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H. Taine, Philosophie de FArt. 2 Bde. 7 Aufl. Paris, 1895.
Leo Tolstoj, Was ist Kunst? Deutsch, Berlin, 1892.
Johannes Volkelt, iEsthetische Zeitfragen, Muenchen, 1895. Deutch, Leipzig,

1902-03.— ^sthetik des Tragischen, Munchen, 1897.— System der ^Esthetik,

Bd. I, Munchen, 1905.
Stephan Witasek, Grundztige der allgemeinen JEsthetik, Leipzig.
Wilheim Wundt, Grundzuge der physiologsichen Psychologic, 1904, Bd. iii. 5

Aufl. Leipzig, 1903.



SHORT PAPERS

A short paper was contributed by Professor A. D. F. Hamlin, of Columbia
University, on the "Sources of Savage Conventional Patterns." The speaker
said that, in the exhibit of the Department of the Interior, two glass cases displayed
side by side the handiwork of the American Indian of one hundred years ago and
of to-day. In the Fine Arts palace the blankets and basketry of the Navahoes
were shown beside the leather work and other handicrafts of white Americans.
In both instances the contrast between the savage and the civilized work em-
phasizes the fact that civilization tends to stifle or destroy the decorative instinct.
The savage art is spontaneous, instructive, unpremeditated. The work of the
civiUzed artist is thoughtful, carefully elaborated, intellectual. Among these
peoples both the crafts and the patterns are traditional, and there is little or no
ambition to iimovate. The forms and combinations we admire in their work are
the result of long-continued processes of eA^olution and elimination in which, as in
the world of living organisms, the fittest have survived. The structure of savage
patterns is almost always extremely simple. There are three theories advanced to
account for them: that they were invented out of hand; that they were evolved
out of the technical processes, tools, and materials of primitive industry; that
they are descended from fetish or animistic representations of natural forms.
The first is the common view of laymen; the second was first expressed (though
chiefly with reference to civilized art) by Semper; and the third is widely enter-
tained by anthropologists.

The savage instinct for decoration has probably developed from primitive
animism — from that fear of the powers of natm-e, and that confounding of the
animate and inanimate world which is universally recognized as a primitive
trait. But once awakened in even the slightest degree, it has found exercise in
the operations of primitive industry, and given existence to a long series of repeti-
tive forms produced in weaving, basketry, string-lashing, and carving. The two
classes of patterns thus originated — those derived from the imitation of nature
under fetish ideas, and those derived from technical processes — have invariably
converged, overlapping at last in many forms of decorative art, so that the real
origin of a given pattern may be dual. Myths have in^^ariably arisen to explain
the origin of the technical patterns, which have received magic significance and
names, in accordance with savage tendency to assign magical powers to all visible
or at least to all valued objects: all savage art is talismanic. One ought to be
cautious about dogmatizing as to origins in dealing with savage art, because both
the phenomenon of what I call convergence in ornament evolution, and that of
the myths, poetic faculty, and habit among savages, tend to confuse and obscure
the real origin of the patterns with which they deal. And finally, for the artist
as distinguished from the archaeologist and the theorist, the real lesson of savage
art is not in its origins, but in its products; in the strength, simplicity, admirable
distribution, and high decorative effects of poor and despised peoples. Savage
all-over patterns and Greek carved ornament and decorative sculpture represent
the opposed poles of decorative design, and both are of fundamental value as
objects of study for the designer.



BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY

PREPARED THROUGH THE COURTESY OF DR. RALPH BARTON PERRY,
OF HARVARD UNIVERSITY

HISTORY OF PHILOSOPHY

BouiLLiER, F., Philosophie Cartesienne.
Burnet, J., Early Greek Philosophy.
Erdmann, J. E., Geschichte der Philosophie.
EucKEN, R. , Lebensanschauungen der grossen Denker.
Fairbanks, A., The First Philosophers of Greece.
Falkenberg, R., Geschichte der neueren Philosophie.
Fischer, K., Geschichte der neueren Philosophie.
Gomperz, Th., Greek Thinkers.
HoPFDiNG, H., Geschichte der neueren Philosophie.
Levy-Bruhl, Histoire de la philosophie modeme.
Royce, J., Spirit of Modern Philosophy.
SiDGwicK, H., History of Ethics.
Turner, W., History of Philosophy.
Ueberweg, F. , Gescliichte der Philosophie.
Weber, A., Histoire de la philosophie europeenne.
Windelband, W., Geschichte der Philosophie.

