THE VALUE OF MATHEMATICS
=401.= Mathematics because of its nature and structure is
peculiarly fitted for high school instruction [Gymnasiallehrfach].
Especially the higher mathematics, even if presented only in its
elements, combines within itself all those qualities which are
demanded of a secondary subject. It engages, it fructifies, it
quickens, compels attention, is as circumspect as inventive,
induces courage and self-confidence as well as modesty and
submission to truth. It yields the essence and kernel of all
things, is brief in form and overflows with its wealth of content.
It discloses the depth and breadth of the law and spiritual
element behind the surface of phenomena; it impels from point to
point and carries within itself the incentive toward progress; it
stimulates the artistic perception, good taste in judgment and
execution, as well as the scientific comprehension of things.
Mathematics, therefore, above all other subjects, makes the
student lust after knowledge, fills him, as it were, with a
longing to fathom the cause of things and to employ his own powers
independently; it collects his mental forces and concentrates them
on a single point and thus awakens the spirit of individual
inquiry, self-confidence and the joy of doing; it fascinates
because of the view-points which it offers and creates certainty
and assurance, owing to the universal validity of its methods.
Thus, both what he receives and what he himself contributes toward
the proper conception and solution of a problem, combine to mature
the student and to make him skillful, to lead him away from the
surface of things and to exercise him in the perception of their
essence. A student thus prepared thirsts after knowledge and is
ready for the university and its sciences. Thus it appears, that
higher mathematics is the best guide to philosophy and to the
philosophic conception of the world (considered as a self-contained
whole) and of one’s own being.--DILLMANN, E.
_Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 40._
=402.= These Disciplines [mathematics] serve to inure and
corroborate the Mind to a constant Diligence in Study; to undergo
the Trouble of an attentive Meditation, and cheerfully contend
with such Difficulties as lie in the Way. They wholly deliver us
from a credulous Simplicity, most strongly fortify us against
the Vanity of Scepticism, effectually restrain from a rash
Presumption, most easily incline us to a due Assent, perfectly
subject us to the Government of right Reason, and inspire us with
Resolution to wrestle against the unjust Tyranny of false
Prejudices. If the Fancy be unstable and fluctuating, it is to be
poised by this Ballast, and steadied by this Anchor, if the Wit
be blunt it is sharpened upon this Whetstone; if luxuriant it is
pared by this Knife; if headstrong it is restrained by this
Bridle; and if dull it is roused by this Spur. The Steps are
guided by no Lamp more clearly through the dark Mazes of Nature,
by no Thread more surely through the intricate Labyrinths of
Philosophy, nor lastly is the Bottom of Truth sounded more
happily by any other Line. I will not mention how plentiful a
Stock of Knowledge the Mind is furnished from these, with what
wholesome Food it is nourished, and what sincere Pleasure it
enjoys. But if I speak farther, I shall neither be the only
Person, nor the first, who affirms it; that while the Mind is
abstracted and elevated from sensible Matter, distinctly views
pure Forms, conceives the Beauty of Ideas, and investigates the
Harmony of Proportions; the Manners themselves are sensibly
corrected and improved, the Affections composed and rectified,
the Fancy calmed and settled, and the Understanding raised and
excited to more divine Contemplation. All which I might defend by
Authority, and confirm by the Suffrages of the greatest
Philosophers.--BARROW, ISAAC.
_Prefatory Oration: Mathematical
Lectures (London, 1734), p. 31._
=403.= No school subject so readily furnishes tasks whose purpose
can be made so clear, so immediate and so appealing to the sober
second-thought of the immature learner as the right sort of
elementary school mathematics.--MYERS, GEORGE.
_Arithmetic in Public School Education
(Chicago, 1911), p. 8._
=404.= Mathematics is a type of thought which seems ingrained in
the human mind, which manifests itself to some extent with even
the primitive races, and which is developed to a high degree with
the growth of civilization.... A type of thought, a body of
results, so essentially characteristic of the human mind, so
little influenced by environment, so uniformly present in every
civilization, is one of which no well-informed mind today can be
ignorant.--YOUNG, J. W. A.
_The Teaching of Mathematics (London,
1907), p. 14._
=405.= Probably among all the pursuits of the University,
mathematics pre-eminently demand self-denial, patience, and
perseverance from youth, precisely at that period when they have
liberty to act for themselves, and when on account of obvious
temptations, habits of restraint and application are peculiarly
valuable.--TODHUNTER, ISAAC.
_The Conflict of Studies and other
Essays (London, 1873), p. 12._
=406.= Mathematics renders its best service through the immediate
furthering of rigorous thought and the spirit of invention.
--HERBART J. F.
_Mathematischer Lehrplan für
Realschulen: Werke [Kehrbach]
(Langensalza, 1890), Bd. 5, p. 170._
=407.= It seems to me that the older subjects, classics and
mathematics, are strongly to be recommended on the ground of the
accuracy with which we can compare the relative performance of
the students. In fact the definiteness of these subjects is
obvious, and is commonly admitted. There is however another
advantage, which I think belongs in general to these subjects,
that the examinations can be brought to bear on what is really
most valuable in these subjects.--TODHUNTER, ISAAC.
