MEMORABILIA MATHEMATICA
THE MACMILLAN COMPANY
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MEMORABILIA MATHEMATICA
OR
THE PHILOMATH’S QUOTATION-BOOK
BY
ROBERT EDOUARD MORITZ, PH. D., PH. N. D.
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF WASHINGTON
New York
THE MACMILLAN COMPANY
1914
_All rights reserved_
COPYRIGHT, 1914, BY
ROBERT EDOUARD MORITZ
PREFACE
Every one knows that the fine phrase “God geometrizes” is
attributed to Plato, but few know where this famous passage is
found, or the exact words in which it was first expressed. Those
who, like the author, have spent hours and even days in the
search of the exact statements, or the exact references, of
similar famous passages, will not question the timeliness and
usefulness of a book whose distinct purpose it is to bring
together into a single volume exact quotations, with their exact
references, bearing on one of the most time-honored, and even
today the most active and most fruitful of all the sciences, the
queen-mother of all the sciences, that is, mathematics.
It is hoped that the present volume will prove indispensable to
every teacher of mathematics, to every writer on mathematics,
and that the student of mathematics and the related sciences
will find its perusal not only a source of pleasure but of
encouragement and inspiration as well. The layman will find it a
repository of useful information covering a field of knowledge
which, owing to the unfamiliar and hence repellant character
of the language employed by mathematicians, is peculiarly
inaccessible to the general reader. No technical processes or
technical facility is required to understand and appreciate the
wealth of ideas here set forth in the words of the world’s great
thinkers.
No labor has been spared to make the present volume worthy of a
place among collections of a like kind in other fields. Ten years
have been devoted to its preparation, years, which if they could
have been more profitably, could scarcely have been more
pleasurably employed. As a result there have been brought
together over one thousand more or less familiar passages
pertaining to mathematics, by poets, philosophers, historians,
statesmen, scientists, and mathematicians. These have been
gathered from over three hundred authors, and have been grouped
under twenty heads, and cross indexed under nearly seven hundred
topics.
The author’s original plan was to give foreign quotations both in
the original and in translation, but with the growth of material
this plan was abandoned as infeasible. It was thought to serve
the best interest of the greater number of English readers to
give translations only, while preserving the references to the
original sources, so that the student or critical reader may
readily consult the original of any given extract. In cases where
the translation is borrowed the translator’s name is inserted in
brackets [] immediately after the author’s name. Brackets are
also used to indicate inserted words or phrases made necessary to
bring out the context.
The absence of similar English works has made the author’s work
largely that of the pioneer. Rebière’s “Mathématiques et
Mathématiciens” and Ahrens’ “Scherz und Ernst in der Mathematik”
have indeed been frequently consulted but rather with a view to
avoid overlapping than to receive aid. Thus certain topics as the
correspondence of German and French mathematicians, so excellently
treated by Ahrens, have purposely been omitted. The repetitions
are limited to a small number of famous utterances whose absence
from a work of this kind could scarcely be defended on any grounds.
No one can be more keenly aware of the shortcomings of a work
than its author, for none can have so intimate an acquaintance
with it. Among those of the present work is its incompleteness,
but it should be borne in mind that incompleteness is a necessary
concomitant of every collection of whatever kind. Much less can
completeness be expected in a first collection, made by a single
individual, in his leisure hours, and in a field which is already
boundless and is yet expanding day by day. A collection of great
thoughts, even if complete today, would be incomplete tomorrow.
Again, if some authors are quoted more frequently than others of
greater fame and authority, the reason may be sought not only in
the fact that the writings of some authors peculiarly lent
themselves to quotation, a quality singularly absent in other
writers of the greatest merit and authority, but also in this,
that the greatest freedom has been exercised in the choice of
selections. The author has followed the bent of his own fancy in
collecting whatever seemed to him sufficiently valuable because
of its content, its beauty, its originality, or its terseness, to
deserve a place in a “Memorabilia.”
Great pains has been taken to furnish exact readings and
references. In some cases where a passage could not be traced to
its first source, the secondary source has been given rather than
the reputed source. For the same reason many references are to
later editions rather than to inaccessible first editions.
The author feels confident that this work will be of assistance
to his co-workers in the field of mathematics and allied fields.
If in addition it should aid in a better appreciation of
mathematicians and their work on the part of laymen and students
in other fields, the author’s foremost aim in the preparation of
this work will have been achieved.
ROBERT EDOUARD MORITZ,
_September, 1913_.
CONTENTS
CHAPTER PAGE
I. DEFINITIONS AND OBJECT OF MATHEMATICS 1
II. THE NATURE OF MATHEMATICS 10
III. ESTIMATES OF MATHEMATICS 39
IV. THE VALUE OF MATHEMATICS 49
V. THE TEACHING OF MATHEMATICS 72
VI. STUDY AND RESEARCH IN MATHEMATICS 86
VII. MODERN MATHEMATICS 108
VIII. THE MATHEMATICIAN 121
IX. PERSONS AND ANECDOTES (A-M) 135
X. PERSONS AND ANECDOTES (N-Z) 166
XI. MATHEMATICS AS A FINE ART 181
XII. MATHEMATICS AS A LANGUAGE 194
XIII. MATHEMATICS AND LOGIC 201
XIV. MATHEMATICS AND PHILOSOPHY 209
XV. MATHEMATICS AND SCIENCE 224
XVI. ARITHMETIC 261
XVII. ALGEBRA 275
XVIII. GEOMETRY 292
XIX. THE CALCULUS AND ALLIED TOPICS 323
XX. THE FUNDAMENTAL CONCEPTS OF TIME AND SPACE 345
XXI. PARADOXES AND CURIOSITIES 364
INDEX 385
Alles Gescheite ist schon gedacht worden; man muss nur
versuchen, es noch einmal zu denken.--GOETHE.
_Sprüche in Prosa, Ethisches, I. 1._
A great man quotes bravely, and will not draw on his
invention when his memory serves him with a word as
good.--EMERSON.
_Letters and Social Aims, Quotation and
Originality._
MEMORABILIA MATHEMATICA
MEMORABILIA MATHEMATICA
CHAPTER I
DEFINITIONS AND OBJECT OF MATHEMATICS
=101.= I think it would be desirable that this form of word
[mathematics] should be reserved for the applications of the
science, and that we should use mathematic in the singular to
denote the science itself, in the same way as we speak of logic,
rhetoric, or (own sister to algebra) music.--SYLVESTER, J. J.
_Presidential Address to the British
Association, Exeter British Association
Report (1869); Collected Mathematical
Papers, Vol. 2, p. 659._
=102.= ... all the sciences which have for their end
investigations concerning order and measure, are related to
mathematics, it being of small importance whether this measure be
sought in numbers, forms, stars, sounds, or any other object;
that, accordingly, there ought to exist a general science which
should explain all that can be known about order and measure,
considered independently of any application to a particular
subject, and that, indeed, this science has its own proper name,
consecrated by long usage, to wit, _mathematics_. And a proof
that it far surpasses in facility and importance the sciences
which depend upon it is that it embraces at once all the objects
to which these are devoted and a great many others besides; ....
--DESCARTES.
_Rules for the Direction of the Mind,
Philosophy of D. [Torrey] (New York,
1892), p. 72._
=103.= [Mathematics] has for its object the _indirect_
measurement of magnitudes, and it _purposes to determine
magnitudes by each other, according to the precise relations
which exist between them_.--COMTE.
_Positive Philosophy [Martineau], Bk. 1,
chap. 1._
=104.= The business of concrete mathematics is to discover the
equations which express the mathematical laws of the phenomenon
under consideration; and these equations are the starting-point
of the calculus, which must obtain from them certain quantities
by means of others.--COMTE.
_Positive Philosophy [Martineau], Bk. 1,
chap. 2._
=105.= Mathematics is the science of the connection of
magnitudes. Magnitude is anything that can be put equal or
unequal to another thing. Two things are equal when in every
assertion each may be replaced by the other.--GRASSMANN, HERMANN.
_Stücke aus dem Lehrbuche der
Arithmetik, Werke (Leipzig, 1904), Bd.
2, p. 298._
=106.= Mathematic is either Pure or Mixed: To Pure Mathematic
belong those sciences which handle Quantity entirely severed from
matter and from axioms of natural philosophy. These are two,
Geometry and Arithmetic; the one handling quantity continued, the
other dissevered.... Mixed Mathematic has for its subject some
axioms and parts of natural philosophy, and considers quantity in
so far as it assists to explain, demonstrate and actuate these.
--BACON, FRANCIS.
