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._