MODERN MATHEMATICS
=701.= Surely this is the golden age of mathematics.
--PIERPONT, JAMES.
_History of Mathematics in the
Nineteenth Century; Congress of Arts and
Sciences (Boston and New York, 1905),
Vol. 1, p. 493._
=702.= The golden age of mathematics--that was not the age of Euclid,
it is ours. Ours is the age when no less than six international
congresses have been held in the course of nine years. It is in
our day that more than a dozen mathematical societies contain a
growing membership of more than two thousand men representing the
centers of scientific light throughout the great culture nations
of the world. It is in our time that over five hundred scientific
journals are each devoted in part, while more than two score
others are devoted exclusively, to the publication of mathematics.
It is in our time that the _Jahrbuch über die Fortschritte der
Mathematik_, though admitting only condensed abstracts with
titles, and not reporting on all the journals, has, nevertheless,
grown to nearly forty huge volumes in as many years. It is in our
time that as many as two thousand books and memoirs drop from the
mathematical press of the world in a single year, the estimated
number mounting up to fifty thousand in the last generation.
Finally, to adduce yet another evidence of a similar kind, it
requires not less than seven ponderous tomes of the forthcoming
_Encyclopaedie der Mathematischen Wissenschaften_ to contain, not
expositions, not demonstrations, but merely compact reports and
bibliographic notices sketching developments that have taken place
since the beginning of the nineteenth century.--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 8._
=703.= I have said that mathematics is the oldest of the sciences; a
glance at its more recent history will show that it has the energy
of perpetual youth. The output of contributions to the advance of the
science during the last century and more has been so enormous that it
is difficult to say whether pride in the greatness of achievement in
this subject, or despair at his inability to cope with the multiplicity
of its detailed developments, should be the dominant feeling of the
mathematician. Few people outside of the small circle of mathematical
specialists have any idea of the vast growth of mathematical
literature. The Royal Society Catalogue contains a list of nearly
thirty-nine thousand papers on subjects of Pure Mathematics alone,
which have appeared in seven hundred serials during the nineteenth
century. This represents only a portion of the total output, the very
large number of treatises, dissertations, and monographs published
during the century being omitted.--HOBSON, E. W.
_Presidential Address British
Association for the Advancement of
Science, Section A, (1910); Nature, Vol.
84, p. 285._
=704.= Mathematics is one of the oldest of the sciences; it is
also one of the most active, for its strength is the vigour of
perpetual youth.--FORSYTH, A. R.
_Presidential Address British
Association for the Advancement of
Science, Section A, (1897); Nature, Vol.
56, p. 378._
=705.= The nineteenth century which prides itself upon the
invention of steam and evolution, might have derived a more
legitimate title to fame from the discovery of pure mathematics.
--RUSSELL, BERTRAND.
_International Monthly, Vol. 4 (1901),
p. 83._
=706.= One of the chiefest triumphs of modern mathematics
consists in having discovered what mathematics really is.
--RUSSELL, BERTRAND.
_International Monthly, Vol. 4 (1901),
p. 84._
=707.= Modern mathematics, that most astounding of intellectual
creations, has projected the mind’s eye through infinite time and
the mind’s hand into boundless space.--BUTLER, N. M.
_The Meaning of Education and other
Essays and Addresses (New York, 1905),
p. 44._
=708.= The extraordinary development of mathematics in the last
century is quite unparalleled in the long history of this most
ancient of sciences. Not only have those branches of mathematics
which were taken over from the eighteenth century steadily grown,
but entirely new ones have sprung up in almost bewildering
profusion, and many of them have promptly assumed proportions of
vast extent.--PIERPONT, J.
_The History of Mathematics in the
Nineteenth Century; Congress of Arts and
Sciences (Boston and New York, 1905),
Vol. 1, p. 474._
=709.= The Modern Theory of Functions--that stateliest of all the
pure creations of the human intellect.--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 16._
=710.= If a mathematician of the past, an Archimedes or even a
Descartes, could view the field of geometry in its present
condition, the first feature to impress him would be its lack of
concreteness. There are whole classes of geometric theories which
proceed not only without models and diagrams, but without the
slightest (apparent) use of spatial intuition. In the main
this is due, to the power of the analytic instruments of
investigations as compared with the purely geometric.
