THE MATHEMATICIAN
=801.= The real mathematician is an enthusiast _per se_. Without
enthusiasm no mathematics.--NOVALIS.
_Schriften (Berlin, 1901), Zweiter Teil,
p. 223._
=802.= It is true that a mathematician, who is not somewhat of a
poet, will never be a perfect mathematician.--WEIERSTRASS.
_Quoted by Mittag-Leffler; Compte rendu
du deuxième congrês international des
mathématiciens (Paris, 1902), p. 149._
=803.= The mathematician is perfect only in so far as he is a
perfect being, in so far as he perceives the beauty of truth;
only then will his work be thorough, transparent, comprehensive,
pure, clear, attractive and even elegant. All this is necessary
to resemble _Lagrange_.--GOETHE.
_Wilhelm Meister’s Wanderjahre, Zweites
Buch; Sprüche in Prosa; Natur, VI, 950._
=804.= A thorough advocate in a just cause, a penetrating
mathematician facing the starry heavens, both alike bear the
semblance of divinity.--GOETHE.
_Wilhelm Meister’s Wanderjahre, Zweites
Buch; Sprüche in Prosa; Natur, VI, 947._
=805.= Mathematicians practice absolute freedom.--ADAMS, HENRY.
_A Letter to American Teachers of
History (Washington, 1910), p. 169._
=806.= The mathematical method is the essence of mathematics. He
who fully comprehends the method is a mathematician.--NOVALIS.
_Schriften (Berlin, 1901), Zweiter Teil,
p. 190._
=807.= He who is unfamiliar with mathematics [literally, he who
is a layman in mathematics] remains more or less a stranger to
our time.--DILLMANN, E.
_Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 39._
=808.= Enlist a great mathematician and a distinguished
Grecian; your problem will be solved. Such men can teach in a
dwelling-house as well as in a palace. Part of the apparatus they
will bring; part we will furnish. [Advice given to the Trustees
of Johns Hopkins University on the choice of a professorial
staff.]--GILMAN, D. C.
_Report of the President of Johns
Hopkins University (1888), p. 29._
=809.= Persons, who have a decided mathematical talent,
constitute, as it were, a favored class. They bear the same
relation to the rest of mankind that those who are academically
trained bear to those who are not.--MOEBIUS, P. J.
_Ueber die Anlage zur Mathematik
(Leipzig, 1900), p. 4._
=810.= One may be a mathematician of the first rank without being
able to compute. It is possible to be a great computer without
having the slightest idea of mathematics.--NOVALIS.
_Schriften, Zweiter Teil (Berlin, 1901),
p. 223._
=811.= It has long been a complaint against mathematicians
that they are hard to convince: but it is a far greater
disqualification both for philosophy, and for the affairs of
life, to be too easily convinced; to have too low a standard of
proof. The only sound intellects are those which, in the first
instance, set their standards of proof high. Practice in concrete
affairs soon teaches them to make the necessary abatement: but
they retain the consciousness, without which there is no sound
practical reasoning, that in accepting inferior evidence because
there is no better to be had, they do not by that acceptance
raise it to completeness.--MILL, J. S.
_An Examination of Sir William
Hamilton’s Philosophy (London, 1878), p.
611._
=812.= It is easier to square the circle than to get round a
mathematician.--DE MORGAN, A.
_Budget of Paradoxes (London, 1872), p.
90._
=813.= Mathematicians are like Frenchmen: whatever you say to
them they translate into their own language and forthwith it is
something entirely different.--GOETHE.
_Maximen und Reflexionen, Sechste
Abtheilung._
=814.= What I chiefly admired, and thought altogether
unaccountable, was the strong disposition I observed in them [the
mathematicians of Laputa] towards news and politics; perpetually
inquiring into public affairs; giving their judgments in matters
of state; and passionately disputing every inch of party opinion.
I have indeed observed the same disposition among most of the
mathematicians I have known in Europe, although I could never
discover the least analogy between the two sciences.
--SWIFT, JONATHAN.
