elements, displays a structure identical to the sensitive universe. Similarly, by means of its axiomatic or theoretical foundation, mathematics assimilates various doctrines and serves the instructive purpose, the one set up by the unifying moral universe of concepts." (Dan Barbilian, "The Autobiography of the Scientist", 1940)

"Mathematics is one component of any plan for liberal education. Mother of all the sciences, it is a builder of the imagination, a weaver of patterns of sheer thought, an intuitive dreamer, a poet. The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality. Through countless dimensions, riding high the winds of intellectual adventure and filled with the zest of discovery, the mathematician tracks the heavens for harmony and eternal verity. There is not wholly unexpected surprise, but surprise nevertheless, that mathematics has direct application to the physical world about us. For mathematics, in a wilderness of tragedy and change, is a creature of the mind, born to the cry of humanity in search of an invariant reality, immutable in substance, unalterable with time. Mathematics is an infinity of flexibles forcing pure thought into a cosmos. It is an arc of austerity cutting realms of reason with geodesic grandeur. Mathematics is crystallized clarity, precision personified, beauty distilled and rigorously sublimated. The life of the spirit is a life of thought; the ideal of thought is truth; everlasting truth is the goal of mathematics." (Cletus O Oakley, "Mathematics", The American Mathematical Monthly, 1949)

"The structures with which mathematics deals are more like lace, the leaves of trees, and the play of light and shadow on a human face, than they are like buildings and machines, the least of their representatives. The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music. The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1975)

"The theory of number is the epipoem of mathematics." (Scott Buchanan, "Poetry and Mathematics", 1975)

"To survive, mathematical ideas must be beautiful, they must be seductive, and they must be illuminating, they must help us to understand, they must inspire us. […] Part of that beauty, an essential part, is the clarity and sharpness that the mathematical way of thinking about things promotes and achieves. Yes, there are also mystic and poetic ways of relating to the world, and to create a new math theory, or to discover new mathematics, you have to feel comfortable with vague, unformed, embryonic ideas, even as you try to sharpen them."  (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"The relationship of math to the real world has been a conundrum for philosophers for centuries, but it is also an inspiration for poets. The patterns of mathematics inhabit a liminal space - they were initially derived from the natural world and yet seem to exist in a separate, self-contained system standing apart from that world. This makes them a source of potential metaphor: mapping back and forth between the world of personal experience and the world of mathematical patterns opens the door to novel connections." (Alice Major, "Mapping from e to Metaphor", 2018)

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Rémy de Gourmont - Collected Quotes

"Art must break the chains, all rules and formulas." (Rémy de Gourmont, "Le Chemin de Velours", 1902)

"We write as we feel, as we think, with our entire body." (Rémy de Gourmont, "Le Problème du Style", 1902)

"Art includes everything that stimulates the desire to live; science, everything that sharpens the desire to know. Art, even the most disinterested, the most disembodied, is the auxiliary of life. Born of the sensibility, it sows and creates it in its turn. It is the flower of life and, as seed, it gives back life. Science, or to use a broader term, knowledge, has its end in itself, apart from any idea of life and propagation of the species." (Rémy de Gourmont, "Art and Science", cca. 1905-1909)

"Science is the only truth and it is the great lie. It knows nothing, and people think it knows everything. It is misrepresented. People think that science is electricity, automobilism, and dirigible balloons. It is something very different. It is life devouring itself. It is the sensibility transformed into intelligence. It is the need to know stifling the need to live. It is the genius of knowledge vivisecting the vital genius." (Rémy de Gourmont, "Art and Science", cca. 1905-1909)

"The terrible thing about the quest for truth is that you find it." (Rémy de Gourmont, "Philosophic Nights in Paris", 1920)

"The human mind is so complex and things are so tangled up with each other that, to explain a blade of straw, one would have to take to pieces an entire universe. A definition is a sack of flour compressed into a thimble." (Rémy de Gourmont, "Decadence and Other Essays on the Culture of Ideas", 1921)

"We live less and less, and we learn more and more. Sensibility is surrendering to intelligence." (Rémy  de Gourmont, "Decadence and Other Essays on the Culture of Ideas", 1921)

"Man has made use of his intelligence; he invented stupidity." (Rémy de Gourmont)

Out of Context: On Art (Definitions)

"Now nature is not at variance with art, nor art with nature, they being both servants of his providence: art is the perfection of nature; [...]" (Sir Thomas Browne, Religio Medici, 1643)

"On the same terms, therefore, as art is attained to, is all knowledge and science acquired; for as art is a habit with reference to things to be done, so is science a habit in respect to things to be known; [...]" (William Harvey, Anatomical Exercises on the Generation of Animals, 1651)

"Art is only Nature operating with the aid of the instruments she has made."  (Paul-Henri T d'Holbach [Baron d'Holbach], "The System of Nature, Or, Laws of the Moral and Physical World", 1770)

"Art is the most intense mode of individualism that the world has known." (Oscar Wilde, "The Soul of Man Under Socialism", 1891)

"Art is a human activity consisting in this, that one consciously, by means of certain external symbols, conveys to others the feelings one has experienced, whereby people so infected by these feelings, also experience them." (Leo Tolstoy, "What is Art?", 1897)