--KASNER, EDWARD.
_The Present Problems in Geometry;
Bulletin American Mathematical Society,
1905, p. 285._
=711.= In Euclid each proposition stands by itself; its
connection with others is never indicated; the leading ideas
contained in its proof are not stated; general principles do not
exist. In modern methods, on the other hand, the greatest
importance is attached to the leading thoughts which pervade the
whole; and general principles, which bring whole groups of
theorems under one aspect, are given rather than separate
propositions. The whole tendency is toward generalization. A
straight line is considered as given in its entirety, extending
both ways to infinity, while Euclid is very careful never to
admit anything but finite quantities. The treatment of the
infinite is in fact another fundamental difference between the
two methods. Euclid avoids it, in modern mathematics it is
systematically introduced, for only thus is generality obtained.
--CAYLEY, ARTHUR.
_Encyclopedia Britannica (9th edition),
Article “Geometry.”_
=712.= This is one of the greatest advantages of modern geometry
over the ancient, to be able, through the consideration of
positive and negative quantities, to include in a single
enunciation the several cases which the same theorem may present
by a change in the relative position of the different parts of a
figure. Thus in our day the nine principal problems and the
numerous particular cases, which form the object of eighty-three
theorems in the two books _De sectione determinata_ of Appolonius
constitute only one problem which is resolved by a single
equation.--CHASLES, M.
_Histoire de la Géométrie, chap. 1,
sect. 35._
=713.= Euclid always contemplates a straight line as drawn
between two definite points, and is very careful to mention when
it is to be produced beyond this segment. He never thinks of the
line as an entity given once for all as a whole. This careful
definition and limitation, so as to exclude an infinity not
immediately apparent to the senses, was very characteristic of
the Greeks in all their many activities. It is enshrined in the
difference between Greek architecture and Gothic architecture,
and between Greek religion and modern religion. The spire of a
Gothic cathedral and the importance of the unbounded straight
line in modern Geometry are both emblematic of the transformation
of the modern world.--WHITEHEAD, A. N.
_Introduction to Mathematics (New York,
1911), p. 119._
=714.= The geometrical problems and theorems of the Greeks always
refer to definite, oftentimes to rather complicated figures. Now
frequently the points and lines of such a figure may assume very
many different relative positions; each of these possible cases
is then considered separately. On the contrary, present day
mathematicians generate their figures one from another, and are
accustomed to consider them subject to variation; in this manner
they unite the various cases and combine them as much as possible
by employing negative and imaginary magnitudes. For example, the
problems which Appolonius treats in his two books _De sectione
rationis_, are solved today by means of a single, universally
applicable construction; Apollonius, on the contrary, separates
it into more than eighty different cases varying only in
position. Thus, as Hermann Hankel has fittingly remarked, the
ancient geometry sacrifices to a seeming simplicity the true
simplicity which consists in the unity of principles; it attained
a trivial sensual presentability at the cost of the recognition
of the relations of geometric forms in all their changes and in
all the variations of their sensually presentable positions.
--REYE, THEODORE.
_Die synthetische Geometrie im Altertum
und in der Neuzeit; Jahresbericht der
Deutschen Mathematiker Vereinigung, Bd.
2, pp. 346-347._
=715.= It is known that the mathematics prescribed for the high
school [Gymnasien] is essentially Euclidean, while it is modern
mathematics, the theory of functions and the infinitesimal
calculus, which has secured for us an insight into the mechanism
and laws of nature. Euclidean mathematics is indeed, a prerequisite
for the theory of functions, but just as one, though he has
learned the inflections of Latin nouns and verbs, will not thereby
be enabled to read a Latin author much less to appreciate the
beauties of a Horace, so Euclidean mathematics, that is the
mathematics of the high school, is unable to unlock nature and
her laws. Euclidean mathematics assumes the completeness and
invariability of mathematical forms; these forms it describes with
appropriate accuracy and enumerates their inherent and related
properties with perfect clearness, order, and completeness, that
is, Euclidean mathematics operates on forms after the manner that
anatomy operates on the dead body and its members.