_Gulliver’s Travels, Part 3, chap. 2._
=815.= The great mathematician, like the great poet or naturalist
or great administrator, is born. My contention shall be that
where the mathematic endowment is found, there will usually be
found associated with it, as essential implications in it, other
endowments in generous measure, and that the appeal of the
science is to the whole mind, direct no doubt to the central
powers of thought, but indirectly through sympathy of all,
rousing, enlarging, developing, emancipating all, so that the
faculties of will, of intellect and feeling learn to respond,
each in its appropriate order and degree, like the parts of an
orchestra to the “urge and ardor” of its leader and lord.
--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 22._
=816.= Whoever limits his exertions to the gratification of others,
whether by personal exhibition, as in the case of the actor and of
the mimic, or by those kinds of literary composition which are
calculated for no end but to please or to entertain, renders
himself, in some measure, dependent on their caprices and humours.
The diversity among men, in their judgments concerning the objects
of taste, is incomparably greater than in their speculative
conclusions; and accordingly, a mathematician will publish to the
world a geometrical demonstration, or a philosopher, a process of
abstract reasoning, with a confidence very different from what a
poet would feel, in communicating one of his productions even to a
friend.--STEWART, DUGALD.
_Elements of the Philosophy of the Human
Mind, Part 3, chap. 1, sect. 3._
=817.= Considering that, among all those who up to this time made
discoveries in the sciences, it was the mathematicians alone who
had been able to arrive at demonstrations--that is to say, at
proofs certain and evident--I did not doubt that I should begin
with the same truths that they have investigated, although I had
looked for no other advantage from them than to accustom my mind
to nourish itself upon truths and not to be satisfied with false
reasons.--DESCARTES.
_Discourse upon Method, Part 2;
Philosophy of Descartes [Torrey] (New
York, 1892), p. 48._
=818.= When the late Sophus Lie ... was asked to name the
characteristic endowment of the mathematician, his answer was the
following quaternion: Phantasie, Energie, Selbstvertrauen,
Selbstkritik.--KEYSER, C. J.
_Lectures on Philosophy, Science and Art
(New York, 1908), p. 31._
=819.= The existence of an extensive Science of Mathematics,
requiring the highest scientific genius in those who contributed
to its creation, and calling for the most continued and vigorous
exertion of intellect in order to appreciate it when created,
etc.--MILL, J. S.
_System of Logic, Bk. 2, chap. 4, sect.
4._
=820.= It may be true, that men, who are _mere_ mathematicians,
have certain specific shortcomings, but that is not the fault of
mathematics, for it is equally true of every other exclusive
occupation. So there are _mere_ philologists, _mere_ jurists,
_mere_ soldiers, _mere_ merchants, etc. To such idle talk it
might further be added: that whenever a certain exclusive
occupation is _coupled_ with specific shortcomings, it is
likewise almost certainly divorced from certain _other_
shortcomings.--GAUSS.
_Gauss-Schumacher Briefwechsel, Bd. 4,
(Altona, 1862), p. 387._
=821.= Mathematical studies ... when combined, as they now
generally are, with a taste for physical science, enlarge
infinitely our views of the wisdom and power displayed in the
universe. The very intimate connexion indeed, which, since the
date of the Newtonian philosophy, has existed between the
different branches of mathematical and physical knowledge,
renders such a character as that of a _mere mathematician_ a very
rare and scarcely possible occurrence.--STEWART, DUGALD.
_Elements of the Philosophy of the Human
Mind, part 3, chap. 1, sect. 3._
=822.= Once when lecturing to a class he [Lord Kelvin] used the
word “mathematician,” and then interrupting himself asked his
class: “Do you know what a mathematician is?” Stepping to the
blackboard he wrote upon it:--
/+∞
| -x²
| e dx = √π
|
/-∞
Then putting his finger on what he had written, he turned to his
class and said: “A mathematician is one to whom _that_ is as
obvious as that twice two makes four is to you. Liouville was a
mathematician.--THOMPSON, S. P.
_Life of Lord Kelvin (London, 1910), p.