_De Augmentis, Bk. 3; Advancement of
Learning, Bk. 2._
=107.= The ideas which these sciences, Geometry, Theoretical
Arithmetic and Algebra involve extend to all objects and changes
which we observe in the external world; and hence the consideration
of mathematical relations forms a large portion of many of the
sciences which treat of the phenomena and laws of external nature,
as Astronomy, Optics, and Mechanics. Such sciences are hence often
termed _Mixed Mathematics_, the relations of space and number
being, in these branches of knowledge, combined with principles
collected from special observation; while Geometry, Algebra, and
the like subjects, which involve no result of experience, are
called _Pure Mathematics_.--WHEWELL, WILLIAM.
_The Philosophy of the Inductive
Sciences, Part 1, Bk. 2, chap. I, sect.
4. (London, 1858)._
=108.= Higher Mathematics is the art of reasoning about numerical
relations between natural phenomena; and the several sections of
Higher Mathematics are different modes of viewing these
relations.--MELLOR, J. W.
_Higher Mathematics for Students of
Chemistry and Physics (New York, 1902),
Prologue._
=109.= Number, place, and combination ... the three intersecting
but distinct spheres of thought to which all mathematical ideas
admit of being referred.--SYLVESTER, J. J.
_Philosophical Magazine, Vol. 24 (1844),
p. 285; Collected Mathematical Papers,
Vol. 1, p. 91._
=110.= There are three ruling ideas, three so to say, spheres of
thought, which pervade the whole body of mathematical science, to
some one or other of which, or to two or all three of them
combined, every mathematical truth admits of being referred;
these are the three cardinal notions, of Number, Space and Order.
Arithmetic has for its object the properties of number in the
abstract. In algebra, viewed as a science of operations, order is
the predominating idea. The business of geometry is with the
evolution of the properties of space, or of bodies viewed as
existing in space.--SYLVESTER, J. J.
_A Probationary Lecture on Geometry,
York British Association Report (1844),
Part 2; Collected Mathematical Papers,
Vol. 2, p. 5._
=111.= The object of pure mathematics is those relations which
may be conceptually established among any conceived elements
whatsoever by assuming them contained in some ordered manifold;
the law of order of this manifold must be subject to our choice;
the latter is the case in both of the only conceivable kinds of
manifolds, in the discrete as well as in the continuous.
--PAPPERITZ, E.
_Über das System der rein mathematischen
Wissenschaften, Jahresbericht der
Deutschen Mathematiker-Vereinigung, Bd.
1, p. 36._
=112.= Pure mathematics is not concerned with magnitude. It is
merely the doctrine of notation of relatively ordered thought
operations which have become mechanical.--NOVALIS.
_Schriften (Berlin, 1901), Zweiter Teil,
p. 282._
=113.= Any conception which is definitely and completely determined
by means of a finite number of specifications, say by assigning a
finite number of elements, is a mathematical conception. Mathematics
has for its function to develop the consequences involved in the
definition of a group of mathematical conceptions. Interdependence
and mutual logical consistency among the members of the group are
postulated, otherwise the group would either have to be treated as
several distinct groups, or would lie beyond the sphere of
mathematics.--CHRYSTAL, GEORGE.
_Encyclopedia Britannica (9th edition),
Article “Mathematics.”_
=114.= The purely formal sciences, logic and mathematics, deal
with those relations which are, or can be, independent of the
particular content or the substance of objects. To mathematics in
particular fall those relations between objects which involve the
concepts of magnitude, of measure and of number.--HANKEL, HERMANN.
_Theorie der Complexen Zahlensysteme,
(Leipzig, 1867), p. 1._
=115.= _Quantity is that which is operated with according to
fixed mutually consistent laws._ Both operator and operand must
derive their meaning from the laws of operation. In the case of
ordinary algebra these are the three laws already indicated [the
commutative, associative, and distributive laws], in the algebra
of quaternions the same save the law of commutation for
multiplication and division, and so on. It may be questioned
whether this definition is sufficient, and it may be objected
that it is vague; but the reader will do well to reflect that any
definition must include the linear algebras of Peirce, the
algebra of logic, and others that may be easily imagined,
although they have not yet been developed. This general
definition of quantity enables us to see how operators may be
treated as quantities, and thus to understand the rationale of
the so called symbolical methods.--CHRYSTAL, GEORGE.
_Encyclopedia Britannica (9th edition),
Article “Mathematics.”_
=116.= Mathematics--in a strict sense--is the abstract science
which investigates deductively the conclusions implicit in the
elementary conceptions of spatial and numerical relations.
--MURRAY, J. A. H.
_A New English Dictionary._
=117.= Everything that the greatest minds of all times have
accomplished toward the _comprehension of forms_ by means of
concepts is gathered into one great science, _mathematics_.
--HERBART, J. F.
_Pestalozzi’s Idee eines A B C der
Anschauung, Werke [Kehrbach],
(Langensalza, 1890), Bd. 1, p. 163._
=118.= Perhaps the least inadequate description of the general scope
of modern Pure Mathematics--I will not call it a definition--would
be to say that it deals with _form_, in a very general sense of the
term; this would include algebraic form, functional relationship,
the relations of order in any ordered set of entities such as
numbers, and the analysis of the peculiarities of form of groups
of operations.--HOBSON, E. W.
_Presidential Address British
Association for the Advancement of
Science (1910); Nature, Vol. 84, p.
287._
=119.= The ideal of mathematics should be to erect a calculus to
facilitate reasoning in connection with every province of
thought, or of external experience, in which the succession of
thoughts, or of events can be definitely ascertained and
precisely stated. So that all serious thought which is not
philosophy, or inductive reasoning, or imaginative literature,
shall be mathematics developed by means of a calculus.
--WHITEHEAD, A. N.
_Universal Algebra (Cambridge, 1898),
Preface._
=120.= Mathematics is the science which draws necessary
conclusions.--PEIRCE, BENJAMIN.
_Linear Associative Algebra, American
Journal of Mathematics, Vol. 4 (1881),
p. 97._
=121.= Mathematics is the universal art apodictic.--SMITH, W. B.
_Quoted by Keyser, C. J. in Lectures on
Science, Philosophy and Art (New York,
1908), p. 13._
=122.= Mathematics in its widest signification is the development
of all types of formal, necessary, deductive reasoning.
--WHITEHEAD, A. N.
_Universal Algebra (Cambridge, 1898),
Preface, p. vi._
=123.= Mathematics in general is fundamentally the science of
self-evident things.--KLEIN, FELIX.
_Anwendung der Differential- und
Integralrechnung auf Geometrie (Leipzig,
1902), p. 26._
=124.= A mathematical science is any body of propositions which
is capable of an abstract formulation and arrangement in such a
way that every proposition of the set after a certain one is a
formal logical consequence of some or all the preceding
propositions. Mathematics consists of all such mathematical
sciences.--YOUNG, CHARLES WESLEY.
_Fundamental Concepts of Algebra and
Geometry (New York, 1911), p. 222._
=125.= Pure mathematics is a collection of hypothetical,
deductive theories, each consisting of a definite system of
primitive, _undefined_, concepts or symbols and primitive,
_unproved_, but self-consistent assumptions (commonly called
axioms) together with their logically deducible consequences
following by rigidly deductive processes without appeal to
intuition.--FITCH, G. D.
_The Fourth Dimension simply Explained
(New York, 1910), p. 58._
=126.= The whole of Mathematics consists in the organization of a
series of aids to the imagination in the process of reasoning.
--WHITEHEAD, A. N.
_Universal Algebra (Cambridge, 1898), p.
12._
=127.= Pure mathematics consists entirely of such asseverations
as that, if such and such a proposition is true of _anything_,
then such and such another proposition is true of that thing. It
is essential not to discuss whether the first proposition is
really true, and not to mention what the anything is of which it
is supposed to be true.... If our hypothesis is about _anything_
and not about some one or more particular things, then our
deductions constitute mathematics. Thus mathematics may be
defined as the subject in which we never know what we are talking
about, nor whether what we are saying is true.--RUSSELL, BERTRAND.
_Recent Work on the Principles of
Mathematics, International Monthly, Vol.
4 (1901), p. 84._
=128.= Pure Mathematics is the class of all propositions of the form
“_p_ implies _q_,” where _p_ and _q_ are propositions containing one
or more variables, the same in the two propositions, and neither _p_
nor _q_ contains any constants except logical constants. And logical
constants are all notions definable in terms of the following:
Implication, the relation of a term to a class of which it is a
member, the notion of _such that_, the notion of relation, and
such further notions as may be involved in the general notion of
propositions of the above form. In addition to these, Mathematics
_uses_ a notion which is not a constituent of the propositions
which it considers--namely, the notion of truth.--RUSSELL, BERTRAND.
_Principles of Mathematics (Cambridge,
1903), p. 1._
=129.= The object of pure Physic is the unfolding of the laws of
the intelligible world; the object of pure Mathematic that of
unfolding the laws of human intelligence.--SYLVESTER, J. J.