Geschichte der alten Philosophie.
Zelleb, E. , Geschichte der griechischen Philosophie.

PHILOSOPHICAL CLASSICS

Abelard, Dialectic.
Anselm, Monologium.
Aristotle, Metaphysics.
De Anima.
Physics.

Nicomachean Ethics.
Bacon, F., Novum Organum.

Berkeley, G. , The Principles of Human Knowledge.
Bruno, G., Dialogi, De la Causa Principio et Uno, etc.
Burnet, J., Early Greek Philosophy; fragments of Heraclitus, Parmenides,

Anaxagoras, etc.
Descartes, R. , Discours de la Methode.

Meditationes de Prima Philosophia.
Duns Scotus, Opus Oxoniense.
FiCHTE, J. G., Wissenschaftslehre.
Hegel, G. W. F., Wissenschaft der Logik.

Encyklopadie.
Hobbes, T., Leviathan.
Hume, D., Enquiry Concerning the Human Understanding.

Enquiry Concerning the Principles of Morals.
Kant, I., Kritik der reinen Vernunft.

Kritik der praktischen Vernunft.
Kritik der Urteilskraft.



450 BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY

Leibniz, G. W., Monadologie.

Th^odicee.
Locke, J., An Essay Concerning Human Understanding.
LoTZE, R. H., Metaphysik.
Lucretius, De Rerum Natura.
Plato, Republic. Phaedo. Theaetetus. Symposium. Phaedrus. Protagoras

(and other dialogues).
Plotinus, Enneades.
St. Augustine, De Civitate Dei.
Schelling, Philosophie der Natur.

Schopenhauer, A., Die Welt als Wille und Vorstellung.
Spencer, H., Synthetic Philosophy.
Spinoza, B., Ethica.
Thomas Aquinas, Summa Theologiae.

INTRODUCTION TO PHILOSOPHY

Baldwin, J. M., Dictionary of Philosophy.

HiBBEN, J. G., Problems of Philosophy.

KuLPE, O., Einleitung in die Philosophie.

Marvin, W. T., Introduction to Philosophy.

Paulsen, F., Einleitung in die Philosophie.

Perry, R. B., Approach to Philosophy.

SiDGwiCK, H., Philosophy, its Scope and Relations.

Stuckenberg, J. H. W., Introduction to the Study of Philosophy.

Watson, J., Outline of Philosophy.

WiNDELBAND, W., Praludien.

METAPHYSICS

AvENARius, R., Ej-itik der reinen Erfahrung.

Bergson, H., Matiere et memoire. '

Bradley, F. H., Appearance and Reality.

Deussen, p., Elements of Metaphysics.

Eucken, R., Der Kampf um einen geistigen Lebensinhalt.

FuLLERTON, G. S., System of Metaphysics.

Hodgson, S., Metaphysics of Experience.

HowisoN, G. H., The Limits of Evolution.

James, W., The Will to Believe.

LiEBMANN, Analysis der Wirklichkeit.

Ormond, a. T., Foundations of Knowledge.

Petzoldt, J. , Philosophie der reinen Erfahrung.

Renouvier, C, Les Dilemmes de la m^taphysique pure.

RicKERT, H., Der Gegenstand der Erkeuntniss.

Riehl, a., Philosophische Kriticismus.

RoYCE, J., The World and the Individual.

Schiller, F. C. S., Humanism.

Seth, a., Man and the Cosmos.

Sturt, H. (editor). Personal Idealism.