"And as the ideal in the whole of Nature moves in an infinite process toward an Absolute Perfection, we may say that art is in strict truth the apotheosis of Nature. Art is thus at once the exaltation of the natural toward its destined supernatural perfection, and the investiture of the Absolute Beauty with the reality of natural existence." (George H Howison, "The Limits of Evolution, and Other Essays, Illustrating the Metaphysical Theory of Personal Idealism", 1901)

"We all know that Art is not truth. Art is a lie that makes us realize truth." (Pablo Picasso, "The Arts", 1923)

"Science, like art, is not a copy of nature but a re-creation of her." (Jacob Bronowski, "The Creative Mind", 1965)

"The role of science, like that of art, is to blend proximate imagery with more distant meaning, the parts we already understand with those given as new into larger patterns that are coherent enough to be acceptable as truth." (Edward O Wilson, "In Search of Nature", 1996)

"Art is a symbol, a thing conjuring up reality in our mental image." (Antoni Tàpies, "Tàpies, Werke auf Papier 1943 – 2003", 2004)

"Art is constructivist in nature, aimed at the deliberate refinement and elaboration of mental models and worldviews." (Mark Turner, "The Artful Mind : cognitive science and the riddle of human creativity", 2006)

"Models need to be judged by what they eliminate as much as by what they include - like stone carving, the art is in removing what you do not need." (John H Miller & Scott E Page, "Complex Adaptive Systems: An Introduction to Computational Models of Social Life", 2007) 

"And art is always a struggle. There is no systematic way to create beautiful and meaningful paintings or sculptures, and there is also no method for producing beautiful and meaningful mathematical arguments." (Paul Lockhart, "Measurement", 2012)

"Art is a lie that makes us realise the truth, at least the truth that is given us to understand." (Pablo Picasso)

"Art is a line around your thoughts." (Gustav Klimt)

"Art is never finished, only abandoned." (Leonardo da Vinci)

"Art is not a handicraft, it is the transmission of feeling the artist has experienced." (Leo Tolstoy)

"Art is not simply an identity of illusion and reality, but a counter-illusion: its world is a material world, but the material of an intelligible spiritual world." (Northrop Frye)

"Art is not what you see, but what you make others see.” (Edgar Degas)

"Art is unquestionably one of the purest and highest elements in human happiness. (John Lubbock)

"At base, science is no more than an investigation of a miracle we can never explain, and art is an interpretation of that miracle." (Ray Bradbury)

"I think that science may be styled the knowledge of universals, or abstract wisdom; and art is science reduced to practice - or science is reason, and art the mechanism of it - and may be called practical science. Science, in fine, is the theorem, and art the problem." (Laurence Sterne)

"Science is what we understand well enough to explain to a computer, Art is all the rest." (Donald E Knuth)

On Facts (1890-1899)

"The study of theory must go hand in hand with that of facts: and for dealing with most modern problems it is modern facts that are of the greatest use." (Alfred Marshall, "Principles of Economics", 1890)

"The graphical method has considerable superiority for the exposition of statistical facts over the tabular. A heavy bank of figures is grievously wearisome to the eye, and the popular mind is as incapable of drawing any useful lessons from it as of extracting sunbeams from cucumbers." (Arthur B Farquhar & Henry Farquhar, "Economic and Industrial Delusions", 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)

"It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts." (Sir Arthur C Doyle, "The Adventures of Sherlock Holmes", 1892)

"The classification of facts, the recognition of their sequence and relative significance is the function of science, and the habit of forming a judgment upon these facts unbiased by personal feeling is characteristic of what may be termed the scientific frame of mind." (Karl Pearson, "The Grammar of Science", 1892)

"The true aim of the teacher must be to impart an appreciation of method and not a knowledge of facts." (Karl Pearson, "The Grammar of Science", 1892)

"Facts are not much use, considered as facts. They bewilder by their number and their apparent incoherency. Let them be digested into theory, however, and brought into mutual harmony, and it is another matter. Theory is of the essence of facts. Without theory scientific knowledge would be only worthy of the mad house." (Oliver Heaviside, "Electromagnetic Theory", 1893)

"Scientific facts accumulate rapidly, and give rise to theories with almost equal rapidity. These theories are often wonderfully enticing, and one is apt to pass from one to another, from theory to theory, without taking care to establish each before passing on to the next, without assuring oneself that the foundation on which one is building is secure. Then comes the crash; the last theory breaks down utterly, and on attempting to retrace our steps to firm ground and start anew, we may find too late that one of the cards, possibly at the very foundation of the pagoda, is either faultily placed or in itself defective, and that this blemish easily remedied if detected in time has, neglected, caused the collapse of the whole structure on whose erection so much skill and perseverance have been spent." (Arthur M Marshall, 1894)

"Without a theory all our knowledge of nature would be reduced to a mere inventory of the results of observation. Every scientific theory must be regarded as an effort of the human mind to grasp the truth, and as long as it is consistent with the facts, it forms a chain by which they are linked together and woven into harmony." (Thomas Preston, "The Theory of Heat", 1894)

