MATHEMATICS AS A LANGUAGE
=1201.= The new mathematics is a sort of supplement to language,
affording a means of thought about form and quantity and a means
of expression, more exact, compact, and ready than ordinary
language. The great body of physical science, a great deal of the
essential facts of financial science, and endless social and
political problems are only accessible and only thinkable to
those who have had a sound training in mathematical analysis, and
the time may not be very remote when it will be understood that
for complete initiation as an efficient citizen of one of the new
great complex world wide states that are now developing, it is as
necessary to be able to compute, to think in averages and maxima
and minima, as it is now to be able to read and to write.
--WELLS, H. G.
_Mankind in the Making (London, 1904),
pp. 191-192._
=1202.= Mathematical language is not only the simplest and most
easily understood of any, but the shortest also.--BROUGHAM, H. L.
_Works (Edinburgh, 1872), Vol. 7, p.
317._
=1203.= Mathematics is the science of definiteness, the necessary
vocabulary of those who know.--WHITE, W. F.
_A Scrap-book of Elementary Mathematics
(Chicago, 1908), p. 7._
=1204.= Mathematics, too, is a language, and as concerns its
structure and content it is the most perfect language which
exists, superior to any vernacular; indeed, since it is
understood by every people, mathematics may be called the
language of languages. Through it, as it were, nature herself
speaks; through it the Creator of the world has spoken, and
through it the Preserver of the world continues to speak.
--DILLMANN, C.
_Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 5._
=1205.= Would it sound too presumptuous to speak of perception as
a quintessence of sensation, language (that is, communicable
thought) of perception, mathematics of language? We should then
have four terms differentiating from inorganic matter and from
each other the Vegetable, Animal, Rational, and Super-sensual
modes of existence.--SYLVESTER, J. J.
_Presidential Address, British
Association; Collected Mathematical
Papers, Vol. 2, p. 652._
=1206.= Little could Plato have imagined, when, indulging his
instinctive love of the true and beautiful for their own sakes,
he entered upon these refined speculations and revelled in a
world of his own creation, that he was writing the grammar of the
language in which it would be demonstrated in after ages that the
pages of the universe are written.--SYLVESTER, J. J.
_A Probationary Lecture on Geometry;
Collected Mathematical Papers, Vol. 2,
p. 7._
=1207.= It is the symbolic language of mathematics only which has
yet proved sufficiently accurate and comprehensive to demand
familiarity with this conception of an inverse process.
--VENN, JOHN.
_Symbolic Logic (London and New York,
1894), p. 74._
=1208.= Without this language [mathematics] most of the intimate
analogies of things would have remained forever unknown to us;
and we should forever have been ignorant of the internal harmony
of the world, which is the only true objective reality....
This harmony ... is the sole objective reality, the only truth we
can attain; and when I add that the universal harmony of the
world is the source of all beauty, it will be understood what
price we should attach to the slow and difficult progress which
little by little enables us to know it better.--POINCARÉ, H.
_The Value of Science [Halsted] Popular
Science Monthly, 1906, pp. 195-196._
=1209.= The most striking characteristic of the written language
of algebra and of the higher forms of the calculus is the
sharpness of definition, by which we are enabled to reason upon
the symbols by the mere laws of verbal logic, discharging our
minds entirely of the meaning of the symbols, until we have
reached a stage of the process where we desire to interpret our
results. The ability to attend to the symbols, and to perform the
verbal, visible changes in the position of them permitted by the
logical rules of the science, without allowing the mind to be
perplexed with the meaning of the symbols until the result is
reached which you wish to interpret, is a fundamental part of
what is called analytical power. Many students find themselves
perplexed by a perpetual attempt to interpret not only the
result, but each step of the process. They thus lose much of the
benefit of the labor-saving machinery of the calculus and are,
indeed, frequently incapacitated for using it.--HILL, THOMAS.
_Uses of Mathesis; Bibliotheca Sacra,
Vol. 32, p. 505._
=1210.= The prominent reason why a mathematician can be judged by
none but mathematicians, is that he uses a peculiar language. The
language of mathesis is special and untranslatable. In its
simplest forms it can be translated, as, for instance, we say a
right angle to mean a square corner. But you go a little higher
in the science of mathematics, and it is impossible to dispense
with a peculiar language. It would defy all the power of Mercury
himself to explain to a person ignorant of the science what is
meant by the single phrase “functional exponent.” How much more
impossible, if we may say so, would it be to explain a whole
treatise like Hamilton’s Quaternions, in such a wise as to make
it possible to judge of its value! But to one who has learned
this language, it is the most precise and clear of all modes of
expression. It discloses the thought exactly as conceived by the
writer, with more or less beauty of form, but never with
obscurity. It may be prolix, as it often is among French writers;
may delight in mere verbal metamorphoses, as in the Cambridge
University of England; or adopt the briefest and clearest forms,
as under the pens of the geometers of our Cambridge; but it
always reveals to us precisely the writer’s thought.
--HILL, THOMAS.
_North American Review, Vol. 85, pp.
224-225._
=1211.= The domain, over which the language of analysis extends
its sway, is, indeed, relatively limited, but within this domain
it so infinitely excels ordinary language that its attempt to
follow the former must be given up after a few steps. The
mathematician, who knows how to think in this marvelously
condensed language, is as different from the mechanical computer
as heaven from earth.--PRINGSHEIM, A.
