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Home > Category Index for Science Quotations > Category Index M > Category: Mathematics As A Fine Art

Mathematics As A Fine Art Quotes (23 quotes)

Ultima se tangunt. How expressive, how nicely characterizing withal is mathematics! As the musician recognizes Mozart, Beethoven, Schubert in the first chords, so the mathematician would distinguish his Cauchy, Gauss, Jacobi, Helmholtz in a few pages.
In Ceremonial Speech (15 Nov 1887) celebrating the 301st anniversary of the Karl-Franzens-University Graz. Published as Gustav Robert Kirchhoff: Festrede zur Feier des 301. Gründungstages der Karl-Franzens-Universität zu Graz (1888), 29, as translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 186-187. From the original German, “Ultima se tangunt. Und wie ausdrucksfähig, wie fein charakterisirend ist dabei die Mathematik. Wie der Musiker bei den ersten Tacten Mozart, Beethoven, Schubert erkennt, so würde der Mathematiker nach wenig Seiten, seinen Cauchy, Gauss, Jacobi, Helmholtz unterscheiden.” [The Latin words translate as “the final touch”. —Webmaster]
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A peculiar beauty reigns in the realm of mathematics, a beauty which resembles not so much the beauty of art as the beauty of nature and which affects the reflective mind, which has acquired an appreciation of it, very much like the latter.
From Berliner Monatsberichte (1867), 395. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 185.
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Among all highly civilized peoples the golden age of art has always been closely coincident with the golden age of the pure sciences, particularly with mathematics, the most ancient among them.
This coincidence must not be looked upon as accidental, but as natural, due to an inner necessity. Just as art can thrive only when the artist, relieved of the anxieties of existence, can listen to the inspirations of his spirit and follow in their lead, so mathematics, the most ideal of the sciences, will yield its choicest blossoms only when life’s dismal phantom dissolves and fades away, when the striving after naked truth alone predominates, conditions which prevail only in nations while in the prime of their development.
From Die Entwickelung der Mathematik im Zusammenhange mit der Ausbreitung der Kultur (1893), 4. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 191-192. From the original German, “Bei allen Kulturvölkern ist die Blüthezeit der Kunst auch immer zeitlich eng verbunden mit einer Blüthezeit der reinen Wissenschaften, insbesondere der ältesten unter ihnen, der Mathematik.
Dieses Zusammentreffen dürfte auch nicht ein zufälliges, sondern ein natürliches, ein Ergebniss innerer Notwendigkeit sein. Wie die Kunst nur gedeihen kann, wenn der Künstler, unbekümmert um die Bedrängnisse des Daseins, den Eingebungen seines Geistes lauschen und ihnen folgen kann, so kann die idealste Wissenschaft, die Mathematik, erst dann ihre schönsten Blüthen treiben, wenn des Erdenlebens schweres Traumbild sinkt und sinkt und sinkt, wenn das Streben nach der nackten Wahrheit allein bestimmend ist, was nur bei Nationen in der Vollkraft ihrer Entwickelung vorkommt.”
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Among the memoirs of Kirchhoff are some of uncommon beauty. … Can anything be beautiful, where the author has no time for the slightest external embellishment?—But—; it is this very simplicity, the indispensableness of each word, each letter, each little dash, that among all artists raises the mathematician nearest to the World-creator; it establishes a sublimity which is equalled in no other art, something like it exists at most in symphonic music. The Pythagoreans recognized already the similarity between the most subjective and the most objective of the arts.