_On a theorem, connected with Newton’s
Rule, etc., Collected Mathematical
Papers, Vol. 3, p. 424._
=130.= First of all, we ought to observe, that mathematical
propositions, properly so called, are always judgments _a
priori,_ and not empirical, because they carry along with them
necessity, which can never be deduced from experience. If people
should object to this, I am quite willing to confine my
statements to pure mathematics, the very concept of which implies
that it does not contain empirical, but only pure knowledge _a
priori_.--KANT, IMMANUEL.
_Critique of Pure Reason [Müller], (New
York, 1900), p. 720._
=131.= Mathematics, the science of the ideal, becomes the means
of investigating, understanding and making known the world of the
real. The complex is expressed in terms of the simple. From one
point of view mathematics may be defined as the science of
successive substitutions of simpler concepts for more complex....
--WHITE, WILLIAM F.
_A Scrap-book of Elementary Mathematics,
(Chicago, 1908), p. 215._
=132.= The critical mathematician has abandoned the search for
truth. He no longer flatters himself that his propositions are or
can be known to him or to any other human being to be true;
and he contents himself with aiming at the correct, or the
consistent. The distinction is not annulled nor even blurred by
the reflection that consistency contains immanently a kind of
truth. He is not absolutely certain, but he believes profoundly
that it is possible to find various sets of a few propositions
each such that the propositions of each set are compatible, that
the propositions of each such set imply other propositions, and
that the latter can be deduced from the former with certainty.
That is to say, he believes that there are systems of coherent or
consistent propositions, and he regards it his business to
discover such systems. Any such system is a branch of mathematics.
--KEYSER, C. J.
_Science, New Series, Vol. 35, p. 107._
=133.= [Mathematics is] the study of ideal constructions (often
applicable to real problems), and the discovery thereby of
relations between the parts of these constructions, before
unknown.--PEIRCE, C. S.
_Century Dictionary, Article
“Mathematics.”_
=134.= Mathematics is that form of intelligence in which we bring
the objects of the phenomenal world under the control of the
conception of quantity. [Provisional definition.]--HOWISON, G. H.
_The Departments of Mathematics, and
their Mutual Relations; Journal of
Speculative Philosophy, Vol. 5, p. 164._
=135.= Mathematics is the science of the functional laws and
transformations which enable us to convert figured extension and
rated motion into number.--HOWISON, G. H.
_The Departments of Mathematics, and
their Mutual Relations; Journal of
Speculative Philosophy, Vol. 5, p. 170._
CHAPTER II
THE NATURE OF MATHEMATICS
=201.= Mathematics, from the earliest times to which the history
of human reason can reach, has followed, among that wonderful
people of the Greeks, the safe way of science. But it must not be
supposed that it was as easy for mathematics as for logic, in
which reason is concerned with itself alone, to find, or rather
to make for itself that royal road. I believe, on the contrary,
that there was a long period of tentative work (chiefly still
among the Egyptians), and that the change is to be ascribed to a
_revolution_, produced by the happy thought of a single man,
whose experiments pointed unmistakably to the path that had to be
followed, and opened and traced out for the most distant times
the safe way of a science. The history of that intellectual
revolution, which was far more important than the passage round
the celebrated Cape of Good Hope, and the name of its fortunate
author, have not been preserved to us.... A new light flashed on
the first man who demonstrated the properties of the isosceles
triangle (whether his name was _Thales_ or any other name), for
he found that he had not to investigate what he saw in the
figure, or the mere concepts of that figure, and thus to learn
its properties; but that he had to produce (by construction) what
he had himself, according to concepts _a priori_, placed into
that figure and represented in it, so that, in order to know
anything with certainty _a priori_, he must not attribute to that
figure anything beyond what necessarily follows from what he has
himself placed into it, in accordance with the concept.
--KANT, IMMANUEL.
_Critique of Pure Reason, Preface to the
Second Edition [Müller], (New York,
1900), p. 690._
=202.= [When followed in the proper spirit], there is no study in
the world which brings into more harmonious action all the
faculties of the mind than the one [mathematics] of which I
stand here as the humble representative and advocate. There is
none other which prepares so many agreeable surprises for its
followers, more wonderful than the transformation scene of a
pantomime, or, like this, seems to raise them, by successive
steps of initiation to higher and higher states of conscious
intellectual being.--SYLVESTER, J. J.
_A Plea for the Mathematician, Nature,
Vol. 1, p. 261._
=203.= Thought-economy is most highly developed in mathematics,
that science which has reached the highest formal development,
and on which natural science so frequently calls for assistance.
Strange as it may seem, the strength of mathematics lies in the
avoidance of all unnecessary thoughts, in the utmost economy of
thought-operations. The symbols of order, which we call numbers,
form already a system of wonderful simplicity and economy. When
in the multiplication of a number with several digits we employ
the multiplication table and thus make use of previously
accomplished results rather than to repeat them each time, when
by the use of tables of logarithms we avoid new numerical
calculations by replacing them by others long since performed,
when we employ determinants instead of carrying through from the
beginning the solution of a system of equations, when we
decompose new integral expressions into others that are
familiar,--we see in all this but a faint reflection of the
intellectual activity of a _Lagrange_ or _Cauchy_, who with the
keen discernment of a military commander marshalls a whole troop
of completed operations in the execution of a new one.--MACH, E.
_Populär-wissenschafliche Vorlesungen
(1908), pp. 224-225._
=204.= Pure mathematics proves itself a royal science both
through its content and form, which contains within itself the
cause of its being and its methods of proof. For in complete
independence mathematics creates for itself the object of which
it treats, its magnitudes and laws, its formulas and symbols.
--DILLMANN, E.
_Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 94._
=205.= The essence of mathematics lies in its freedom.
--CANTOR, GEORGE.
_Mathematische Annalen, Bd. 21, p. 564._
=206.= Mathematics pursues its own course unrestrained, not
indeed with an unbridled licence which submits to no laws, but
rather with the freedom which is determined by its own nature and
in conformity with its own being.--HANKEL, HERMANN.
_Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884),
p. 16._
=207.= Mathematics is perfectly free in its development and is
subject only to the obvious consideration, that its concepts must
be free from contradictions in themselves, as well as definitely
and orderly related by means of definitions to the previously
existing and established concepts.--CANTOR, GEORGE.
_Grundlagen einer allgemeinen
Manigfaltigkeitslehre (Leipzig, 1883),
Sect. 8._
=208.= Mathematicians assume the right to choose, within the
limits of logical contradiction, what path they please in
reaching their results.--ADAMS, HENRY.
_A Letter to American Teachers of
History (Washington, 1910),
Introduction, p. v._
=209.= Mathematics is the predominant science of our time; its
conquests grow daily, though without noise; he who does not
employ it for himself, will some day find it employed against
himself.--HERBART, J. F.
_Werke [Kehrbach] (Langensalza, 1890),
Bd. 5, p. 105._
=210.= Mathematics is not the discoverer of laws, for it is not
induction; neither is it the framer of theories, for it is not
hypothesis; but it is the judge over both, and it is the arbiter
to which each must refer its claims; and neither law can rule nor
theory explain without the sanction of mathematics.
--PEIRCE, BENJAMIN.
_Linear Associative Algebra, American
Journal of Mathematics, Vol. 4 (1881),
p. 97._
=211.= Mathematics is a science continually expanding; and its
growth, unlike some political and industrial events, is attended
by universal acclamation.--WHITE, H. S.
_Congress of Arts and Sciences (Boston
and New York, 1905), Vol. 1, p. 455._
=212.= Mathematics accomplishes really nothing outside of the
realm of magnitude; marvellous, however, is the skill with which
it masters magnitude wherever it finds it. We recall at once the
network of lines which it has spun about heavens and earth; the
system of lines to which azimuth and altitude, declination and
right ascension, longitude and latitude are referred; those
abscissas and ordinates, tangents and normals, circles of
curvature and evolutes; those trigonometric and logarithmic
functions which have been prepared in advance and await
application. A look at this apparatus is sufficient to show that
mathematicians are not magicians, but that everything is
accomplished by natural means; one is rather impressed by the
multitude of skilful machines, numerous witnesses of a manifold
and intensely active industry, admirably fitted for the
acquisition of true and lasting treasures.--HERBART, J. F.
_Werke [Kehrbach] (Langensalza, 1890),
Bd. 5, p. 101._
=213.= They [mathematicians] only take those things into
consideration, of which they have clear and distinct ideas,
designating them by proper, adequate, and invariable names, and
premising only a few axioms which are most noted and certain to
investigate their affections and draw conclusions from them, and
agreeably laying down a very few hypotheses, such as are in the
highest degree consonant with reason and not to be denied by
anyone in his right mind. In like manner they assign generations
or causes easy to be understood and readily admitted by all, they
preserve a most accurate order, every proposition immediately
following from what is supposed and proved before, and reject all
things howsoever specious and probable which can not be inferred
and deduced after the same manner.--BARROW, ISAAC.