1139._
=823.= It is not surprising, in view of the polydynamic constitution
of the genuinely mathematical mind, that many of the major heroes
of the science, men like Desargues and Pascal, Descartes and
Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford,
Riemann and Salmon and Plücker and Poincaré, have attained to high
distinction in other fields not only of science but of philosophy
and letters too. And when we reflect that the very greatest
mathematical achievements have been due, not alone to the peering,
microscopic, histologic vision of men like Weierstrass, illuminating
the hidden recesses, the minute and intimate structure of logical
reality, but to the larger vision also of men like Klein who
survey the kingdoms of geometry and analysis for the endless
variety of things that nourish there, as the eye of Darwin ranged
over the flora and fauna of the world, or as a commercial monarch
contemplates its industry, or as a statesman beholds an empire;
when we reflect not only that the Calculus of Probability is a
creation of mathematics but that the master mathematician is
constantly required to exercise judgment--judgment, that is, in
matters not admitting of certainty--balancing probabilities not
yet reduced nor even reducible perhaps to calculation; when we
reflect that he is called upon to exercise a function analogous to
that of the comparative anatomist like Cuvier, comparing theories
and doctrines of every degree of similarity and dissimilarity of
structure; when, finally, we reflect that he seldom deals with a
single idea at a time, but is for the most part engaged in
wielding organized hosts of them, as a general wields at once the
division of an army or as a great civil administrator directs from
his central office diverse and scattered but related groups of
interests and operations; then, I say, the current opinion that
devotion to mathematics unfits the devotee for practical affairs
should be known for false on _a priori_ grounds. And one should
be thus prepared to find that as a fact Gaspard Monge, creator of
descriptive geometry, author of the classic “Applications de
l’analyse à la géométrie”; Lazare Carnot, author of the celebrated
works, “Géométrie de position,” and “Réflections sur la Métaphysique
du Calcul infinitesimal”; Fourier, immortal creator of the
“Théorie analytique de la chaleur”; Arago, rightful inheritor of
Monge’s chair of geometry; Poncelet, creator of pure projective
geometry; one should not be surprised, I say, to find that these
and other mathematicians in a land sagacious enough to invoke
their aid, rendered, alike in peace and in war, eminent public
service.--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), pp. 32-33._
=824.= If in Germany the goddess _Justitia_ had not the
unfortunate habit of depositing the ministerial portfolios only
in the cradles of her own progeny, who knows how many a German
mathematician might not also have made an excellent minister.
--PRINGSHEIM, A.
_Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 13 (1904),
p. 372._
=825.= We pass with admiration along the great series of
mathematicians, by whom the science of theoretical mechanics has
been cultivated, from the time of Newton to our own. There is no
group of men of science whose fame is higher or brighter. The
great discoveries of Copernicus, Galileo, Newton, had fixed all
eyes on those portions of human knowledge on which their
successors employed their labors. The certainty belonging to this
line of speculation seemed to elevate mathematicians above the
students of other subjects; and the beauty of mathematical
relations and the subtlety of intellect which may be shown in
dealing with them, were fitted to win unbounded applause. The
successors of Newton and the Bernoullis, as Euler, Clairaut,
D’Alembert, Lagrange, Laplace, not to introduce living names,
have been some of the most remarkable men of talent which the
world has seen.--WHEWELL, W.
_History of the Inductive Sciences, Vol.
1, Bk. 4, chap. 6, sect. 6._
=826.= The persons who have been employed on these problems of
applying the properties of matter and the laws of motion to the
explanation of the phenomena of the world, and who have brought
to them the high and admirable qualities which such an office
requires, have justly excited in a very eminent degree the
admiration which mankind feels for great intellectual powers.
Their names occupy a distinguished place in literary history; and
probably there are no scientific reputations of the last century
higher, and none more merited, than those earned by great
mathematicians who have laboured with such wonderful success in
unfolding the mechanism of the heavens; such for instance as
D’Alembert, Clairaut, Euler, Lagrange, Laplace.--WHEWELL, W.
