CHAPTER XI

MATHEMATICS AS A FINE ART

=1101.= The world of idea which it discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.--SYLVESTER, J. J.

_Presidential Address, British Association Report (1869); Collected Mathematical Papers, Vol. 2, p. 659._

=1102.= Mathematics has a triple end. It should furnish an instrument for the study of nature. Furthermore it has a philosophic end, and, I venture to say, an end esthetic. It ought to incite the philosopher to search into the notions of number, space, and time; and, above all, adepts find in mathematics delights analogous to those that painting and music give. They admire the delicate harmony of number and of forms; they are amazed when a new discovery discloses for them an unlooked for perspective; and the joy they thus experience, has it not the esthetic character although the senses take no part in it? Only the privileged few are called to enjoy it fully, it is true; but is it not the same with all the noblest arts? Hence I do not hesitate to say that mathematics deserves to be cultivated for its own sake, and that the theories not admitting of application to physics deserve to be studied as well as others.--POINCARÉ, HENRI.

_The Relation of Analysis and Mathematical Physics; Bulletin American Mathematical Society, Vol. 4 (1899), p. 248._

=1103.= 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.

--BÔCHER, MAXIME.

_Fundamental Conceptions and Methods in Mathematics; Bulletin American Mathematical Society, Vol. 9 (1904), p. 133._

=1104.= Mathematics, rightly viewed, possesses not only truth, but supreme beauty--a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learned as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the real and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the natural world.

--RUSSELL, BERTRAND.

_The Study of Mathematics: Philosophical Essays (London, 1910), p. 73._

=1105.= It was not alone the striving for universal culture which attracted the great masters of the Renaissance, such as Brunellesco, Leonardo de Vinci, Raphael, Michael Angelo and especially Albrecht Dürer, with irresistible power to the mathematical sciences. They were conscious that, with all the freedom of the individual phantasy, art is subject to necessary laws, and conversely, with all its rigor of logical structure, mathematics follows esthetic laws.--RUDIO, F.

_Virchow-Holtzendorf: Sammlung gemeinverständliche wissenschaftliche Vorträge, Heft 142, p. 19._

=1106.= 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.

--SYLVESTER, J. J.

_Proof of the hitherto undemonstrated Fundamental Theorem of Invariants: Collected Mathematical Papers, Vol. 3, p. 123._

=1107.= 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.--MOEBIUS, P. J.

_Ueber die Anlage zur Mathematik (Leipzig, 1900), p. 5._

=1108.= 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.--PRINGSHEIM, A.

_Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 32, p. 381._

=1109.= Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, Sophus Lie, and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, in conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination.--HOBSON, E. W.

_Presidential Address British Association for the Advancement of Science (1910) Nature, Vol. 84, p. 290._

=1110.= Mathematics has beauties of its own--a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found elsewhere only in the works of the greatest beauty. It was a felicitous expression of Goethe’s to call a noble cathedral “frozen music,” but it might even better be called “petrified mathematics.” The beauties of mathematics--of simplicity, of symmetry, of completeness--can and should be exemplified even to young children. When this subject is properly and concretely presented, the mental emotion should be that of enjoyment of beauty, not that of repulsion from the ugly and the unpleasant.--YOUNG, J. W. A.

_The Teaching of Mathematics (New York, 1907), p. 44._

=1111.= 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.

--KUMMER, E. E.

_Berliner Monatsberichte (1867), p. 395._

=1112.= 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.--ARBUTHNOT, JOHN.

_Usefulness of Mathematical Learning._

=1113.= 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.--LAMPE, E.

_Die Entwickelung der Mathematik, etc. (Berlin, 1893), p. 4._

=1114.= As pure truth is the polar star of our science [mathematics], so it is the great advantage of our science over others that it awakens more easily the love of truth in our pupils.... If Hegel justly said, “Whoever does not know the works of the ancients, has lived without knowing _beauty_,” Schellbach responds with equal right, “Who does not know mathematics, and the results of recent scientific investigation, dies without knowing _truth_.”--SIMON, MAX.

_Quoted in J. W. A. Young: Teaching of Mathematics (New York, 1907), p. 44._

=1115.= Büchsel in his reminiscences from the life of a country parson relates that he sought his recreation in Lacroix’s Differential Calculus and thus found intellectual refreshment for his calling. Instances like this make manifest the great advantage which occupation with mathematics affords to one who lives remote from the city and is compelled to forego the pleasures of art. The entrancing charm of mathematics, which captivates every one who devotes himself to it, and which is comparable to the fine frenzy under whose ban the poet completes his work, has ever been incomprehensible to the spectator and has often caused the enthusiastic mathematician to be held in derision. A classic illustration is the example of Archimedes, ....--LAMPE, E.

_Die Entwickelung der Mathematik, etc. (Berlin 1893), p. 22._

=1116.= Among the memoirs of Kirchhoff are some of uncommon beauty. Beauty, I hear you ask, do not the Graces flee where integrals stretch forth their necks? Can anything be beautiful, where the author has no time for the slightest external embellishment?... Yet 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.... _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. Extreme external elegance, sometimes a somewhat weak skeleton of conclusions characterizes the French; the English, above all Maxwell, are distinguished by the greatest dramatic bulk. 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....