_Mathematical Lectures (London, 1734),
p. 66._
=214.= The dexterous management of terms and being able to _fend_
and _prove_ with them, I know has and does pass in the world for
a great part of learning; but it is learning distinct from
knowledge, for knowledge consists only in perceiving the
habitudes and relations of ideas one to another, which is done
without words; the intervention of sounds helps nothing to it.
And hence we see that there is least use of distinction where
there is most knowledge: I mean in mathematics, where men have
determined ideas with known names to them; and so, there being no
room for equivocations, there is no need of distinctions.
--LOCKE, JOHN.
_Conduct of the Understanding, Sect.
31._
=215.= In mathematics it [sophistry] had no place from the
beginning: Mathematicians having had the wisdom to define
accurately the terms they use, and to lay down, as axioms, the
first principles on which their reasoning is grounded. Accordingly
we find no parties among mathematicians, and hardly any disputes.
--REID, THOMAS.
_Essays on the Intellectual Powers of
Man, Essay 1, chap. 1._
=216.= In most sciences one generation tears down what another
has built and what one has established another undoes. In
Mathematics alone each generation builds a new story to the old
structure.--HANKEL, HERMANN.
_Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884),
p. 25._
=217.= Mathematics, the priestess of definiteness and
clearness.--HERBART, J. F.
_Werke [Kehrbach] (Langensalza, 1890),
Bd. 1, p. 171._
=218.= ... mathematical analysis is co-extensive with nature
itself, it defines all perceivable relations, measures times,
spaces, forces, temperatures; it is a difficult science which
forms but slowly, but preserves carefully every principle once
acquired; it increases and becomes stronger incessantly amidst
all the changes and errors of the human mind.
Its chief attribute is clearness; it has no means for expressing
confused ideas. It compares the most diverse phenomena and
discovers the secret analogies which unite them. If matter
escapes us, as that of air and light because of its extreme
tenuity, if bodies are placed far from us in the immensity of
space, if man wishes to know the aspect of the heavens at
successive periods separated by many centuries, if gravity and
heat act in the interior of the solid earth at depths which will
forever be inaccessible, mathematical analysis is still able to
trace the laws of these phenomena. It renders them present and
measurable, and appears to be the faculty of the human mind
destined to supplement the brevity of life and the imperfection
of the senses, and what is even more remarkable, it follows the
same course in the study of all phenomena; it explains them in
the same language, as if in witness to the unity and simplicity
of the plan of the universe, and to make more manifest the
unchangeable order which presides over all natural causes.
--FOURIER, J.
_Théorie Analytique de la Chaleur,
Discours Préliminaire._
=219.= Let us now declare the means whereby our understanding can
rise to knowledge without fear of error. There are two such
means: intuition and deduction. By intuition I mean not the
varying testimony of the senses, nor the deductive judgment of
imagination naturally extravagant, but the conception of an
attentive mind so distinct and so clear that no doubt remains to
it with regard to that which it comprehends; or, what amounts to
the same thing, the self-evidencing conception of a sound and
attentive mind, a conception which springs from the light of
reason alone, and is more certain, because more simple, than
deduction itself....
It may perhaps be asked why to intuition we add this other mode
of knowing, by deduction, that is to say, the process which, from
something of which we have certain knowledge, draws consequences
which necessarily follow therefrom. But we are obliged to admit
this second step; for there are a great many things which,
without being evident of themselves, nevertheless bear the
marks of certainty if only they are deduced from true and
incontestable principles by a continuous and uninterrupted
movement of thought, with distinct intuition of each thing; just
as we know that the last link of a long chain holds to the first,
although we can not take in with one glance of the eye the
intermediate links, provided that, after having run over them in
succession, we can recall them all, each as being joined to its
fellows, from the first up to the last. Thus we distinguish
intuition from deduction, inasmuch as in the latter case there is
conceived a certain progress or succession, while it is not so in
the former;... whence it follows that primary propositions,
derived immediately from principles, may be said to be known,
according to the way we view them, now by intuition, now by
deduction; although the principles themselves can be known only
by intuition, the remote consequences only by deduction.
--DESCARTES.
_Rules for the Direction of the Mind,
Philosophy of D. [Torrey] (New York,
1892), pp. 64, 65._
=220.= Analysis and natural philosophy owe their most important
discoveries to this fruitful means, which is called induction.
Newton was indebted to it for his theorem of the binomial and the
principle of universal gravity.--LAPLACE.
_A Philosophical Essay on Probabilities
[Truscott and Emory] (New York 1902), p.
176._
=221.= There is in every step of an arithmetical or algebraical
calculation a real induction, a real inference from facts to facts,
and what disguises the induction is simply its comprehensive
nature, and the consequent extreme generality of its language.
--MILL, J. S.
_System of Logic, Bk. 2, chap. 6, 2._
=222.= It would appear that Deductive and Demonstrative Sciences
are all, without exception, Inductive Sciences: that their
evidence is that of experience, but that they are also, in virtue
of the peculiar character of one indispensable portion of the
general formulae according to which their inductions are made,
Hypothetical Sciences. Their conclusions are true only upon
certain suppositions, which are, or ought to be, approximations
to the truth, but are seldom, if ever, exactly true; and to this
hypothetical character is to be ascribed the peculiar certainty,
which is supposed to be inherent in demonstration.--MILL, J. S.
_System of Logic, Bk. 2, chap. 6, 1._
=223.= The peculiar character of mathematical truth is, that it
is necessarily and inevitably true; and one of the most important
lessons which we learn from our mathematical studies is a
knowledge that there are such truths, and a familiarity with
their form and character.
This lesson is not only lost, but read backward, if the student
is taught that there is no such difference, and that mathematical
truths themselves are learned by experience.--WHEWELL, W.
_Thoughts on the Study of Mathematics.
Principles of English University
Education (London, 1838)._
=224.= These sciences, Geometry, Theoretical Arithmetic and Algebra,
have no principles besides definitions and axioms, and no process
of proof but _deduction_; this process, however, assuming a most
remarkable character; and exhibiting a combination of simplicity
and complexity, of rigour and generality, quite unparalleled in
other subjects.--WHEWELL, W.
_The Philosophy of the Inductive
Sciences, Part 1, Bk. 2, chap. 1, sect.
2 (London, 1858)._
=225.= The apodictic quality of mathematical thought, the certainty
and correctness of its conclusions, are due, not to a special mode
of ratiocination, but to the character of the concepts with which
it deals. What is that distinctive characteristic? I answer:
_precision_, _sharpness_, _completeness_,[1] of definition. But
how comes your mathematician by such completeness? There is no
mysterious trick involved; some ideas admit of such precision,
others do not; and the mathematician is one who deals with those
that do.--KEYSER, C. J.
_The Universe and Beyond; Hibbert
Journal, Vol. 3 (1904-1905), p. 309._
[1] i.e., in terms of the absolutely clear and
_in_definable.
=226.= The reasoning of mathematicians is founded on certain and
infallible principles. Every word they use conveys a determinate
idea, and by accurate definitions they excite the same ideas in
the mind of the reader that were in the mind of the writer. When
they have defined the terms they intend to make use of, they
premise a few axioms, or self-evident principles, that every one
must assent to as soon as proposed. They then take for granted
certain postulates, that no one can deny them, such as, that a
right line may be drawn from any given point to another, and from
these plain, simple principles they have raised most astonishing
speculations, and proved the extent of the human mind to be more
spacious and capacious than any other science.--ADAMS, JOHN.
_Diary, Works (Boston, 1850), Vol. 2, p.
21._
=227.= It may be observed of mathematicians that they only meddle
with such things as are certain, passing by those that are
doubtful and unknown. They profess not to know all things,
neither do they affect to speak of all things. What they know to
be true, and can make good by invincible arguments, that they
publish and insert among their theorems. Of other things they
are silent and pass no judgment at all, choosing rather to
acknowledge their ignorance, than affirm anything rashly. They
affirm nothing among their arguments or assertions which is not
most manifestly known and examined with utmost rigour, rejecting
all probable conjectures and little witticisms. They submit
nothing to authority, indulge no affection, detest subterfuges of
words, and declare their sentiments, as in a court of justice,
_without passion, without apology_; knowing that their reasons,
as Seneca testifies of them, are not brought to _persuade_, but
to compel.--BARROW, ISAAC.
_Mathematical Lectures (London, 1734),
p. 64._
=228.= What is exact about mathematics but exactness? And is not
this a consequence of the inner sense of truth?--GOETHE.
_Sprüche in Prosa, Natur, 6, 948._
=229.= ... the three positive characteristics that distinguish
mathematical knowledge from other knowledge ... may be briefly
expressed as follows: first, mathematical knowledge bears more
distinctly the imprint of truth on all its results than any other
kind of knowledge; secondly, it is always a sure preliminary step
to the attainment of other correct knowledge; thirdly, it has no
need of other knowledge.--SCHUBERT, H.