_Conflict of Studies and other Essays
(London, 1873), pp. 6, 7._
=408.= It is better to teach the child arithmetic and Latin
grammar than rhetoric and moral philosophy, because they require
exactitude of performance it is made certain that the lesson is
mastered, and that power of performance is worth more than
knowledge.--EMERSON, R. W.
_Lecture on Education._
=409.= Besides accustoming the student to demand complete proof,
and to know when he has not obtained it, mathematical studies are
of immense benefit to his education by habituating him to
precision. It is one of the peculiar excellencies of mathematical
discipline, that the mathematician is never satisfied with
_à peu près._ He requires the exact truth. Hardly any of the
non-mathematical sciences, except chemistry, has this advantage.
One of the commonest modes of loose thought, and sources of error
both in opinion and in practice, is to overlook the importance of
quantities. Mathematicians and chemists are taught by the whole
course of their studies, that the most fundamental difference of
quality depends on some very slight difference in proportional
quantity; and that from the qualities of the influencing
elements, without careful attention to their quantities, false
expectation would constantly be formed as to the very nature and
essential character of the result produced.--MILL, J. S.
_An Examination of Sir William
Hamilton’s Philosophy (London, 1878), p.
611._
=410.= In mathematics I can report no deficience, except it be
that men do not sufficiently understand the excellent use of the
Pure Mathematics, in that they do remedy and cure many defects in
the wit and faculties intellectual. For if the wit be too dull,
they sharpen it; if too wandering, they fix it; if too inherent
in the senses, they abstract it. So that as tennis is a game of
no use in itself, but of great use in respect it maketh a quick
eye and a body ready to put itself into all positions; so in the
Mathematics, that use which is collateral and intervenient is no
less worthy than that which is principal and intended.--BACON, LORD.
_De Augmentis, Bk. 3; Advancement of
Learning, Bk. 2._
=411.= If a man’s wit be wandering, let him study mathematics;
for in demonstrations, if his wit be called away never so little,
he must begin again.--BACON, LORD.
_Essays: On Studies._
=412.= If one be bird-witted, that is easily distracted and
unable to keep his attention as long as he should, mathematics
provides a remedy; for in them if the mind be caught away but a
moment, the demonstration has to be commenced anew.--BACON, LORD.
_De Augmentis, Bk. 6; Advancement of
Learning, Bk. 2._
=413.= The metaphysical philosopher from his point of view
recognizes mathematics as an instrument of education, which
strengthens the power of attention, develops the sense of order
and the faculty of construction, and enables the mind to grasp
under the simple formulae the quantitative differences of
physical phenomena.--JOWETT, B.
_Dialogues of Plato (New York, 1897),
Vol. 2, p. 78._
=414.= Nor do I know any study which can compete with mathematics
in general in furnishing matter for severe and continued thought.
Metaphysical problems may be even more difficult; but then they
are far less definite, and, as they rarely lead to any precise
conclusion, we miss the power of checking our own operations, and
of discovering whether we are thinking and reasoning or merely
fancying and dreaming.--TODHUNTER, ISAAC.
_Conflict of Studies (London, 1873), p.
13._
=415.= Another great and special excellence of mathematics is
that it demands earnest voluntary exertion. It is simply
impossible for a person to become a good mathematician by the
happy accident of having been sent to a good school; this may
give him a preparation and a start, but by his own individual
efforts alone can he reach an eminent position.--TODHUNTER, ISAAC.
_Conflict of Studies (London, 1873), p.
2._
=416.= The faculty of resolution is possibly much invigorated by
mathematical study, and especially by that highest branch of it
which, unjustly, merely on account of its retrograde operations,
has been called, as if par excellence, analysis.--POE, E. A.
_The Murders in Rue Morgue._
=417.= He who gives a portion of his time and talent to the
investigation of mathematical truth will come to all other
questions with a decided advantage over his opponents. He will be
in argument what the ancient Romans were in the field: to them
the day of battle was a day of comparative recreation, because
they were ever accustomed to exercise with arms much heavier than
they fought; and reviews differed from a real battle in two
respects: they encountered more fatigue, but the victory was
bloodless.--COLTON, C. C.
_Lacon (New York, 1866)._
=418.= Mathematics is the study which forms the foundation of the
course [West Point Military Academy]. This is necessary, both to
impart to the mind that combined strength and versatility, the
peculiar vigor and rapidity of comparison necessary for military
action, and to pave the way for progress in the higher military
sciences.
_Congressional Committee on Military
Affairs, 1834; U. S. Bureau of
Education, Bulletin 1912, No. 2, p. 10._
=419.= Mathematics, among all school subjects, is especially
adapted to further clearness, definite brevity and precision in
expression, although it offers no exercise in flights of
rhetoric. This is due in the first place to the logical rigour
with which it develops thought, avoiding every departure from the
shortest, most direct way, never allowing empty phrases to enter.