_Jahresberichte der Deutschen
Mathematiker Vereinigung, Bd. 13, p.
367._
=1212.= The results of systematic symbolical reasoning must
_always_ express general truths, by their nature; and do not, for
their justification, require each of the steps of the process to
represent some definite operation upon quantity. The _absolute
universality of the interpretation of symbols_ is the fundamental
principle of their use.--WHEWELL, WILLIAM.
_The Philosophy of the Inductive
Sciences, Part I, Bk. 2, chap. 12, sect.
2 (London, 1858)._
=1213.= Anyone who understands algebraic notation, reads at a
glance in an equation results reached arithmetically only with
great labour and pains.--COURNOT, A.
_Theory of Wealth [N. T. Bacon], (New
York, 1897), p. 4._
=1214.= As arithmetic and algebra are sciences of great
clearness, certainty, and extent, which are immediately
conversant about signs, upon the skilful use whereof they
entirely depend, so a little attention to them may possibly help
us to judge of the progress of the mind in other sciences, which,
though differing in nature, design, and object, may yet agree in
the general methods of proof and inquiry.--BERKELEY, GEORGE.
_Alciphron, or the Minute Philosopher,
Dialogue 7, sect. 12._
=1215.= In general the position as regards all such new calculi
is this--That one cannot accomplish by them anything that could
not be accomplished without them. However, the advantage is,
that, provided such a calculus corresponds to the inmost
nature of frequent needs, anyone who masters it thoroughly is
able--without the unconscious inspiration of genius which no one
can command--to solve the respective problems, yea, to solve them
mechanically in complicated cases in which, without such aid,
even genius becomes powerless. Such is the case with the
invention of general algebra, with the differential calculus, and
in a more limited region with Lagrange’s calculus of variations,
with my calculus of congruences, and with Möbius’s calculus. Such
conceptions unite, as it were, into an organic whole countless
problems which otherwise would remain isolated and require for
their separate solution more or less application of inventive
genius.--GAUSS, C. J.
_Werke, Bd. 8, p. 298._
=1216.= The invention of what we may call primary or fundamental
notation has been but little indebted to analogy, evidently owing
to the small extent of ideas in which comparison can be made
useful. But at the same time analogy should be attended to, even
if for no other reason than that, by making the invention of
notation an art, the exertion of individual caprice ceases to be
allowable. Nothing is more easy than the invention of notation,
and nothing of worse example and consequence than the confusion
of mathematical expressions by unknown symbols. If new notation
be advisable, permanently or temporarily, it should carry with it
some mark of distinction from that which is already in use,
unless it be a demonstrable extension of the latter.
--DE MORGAN, A.
_Calculus of Functions; Encyclopedia
Metropolitana, Addition to Article 26._
=1217.= Before the introduction of the Arabic notation,
multiplication was difficult, and the division even of integers
called into play the highest mathematical faculties. Probably
nothing in the modern world could have more astonished a Greek
mathematician than to learn that, under the influence of
compulsory education, the whole population of Western Europe,
from the highest to the lowest, could perform the operation of
division for the largest numbers. This fact would have seemed to
him a sheer impossibility.... Our modern power of easy reckoning
with decimal fractions is the most miraculous result of a perfect
notation.--WHITEHEAD, A. N.
_Introduction to Mathematics (New York,
1911), p. 59._
=1218.= Mathematics is often considered a difficult and
mysterious science, because of the numerous symbols which it
employs. Of course, nothing is more incomprehensible than a
symbolism which we do not understand. Also a symbolism, which we
only partially understand and are unaccustomed to use, is
difficult to follow. In exactly the same way the technical terms
of any profession or trade are incomprehensible to those who have
never been trained to use them. But this is not because they are
difficult in themselves. On the contrary they have invariably
been introduced to make things easy. So in mathematics, granted
that we are giving any serious attention to mathematical ideas,
the symbolism is invariably an immense simplification.
--WHITEHEAD, A. N.
_Introduction to Mathematics (New York,
1911), pp. 59-60._
=1219.= Symbolism is useful because it makes things difficult.
Now in the beginning everything is self-evident, and it is hard
to see whether one self-evident proposition follows from another
or not. Obviousness is always the enemy to correctness. Hence we
must invent a new and difficult symbolism in which nothing is
obvious.... Thus the whole of Arithmetic and Algebra has
been shown to require three indefinable notions and five
indemonstrable propositions.--RUSSELL, BERTRAND.
_International Monthly, 1901, p. 85._
=1220.= The employment of mathematical symbols is perfectly
natural when the relations between magnitudes are under
discussion; and even if they are not rigorously necessary, it
would hardly be reasonable to reject them, because they are not
equally familiar to all readers and because they have sometimes
been wrongly used, if they are able to facilitate the exposition
of problems, to render it more concise, to open the way to more
extended developments, and to avoid the digressions of vague
argumentation.--COURNOT, A.
_Theory of Wealth [N. T. Bacon], (New
York, 1897), pp. 3-4._
=1221.= An all-inclusive geometrical symbolism, such as Hamilton
and Grassmann conceived of, is impossible.--BURKHARDT, H.
_Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 5, p. 52._
=1222.= The language of analysis, most perfect of all, being in
itself a powerful instrument of discoveries, its notations,
especially when they are necessary and happily conceived, are so
many germs of new calculi.--LAPLACE.
_Oeuvres, t. 7 (Paris, 1896), p. xl._