_Astronomy and General Physics (London,
1833), Bk. 3, chap. 4, p. 327._
=827.= Two extreme views have always been held as to the use of
mathematics. To some, mathematics is only measuring and
calculating instruments, and their interest ceases as soon as
discussions arise which cannot benefit those who use the
instruments for the purposes of application in mechanics,
astronomy, physics, statistics, and other sciences. At the other
extreme we have those who are animated exclusively by the love of
pure science. To them pure mathematics, with the theory of
numbers at the head, is the only real and genuine science, and
the applications have only an interest in so far as they contain
or suggest problems in pure mathematics.
Of the two greatest mathematicians of modern times, Newton and
Gauss, the former can be considered as a representative of the
first, the latter of the second class; neither of them was
exclusively so, and Newton’s inventions in the science of pure
mathematics were probably equal to Gauss’s work in applied
mathematics. Newton’s reluctance to publish the method of
fluxions invented and used by him may perhaps be attributed to
the fact that he was not satisfied with the logical foundations
of the Calculus; and Gauss is known to have abandoned his
electro-dynamic speculations, as he could not find a satisfying
physical basis....
Newton’s greatest work, the “Principia”, laid the foundation of
mathematical physics; Gauss’s greatest work, the “Disquisitiones
Arithmeticae”, that of higher arithmetic as distinguished from
algebra. Both works, written in the synthetic style of the
ancients, are difficult, if not deterrent, in their form, neither
of them leading the reader by easy steps to the results. It took
twenty or more years before either of these works received due
recognition; neither found favour at once before that great
tribunal of mathematical thought, the Paris Academy of Sciences....
The country of Newton is still pre-eminent for its culture of
mathematical physics, that of Gauss for the most abstract work in
mathematics.--MERZ, J. T.
_History of European Thought in the
Nineteenth Century (Edinburgh and
London, 1903), p. 630._
=828.= As there is no study which may be so advantageously
entered upon with a less stock of preparatory knowledge than
mathematics, so there is none in which a greater number of
uneducated men have raised themselves, by their own exertions, to
distinction and eminence.... Many of the intellectual defects
which, in such cases, are commonly placed to the account of
mathematical studies, ought to be ascribed to the want of a
liberal education in early youth.--STEWART, DUGALD.
_Elements of the Philosophy of the Human
Mind, Part 3, chap. 1, sect. 3._
=829.= I know, indeed, and can conceive of no pursuit so
antagonistic to the cultivation of the oratorical faculty ... as
the study of Mathematics. An eloquent mathematician must, from
the nature of things, ever remain as rare a phenomenon as a
talking fish, and it is certain that the more anyone gives
himself up to the study of oratorical effect the less will he
find himself in a fit state to mathematicize. It is the constant
aim of the mathematician to reduce all his expressions to their
lowest terms, to retrench every superfluous word and phrase, and
to condense the Maximum of meaning into the Minimum of language.
He has to turn his eye ever inwards, to see everything in its
dryest light, to train and inure himself to a habit of internal
and impersonal reflection and elaboration of abstract thought,
which makes it most difficult for him to touch or enlarge upon
any of those themes which appeal to the emotional nature of his
fellow-men. When called upon to speak in public he feels as a man
might do who has passed all his life in peering through a
microscope, and is suddenly called upon to take charge of a
astronomical observatory. He has to get out of himself, as it
were, and change the habitual focus of his vision.
--SYLVESTER, J. J.
_Baltimore Address; Mathematical Papers,
Vol. 3, pp. 72-73._
=830.= An accomplished mathematician, i.e. a most wretched
orator.--BARROW, ISAAC.
_Mathematical Lectures (London, 1734),
p. 32._
=831.= _Nemo mathematicus genium indemnatus habebit._ [No
mathematician[2] is esteemed a genius until condemned.]
_Juvenal, Liberii, Satura VI, 562._
[2] Used here in the sense of astrologer, or
soothsayer.