_Mathematical Essays and Recreations
(Chicago, 1898), p. 35._
=230.= It is now necessary to indicate more definitely the reason
why mathematics not only carries conviction in itself, but also
transmits conviction to the objects to which it is applied. The
reason is found, first of all, in the perfect precision with
which the elementary mathematical concepts are determined; in
this respect each science must look to its own salvation.... But
this is not all. As soon as human thought attempts long chains of
conclusions, or difficult matters generally, there arises not
only the danger of error but also the suspicion of error, because
since all details cannot be surveyed with clearness at the same
instant one must in the end be satisfied with a _belief_ that
nothing has been overlooked from the beginning. Every one knows
how much this is the case even in arithmetic, the most elementary
use of mathematics. No one would imagine that the higher parts of
mathematics fare better in this respect; on the contrary, in more
complicated conclusions the uncertainty and suspicion of hidden
errors increases in rapid progression. How does mathematics
manage to rid itself of this inconvenience which attaches to it
in the highest degree? By making proofs more rigorous? By giving
new rules according to which the old rules shall be applied? Not
in the least. A very great uncertainty continues to attach to the
result of each single computation. But there are checks. In the
realm of mathematics each point may be reached by a hundred
different ways; and if each of a hundred ways leads to the same
point, one may be sure that the right point has been reached. A
calculation without a check is as good as none. Just so it is
with every isolated proof in any speculative science whatever;
the proof may be ever so ingenious, and ever so perfectly true
and correct, it will still fail to convince permanently. He will
therefore be much deceived, who, in metaphysics, or in psychology
which depends on metaphysics, hopes to see his greatest care in
the precise determination of the concepts and in the logical
conclusions rewarded by conviction, much less by success in
transmitting conviction to others. Not only must the conclusions
support each other, without coercion or suspicion of subreption,
but in all matters originating in experience, or judging
concerning experience, the results of speculation must be
verified by experience, not only superficially, but in countless
special cases.--HERBART, J. F.
_Werke [Kehrbach] (Langensalza, 1890),
Bd. 5, p. 105._
=231.= [In mathematics] we behold the conscious logical activity
of the human mind in its purest and most perfect form. Here we
learn to realize the laborious nature of the process, the great
care with which it must proceed, the accuracy which is necessary
to determine the exact extent of the general propositions arrived
at, the difficulty of forming and comprehending abstract concepts;
but here we learn also to place confidence in the certainty, scope
and fruitfulness of such intellectual activity.--HELMHOLTZ, H.
_Ueber das Verhältniss der
Naturwissenschaften zur Gesammtheit der
Wissenschaft, Vorträge und Reden, Bd. 1
(1896), p. 176._
=232.= It is true that mathematics, owing to the fact that its
whole content is built up by means of purely logical deduction
from a small number of universally comprehended principles,
has not unfittingly been designated as the science of the
_self-evident_ [Selbstverständlichen]. Experience however, shows
that for the majority of the cultured, even of scientists,
mathematics remains the science of the _incomprehensible_
[Unverständlichen].--PRINGSHEIM, ALFRED.
_Ueber Wert und angeblichen Unwert der
Mathematik, Jahresbericht der Deutschen
Mathematiker Vereinigung (1904), p.
357._
=233.= Mathematical reasoning is deductive in the sense that it
is based upon definitions which, as far as the validity of the
reasoning is concerned (apart from any existential import), needs
only the test of self-consistency. Thus no external verification
of definitions is required in mathematics, as long as it is
considered merely as mathematics.--WHITEHEAD, A. N.
_Universal Algebra (Cambridge, 1898),
Preface, p. vi._
=234.= The mathematician pays not the least regard either to
testimony or conjecture, but deduces everything by demonstrative
reasoning, from his definitions and axioms. Indeed, whatever is
built upon conjecture, is improperly called science; for
conjecture may beget opinion, but cannot produce knowledge.
--REID, THOMAS.
_Essays on the Intellectual Powers of
Man, Essay 1, chap. 3._
=235.= ... for the saving the long progression of the thoughts to
remote and first principles in every case, the mind should
provide itself several stages; that is to say, intermediate
principles, which it might have recourse to in the examining
those positions that come in its way. These, though they are not
self-evident principles, yet, if they have been made out from
them by a wary and unquestionable deduction, may be depended on
as certain and infallible truths, and serve as unquestionable
truths to prove other points depending upon them, by a nearer and
shorter view than remote and general maxims.... And thus
mathematicians do, who do not in every new problem run it back to
the first axioms through all the whole train of intermediate
propositions. Certain theorems that they have settled to
themselves upon sure demonstration, serve to resolve to them
multitudes of propositions which depend on them, and are as
firmly made out from thence as if the mind went afresh over every
link of the whole chain that tie them to first self-evident
principles.--LOCKE, JOHN.
_The Conduct of the Understanding, Sect.
21._
=236.= Those intervening ideas, which serve to show the agreement
of any two others, are called _proofs_; and where the agreement or
disagreement is by this means plainly and clearly perceived, it is
called _demonstration_; it being _shown_ to the understanding, and
the mind made to see that it is so. A quickness in the mind to
find out these intermediate ideas, (that shall discover the
agreement or disagreement of any other) and to apply them right,
is, I suppose, that which is called _sagacity_.--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 6, chaps. 2, 3._
=237.= ... the speculative propositions of mathematics do not
relate to _facts_; ... all that we are convinced of by any
demonstration in the science, is of a necessary connection
subsisting between certain suppositions and certain conclusions.
When we find these suppositions actually take place in a
particular instance, the demonstration forces us to apply the
conclusion. Thus, if I could form a triangle, the three sides of
which were accurately mathematical lines, I might affirm of this
individual figure, that its three angles are equal to two right
angles; but, as the imperfection of my senses puts it out of my
power to be, in any case, _certain_ of the exact correspondence
of the diagram which I delineate, with the definitions given in
the elements of geometry, I never can apply with confidence to a
particular figure, a mathematical theorem. On the other hand, it
appears from the daily testimony of our senses that the
speculative truths of geometry may be applied to material objects
with a degree of accuracy sufficient for the purposes of life;
and from such applications of them, advantages of the most
important kind have been gained to society.--STEWART, DUGALD.
_Elements of the Philosophy of the Human
Mind, Part 3, chap. 1, sect. 3._
=238.= No process of sound reasoning can establish a result not
contained in the premises.--MELLOR, J. W.
_Higher Mathematics for Students of
Chemistry and Physics (New York, 1902),
p. 2._
=239.= ... we cannot get more out of the mathematical mill than
we put into it, though we may get it in a form infinitely more
useful for our purpose.--HOPKINSON, JOHN.
_James Forrest Lecture, 1894._
=240.= The iron labor of conscious logical reasoning demands
great perseverance and great caution; it moves on but slowly, and
is rarely illuminated by brilliant flashes of genius. It knows
little of that facility with which the most varied instances come
thronging into the memory of the philologist or historian. Rather
is it an essential condition of the methodical progress of
mathematical reasoning that the mind should remain concentrated
on a single point, undisturbed alike by collateral ideas on the
one hand, and by wishes and hopes on the other, and moving on
steadily in the direction it has deliberately chosen.
--HELMHOLTZ, H.
_Ueber das Verhältniss der
Naturwissenschaften zur Gesammtheit der
Wissenschaft, Vorträge und Reden, Bd. 1
(1896), p. 178._
=241.= If it were always necessary to reduce everything to
intuitive knowledge, demonstration would often be insufferably
prolix. This is why mathematicians have had the cleverness to
divide the difficulties and to demonstrate separately the
intervening propositions. And there is art also in this; for as
the mediate truths (which are called _lemmas_, since they appear
to be a digression) may be assigned in many ways, it is well, in
order to aid the understanding and memory, to choose of them
those which greatly shorten the process, and appear memorable and
worthy in themselves of being demonstrated. But there is another
obstacle, viz.: that it is not easy to demonstrate all the
axioms, and to reduce demonstrations wholly to intuitive
knowledge. And if we had chosen to wait for that, perhaps we
should not yet have the science of geometry.--LEIBNITZ, G. W.
_New Essay on Human Understanding
[Langley], Bk. 4, chaps. 2, 8._
=242.= In Pure Mathematics, where all the various truths are
necessarily connected with each other, (being all necessarily
connected with those _hypotheses_ which are the principles of the
science), an arrangement is beautiful in proportion as the
principles are few; and what we admire perhaps chiefly in the
science, is the astonishing variety of consequences which may be
demonstrably deduced from so small a number of premises.
--STEWART, DUGALD.
_The Elements of the Philosophy of the
Human Mind, Part 3, chap. 1, sect. 3._
=243.= Whenever ... a controversy arises in mathematics, the
issue is not whether a thing is true or not, but whether the
proof might not be conducted more simply in some other way, or
whether the proposition demonstrated is sufficiently important
for the advancement of the science as to deserve especial
enunciation and emphasis, or finally, whether the proposition is
not a special case of some other and more general truth which is
as easily discovered.--SCHUBERT, H.