Other subjects excel in the development of expression in other
respects: translation from foreign languages into the mother
tongue gives exercise in finding the proper word for the given
foreign word and gives knowledge of laws of syntax, the study of
poetry and prose furnish fit patterns for connected presentation
and elegant form of expression, composition is to exercise the
pupil in a like presentation of his own or borrowed thoughts and
their development, the natural sciences teach description of
natural objects, apparatus and processes, as well as the
statement of laws on the grounds of immediate sense-perception.
But all these aids for exercise in the use of the mother tongue,
each in its way valuable and indispensable, do not guarantee, in
the same manner as mathematical training, the exclusion of words
whose concepts, if not entirely wanting, are not sufficiently
clear. They do not furnish in the same measure that which the
mathematician demands particularly as regards precision of
expression.--REIDT, F.
_Anleitung zum mathematischen Unterricht
in höheren Schulen (Berlin, 1906), p.
17._
=420.= One rarely hears of the mathematical recitation as a
preparation for public speaking. Yet mathematics shares with
these studies [foreign languages, drawing and natural science]
their advantages, and has another in a higher degree than either
of them.
Most readers will agree that a prime requisite for healthful
experience in public speaking is that the attention of the
speaker and hearers alike be drawn wholly away from the speaker
and concentrated upon the thought. In perhaps no other classroom
is this so easy as in the mathematical, where the close
reasoning, the rigorous demonstration, the tracing of necessary
conclusions from given hypotheses, commands and secures the
entire mental power of the student who is explaining, and of his
classmates. In what other circumstances do students feel so
instinctively that manner counts for so little and mind for so
much? In what other circumstances, therefore, is a simple,
unaffected, easy, graceful manner so naturally and so healthfully
cultivated? Mannerisms that are mere affectation or the result of
bad literary habit recede to the background and finally
disappear, while those peculiarities that are the expression of
personality and are inseparable from its activity continually
develop, where the student frequently presents, to an audience of
his intellectual peers, a connected train of reasoning....
One would almost wish that our institutions of the science and
art of public speaking would put over their doors the motto that
Plato had over the entrance to his school of philosophy: “Let no
one who is unacquainted with geometry enter here.”--WHITE, W. F.
_A Scrap-book of Elementary Mathematics
(Chicago, 1908), p. 210._
=421.= The training which mathematics gives in working with
symbols is an excellent preparation for other sciences; ... the
world’s work requires constant mastery of symbols.--YOUNG, J. W. A.
_The Teaching of Mathematics (New York,
1907), p. 42._
=422.= One striking peculiarity of mathematics is its unlimited
power of evolving examples and problems. A student may read a
book of Euclid, or a few chapters of Algebra, and within that
limited range of knowledge it is possible to set him exercises as
real and as interesting as the propositions themselves which he
has studied; deductions which might have pleased the Greek
geometers, and algebraic propositions which Pascal and Fermat
would not have disdained to investigate.--TODHUNTER, ISAAC.
_Private Study of Mathematics: Conflict
of Studies and other Essays (London,
1873), p. 82._
=423.= Would you have a man reason well, you must use him to it
betimes; exercise his mind in observing the connection between
ideas, and following them in train. Nothing does this better than
mathematics, which therefore, I think should be taught to all who
have the time and opportunity, not so much to make them
mathematicians, as to make them reasonable creatures; for though
we all call ourselves so, because we are born to it if we please,
yet we may truly say that nature gives us but the seeds of it,
and we are carried no farther than industry and application have
carried us.--LOCKE, JOHN.
_Conduct of the Understanding, Sect. 6._
=424.= Secondly, the study of mathematics would show them the
necessity there is in reasoning, to separate all the distinct
ideas, and to see the habitudes that all those concerned in the
present inquiry have to one another, and to lay by those which
relate not to the proposition in hand, and wholly to leave them
out of the reckoning. This is that which, in other respects
besides quantity is absolutely requisite to just reasoning,
though in them it is not so easily observed and so carefully
practised. In those parts of knowledge where it is thought
demonstration has nothing to do, men reason as it were in a lump;
and if upon a summary and confused view, or upon a partial
consideration, they can raise the appearance of a probability,
they usually rest content; especially if it be in a dispute where
every little straw is laid hold on, and everything that can but
be drawn in any way to give color to the argument is advanced
with ostentation. But that mind is not in a posture to find truth
that does not distinctly take all the parts asunder, and,
omitting what is not at all to the point, draws a conclusion from
the result of all the particulars which in any way influence it.
--LOCKE, JOHN.