On the other hand, the mathematics of variable
magnitudes--function theory or analysis--considers mathematical
forms in their genesis. By writing the equation of the parabola,
we express its law of generation, the law according to which the
variable point moves. The path, produced before the eyes of the
student by a point moving in accordance to this law, is the
parabola.
If, then, Euclidean mathematics treats space and number forms
after the manner in which anatomy treats the dead body, modern
mathematics deals, as it were, with the living body, with growing
and changing forms, and thus furnishes an insight, not only into
nature as she is and appears, but also into nature as she
generates and creates,--reveals her transition steps and in so
doing creates a mind for and understanding of the laws of
becoming. Thus modern mathematics bears the same relation to
Euclidean mathematics that physiology or biology ... bears to
anatomy. But it is exactly in this respect that our view of
nature is so far above that of the ancients; that we no longer
look on nature as a quiescent complete whole, which compels
admiration by its sublimity and wealth of forms, but that we
conceive of her as a vigorous growing organism, unfolding
according to definite, as delicate as far-reaching, laws; that we
are able to lay hold of the permanent amidst the transitory, of
law amidst fleeting phenomena, and to be able to give these their
simplest and truest expression through the mathematical formulas.
--DILLMANN, E.
_Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 37._
=716.= The Excellence of _Modern Geometry_ is in nothing more
evident, than in those full and adequate Solutions it gives to
Problems; representing all possible Cases in one view, and in one
general Theorem many times comprehending whole Sciences; which
deduced at length into Propositions, and demonstrated after the
manner of the _Ancients_, might well become the subjects of large
Treatises: For whatsoever Theorem solves the most complicated
Problem of the kind, does with a due Reduction reach all the
subordinate Cases.--HALLEY, E.
_An Instance of the Excellence of Modern
Algebra, etc.; Philosophical
Transactions, 1694, p. 960._
=717.= One of the most conspicuous and distinctive features of
mathematical thought in the nineteenth century is its critical
spirit. Beginning with the calculus, it soon permeates all analysis,
and toward the close of the century it overhauls and recasts the
foundations of geometry and aspires to further conquests in
mechanics and in the immense domains of mathematical physics....
A searching examination of the foundations of arithmetic and the
calculus has brought to light the insufficiency of much of the
reasoning formerly considered as conclusive.--PIERPONT, J.
_History of Mathematics in the
Nineteenth Century; Congress of Arts and
Sciences (Boston and New York, 1905),
Vol. 1, p. 482._
=718.= If we compare a mathematical problem with an immense rock,
whose interior we wish to penetrate, then the work of the Greek
mathematicians appears to us like that of a robust stonecutter,
who, with indefatigable perseverance, attempts to demolish the
rock gradually from the outside by means of hammer and chisel;
but the modern mathematician resembles an expert miner, who first
constructs a few passages through the rock and then explodes it
with a single blast, bringing to light its inner treasures.
--HANKEL, HERMANN.
_Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884),
p. 9._
=719.= All the modern higher mathematics is based on a calculus
of operations, on laws of thought. All mathematics, from the
first, was so in reality; but the evolvers of the modern higher
calculus have known that it is so. Therefore elementary teachers
who, at the present day, persist in thinking about algebra and
arithmetic as dealing with laws of number, and about geometry as
dealing with laws of surface and solid content, are doing the
best that in them lies to put their pupils on the wrong track for
reaching in the future any true understanding of the higher
algebras. Algebras deal not with laws of number, but with such
laws of the human thinking machinery as have been discovered in
the course of investigations on numbers. Plane geometry deals
with such laws of thought as were discovered by men intent on
finding out how to measure surface; and solid geometry with such
additional laws of thought as were discovered when men began to
extend geometry into three dimensions.--BOOLE M. E.
_Logic of Arithmetic (Oxford, 1903),
Preface, pp. 18-19._
=720.= It is not only a decided preference for synthesis and a
complete denial of general methods which characterizes the
ancient mathematics as against our newer science [modern
mathematics]: besides this external formal difference there is
another real, more deeply seated, contrast, which arises from the
different attitudes which the two assumed relative to the use of
the concept of _variability_. For while the ancients, on account
of considerations which had been transmitted to them from the
philosophic school of the Eleatics, never employed the concept of
motion, the spatial expression for variability, in their rigorous
system, and made incidental use of it only in the treatment of
phonoromically generated curves, modern geometry dates from the
instant that Descartes left the purely algebraic treatment of
equations and proceeded to investigate the variations which an
algebraic expression undergoes when one of its variables assumes
a continuous succession of values.--HANKEL, HERMANN.