_Mathematical Essays and Recreations
(Chicago, 1898), p. 28._
=244.= ... just as the astronomer, the physicist, the geologist,
or other student of objective science looks about in the world of
sense, so, not metaphorically speaking but literally, the mind of
the mathematician goes forth in the universe of logic in quest of
the things that are there; exploring the heights and depths for
facts--ideas, classes, relationships, implications, and the rest;
observing the minute and elusive with the powerful microscope of
his Infinitesimal Analysis; observing the elusive and vast with
the limitless telescope of his Calculus of the Infinite; making
guesses regarding the order and internal harmony of the data
observed and collocated; testing the hypotheses, not merely by
the complete induction peculiar to mathematics, but, like his
colleagues of the outer world, resorting also to experimental tests
and incomplete induction; frequently finding it necessary, in view
of unforeseen disclosures, to abandon one hopeful hypothesis or to
transform it by retrenchment or by enlargement:--thus, in his own
domain, matching, point for point, the processes, methods and
experience familiar to the devotee of natural science.
--KEYSER, CASSIUS J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 26._
=245.= That mathematics “do not cultivate the power of
generalization,” ... will be admitted by no person of competent
knowledge, except in a very qualified sense. The generalizations
of mathematics, are, no doubt, a different thing from the
generalizations of physical science; but in the difficulty of
seizing them, and the mental tension they require, they are no
contemptible preparation for the most arduous efforts of the
scientific mind. Even the fundamental notions of the higher
mathematics, from those of the differential calculus upwards are
products of a very high abstraction.... To perceive the
mathematical laws common to the results of many mathematical
operations, even in so simple a case as that of the binomial
theorem, involves a vigorous exercise of the same faculty which
gave us Kepler’s laws, and rose through those laws to the theory
of universal gravitation. Every process of what has been called
Universal Geometry--the great creation of Descartes and his
successors, in which a single train of reasoning solves whole
classes of problems at once, and others common to large groups
of them--is a practical lesson in the management of wide
generalizations, and abstraction of the points of agreement from
those of difference among objects of great and confusing
diversity, to which the purely inductive sciences cannot furnish
many superior. Even so elementary an operation as that of
abstracting from the particular configuration of the triangles or
other figures, and the relative situation of the particular lines
or points, in the diagram which aids the apprehension of a common
geometrical demonstration, is a very useful, and far from being
always an easy, exercise of the faculty of generalization so
strangely imagined to have no place or part in the processes of
mathematics.--MILL, JOHN STUART.
_An Examination of Sir William
Hamilton’s Philosophy (London, 1878),
pp. 612, 613._
=246.= When the greatest of American logicians, speaking of the
powers that constitute the born geometrician, had named
Conception, Imagination, and Generalization, he paused. Thereupon
from one of the audience there came the challenge, “What of
reason?” The instant response, not less just than brilliant, was:
“Ratiocination--that is but the smooth pavement on which the
chariot rolls.”--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 31._
=247.= ... the reasoning process [employed in mathematics] is not
different from that of any other branch of knowledge, ... but
there is required, and in a great degree, that attention of mind
which is in some part necessary for the acquisition of all
knowledge, and in this branch is indispensably necessary. This
must be given in its fullest intensity; ... the other elements
especially characteristic of a mathematical mind are quickness
in perceiving logical sequence, love of order, methodical
arrangement and harmony, distinctness of conception.--PRICE, B.
_Treatise on Infinitesimal Calculus
(Oxford, 1868), Vol. 3, p. 6._
=248.= Histories make men wise; poets, witty; the mathematics,
subtile; natural philosophy, deep; moral, grave; logic and
rhetoric, able to contend.--BACON, FRANCIS.
_Essays, Of Studies._
=249.= The Mathematician deals with two properties of objects
only, number and extension, and all the inductions he wants have
been formed and finished ages ago. He is now occupied with
nothing but deduction and verification.--HUXLEY, T. H.
_On the Educational Value of the Natural
History Sciences; Lay Sermons, Addresses
and Reviews; (New York, 1872), p. 87._
=250.= [Mathematics] is that [subject] which knows nothing of
observation, nothing of experiment, nothing of induction, nothing
of causation.--HUXLEY, T. H.
_The Scientific Aspects of Positivism,
Fortnightly Review (1898); Lay Sermons,
Addresses and Reviews, (New York, 1872),
p. 169._
=251.= We are told that “Mathematics is that study which knows
nothing of observation, nothing of experiment, nothing of induction,
nothing of causation.” I think no statement could have been made
more opposite to the facts of the case; that mathematical analysis
is constantly invoking the aid of new principles, new ideas, and
new methods, not capable of being defined by any form of words,
but springing direct from the inherent powers and activities of
the human mind, and from continually renewed introspection of that
inner world of thought of which the phenomena are as varied and
require as close attention to discern as those of the outer physical
world (to which the inner one in each individual man may, I think,
be conceived to stand somewhat in the same relation of correspondence
as a shadow to the object from which it is projected, or as the
hollow palm of one hand to the closed fist which it grasps of the
other), that it is unceasingly calling forth the faculties of
observation and comparison, that one of its principal weapons is
induction, that it has frequent recourse to experimental trial and
verification, and that it affords a boundless scope for the exercise
of the highest efforts of the imagination and invention.
--SYLVESTER, J. J.
_Presidential Address to British
Association, Exeter British Association
Report (1869), pp. 1-9.; Collected
Mathematical Papers, Vol. 2, p. 654._
=252.= The actual evolution of mathematical theories proceeds by a
process of induction strictly analogous to the method of induction
employed in building up the physical sciences; observation,
comparison, classification, trial, and generalisation are
essential in both cases. Not only are special results, obtained
independently of one another, frequently seen to be really
included in some generalisation, but branches of the subject
which have been developed quite independently of one another are
sometimes found to have connections which enable them to be
synthesised in one single body of doctrine. The essential nature
of mathematical thought manifests itself in the discernment of
fundamental identity in the mathematical aspects of what are
superficially very different domains. A striking example of this
species of immanent identity of mathematical form was exhibited by
the discovery of that distinguished mathematician ... Major MacMahon,
that all possible Latin squares are capable of enumeration by the
consideration of certain differential operators. Here we have a
case in which an enumeration, which appears to be not amenable to
direct treatment, can actually be carried out in a simple manner
when the underlying identity of the operation is recognised with
that involved in certain operations due to differential operators,
the calculus of which belongs superficially to a wholly different
region of thought from that relating to Latin squares.
--HOBSON, E. W.
_Presidential Address British
Association for the Advancement of
Science (1910); Nature, Vol. 84, p.
290._
=253.= It has been asserted ... that the power of observation is not
developed by mathematical studies; while the truth is, that; from
the most elementary mathematical notion that arises in the mind of
a child to the farthest verge to which mathematical investigation
has been pushed and applied, this power is in constant exercise.
By observation, as here used, can only be meant the fixing of the
attention upon objects (physical or mental) so as to note distinctive
peculiarities--to recognize resemblances, differences, and other
relations. Now the first mental act of the child recognizing the
distinction between _one_ and more than one, between _one_ and _two_,
_two_ and _three_, etc., is exactly this. So, again, the first
geometrical notions are as pure an exercise of this power as can be
given. To know a straight line, to distinguish it from a curve; to
recognize a triangle and distinguish the several forms--what are these,
and all perception of form, but a series of observations? Nor is it
alone in securing these fundamental conceptions of number and form that
observation plays so important a part. The very genius of the common
geometry as a method of reasoning--a system of investigation--is,
that it is but a series of observations. The figure being before the
eye in actual representation, or before the mind in conception, is so
closely scrutinized, that all its distinctive features are perceived;
auxiliary lines are drawn (the imagination leading in this), and a new
series of inspections is made; and thus, by means of direct, simple
observations, the investigation proceeds. So characteristic of common
geometry is this method of investigation, that Comte, perhaps the
ablest of all writers upon the philosophy of mathematics, is disposed
to class geometry, as to its method, with the natural sciences, being
based upon observation. Moreover, when we consider applied mathematics,
we need only to notice that the exercise of this faculty is so
essential, that the basis of all such reasoning, the very material
with which we build, have received the name _observations_. Thus we
might proceed to consider the whole range of the human faculties, and
find for the most of them ample scope for exercise in mathematical
studies. Certainly, the _memory_ will not be found to be neglected.
The very first steps in number--counting, the multiplication table,
etc., make heavy demands on this power; while the higher branches
require the memorizing of formulas which are simply appalling to the
uninitiated. So the _imagination_, the creative faculty of the mind,
has constant exercise in all original mathematical investigations,
from the solution of the simplest problems to the discovery of the
most recondite principle; for it is not by sure, consecutive steps,
as many suppose, that we advance from the known to the unknown. The
imagination, not the logical faculty, leads in this advance. In fact,
practical observation is often in advance of logical exposition.