_Conduct of the Understanding, Sect. 7._
=425.= I have before mentioned mathematics, wherein algebra gives
new helps and views to the understanding. If I propose these it
is not to make every man a thorough mathematician or deep
algebraist; but yet I think the study of them is of infinite use
even to grown men; first by experimentally convincing them, that
to make anyone reason well, it is not enough to have parts
wherewith he is satisfied, and that serve him well enough in his
ordinary course. A man in those studies will see, that however
good he may think his understanding, yet in many things, and
those very visible, it may fail him. This would take off that
presumption that most men have of themselves in this part; and
they would not be so apt to think their minds wanted no helps to
enlarge them, that there could be nothing added to the acuteness
and penetration of their understanding.--LOCKE, JOHN.
_The Conduct of the Understanding, Sect. 7._
=426.= I have mentioned mathematics as a way to settle in the mind
a habit of reasoning closely and in train; not that I think it
necessary that all men should be deep mathematicians, but that,
having got the way of reasoning which that study necessarily
brings the mind to, they might be able to transfer it to other
parts of knowledge, as they shall have occasion. For in all sorts
of reasoning, every single argument should be managed as a
mathematical demonstration; the connection and dependence of ideas
should be followed till the mind is brought to the source on which
it bottoms, and observes the coherence all along; ....--LOCKE, JOHN.
_The Conduct of the Understanding, Sect.
7._
=427.= As an exercise of the reasoning faculty, pure mathematics
is an admirable exercise, because it consists of _reasoning_
alone, and does not encumber the student with an exercise of
_judgment_: and it is well to begin with learning one thing at a
time, and to defer a combination of mental exercises to a later
period.--WHATELY, R.
_Annotations to Bacon’s Essays (Boston,
1873), Essay 1, p. 493._
=428.= It hath been an old remark, that Geometry is an excellent
Logic. And it must be owned that when the definitions are clear;
when the postulata cannot be refused, nor the axioms denied; when
from the distinct contemplation and comparison of figures, their
properties are derived, by a perpetual well-connected chain of
consequences, the objects being still kept in view, and the
attention ever fixed upon them; there is acquired a habit of
reasoning, close and exact and methodical; which habit strengthens
and sharpens the mind, and being transferred to other subjects is
of general use in the inquiry after truth.--BERKELY, GEORGE.
_The Analyst, 2; Works (London, 1898),
Vol. 3, p. 10._
=429.= Suppose then I want to give myself a little training in the
art of reasoning; suppose I want to get out of the region of
conjecture and probability, free myself from the difficult task of
weighing evidence, and putting instances together to arrive at
general propositions, and simply desire to know how to deal with
my general propositions when I get them, and how to deduce right
inferences from them; it is clear that I shall obtain this sort of
discipline best in those departments of thought in which the first
principles are unquestionably true. For in all our thinking, if we
come to erroneous conclusions, we come to them either by accepting
false premises to start with--in which case our reasoning, however
good, will not save us from error; or by reasoning badly, in which
case the data we start from may be perfectly sound, and yet
our conclusions may be false. But in the mathematical or pure
sciences,--geometry, arithmetic, algebra, trigonometry, the calculus
of variations or of curves,--we know at least that there is not,
and cannot be, error in our first principles, and we may therefore
fasten our whole attention upon the processes. As mere exercises
in logic, therefore, these sciences, based as they all are on
primary truths relating to space and number, have always been
supposed to furnish the most exact discipline. When Plato wrote
over the portal of his school. “Let no one ignorant of geometry
enter here,” he did not mean that questions relating to lines and
surfaces would be discussed by his disciples. On the contrary, the
topics to which he directed their attention were some of the
deepest problems,--social, political, moral,--on which the mind
could exercise itself. Plato and his followers tried to think out
together conclusions respecting the being, the duty, and the
destiny of man, and the relation in which he stood to the gods and
to the unseen world. What had geometry to do with these things?
Simply this: That a man whose mind has not undergone a rigorous
training in systematic thinking, and in the art of drawing legitimate
inferences from premises, was unfitted to enter on the discussion
of these high topics; and that the sort of logical discipline
which he needed was most likely to be obtained from geometry--the
only mathematical science which in Plato’s time had been formulated
and reduced to a system. And we in this country [England] have
long acted on the same principle. Our future lawyers, clergy, and
statesmen are expected at the University to learn a good deal
about curves, and angles, and numbers and proportions; not because
these subjects have the smallest relation to the needs of their
lives, but because in the very act of learning them they are
likely to acquire that habit of steadfast and accurate thinking,
which is indispensable to success in all the pursuits of life.
--FITCH, J. C.