_Untersuchungen über die unendlich oft
oszillierenden und unstetigen
Functionen; Ostwald’s Klassiker der
exacten Wissenschaften, No. 153, pp.
44-45._
=721.= Without doubt one of the most characteristic features of
mathematics in the last century is the systematic and universal
use of the complex variable. Most of its great theories received
invaluable aid from it, and many owe their very existence to it.
--PIERPONT, J.
_History of Mathematics in the
Nineteenth Century; Congress of Arts and
Sciences (Boston and New York, 1905),
Vol. 1, p. 474._
=722.= The notion, which is really the fundamental one (and I
cannot too strongly emphasise the assertion), underlying and
pervading the whole of modern analysis and geometry, is that of
imaginary magnitude in analysis and of imaginary space in
geometry.--CAYLEY, ARTHUR.
_Presidential Address; Collected Works,
Vol. 11, p. 434._
=723.= The solution of the difficulties which formerly surrounded
the mathematical infinite is probably the greatest achievement of
which our age has to boast.--RUSSELL, BERTRAND.
_The Study of Mathematics; Philosophical
Essays (London, 1910), p. 77._
=724.= Induction and analogy are the special characteristics of
modern mathematics, in which theorems have given place to
theories and no truth is regarded otherwise than as a link in an
infinite chain. “_Omne exit in infinitum_” is their favorite
motto and accepted axiom.--SYLVESTER, J. J.
_A Plea for the Mathematician; Nature,
Vol. 1, p. 261._
=725.= The conception of correspondence plays a great part in
modern mathematics. It is the fundamental notion in the science
of order as distinguished from the science of magnitude. If the
older mathematics were mostly dominated by the needs of
mensuration, modern mathematics are dominated by the conception
of order and arrangement. It may be that this tendency of thought
or direction of reasoning goes hand in hand with the modern
discovery in physics, that the changes in nature depend not only
or not so much on the quantity of mass and energy as on their
distribution or arrangement.--MERZ, J. T.
_History of European Thought in the
Nineteenth Century (Edinburgh and
London, 1903), p. 736._
=726.= Now this establishment of correspondence between two
aggregates and investigation of the propositions that are carried
over by the correspondence may be called the central idea of
modern mathematics.--CLIFFORD, W. K.
_Philosophy of the Pure Sciences;
Lectures and Essays (London, 1901), Vol.
1, p. 402._
=727.= In our century the conceptions substitution and
substitution group, transformation and transformation group,
operation and operation group, invariant, differential invariant
and differential parameter, appear more and more clearly as the
most important conceptions of mathematics.--LIE, SOPHUS.
_Leipziger Berichte, No. 47 (1895), p.
261._
=728.= Generality of points of view and of methods, precision and
elegance in presentation, have become, since Lagrange, the common
property of all who would lay claim to the rank of scientific
mathematicians. And, even if this generality leads at times to
abstruseness at the expense of intuition and applicability, so
that general theorems are formulated which fail to apply to
a single special case, if furthermore precision at times
degenerates into a studied brevity which makes it more difficult
to read an article than it was to write it; if, finally, elegance
of form has well-nigh become in our day the criterion of the
worth or worthlessness of a proposition,--yet are these
conditions of the highest importance to a wholesome development,
in that they keep the scientific material within the limits which
are necessary both intrinsically and extrinsically if mathematics
is not to spend itself in trivialities or smother in profusion.
--HANKEL, HERMANN.
_Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884),
pp. 14-15._
=729.= The development of abstract methods during the past few
years has given mathematics a new and vital principle which
furnishes the most powerful instrument for exhibiting the
essential unity of all its branches.--YOUNG, J. W.
_Fundamental Concepts of Algebra and
Geometry (New York, 1911), p. 225._
=730.= Everybody praises the incomparable power of the
mathematical method, but so is everybody aware of its
incomparable unpopularity.--ROSANES, J.
_Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 13, p.