In Ceremonial Speech (15 Nov 1887) celebrating the 301st anniversary of the Karl-Franzens-University Graz. Published as Gustav Robert Kirchhoff: Festrede zur Feier des 301. Gründungstages der Karl-Franzens-Universität zu Graz (1888), 28-29, as translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 186. From the original German, “Gerade unter den zuletzt erwähnten Abhandlungen Kirchhoff’s sind einige von ungewöhnlicher Schönheit. … kann etwas schön sein, wo dem Autor auch zur kleinsten äusseren Ausschmückung die Zeit fehlt?–Doch–; gerade durch diese Einfachheit, durch diese Unentbehrlichkeit jedes Wortes, jedes Buchstaben, jedes Strichelchens kömmt der Mathematiker unter allen Künstlern dem Weltenschöpfer am nächsten; sie begründet eine Erhabenheit, die in keiner Kunst ein Gleiches,–Aehnliches höchstens in der symphonischen Musik hat. Erkannten doch schon die Pythagoräer die Aehnlichkeit der subjectivsten und der objectivsten der Künste.”
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However far the calculating reason of the mathematician may seem separated from the bold flight of the artist’s phantasy, it must be remembered that these expressions are but momentary images snatched arbitrarily from among the activities of both. In the projection of new theories the mathematician needs as bold and creative a phantasy as the productive artist, and in the execution of the details of a composition the artist too must calculate dispassionately the means which are necessary for the successful consummation of the parts. Common to both is the creation, the generation, of forms out of mind.
From Die Entwickelung der Mathematik im Zusammenhange mit der Ausbreitung der Kultur (1893), 4. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 185. From the original German, “Wie weit auch der rechnende Verstand des Mathematikers von dem kühnen Fluge der Phantasie des Künstlers getrennt zu sein scheint, so bezeichnen diese Ausdrücke doch blosse Augenblicksbilder, die willkürlich aus der Thätigkeit Beider herausgerissen sind. Bei dem Entwurfe neuer Theorieen bedarf der Mathematiker einer ebenso kühnen und schöpferischen Phantasie wie der schaffende Künstler, und bei der Ausführung der Einzelheiten eines Werkes muss auch der Künstler kühl alle Mittel berechnen, welche zum Gelingen der Theile erforderlich sind. Gemeinsam ist Beiden die Hervorbringung, die Erzeugung der Gebilde aus dem Geiste.”
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However far the mathematician’s calculating senses seem to be separated from the audacious flight of the artist’s imagination, these manifestations refer to mere instantaneous images, which have been arbitrarily torn from the operation of both. In designing new theories, the mathematician needs an equally bold and inspired imagination as creative as the artist, and in carrying out the details of a work the artist must unemotionally reckon all the resources necessary for the success of the parts. Common to both is the fabrication, the creation of the structure from the intellect.
From Die Entwickelung der Mathematik im Zusammenhange mit der Ausbreitung der Kultur (1893), 4. Translated by Webmaster using online resources. From the original German, “Wie weit auch der rechnende Verstand des Mathematikers von dem kühnen Fluge der Phantasie des Künstlers getrennt zu sein scheint, so bezeichnen diese Ausdrücke doch blosse Augenblicksbilder, die willkürlich aus der Thätigkeit Beider herausgerissen sind. Bei dem Entwurfe neuer Theorieen bedarf der Mathematiker einer ebenso kühnen und schöpferischen Phantasie wie der schaffende Künstler, und bei der Ausführung der Einzelheiten eines Werkes muss auch der Künstler kühl alle Mittel berechnen, welche zum Gelingen der Theile erforderlich sind. Gemeinsam ist Beiden die Hervorbringung, die Erzeugung der Gebilde aus dem Geiste.”
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I like to look at mathematics almost more as an art than as a science; for the activity of the mathematician, constantly creating as he is, guided though not controlled by the external world of the senses, bears a resemblance, not fanciful I believe but real, to the activity of an artist, of a painter let us say. Rigorous deductive reasoning on the part of the mathematician may be likened here to technical skill in drawing on the part of the painter. Just as no one can become a good painter without a certain amount of skill, so no one can become a mathematician without the power to reason accurately up to a certain point. Yet these qualities, fundamental though they are, do not make a painter or mathematician worthy of the name, nor indeed are they the most important factors in the case. Other qualities of a far more subtle sort, chief among which in both cases is imagination, go to the making of a good artist or good mathematician.