"The first step, whenever a practical problem is set before a mathematician, is to form the mathematical hypothesis. It is neither needful nor practical that we should take account of the details of the structure as it will exist. We have to reason about a skeleton diagram in which bearings are reduced to points, pieces to lines, etc. and [in] which it is supposed that certain relations between motions are absolutely constrained, irrespective of forces. Some writers call such a hypothesis a fiction, and say that the mathematician does not solve the real problem, but only a fictitious one. That is one way of looking at the matter, to which I have no objection to make: only, I notice, that in precisely the same sense in which the mathematical hypothesis is 'false', so also is this statement 'false', that it is false. Namely, both representations are false in the sense that they omit subsidiary elements of the fact, provided that element of the case can be said to be subsidiary which those writers overlook, namely, that the skeleton diagram is true in the only sense in which from the nature of things any mental representation, or understanding, of the brute existent can be true. For every possible conception, by the very nature of thought, involves generalization; now generalization omits, means to omit, and professes to omit, the differences between the facts generalized." (Charles S Peirce, "Report on Live Loads", cca. 1895)

"The world is chiefly a mental fact. From mind it receives the forms of time and space, the principle of causality, color, warmth, and beauty. Were there no mind, there would be no world." (John L Spalding, "Means and Ends of Education", 1895)

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

"Mathematics is the most abstract of all the sciences. For it makes no external observations, nor asserts anything as a real fact. When the mathematician deals with facts, they become for him mere ‘hypotheses’; for with their truth he refuses to concern himself. The whole science of mathematics is a science of hypotheses; so that nothing could be more completely abstracted from concrete reality." (Charles S Peirce, "The Regenerated Logic", The Monist Vol. 7 (1), 1896)

"Round about the accredited and orderly facts of every science there ever fl oats a sort of dust-cloud of exceptional observations, of occurrences minute and irregular and seldom met with, which it always proves more easy to ignore than to attend to […]" (William James, "The Will to Believe", 1896)

"Science like life feeds on its own decay. New facts burst old rules; then newly developed concepts bind old and new together into a reconciling law." (William James, "The Will to Believe and Other Essays in Popular Philosophy", 1896)

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

"To use an old analogy - and here we can hardly go except upon analogy - while the building of Nature is growing spontaneously from within, the model of it, which we seek to construct in our descriptive science, can only be constructed by means of scaffolding from without, a scaffolding of hypotheses. While in the real building all is continuous, in our model there are detached parts which must be connected with the rest by temporary ladders and passages, or which must be supported till we can see how to fill in the understructure. To give the hypotheses equal validity with facts is to confuse the temporary scaffolding with the building itself." (John H Poynting, 1899)

On Axioms (1600-1699)

"It cannot be that axioms established by argumentation should avail for the discovery of new works, since the subtlety of nature is greater many times over than the subtlety of argument. But axioms duly and orderly formed from particulars easily discover the way to new particulars, and thus render sciences active." (Francis Bacon, "Novum Organum", 1620)

"There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried." (Francis Bacon, "Novum Organum", 1620)

"We must first, by every kind of experiment, elicit the discovery of causes and true axioms, and seek for experiments which may afford light rather than profit." (Francis Bacon, "Novum Organum", 1620)

"Rules for Axioms. I. Not to omit any necessary principle without asking whether it is admittied, however clear and evident it may be. II. Not to demand, in axioms, any but things that are perfectly evident in themselves." (Blaise Pascal, "The Art of Persuasion", cca. 1658)

"For it is unquestionable that it is no great error to define and clearly explain things, although very clear of themselves, nor to omit to require in advance axioms which cannot be refused in the place where they are necessary; nor lastly to prove propositions that would be admitted without proof." (Blaise Pascal, "The Art of Persuasion",  cca. 1658)

"To prove all propositions, and to employ nothing for their proof but axioms fully evident of themselves, or propositions already demonstrated or admitted; Never to take advantage of the ambiguity of terms by failing mentally to substitute definitions that restrict or explain them." (Blaise Pascal, "The Art of Persuasion", cca. 1658)

"This art, which I call the art of persuading, and which, properly speaking, is simply the process of perfect methodical proofs, consists of three essential parts: of defining the terms of which we should avail ourselves by clear definitions, of proposing principles of evident axioms to prove the thing in question; and of always mentally substituting in the demonstrations the definition in the place of the thing defined." (Blaise Pascal, "The Art of Persuasion", cca. 1658)

"Mathematicians who are only mathematicians have exact minds, provided all things are explained to them by means of definitions and axioms; otherwise they are inaccurate and insufferable, for they are only right when the principles are quite clear." (Blaise Pascal, "Pensées", 1670)

"Rules necessary for definitions. Not to leave any terms at all obscure or ambiguous without definition; Not to employ in definitions any but terms perfectly known or already explained. […] A few rules include all that is necessary for the perfection of the definitions, the axioms, and the demonstrations, and consequently of the entire method of the geometrical proofs of the art of persuading." (Blaise Pascal, "Pensées", 1670)

On Technology I

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

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

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

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

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

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

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

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

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

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