17._
=731.= Indeed the modern developments of mathematics constitute
not only one of the most impressive, but one of the most
characteristic, phenomena of our age. It is a phenomenon,
however, of which the boasted intelligence of a “universalized”
daily press seems strangely unaware; and there is no other great
human interest, whether of science or of art, regarding which the
mind of the educated public is permitted to hold so many
fallacious opinions and inferior estimates.--KEYSER, C. J.
_Lectures on Science, Philosophy and
Arts (New York, 1908), p. 8._
=732.= It may be asserted without exaggeration that the domain of
mathematical knowledge is the only one of which our otherwise
omniscient journalism has not yet possessed itself.
--PRINGSHEIM, ALFRED.
_Ueber Wert und angeblichen Unwert der
Mathematik; Jahresbericht der Deutschen
Mathematiker Vereinigung, (1904) p.
357._
=733.= [The] inaccessibility of special fields of mathematics,
except by the regular way of logically antecedent acquirements,
renders the study discouraging or hateful to weak or indolent
minds.--LEFEVRE, ARTHUR.
_Number and its Algebra (Boston, 1903),
sect. 223._
=734.= The majority of mathematical truths now possessed by us
presuppose the intellectual toil of many centuries. A mathematician,
therefore, who wishes today to acquire a thorough understanding of
modern research in this department, must think over again in
quickened tempo the mathematical labors of several centuries. This
constant dependence of new truths on old ones stamps mathematics
as a science of uncommon exclusiveness and renders it generally
impossible to lay open to uninitiated readers a speedy path to the
apprehension of the higher mathematical truths. For this reason,
too, the theories and results of mathematics are rarely adapted
for popular presentation.... This same inaccessibility of
mathematics, although it secures for it a lofty and aristocratic
place among the sciences, also renders it odious to those who have
never learned it, and who dread the great labor involved in
acquiring an understanding of the questions of modern mathematics.
Neither in the languages nor in the natural sciences are the
investigations and results so closely interdependent as to make it
impossible to acquaint the uninitiated student with single
branches or with particular results of these sciences, without
causing him to go through a long course of preliminary study.
--SCHUBERT, H.
_Mathematical Essays and Recreations
(Chicago, 1898), p. 32._
=735.= Such is the character of mathematics in its profounder
depths and in its higher and remoter zones that it is well nigh
impossible to convey to one who has not devoted years to its
exploration a just impression of the scope and magnitude of the
existing body of the science. An imagination formed by other
disciplines and accustomed to the interests of another field may
scarcely receive suddenly an apocalyptic vision of that infinite
interior world. But how amazing and how edifying were such a
revelation, if it only could be made.--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 6._
=736.= It is not so long since, during one of the meetings of the
Association, one of the leading English newspapers briefly
described a sitting of this Section in the words, “Saturday
morning was devoted to pure mathematics, and so there was nothing
of any general interest:” still, such toleration is better than
undisguised and ill-informed hostility.--FORSYTH, A. R.
_Report of the 67th meeting of the
British Association for the Advancement
of Science._
=737.= The science [of mathematics] has grown to such vast
proportion that probably no living mathematician can claim to
have achieved its mastery as a whole.--WHITEHEAD, A. N.
_An Introduction to Mathematics (New
York, 1911), p. 252._
=738.= There is perhaps no science of which the development has
been carried so far, which requires greater concentration and
will power, and which by the abstract height of the qualities
required tends more to separate one from daily life.
_Provisional Report of the American
Subcommittee of the International
Commission on the Teaching of
Mathematics; Bulletin American Society
(1910), p. 97._
=739.= Angling may be said to be so like the mathematics, that it
can never be fully learnt.--WALTON, ISAAC.
_The Complete Angler, Preface._
=740.= The flights of the imagination which occur to the pure
mathematician are in general so much better described in his
formulæ than in words, that it is not remarkable to find the
subject treated by outsiders as something essentially cold and
uninteresting--... the only successful attempt to invest
mathematical reasoning with a halo of glory--that made in this
section by Prof. Sylvester--is known to a comparative few, ....
--TAIT, P. G.
_Presidential Address British
Association for the Advancement of
Science (1871); Nature Vol. 4, p. 271._