From 'Fundamental Conceptions and Methods in Mathematics', Bulletin American Mathematical Society (1904), 9, 133. As cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 182.
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I venture to assert that the feelings one has when the beautiful symbolism of the infinitesimal calculus first gets a meaning, or when the delicate analysis of Fourier has been mastered, or while one follows Clerk Maxwell or Thomson into the strange world of electricity, now growing so rapidly in form and being, or can almost feel with Stokes the pulsations of light that gives nature to our eyes, or track with Clausius the courses of molecules we can measure, even if we know with certainty that we can never see them I venture to assert that these feelings are altogether comparable to those aroused in us by an exquisite poem or a lofty thought.
In paper (May 1891) read before Bath Branch of the Teachers’ Guild, published in The Practical Teacher (July 1891), reprinted as 'Geometry', in Frederic Spencer, Chapters on the Aims and Practice of Teaching (1897), 194.
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It is an open secret to the few who know it, but a mystery and stumbling block to the many, that Science and Poetry are own sisters; insomuch that in those branches of scientific inquiry which are most abstract, most formal, and most remote from the grasp of the ordinary sensible imagination, a higher power of imagination akin to the creative insight of the poet is most needed and most fruitful of lasting work.
From Introduction written for William Kingdon Clifford, Clifford’s Lectures and Essays (1879), Vol. 1, 1.
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It is as great a mistake to maintain that a high development of the imagination is not essential to progress in mathematical studies as to hold with Ruskin and others that science and poetry are antagonistic pursuits.
In Sphere of Science (1898), 107.
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It is with mathematics not otherwise than it is with music, painting or poetry. Anyone can become a lawyer, doctor or chemist, and as such may succeed well, provided he is clever and industrious, but not every one can become a painter, or a musician, or a mathematician: general cleverness and industry alone count here for nothing.
In Ueber die Anlage zur Mathematik (1900), 5. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 184.
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Just as the musician is able to form an acoustic image of a composition which he has never heard played by merely looking at its score, so the equation of a curve, which he has never seen, furnishes the mathematician with a complete picture of its course. Yea, even more: as the score frequently reveals to the musician niceties which would escape his ear because of the complication and rapid change of the auditory impressions, so the insight which the mathematician gains from the equation of a curve is much deeper than that which is brought about by a mere inspection of the curve.
In Jahresbericht der Deutschen Mathematiker Vereiningung, 13, 864. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 190
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Kirchhoff’s whole tendency, and its true counterpart, the form of his presentation, was different [from Maxwell’s “dramatic bulk”]. … He is characterized by the extreme precision of his hypotheses, minute execution, a quiet rather than epic development with utmost rigor, never concealing a difficulty, always dispelling the faintest obscurity. … he resembled Beethoven, the thinker in tones. — He who doubts that mathematical compositions can be beautiful, let him read his memoir on Absorption and Emission … or the chapter of his mechanics devoted to Hydrodynamics.
In Ceremonial Speech (15 Nov 1887) celebrating the 301st anniversary of the Karl-Franzens-University Graz. Published as Gustav Robert Kirchhoff: Festrede zur Feier des 301. Gründungstages der Karl-Franzens-Universität zu Graz (1888), 30, as translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 187. From the original German, “Kirchhoff … seine ganze Richtung war eine andere, und ebenso auch deren treues Abbild, die Form seiner Darstellung. … Ihn charakterisirt die schärfste Präcisirung der Hypothesen, feine Durchfeilung, ruhige mehr epische Fortentwicklung mit eiserner Consequenz ohne Verschweigung irgend einer Schwierigkeit, unter Aufhellung des leisesten Schattens. … er glich dem Denker in Tönen: Beethoven. – Wer in Zweifel zieht, dass mathematische Werke künstlerisch schön sein können, der lese seine Abhandlung über Absorption und Emission oder den der Hydrodynamik gewidmeten Abschnitt seiner Mechanik.” The memoir reference is Gesammelte Abhandlungen (1882), 571-598.