=832.= Taking ... the mathematical faculty, probably fewer than
one in a hundred really possess it, the great bulk of the
population having no natural ability for the study, or feeling
the slightest interest in it.[3] And if we attempt to measure the
amount of variation in the faculty itself between a first-class
mathematician and the ordinary run of people who find any kind of
calculation confusing and altogether devoid of interest, it is
probable that the former could not be estimated at less than a
hundred times the latter, and perhaps a thousand times would more
nearly measure the difference between them.--WALLACE, A. R.
_Darwinism, chap. 15._
[3] This is the estimate furnished me by two
mathematical masters in one of our great
public schools of the proportion of boys who
have any special taste or capacity for
mathematical studies. Many more, of course,
can be drilled into a fair knowledge of
elementary mathematics, but only this small
proportion possess the natural faculty which
renders it possible for them ever to rank
high as mathematicians, to take any pleasure
in it, or to do any original mathematical work.
=833.= ... the present gigantic development of the mathematical
faculty is wholly unexplained by the theory of natural selection,
and must be due to some altogether distinct cause.--WALLACE, A. R.
_Darwinism, chap. 15._
=834.= Dr. Wallace, in his “Darwinism”, declares that he can find
no ground for the existence of pure scientists, especially
mathematicians, on the hypothesis of natural selection. If we put
aside the fact that great power in theoretical science is
correlated with other developments of increasing brain-activity,
we may, I think, still account for the existence of pure
scientists as Dr. Wallace would himself account for that of
worker-bees. Their function may not fit them individually to
survive in the struggle for existence, but they are a source of
strength and efficiency to the society which produces them.
--PEARSON, KARL.
_Grammar of Science (London, 1911), Part
1, p. 221._
=835.= It is only in mathematics, and to some extent in poetry,
that originality may be attained at an early age, but even then
it is very rare (Newton and Keats are examples), and it is not
notable until adolescence is completed.--ELLIS, HAVELOCK.
_A Study of British Genius (London,
1904), p. 142._
=836.= The Anglo-Dane appears to possess an aptitude for
mathematics which is not shared by the native of any other
English district as a whole, and it is in the exact sciences that
the Anglo-Dane triumphs.[4]--ELLIS, HAVELOCK.
_A Study of British Genius (London,
1904), p. 69._
[4] The mathematical tendencies of Cambridge are
due to the fact that Cambridge drains the
ability of nearly the whole Anglo-Danish
district.
=837.= In the whole history of the world there was never a race
with less liking for abstract reasoning than the Anglo-Saxon....
Common-sense and compromise are believed in, logical deductions
from philosophical principles are looked upon with suspicion, not
only by legislators, but by all our most learned professional
men.--PERRY, JOHN.
_The Teaching of Mathematics (London,
1902), pp. 20-21._
=838.= The degree of exactness of the intuition of space may be
different in different individuals, perhaps even in different
races. It would seem as if a strong naïve space-intuition were an
attribute pre-eminently of the Teutonic race, while the critical,
purely logical sense is more fully developed in the Latin and
Hebrew races. A full investigation of this subject, somewhat on
the lines suggested by _Francis Galton_ in his researches on
heredity, might be interesting.--KLEIN, FELIX.
_The Evanston Colloquium Lectures (New
York, 1894), p. 46._
=839.= This [the fact that the pursuit of mathematics brings into
harmonious action all the faculties of the human mind] accounts
for the extraordinary longevity of all the greatest masters of
the Analytic art, the Dii Majores of the mathematical Pantheon.
