"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.
"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)
"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)
"Here is another disguise of π: Consider a river flowing in uniformly erodible sand under the influence of a gentle slope. Theory predicts that over time the river’s actual length divided by the straight-line distance between its beginning and end will tend toward π. If you guessed that the circle might be a cause, you would be right."
"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)
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