Kirchhoff’s whole tendency, and its true counterpart, the form of his presentation, was different.... 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. To return once more to my allegory, 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 (Gesammelte Abhandlungen, Leipzig, 1882, p. 571-598) or the chapter of his mechanics devoted to Hydrodynamics.--BOLTZMANN, L.

_Gustav Robert Kirchhoff (Leipzig 1888), pp. 28-30._

=1117.=

On poetry and geometric truth, And their high privilege of lasting life, From all internal injury exempt, I mused; upon these chiefly: and at length, My senses yielding to the sultry air, Sleep seized me, and I passed into a dream. --WORDSWORTH.

_The Prelude, Bk. 5._

=1118.= Geometry seems to stand for all that is practical, poetry for all that is visionary, but in the kingdom of the imagination you will find them close akin, and they should go together as a precious heritage to every youth.--MILNER, FLORENCE.

_School Review, 1898, p. 114._

=1119.= The beautiful has its place in mathematics as elsewhere. The prose of ordinary intercourse and of business correspondence might be held to be the most practical use to which language is put, but we should be poor indeed without the literature of imagination. Mathematics too has its triumphs of the creative imagination, its beautiful theorems, its proofs and processes whose perfection of form has made them classic. He must be a “practical” man who can see no poetry in mathematics.

--WHITE, W. F.

_A Scrap-book of Elementary Mathematics (Chicago, 1908), p. 208._

=1120.= 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.--WORKMAN, W. P.

_F. Spencer: Aim and Practice of Teaching (New York, 1897), p. 194._

=1121.= 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.--POLLOCK, F.

_Clifford’s Lectures and Essays (New York, 1901), Vol. 1, Introduction, p. 1._

=1122.= 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.--HOFFMAN, F. S.

_Sphere of Science (London, 1898), p. 107._

=1123.= We have heard much about the poetry of mathematics, but very little of it has as yet been sung. The ancients had a juster notion of their poetic value than we. The most distinct and beautiful statements of any truth must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both.--THOREAU, H. D.

_A Week on the Concord and Merrimac Rivers (Boston, 1893), p. 477._

=1124.= 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.--EMERSON, R. W.

_Society and Solitude, Chap. 7, Works and Days._

=1125.= Mathesis and Poetry are ... the utterance of the same power of imagination, only that in the one case it is addressed to the head, and in the other, to the heart.--HILL, THOMAS.

_North American Review, Vol. 85, p. 230._

=1126.= The Mathematics are usually considered as being the very antipodes of Poesy. Yet Mathesis and Poesy are of the closest kindred, for they are both works of the imagination. Poesy is a creation, a making, a fiction; and the Mathematics have been called, by an admirer of them, the sublimest and most stupendous of fictions. It is true, they are not only μάθησις, learning, but ποίησις, a creation.--HILL, THOMAS.

_North American Review, Vol. 85, p. 229._

=1127.=

Music and poesy used to quicken you: The mathematics, and the metaphysics, Fall to them as you find your stomach serves you. No profit grows, where is no pleasure ta’en:-- In brief, sir, study what you most affect. --SHAKESPEARE.

_Taming of the Shrew, Act 1, Scene 1._

=1128.= Music has much resemblance to algebra.--NOVALIS.

_Schriften, Teil 2 (Berlin, 1901), p. 549._

=1129.=

I do present you with a man of mine, Cunning in music and in mathematics, To instruct her fully in those sciences, Whereof, I know, she is not ignorant. --SHAKESPEARE.

_Taming of the Shrew, Act 2, Scene 1._

=1130.= 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.”--CAJORI, F.

_History of Mathematics (New York, 1897), p. 67._

=1131.= 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!--SYLVESTER, J. J.

_On Newton’s Rule for the Discovery of Imaginary Roots; Collected Mathematical Papers, Vol. 2, p. 419._

=1132.= 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.--PRINGSHEIM, A.

_Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 13, p. 364._

=1133.= 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.--HELMHOLTZ, H.

_Vorträge und Reden, Bd. 1 (Braunschweig, 1884), p. 82._

=1134.= 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.--LAMPE, E.

_Die Entwickelung der Mathematik etc. (Berlin, 1893), p. 4._

=1135.= 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.

--WALLACE, A. R.

_Darwinism, Chap. 15._

=1136.= In my opinion, there is absolutely no trustworthy proof that talents have been improved by their exercise through the course of a long series of generations. The Bach family shows that musical talent, and the Bernoulli family that mathematical power, can be transmitted from generation to generation, but this teaches us nothing as to the origin of such talents. In both families the high-watermark of talent lies, not at the end of the series of generations, as it should do if the results of practice are transmitted, but in the middle. Again, talents frequently appear in some member of a family which has not been previously distinguished.

Gauss was not the son of a mathematician; Handel’s father was a surgeon, of whose musical powers nothing is known; Titian was the son and also the nephew of a lawyer, while he and his brother, Francesco Vecellio, were the first painters in a family which produced a succession of seven other artists with diminishing talents. These facts do not, however, prove that the condition of the nerve-tracts and centres of the brain, which determine the specific talent, appeared for the first time in these men: the appropriate condition surely existed previously in their parents, although it did not achieve expression. They prove, as it seems to me, that a high degree of endowment in a special direction, which we call talent, cannot have arisen from the experience of previous generations, that is, by the exercise of the brain in the same specific direction.--WEISMANN, AUGUST.

_Essays upon Heredity [A. E. Shipley], (Oxford, 1891), Vol. 1, p. 97._