_Lectures on Teaching (New York, 1906),
pp. 291-292._
=430.= It is admitted by all that a finished or even a competent
reasoner is not the work of nature alone; the experience of every
day makes it evident that education develops faculties which
would otherwise never have manifested their existence. It is,
therefore, as necessary to _learn to reason_ before we can expect
to be able to reason, as it is to learn to swim or fence, in
order to attain either of those arts. Now, something must be
reasoned upon, it matters not much what it is, provided it can be
reasoned upon with certainty. The properties of mind or matter,
or the study of languages, mathematics, or natural history, may
be chosen for this purpose. Now of all these, it is desirable to
choose the one which admits of the reasoning being verified, that
is, in which we can find out by other means, such as measurement
and ocular demonstration of all sorts, whether the results are
true or not. When the guiding property of the loadstone was first
ascertained, and it was necessary to learn how to use this new
discovery, and to find out how far it might be relied on, it
would have been thought advisable to make many passages between
ports that were well known before attempting a voyage of
discovery. So it is with our reasoning faculties: it is desirable
that their powers should be exerted upon objects of such a
nature, that we can tell by other means whether the results which
we obtain are true or false, and this before it is safe to trust
entirely to reason. Now the mathematics are peculiarly well
adapted for this purpose, on the following grounds:
1. Every term is distinctly explained, and has but one meaning,
and it is rarely that two words are employed to mean the same
thing.
2. The first principles are self-evident, and, though derived
from observation, do not require more of it than has been made by
children in general.
3. The demonstration is strictly logical, taking nothing for
granted except self-evident first principles, resting nothing
upon probability, and entirely independent of authority and
opinion.
4. When the conclusion is obtained by reasoning, its truth or
falsehood can be ascertained, in geometry by actual measurement,
in algebra by common arithmetical calculation. This gives
confidence, and is absolutely necessary, if, as was said before,
reason is not to be the instructor, but the pupil.
5. There are no words whose meanings are so much alike that the
ideas which they stand for may be confounded. Between the meaning
of terms there is no distinction, except a total distinction, and
all adjectives and adverbs expressing difference of degrees are
avoided.--DE MORGAN, AUGUSTUS.
_On the Study and Difficulties of
Mathematics (Chicago, 1898), chap. 1._
=431.= The instruction of children should aim gradually to
combine knowing and doing [Wissen und Können]. Among all sciences
mathematics seems to be the only one of a kind to satisfy this
aim most completely.--KANT, IMMANUEL.
_Werke [Rosenkranz und Schubert], Bd. 9
(Leipzig, 1838), p. 409._
=432.= Every discipline must be honored for reason other than its
utility, otherwise it yields no enthusiasm for industry.
For both reasons, I consider mathematics the chief subject for
the common school. No more highly honored exercise for the mind
can be found; the buoyancy [Spannkraft] which it produces is even
greater than that produced by the ancient languages, while its
utility is unquestioned.--HERBART, J. F.
_Mathematischer Lehrplan für
Realgymnasien, Werke [Kehrbach],
(Langensalza, 1890), Bd. 5, p. 167._
=433.= The motive for the study of mathematics is insight into the
nature of the universe. Stars and strata, heat and electricity,
the laws and processes of becoming and being, incorporate
mathematical truths. If language imitates the voice of the
Creator, revealing His heart, mathematics discloses His intellect,
repeating the story of how things came into being. And the value
of mathematics, appealing as it does to our energy and to our
honor, to our desire to know the truth and thereby to live as of
right in the household of God, is that it establishes us in
larger and larger certainties. As literature develops emotion,
understanding, and sympathy, so mathematics develops observation,
imagination, and reason.--CHANCELLOR, W. E.
_A Theory of Motives, Ideals and Values
in Education (Boston and New York,
1907), p. 406._
=434.= Mathematics in its pure form, as arithmetic, algebra,
geometry, and the applications of the analytic method, as well as
mathematics applied to matter and force, or statics and dynamics,
furnishes the peculiar study that gives to us, whether as children
or as men, the command of nature in this its quantitative aspect;
mathematics furnishes the instrument, the tool of thought, which
we wield in this realm.--HARRIS, W. T.
_Psychologic Foundations of Education
(New York, 1898), p. 325._
=435.= Little can be understood of even the simplest phenomena of
nature without some knowledge of mathematics, and the attempt to
penetrate deeper into the mysteries of nature compels simultaneous
development of the mathematical processes.--YOUNG, J. W. A.
_The Teaching of Mathematics (New York,
1907), p. 16._
=436.= For many parts of nature can neither be invented with
sufficient subtility nor demonstrated with sufficient perspicuity
nor accommodated unto use with sufficient dexterity, without the
aid and intervening of mathematics.--BACON, LORD.
_De Augmentis, Bk. 2; Advancement of
Learning, Bk. 3._
=437.= I confess, that after I began ... to discern how useful
mathematicks may be made to physicks, I have often wished that I
had employed about the speculative part of geometry, and the
cultivation of the specious Algebra I had been taught very young,
a good part of that time and industry, that I had spent about
surveying and fortification (of which I remember I once wrote an
entire treatise) and other parts of practick mathematicks.
--BOYLE, ROBERT.