Leibnitz lived to the age of 70; Euler to 76; Lagrange to 77;
Laplace to 78; Gauss to 78; Plato, the supposed inventor of the
conic sections, who made mathematics his study and delight, who
called them the handles or aids to philosophy, the medicine of
the soul, and is said never to have let a day go by without
inventing some new theorems, lived to 82; Newton, the crown and
glory of his race, to 85; Archimedes, the nearest akin, probably,
to Newton in genius, was 75, and might have lived on to be 100,
for aught we can guess to the contrary, when he was slain by the
impatient and ill-mannered sergeant, sent to bring him before the
Roman general, in the full vigour of his faculties, and in the
very act of working out a problem; Pythagoras, in whose school, I
believe, the word mathematician (used, however, in a somewhat
wider than its present sense) originated, the second founder of
geometry, the inventor of the matchless theorem which goes by his
name, the pre-cognizer of the undoubtedly mis-called Copernican
theory, the discoverer of the regular solids and the musical
canon who stands at the very apex of this pyramid of fame, (if we
may credit the tradition) after spending 22 years studying in
Egypt, and 12 in Babylon, opened school when 56 or 57 years old
in Magna Græcia, married a young wife when past 60, and died,
carrying on his work with energy unspent to the last, at the age
of 99. The mathematician lives long and lives young; the wings of
his soul do not early drop off, nor do its pores become clogged
with the earthy particles blown from the dusty highways of vulgar
life.--SYLVESTER, J. J.
_Presidential Address to the British
Association; Collected Mathematical
Papers, Vol. 2 (1908), p. 658._
=840.= The game of chess has always fascinated mathematicians,
and there is reason to suppose that the possession of great
powers of playing that game is in many features very much like
the possession of great mathematical ability. There are the
different pieces to learn, the pawns, the knights, the bishops,
the castles, and the queen and king. The board possesses certain
possible combinations of squares, as in rows, diagonals, etc. The
pieces are subject to certain rules by which their motions are
governed, and there are other rules governing the players....
One has only to increase the number of pieces, to enlarge the
field of the board, and to produce new rules which are to govern
either the pieces or the player, to have a pretty good idea of
what mathematics consists.--SHAW, J. B.
_What is Mathematics? Bulletin American
Mathematical Society Vol. 18 (1912), pp.
386-387._
=841.= Every man is ready to join in the approval or condemnation
of a philosopher or a statesman, a poet or an orator, an artist
or an architect. But who can judge of a mathematician? Who will
write a review of Hamilton’s Quaternions, and show us wherein it
is superior to Newton’s Fluxions?--HILL, THOMAS.
_Imagination in Mathematics; North
American Review, Vol. 85, p. 224._
=842.= The pursuit of mathematical science makes its votary
appear singularly indifferent to the ordinary interests and cares
of men. Seeking eternal truths, and finding his pleasures in the
realities of form and number, he has little interest in the
disputes and contentions of the passing hour. His views on social
and political questions partake of the grandeur of his favorite
contemplations, and, while careful to throw his mite of influence
on the side of right and truth, he is content to abide the
workings of those general laws by which he doubts not that the
fluctuations of human history are as unerringly guided as are the
perturbations of the planetary hosts.--HILL, THOMAS.
_Imagination in Mathematics; North
American Review, Vol. 85, p. 227._
=843.= There is something sublime in the secrecy in which the
really great deeds of the mathematician are done. No popular
applause follows the act; neither contemporary nor succeeding
generations of the people understand it. The geometer must be tried
by his peers, and those who truly deserve the title of geometer or
analyst have usually been unable to find so many as twelve living
peers to form a jury. Archimedes so far outstripped his
competitors in the race, that more than a thousand years elapsed
before any man appeared, able to sit in judgment on his work, and
to say how far he had really gone. And in judging of those men
whose names are worthy of being mentioned in connection with
his,--Galileo, Descartes, Leibnitz, Newton, and the mathematicians
created by Leibnitz and Newton’s calculus,--we are forced to
depend upon their testimony of one another. They are too far above
our reach for us to judge of them.--HILL, THOMAS.
_Imagination in Mathematics; North
American Review, Vol. 85, p. 223._
=844.= To think the thinkable--that is the mathematician’s aim.
--KEYSER, C. J.
_The Universe and Beyond; Hibbert
Journal, Vol. 3 (1904-1905), p. 312._
=845.= Every common mechanic has something to say in his craft
about good and evil, useful and useless, but these practical
considerations never enter into the purview of the mathematician.
--ARISTIPPUS THE CYRENAIC.
_Quoted in Hicks, R. D., Stoic and
Epicurean, (New York, 1910) p. 210._