Thus, in the discovery of truth, the imagination habitually presents
hypotheses, and observation supplies facts, which it may require ages
for the tardy reason to connect logically with the known. Of this
truth, mathematics, as well as all other sciences, affords abundant
illustrations. So remarkably true is this, that today it is seriously
questioned by the majority of thinkers, whether the sublimest branch of
mathematics,--the _infinitesimal calculus_--has anything more than an
empirical foundation, mathematicians themselves not being agreed as to
its logical basis. That the imagination, and not the logical faculty,
leads in all original investigation, no one who has ever succeeded in
producing an original demonstration of one of the simpler propositions
of geometry, can have any doubt. Nor are _induction_, _analogy_,
the _scrutinization_ of _premises_ or the _search_ for them, or the
_balancing_ of _probabilities_, spheres of mental operations foreign to
mathematics. No one, indeed, can claim pre-eminence for mathematical
studies in all these departments of intellectual culture, but it may,
perhaps, be claimed that scarcely any department of science affords
discipline to so great a number of faculties, and that none presents so
complete a gradation in the exercise of these faculties, from the first
principles of the science to the farthest extent of its applications,
as mathematics.--OLNEY, EDWARD.
_Kiddle and Schem’s Encyclopedia of
Education, (New York, 1877), Article
“Mathematics.”_
=254.= The opinion appears to be gaining ground that this very
general conception of functionality, born on mathematical ground,
is destined to supersede the narrower notion of causation,
traditional in connection with the natural sciences. As an
abstract formulation of the idea of determination in its most
general sense, the notion of functionality includes and
transcends the more special notion of causation as a one-sided
determination of future phenomena by means of present conditions;
it can be used to express the fact of the subsumption under a
general law of past, present, and future alike, in a sequence of
phenomena. From this point of view the remark of Huxley that
Mathematics “knows nothing of causation” could only be taken to
express the whole truth, if by the term “causation” is understood
“efficient causation.” The latter notion has, however, in recent
times been to an increasing extent regarded as just as irrelevant
in the natural sciences as it is in Mathematics; the idea of
thorough-going determinancy, in accordance with formal law, being
thought to be alone significant in either domain.--HOBSON, E. W.
_Presidential Address British
Association for the Advancement of
Science (1910); Nature, Vol. 84, p.
290._
=255.= Most, if not all, of the great ideas of modern mathematics
have had their origin in observation. Take, for instance, the
arithmetical theory of forms, of which the foundation was laid in
the diophantine theorems of Fermat, left without proof by their
author, which resisted all efforts of the myriad-minded Euler to
reduce to demonstration, and only yielded up their cause of being
when turned over in the blow-pipe flame of Gauss’s transcendent
genius; or the doctrine of double periodicity, which resulted
from the observation of Jacobi of a purely analytical fact of
transformation; or Legendre’s law of reciprocity; or Sturm’s
theorem about the roots of equations, which, as he informed me
with his own lips, stared him in the face in the midst of some
mechanical investigations connected (if my memory serves me
right) with the motion of compound pendulums; or Huyghen’s method
of continued fractions, characterized by Lagrange as one of the
principal discoveries of that great mathematician, and to which
he appears to have been led by the construction of his Planetary
Automaton; or the new algebra, speaking of which one of my
predecessors (Mr. Spottiswoode) has said, not without just reason
and authority, from this chair, “that it reaches out and
indissolubly connects itself each year with fresh branches of
mathematics, that the theory of equations has become almost new
through it, algebraic geometry transfigured in its light, that
the calculus of variations, molecular physics, and mechanics” (he
might, if speaking at the present moment, go on to add the theory
of elasticity and the development of the integral calculus) “have
all felt its influence.”--SYLVESTER, J. J.
_A Plea for the Mathematician, Nature,
Vol. 1, p. 238; Collected Mathematical
Papers, Vol. 2, pp. 655, 656._
=256.= The ability to imagine relations is one of the most
indispensable conditions of all precise thinking. No subject can
be named, in the investigation of which it is not imperatively
needed; but it can be nowhere else so thoroughly acquired as in
the study of mathematics.--FISKE, JOHN.
_Darwinism and other Essays (Boston,
1893), p. 296._
=257.= The great science [mathematics] occupies itself at least
just as much with the power of imagination as with the power of
logical conclusion.--HERBART, F. J.
_Pestalozzi’s Idee eines ABC der
Anschauung. Werke [Kehrbach]
(Langensaltza, 1890), Bd. 1, p. 174._
=258.= The moving power of mathematical invention is not
reasoning but imagination.--DE MORGAN, A.
_Quoted in Graves’ Life of Sir W. R.
Hamilton, Vol. 3 (1889), p. 219._
=259.= There is an astonishing imagination, even in the science
of mathematics.... We repeat, there was far more imagination in
the head of Archimedes than in that of Homer.--VOLTAIRE.
_A Philosophical Dictionary (Boston,
1881), Vol. 3, p. 40. Article
“Imagination.”_
=260.= As the prerogative of Natural Science is to cultivate a
taste for observation, so that of Mathematics is, almost from the
starting point, to stimulate the faculty of invention.
--SYLVESTER, J. J.
_A Plea for the Mathematician, Nature,
Vol. 1, p. 261; Collected Mathematical
Papers, Vol. 2 (Cambridge, 1908), p.
717._
=261.= A marveilous newtrality have these things mathematicall,
and also a strange participation between things supernaturall,
immortall, intellectuall, simple and indivisible, and things
naturall, mortall, sensible, componded and divisible.--DEE, JOHN.
_Euclid (1570), Preface._
=262.= Mathematics stands forth as that which unites, mediates
between Man and Nature, inner and outer world, thought and
perception, as no other subject does.--FROEBEL.
_[Herford translation] (London, 1893),
Vol. 1, p. 84._
=263.= The intrinsic character of mathematical research and
knowledge is based essentially on three properties: first, on its
conservative attitude towards the old truths and discoveries of
mathematics; secondly, on its progressive mode of development,
due to the incessant acquisition of new knowledge on the basis of
the old; and thirdly, on its self-sufficiency and its consequent
absolute independence.--SCHUBERT, H.
_Mathematical Essays and Recreations
(Chicago, 1898), p. 27._
=264.= Our science, in contrast with others, is not founded on a
single period of human history, but has accompanied the
development of culture through all its stages. Mathematics is as
much interwoven with Greek culture as with the most modern
problems in Engineering. She not only lends a hand to the
progressive natural sciences but participates at the same time in
the abstract investigations of logicians and philosophers.
--KLEIN, F.
_Klein und Riecke: Ueber angewandte
Mathematik und Physik (1900), p. 228._
=265.= There is probably no other science which presents such
different appearances to one who cultivates it and to one who
does not, as mathematics. To this person it is ancient,
venerable, and complete; a body of dry, irrefutable, unambiguous
reasoning. To the mathematician, on the other hand, his science
is yet in the purple bloom of vigorous youth, everywhere
stretching out after the “attainable but unattained” and full of
the excitement of nascent thoughts; its logic is beset with
ambiguities, and its analytic processes, like Bunyan’s road, have
a quagmire on one side and a deep ditch on the other and branch
off into innumerable by-paths that end in a wilderness.
--CHAPMAN, C. H.
_Bulletin American Mathematical Society,
Vol. 2 (First series), p. 61._
=266.= Mathematical science is in my opinion an indivisible whole,
an organism whose vitality is conditioned upon the connection of
its parts. For with all the variety of mathematical knowledge, we
are still clearly conscious of the similarity of the logical
devices, the _relationship_ of the _ideas_ in mathematics as a
whole and the numerous analogies in its different departments. We
also notice that, the farther a mathematical theory is developed,
the more harmoniously and uniformly does its construction proceed,
and unsuspected relations are disclosed between hitherto separated
branches of the science. So it happens that, with the extension of
mathematics, its organic character is not lost but manifests
itself the more clearly.--HILBERT, D.
_Mathematical Problems, Bulletin
American Mathematical Society, Vol. 8,
p. 478._
=267.= The mathematics have always been the implacable enemies of
scientific romances.--ARAGO.
_Oeuvres (1855), t. 3, p. 498._
=268.= Those skilled in mathematical analysis know that its
object is not simply to calculate numbers, but that it is also
employed to find the relations between magnitudes which cannot be
expressed in numbers and between functions whose law is not
capable of algebraic expression.--COURNOT, AUGUSTIN.
_Mathematical Theory of the Principles
of Wealth [Bacon, N. T.], (New York,
1897), p. 3._
=269.= Coterminous with space and coeval with time is the Kingdom
of Mathematics; within this range her dominion is supreme;
otherwise than according to her order nothing can exist; in
contradiction to her laws nothing takes place. On her mysterious
scroll is to be found written for those who can read it that
which has been, that which is, and that which is to come.