_The Usefulness of Mathematicks to
Natural Philosophy; Works (London,
1772), Vol. 3, p. 426._
=438.= Mathematics gives the young man a clear idea of demonstration
and habituates him to form long trains of thought and reasoning
methodically connected and sustained by the final certainty of the
result; and it has the further advantage, from a purely moral
point of view, of inspiring an absolute and fanatical respect for
truth. In addition to all this, mathematics, and chiefly algebra
and infinitesimal calculus, excite to a high degree the conception
of the signs and symbols--necessary instruments to extend the
power and reach of the human mind by summarizing an aggregate of
relations in a condensed form and in a kind of mechanical way.
These auxiliaries are of special value in mathematics because they
are there adequate to their definitions, a characteristic which
they do not possess to the same degree in the physical and
mathematical [natural?] sciences.
There are, in fact, a mass of mental and moral faculties that can
be put in full play only by instruction in mathematics; and they
would be made still more available if the teaching was directed
so as to leave free play to the personal work of the student.
--BERTHELOT, M. P. E. M.
_Science as an Instrument of Education;
Popular Science Monthly (1897), p. 253._
=439.= Mathematical knowledge, therefore, appears to us of value
not only in so far as it serves as means to other ends, but for
its own sake as well, and we behold, both in its systematic
external and internal development, the most complete and
purest logical mind-activity, the embodiment of the highest
intellect-esthetics.--PRINGSHEIM, ALFRED.
_Ueber Wert und angeblichen Unwert der
Mathematik; Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 13, p.
381._
=440.= The advantages which mathematics derives from the peculiar
nature of those relations about which it is conversant, from its
simple and definite phraseology, and from the severe logic so
admirably displayed in the concatenation of its innumerable
theorems, are indeed immense, and well entitled to separate and
ample illustration.--STEWART, DUGALD.
_Philosophy of the Human Mind, Part 2,
chap. 2, sect. 3._
=441.= I do not intend to go deeply into the question how far
mathematical studies, as the representatives of conscious logical
reasoning, should take a more important place in school
education. But it is, in reality, one of the questions of the
day. In proportion as the range of science extends, its system
and organization must be improved, and it must inevitably come
about that individual students will find themselves compelled to
go through a stricter course of training than grammar is in a
position to supply. What strikes me in my own experience with
students who pass from our classical schools to scientific and
medical studies, is first, a certain laxity in the application of
strictly universal laws. The grammatical rules, in which they
have been exercised, are for the most part followed by long lists
of exceptions; accordingly they are not in the habit of relying
implicitly on the certainty of a legitimate deduction from a
strictly universal law. Secondly, I find them for the most part
too much inclined to trust to authority, even in cases where they
might form an independent judgment. In fact, in philological
studies, inasmuch as it is seldom possible to take in the whole
of the premises at a glance, and inasmuch as the decision of
disputed questions often depends on an æsthetic feeling for
beauty of expression, or for the genius of the language,
attainable only by long training, it must often happen that the
student is referred to authorities even by the best teachers.
Both faults are traceable to certain indolence and vagueness of
thought, the sad effects of which are not confined to subsequent
scientific studies. But certainly the best remedy for both is to
be found in mathematics, where there is absolute certainty in the
reasoning, and no authority is recognized but that of one’s own
intelligence.--HELMHOLTZ, H.
_On the Relation of Natural Science to
Science in general; Popular Lectures on
Scientific Subjects; Atkinson (New York,
1900), pp. 25-26._
=442.= What renders a problem definite, and what leaves it
indefinite, may best be understood from mathematics. The very
important idea of solving a problem within limits of error is an
element of rational culture, coming from the same source. The art
of totalizing fluctuations by curves is capable of being carried,
in conception, far beyond the mathematical domain, where it is
first learned. The distinction between laws and coefficients
applies in every department of causation. The theory of Probable
Evidence is the mathematical contribution to Logic, and is of
paramount importance.--BAIN, ALEXANDER.
_Education as a Science (New York,
1898), pp. 151-152._
=443.= We receive it as a fact, that some minds are so
constituted as absolutely to require for their nurture the severe
logic of the abstract sciences; that rigorous sequence of ideas
which leads from the premises to the conclusion, by a path,
arduous and narrow, it may be, and which the youthful reason may
find it hard to mount, but where it cannot stray; and on which,
if it move at all, it must move onward and upward.... Even for
intellects of a different character, whose natural aptitude is
for moral evidence and those relations of ideas which are
perceived and appreciated by taste, the study of the exact
sciences may be recommended as the best protection against the
errors into which they are most likely to fall. Although the
study of language is in many respects no mean exercise in logic,
yet it must be admitted that an eminently practical mind is
hardly to be formed without mathematical training.
--EVERETT, EDWARD.
_Orations and Speeches (Boston, 1870),
Vol. 2, p. 510._
=444.= The value of mathematical instruction as a preparation for
those more difficult investigations, consists in the applicability
not of its doctrines but of its methods. Mathematics will ever
remain the past perfect type of the deductive method in general;
and the applications of mathematics to the simpler branches
of physics furnish the only school in which philosophers can
effectually learn the most difficult and important of their art,
the employment of the laws of simpler phenomena for explaining and
predicting those of the more complex. These grounds are quite
sufficient for deeming mathematical training an indispensable
basis of real scientific education, and regarding with Plato, one
who is ἀγεωμέτρητος, as wanting in one of the most essential
qualifications for the successful cultivation of the higher
branches of philosophy.--MILL, J. S.