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Mathematics and music, the most sharply contrasted fields of scientific activity which can be found, and yet related, supporting each other, as if to show forth the secret connection which ties together all the activities of our mind, and which leads us to surmise that the manifestations of the artist’s genius are but the unconscious expressions of a mysteriously acting rationality.
In Vorträge und Reden (1884, 1896), Vol 1, 122. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 191. From the original German, “Mathematik und Musik, der schärfste Gegensatz geistiger Thätigkeit, den man auffinden kann, und doch verbunden, sich unterstützend, als wollten sie die geheime Consequenz nachweisen, die sich durch alle Thätigkeiten unseres Geistes hinzieht, und die auch in den Offenbarungen des künstlerischen Genius uns unbewusste Aeusserungen geheimnissvoll wirkender Vernunftmässigkeit ahnen lässt.”
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Mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not obvious. Truth is the same thing to the understanding as music to the ear and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator as the pleasing of our senses: only in the former case, as the object and faculty are more spiritual, the delight is more pure, free from regret, turpitude, lassitude, and intemperance that commonly attend sensual pleasures.
In An Essay on the Usefulness of Mathematical Learning (1701), 3-4.
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May not Music be described as the Mathematic of sense, Mathematic as Music of the reason? the soul of each the same! Thus the musician feels Mathematic, the mathematician thinks Music, Music the dream, Mathematic the working life each to receive its consummation from the other when the human intelligence, elevated to its perfect type, shall shine forth glorified in some future Mozart-Dirichlet or Beethoven-Gauss a union already not indistinctly foreshadowed in the genius and labours of a Helmholtz!
In paper read 7 Apr 1864, printed in 'Algebraical Researches Containing a Disquisition On Newton’s Rule for the Discovery of Imaginary Roots', Philosophical Transactions of the Royal Society of London (1865), 154, 613, footnote. Also in Collected Mathematical Papers, Vol. 2, 419.
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Music has much resemblance to algebra.
From Ludwig Tieck and Fr. Schlegel (eds.) Novalis schriften (1837), Vol. 2, 313. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 190, from the original German, “Die Musik hat viel Aehnlichkeit mit der Algebra.”
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Saturated with that speculative spirit then pervading the Greek mind, he [Pythagoras] endeavoured to discover some principle of homogeneity in the universe. Before him, the philosophers of the Ionic school had sought it in the matter of things; Pythagoras looked for it in the structure of things. He observed the various numerical relations or analogies between numbers and the phenomena of the universe. Being convinced that it was in numbers and their relations that he was to find the foundation to true philosophy, he proceeded to trace the origin of all things to numbers. Thus he observed that musical strings of equal lengths stretched by weights having the proportion of 1/2, 2/3, 3/4, produced intervals which were an octave, a fifth and a fourth. Harmony, therefore, depends on musical proportion; it is nothing but a mysterious numerical relation. Where harmony is, there are numbers. Hence the order and beauty of the universe have their origin in numbers. There are seven intervals in the musical scale, and also seven planets crossing the heavens. The same numerical relations which underlie the former must underlie the latter. But where number is, there is harmony. Hence his spiritual ear discerned in the planetary motions a wonderful “Harmony of spheres.”
In History of Mathematics (1893), 67.