Everything material which is the subject of knowledge has number,
order, or position; and these are her first outlines for a sketch
of the universe. If our feeble hands cannot follow out the
details, still her part has been drawn with an unerring pen, and
her work cannot be gainsaid. So wide is the range of mathematical
sciences, so indefinitely may it extend beyond our actual powers
of manipulation that at some moments we are inclined to fall down
with even more than reverence before her majestic presence. But
so strictly limited are her promises and powers, about so much
that we might wish to know does she offer no information
whatever, that at other moments we are fain to call her results
but a vain thing, and to reject them as a stone where we had
asked for bread. If one aspect of the subject encourages our
hopes, so does the other tend to chasten our desires, and he is
perhaps the wisest, and in the long run the happiest, among his
fellows, who has learned not only this science, but also the
larger lesson which it directly teaches, namely, to temper our
aspirations to that which is possible, to moderate our desires to
that which is attainable, to restrict our hopes to that of which
accomplishment, if not immediately practicable, is at least
distinctly within the range of conception.--SPOTTISWOODE, W.
_Quoted in Sonnenschein’s Encyclopedia
of Education (London, 1906), p. 208._
=270.= But it is precisely mathematics, and the pure science
generally, from which the general educated public and independent
students have been debarred, and into which they have only rarely
attained more than a very meagre insight. The reason of this is
twofold. In the first place, the ascendant and consecutive
character of mathematical knowledge renders its results
absolutely insusceptible of presentation to persons who are
unacquainted with what has gone before, and so necessitates on
the part of its devotees a thorough and patient exploration of
the field from the very beginning, as distinguished from those
sciences which may, so to speak, be begun at the end, and which
are consequently cultivated with the greatest zeal. The second
reason is that, partly through the exigencies of academic
instruction, but mainly through the martinet traditions of
antiquity and the influence of mediæval logic-mongers, the
great bulk of the elementary text-books of mathematics have
unconsciously assumed a very repellant form,--something similar
to what is termed in the theory of protective mimicry in biology
“the terrifying form.” And it is mainly to this formidableness
and touch-me-not character of exterior, concealing withal a
harmless body, that the undue neglect of typical mathematical
studies is to be attributed.--MCCORMACK, T. J.
_Preface to De Morgan’s Elementary
Illustrations of the Differential and
Integral Calculus (Chicago, 1899)._
=271.= Mathematics in gross, it is plain, are a grievance in
natural philosophy, and with reason: for mathematical proofs,
like diamonds, are hard as well as clear, and will be touched
with nothing but strict reasoning. Mathematical proofs are out of
the reach of topical arguments; and are not to be attacked by the
equivocal use of words or declaration, that make so great a part
of other discourses,--nay, even of controversies.--LOCKE, JOHN.
_Second Reply to the Bishop of
Worcester._
=272.= The belief that mathematics, because it is abstract, because
it is static and cold and gray, is detached from life, is a mistaken
belief. Mathematics, even in its purest and most abstract estate,
is not detached from life. It is just the ideal handling of the
problems of life, as sculpture may idealize a human figure or as
poetry or painting may idealize a figure or a scene. Mathematics
is precisely the ideal handling of the problems of life, and the
central ideas of the science, the great concepts about which its
stately doctrines have been built up, are precisely the chief
ideas with which life must always deal and which, as it tumbles
and rolls about them through time and space, give it its interests
and problems, and its order and rationality. That such is the case
a few indications will suffice to show. The mathematical concepts
of constant and variable are represented familiarly in life by the
notions of fixedness and change. The concept of equation or that
of an equational system, imposing restriction upon variability,
is matched in life by the concept of natural and spiritual law,
giving order to what were else chaotic change and providing partial
freedom in lieu of none at all. What is known in mathematics under
the name of limit is everywhere present in life in the guise of
some ideal, some excellence high-dwelling among the rocks, an
“ever flying perfect” as Emerson calls it, unto which we may
approximate nearer and nearer, but which we can never quite attain,
save in aspiration. The supreme concept of functionality finds its
correlate in life in the all-pervasive sense of interdependence
and mutual determination among the elements of the world. What is
known in mathematics as transformation--that is, lawful transfer
of attention, serving to match in orderly fashion the things of
one system with those of another--is conceived in life as a
process of transmutation by which, in the flux of the world, the
content of the present has come out of the past and in its turn,
in ceasing to be, gives birth to its successor, as the boy is
father to the man and as things, in general, become what they are
not. The mathematical concept of invariance and that of infinitude,
especially the imposing doctrines that explain their meanings and
bear their names--What are they but mathematicizations of that
which has ever been the chief of life’s hopes and dreams, of that
which has ever been the object of its deepest passion and of its
dominant enterprise, I mean the finding of the worth that abides,
the finding of permanence in the midst of change, and the discovery
of a presence, in what has seemed to be a finite world, of being
that is infinite? It is needless further to multiply examples of a
correlation that is so abounding and complete as indeed to suggest
a doubt whether it be juster to view mathematics as the abstract
idealization of life than to regard life as the concrete realization
of mathematics.--KEYSER, C. J.
_The Humanization of the Teaching of
Mathematics; Science, New Series, Vol.
35, pp. 645-646._
=273.= Mathematics, like dialectics, is an organ of the inner
higher sense; in its execution it is an art like eloquence. Both
alike care nothing for the content, to both nothing is of value
but the form. It is immaterial to mathematics whether it
computes pennies or guineas, to rhetoric whether it defends truth
or error.--GOETHE.
_Wilhelm Meisters Wanderjahre, Zweites
Buch._
=274.= The genuine spirit of Mathesis is devout. No intellectual
pursuit more truly leads to profound impressions of the existence
and attributes of a Creator, and to a deep sense of our filial
relations to him, than the study of these abstract sciences. Who
can understand so well how feeble are our conceptions of Almighty
Power, as he who has calculated the attraction of the sun and the
planets, and weighed in his balance the irresistible force of the
lightning? Who can so well understand how confused is our
estimate of the Eternal Wisdom, as he who has traced out the
secret laws which guide the hosts of heaven, and combine the
atoms on earth? Who can so well understand that man is made in
the image of his Creator, as he who has sought to frame new laws
and conditions to govern imaginary worlds, and found his own
thoughts similar to those on which his Creator has acted?
--HILL, THOMAS.
_The Imagination in Mathematics; North
American Review, Vol. 85, p. 226._
=275.= ... what is physical is subject to the laws of
mathematics, and what is spiritual to the laws of God, and the
laws of mathematics are but the expression of the thoughts of
God.--HILL, THOMAS.
_The Uses of Mathesis; Bibliotheca
Sacra, Vol. 32, p. 523._
=276.= It is in the inner world of pure thought, where all
_entia_ dwell, where is every type of order and manner of
correlation and variety of relationship, it is in this infinite
ensemble of eternal verities whence, if there be one cosmos or
many of them, each derives its character and mode of being,--it
is there that the spirit of mathesis has its home and its life.
Is it a restricted home, a narrow life, static and cold and grey
with logic, without artistic interest, devoid of emotion and mood
and sentiment? That world, it is true, is not a world of _solar_
light, not clad in the colours that liven and glorify the things
of sense, but it is an illuminated world, and over it all and
everywhere throughout are hues and tints transcending _sense_,
painted there by radiant pencils of _psychic_ light, the light in
which it lies. It is a silent world, and, nevertheless, in
respect to the highest principle of art--the interpenetration of
content and form, the perfect fusion of mode and meaning--it even
surpasses music. In a sense, it is a static world, but so, too,
are the worlds of the sculptor and the architect. The figures,
however, which reason constructs and the mathematic vision
beholds, transcend the temple and the statue, alike in simplicity
and in intricacy, in delicacy and in grace, in symmetry and in
poise. Not only are this home and this life thus rich in æsthetic
interests, really controlled and sustained by motives of a
sublimed and supersensuous art, but the religious aspiration,
too, finds there, especially in the beautiful doctrine of
invariants, the most perfect symbols of what it seeks--the
changeless in the midst of change, abiding things in a world of
flux, configurations that remain the same despite the swirl and
stress of countless hosts of curious transformations. The domain
of mathematics is the sole domain of certainty. There and there
alone prevail the standards by which every hypothesis respecting
the external universe and all observation and all experiment must
be finally judged. It is the realm to which all speculation and
all thought must repair for chastening and sanitation--the court
of last resort, I say it reverently, for all intellection
whatsoever, whether of demon or man or deity. It is there that
mind as mind attains its highest estate, and the condition of
knowledge there is the ultimate object, the tantalising goal of
the aspiration, the _Anders-Streben_, of all other knowledge of
every kind.--KEYSER, C. J.
_The Universe and Beyond; Hibbert
Journal, Vol. 3 (1904-1905), pp.
313-314._