_System of Logic, Bk. 3, chap. 24, sect.
9._
=445.= This science, Geometry, is one of indispensable use and
constant reference, for every student of the laws of nature; for
the relations of space and number are the _alphabet_ in which
those laws are written. But besides the interest and importance
of this kind which geometry possesses, it has a great and
peculiar value for all who wish to understand the foundations of
human knowledge, and the methods by which it is acquired. For the
student of geometry acquires, with a degree of insight and
clearness which the unmathematical reader can but feebly imagine,
a conviction that there are necessary truths, many of them of a
very complex and striking character; and that a few of the most
simple and self-evident truths which it is possible for the mind
of man to apprehend, may, by systematic deduction, lead to the
most remote and unexpected results.--WHEWELL, WILLIAM.
_The Philosophy of the Inductive
Sciences, Part 1, Bk. 2, chap. 4, sect.
8 (London, 1858)._
=446.= Mathematics, while giving no quick remuneration, like the
art of stenography or the craft of bricklaying, does furnish the
power for deliberate thought and accurate statement, and to speak
the truth is one of the most social qualities a person can
possess. Gossip, flattery, slander, deceit, all spring from a
slovenly mind that has not been trained in the power of truthful
statement, which is one of the highest utilities.--DUTTON, S. T.
_Social Phases of Education in the
School and the Home (London, 1900), p.
30._
=447.= It is from this absolute indifference and tranquility of
the mind, that mathematical speculations derive some of their
most considerable advantages; because there is nothing to
interest the imagination; because the judgment sits free and
unbiased to examine the point. All proportions, every arrangement
of quantity, is alike to the understanding, because the same
truths result to it from all; from greater from lesser, from
equality and inequality.--BURKE, EDMUND.
_On the Sublime and Beautiful, Part 3,
sect. 2._
=448.= Out of the interaction of form and content in mathematics
grows an acquaintance with methods which enable the student to
produce independently within certain though moderate limits, and
to extend his knowledge through his own reflection. The deepening
of the consciousness of the intellectual powers connected with
this kind of activity, and the gradual awakening of the feeling
of intellectual self-reliance may well be considered as the most
beautiful and highest result of mathematical training.
--PRINGSHEIM, ALFRED.
_Ueber Wert und angeblichen Unwert der
Mathematik; Jahresbericht der Deutschen
Mathematiker Vereinigung (1904), p.
374._
=449.= He who would know what geometry is, must venture boldly
into its depths and learn to think and feel as a geometer. I
believe that it is impossible to do this, and to study geometry
as it admits of being studied and am conscious it can be taught,
without finding the reason invigorated, the invention quickened,
the sentiment of the orderly and beautiful awakened and enhanced,
and reverence for truth, the foundation of all integrity of
character, converted into a fixed principle of the mental and
moral constitution, according to the old and expressive adage
“_abeunt studia in mores_.”--SYLVESTER, J. J.
_A probationary Lecture on Geometry;
Collected Mathematical Papers
(Cambridge, 1908), Vol. 2, p. 9._
=450.= Mathematical knowledge adds vigour to the mind, frees it
from prejudice, credulity, and superstition.--ARBUTHNOT, JOHN.
_Usefulness of Mathematical Learning._
=451.= When the boy begins to understand that the visible point
is preceded by an invisible point, that the shortest distance
between two points is conceived as a straight line before it is
ever drawn with the pencil on paper, he experiences a feeling of
pride, of satisfaction. And justly so, for the fountain of all
thought has been opened to him, the difference between the ideal
and the real, _potentia et actu_, has become clear to him;
henceforth the philosopher can reveal him nothing new, as a
geometrician he has discovered the basis of all thought.--GOETHE.
_Sprüche in Prosa, Ethisches, VI, 455._
=452.= In mathematics, ... and in natural philosophy since
mathematics was applied to it, we see the noblest instance of the
force of the human mind, and of the sublime heights to which it
may rise by cultivation. An acquaintance with such sciences
naturally leads us to think well of our faculties, and to
indulge sanguine expectations concerning the improvement of other
parts of knowledge. To this I may add, that, as mathematical
and physical truths are perfectly uninteresting in their
consequences, the understanding readily yields its assent to the
evidence which is presented to it; and in this way may be
expected to acquire the habit of trusting to its own conclusions,
which will contribute to fortify it against the weaknesses of
scepticism, in the more interesting inquiries after moral truth
in which it may afterwards engage.--STEWART, DUGALD.