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Surely the claim of mathematics to take a place among the liberal arts must now be admitted as fully made good. Whether we look at the advances made in modern geometry, in modern integral calculus, or in modern algebra, in each of these three a free handling of the material employed is now possible, and an almost unlimited scope is left to the regulated play of fancy. It seems to me that the whole of aesthetic (so far as at present revealed) may be regarded as a scheme having four centres, which may be treated as the four apices of a tetrahedron, namely Epic, Music, Plastic, and Mathematic. There will be found a common plane to every three of these, outside of which lies the fourth; and through every two may be drawn a common axis opposite to the axis passing through the other two. So far is certain and demonstrable. I think it also possible that there is a centre of gravity to each set of three, and that the line joining each such centre with the outside apex will intersect in a common point the centre of gravity of the whole body of aesthetic; but what that centre is or must be I have not had time to think out.
In 'Proof of the Hitherto Undemonstrated Fundamental Theorem of Invariants', Collected Mathematical Papers (1909), Vol. 3, 123.
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The true mathematician is always a good deal of an artist, an architect, yes, of a poet. Beyond the real world, though perceptibly connected with it, mathematicians have intellectually 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 he who knows it.
In Jahresbericht der Deutschen Mathematiker Vereinigung, 32, 381. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 184.
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Till the fifteenth century little progress appears to have been made in the science or practice of music; but since that era it has advanced with marvelous rapidity, its progress being curiously parallel with that of mathematics, inasmuch as great musical geniuses appeared suddenly among different nations, equal in their possession of this special faculty to any that have since arisen. As with the mathematical so with the musical faculty it is impossible to trace any connection between its possession and survival in the struggle for existence.
In 'Darwinism Applied to Man', Darwinism: An Exposition of the Theory of Natural Selection with Some of Its Applications (1901), Chap. 15, 468.
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We do not listen with the best regard to the verses of a man who is only a poet, nor to his problems if he is only an algebraist; but if a man is at once acquainted with the geometric foundation of things and with their festal splendor, his poetry is exact and his arithmetic musical.
In 'Works and Days', Society and Solitude (1883), Chap. 7, 171.
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Who does not know Maxwell’s dynamic theory of gases? At first there is the majestic development of the variations of velocities, then enter from one side the equations of condition and from the other the equations of central motions, higher and higher surges the chaos of formulas, suddenly four words burst forth: “Put n = 5.” The evil demon V disappears like the sudden ceasing of the basso parts in music, which hitherto wildly permeated the piece; what before seemed beyond control is now ordered as by magic. There is no time to state why this or that substitution was made, he who cannot feel the reason may as well lay the book aside; Maxwell is no program-musician who explains the notes of his composition. Forthwith the formulas yield obediently result after result, until the temperature-equilibrium of a heavy gas is reached as a surprising final climax and the curtain drops.
In Ceremonial Speech (15 Nov 1887) celebrating the 301st anniversary of the Karl-Franzens-University Graz. Published as Gustav Robert Kirchhoff: Festrede zur Feier des 301. Gründungstages der Karl-Franzens-Universität zu Graz (1888), 29-30, as translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 187. From the original German, “Wer kennt nicht seine dynamische Gastheorie? – Zuerst entwickeln sich majestätisch die Variationen der Geschwindigkeiten, dann setzen von der einen Seite die Zustands-Gleichungen, von der anderen die Gleichungen der Centralbewegung ein, immer höher wogt das Chaos der Formeln; plötzlich ertönen die vier Worte: „Put n=5.“Der böse Dämon V verschwindet, wie in der Musik eine wilde, bisher alles unterwühlende Figur der Bässe plötzlich verstummt; wie mit einem Zauberschlage ordnet sich, was früher unbezwingbar schien. Da ist keine Zeit zu sagen, warum diese oder jene Substitution gemacht wird; wer das nicht fühlt, lege das Buch weg; Maxwell ist kein Programmmusiker, der über die Noten deren Erklärung setzen muss. Gefügig speien nun die Formeln Resultat auf Resultat aus, bis überraschend als Schlusseffect noch das Wärme-Gleichgewicht eines schweren Gases gewonnen wird und der Vorhang sinkt.” A condensed alternate translation also appears on the Ludwig Boltzmann Quotes page of this website.
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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