_Philosophy of the Human Mind, Part 3,
chap. 1, sect. 3._
=453.= Those that can readily master the difficulties of
Mathematics find a considerable charm in the study, sometimes
amounting to fascination. This is far from universal; but the
subject contains elements of strong interest of a kind that
constitutes the pleasures of knowledge. The marvellous devices
for solving problems elate the mind with the feeling of
intellectual power; and the innumerable constructions of the
science leave us lost in wonder.--BAIN, ALEXANDER.
_Education as a Science (New York,
1898), p. 153._
=454.= Thinking is merely the comparing of ideas, discerning
relations of likeness and of difference between ideas, and
drawing inferences. It is seizing general truths on the basis of
clearly apprehended particulars. It is but generalizing and
particularizing. Who will deny that a child can deal profitably
with sequences of ideas like: How many marbles are 2 marbles and
3 marbles? 2 pencils and 3 pencils? 2 balls and 3 balls? 2
children and 3 children? 2 inches and 3 inches? 2 feet and 3
feet? 2 and 3? Who has not seen the countenance of some little
learner light up at the end of such a series of questions with
the exclamation, “Why it’s always that way. Isn’t it?” This is
the glow of pleasure that the generalizing step always affords
him who takes the step himself. This is the genuine life-giving
joy which comes from feeling that one can successfully take this
step. The reality of such a discovery is as great, and the
lasting effect upon the mind of him that makes it is as sure as
was that by which the great Newton hit upon the generalization of
the law of gravitation. It is through these thrills of discovery
that love to learn and intellectual pleasure are begotten and
fostered. Good arithmetic teaching abounds in such opportunities.
--MYERS, GEORGE.
_Arithmetic in Public Education
(Chicago), p. 13._
=455.= A _general course_ in mathematics should be required of
all officers for its practical value, but no less for its
educational value in training the mind to logical forms of
thought, in developing the sense of absolute truthfulness,
together with a confidence in the accomplishment of definite
results by definite means.--ECHOLS, C. P.
_Mathematics at West Point and
Annapolis; U. S. Bureau of Education,
Bulletin 1912, No. 2, p. 11._
=456.= Exercise in the most rigorous thinking that is possible
will of its own accord strengthen the sense of truth and right,
for each advance in the ability to distinguish between correct
and false thoughts, each habit making for rigour in thought
development will increase in the sound pupil the ability and the
wish to ascertain what is right in life and to defend it.
--REIDT, F.
_Anleitung zum mathematischen Unterricht
in den höheren Schulen (Berlin, 1906),
p. 28._
=457.= I do not maintain that the _chief value_ of the study of
arithmetic consists in the lessons of morality that arise from
this study. I claim only that, to be impressed from day to day,
that there is something _that is right_ as an answer to the
questions with which one is _able_ to grapple, and that there is
a wrong answer--that there are ways in which the right answer can
be established as right, that these ways automatically reject
error and slovenliness, and that the learner is able himself to
manipulate these ways and to arrive at the establishment of the
true as opposed to the untrue, this relentless hewing _to_ the
line and stopping _at_ the line, must color distinctly the
thought life of the pupil with more than a tinge of morality....
To be neighborly with truth, to feel one’s self somewhat facile
in ways of recognizing and establishing what is right, what is
correct, to find the wrong persistently and unfailingly rejected
as of no value, to feel that one can apply these ways for
himself, that one can think and work independently, have a real,
a positive, and a purifying effect upon moral character. They are
the quiet, steady undertones of the work that always appeal to
the learner for the sanction of his best judgment, and these are
the really significant matters in school work. It is not the
noise and bluster, not even the dramatics or the polemics from
the teacher’s desk, that abide longest and leave the deepest and
stablest imprint upon character. It is these still, small voices
that speak unmistakably for the right and against the wrong and
the erroneous that really form human character. When the school
subjects are arranged on the basis of the degree to which they
contribute to the moral upbuilding of human character good
arithmetic will be well up the list.--MYERS, GEORGE.
_Arithmetic in Public Education
(Chicago), p. 18._
=458.= In destroying the predisposition to anger, science of all
kind is useful; but the mathematics possess this property in the
most eminent degree.--DR. RUSH.
_Quoted in Day’s Collacon (London, no
date)._
=459.= The mathematics are the friends to religion, inasmuch as
they charm the passions, restrain the impetuosity of the
imagination, and purge the mind from error and prejudice. Vice is
error, confusion and false reasoning; and all truth is more or
less opposite to it. Besides, mathematical truth may serve for a
pleasant entertainment for those hours which young men are apt to
throw away upon their vices; the delightfulness of them being
such as to make solitude not only easy but desirable.
--ARBUTHNOT, JOHN.
_Usefulness of Mathematical Learning._
=460.= There is no prophet which preaches the superpersonal God
more plainly than mathematics.--CARUS, PAUL.
_Reflections on Magic Squares; Monist
(1906), p. 147._
=461.= Mathematics must subdue the flights of our reason; they
are the staff of the blind; no one can take a step without them;
and to them and experience is due all that is certain in physics.
--VOLTAIRE.
_Oeuvres Complètes (Paris, 1880), t. 35,
p. 219._