Taylor, A. E. , Elements of Metaphysics.

VoLKELT, J., Erfahrung imd Denken.

WiNDELBAND, W., Praludien.

WuNDT, W., System der Philosophie.



BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY 451



PHILOSOPHY OF RELIGION

BoussET, W., Das Wesen der Religion, dargestellt in ihrer Geschichte.

Caird, E., The Evolution of Religion.

DoRNER, A., Religionsphilosophie.

EucKEN, R., Der Wahrheitsgehalt der Religion.

Everett, C. C, The Psychological Elements of Religious Eaith.

Hartmann, von, E., Religionsphilosophie.

HoFFDiNG, H., Religionsphilosophie.

■James, W., Varieties of Religious Experience.

Martineau, J., A Study of Religion, its Sources and Contents.

MiJLLER, M., ELnleitung in die vergleichende Religionswissenschaft.

Pfleiderer, 0., Religionsphilosophie auf geschichtelichen Grundlage.

Rauenhopp, Religionsphilosophie.

Royce, J., The Religious Aspect of Philosophy.

Sabatier, a., Religionsphilosophie auf psychologischen und geschichtUchen

Grundlage.
Saussaye, Lehrbuch der Religionsgeschichte.
Seydel, R., Religionsphilosophie.
Teichmuller, G., Religionsphilosophie.
Tiele, C. p., Grundziige der Religionswissenschaft.

LOGIC

Bradley, F. H., The Principles of Logic.

Bosanquet, B., Logic.

Cohen, H., Die Logik der reinen Erkenntniss.

Dewey, J., Studies in Logical Theory.

Erdmann, B., Logik.

Hibben, J. G., Logic.

HoBHOUSE, L. T., Theory of Knowledge.

HussERL, Logische Untersuchimgen.

LoTZE, R. H., Grimdzuge der Logik.

ScHUPPE, W., Erkenntnisstheoretische Logik.

SiGw.iRT, C, Logik.

Wundt, W., Logik.

METHODOLOGY OF SCIENCE

Cantor, G., Grundlagen einer allgemeinen Mannigfaltigkeitslehre.

Dedekind, R., Was sind und was so Hen die Zahlen?

Hertz, H., Die Principien der Mechanik.

Jevons, W. S., Principles of Science.

Mach, E., Die Analyse der Empfindung.

MiJNSTERBERG, H., Grundziige der Psychologie.

Natorp, p., Einleitung in die Psychologie.

OsTWALD, W., Vorlesungen tiber Naturphilosophie.

Pearson, K., Grammar of Science.

PoiNCARE, H., La Science et I'Hypoth^se.

RiCKERT, H., Die Grenzen der naturwissenschaftlichen Begriffsbildung.

RoYCE, J., The World and the Individual, Second Series.

Russell, B., The Principles of Mathematics.

Ward, J., Naturalism and Agnosticism.

Windelband, W., Geschichte und Naturwissenschaft.



452 BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY



ETHICS

Alexander, S., Moral Order and Progress.

Bradley, F. H., Ethical Studies.

Cohen, H., Ethik des reinen Willens.

GiZYCKi, G., Grundziige der Moral.

Green, T. H., Prolegomena to Ethics.

Guy Air, M. J., Esqiiisse d'une morale sans obligation ni sanction.

Ladd, G. T., Philosophy of Conduct.

Martineau, J., Types of Ethical Theory.

Mezes, S. E., Ethics, Descriptive and Explanatory.

Moore, G. E., Principia Ethica.

Palmer, G. H., The Nature of Goodness.

Paulsen, F., System der Ethik.

RoYCE, J., studies of Good and Evil.

Seth, J., Principles of Ethics.

Sidgwick, H., Methods of Ethics.

Simmel, G., Einleitung in die Moralwissenschaft.

SoRLEY, W. R., Ethics of Naturalism.

Spencer, H., Principles of Ethics.

Stephen, L., Science of Ethics.

Taylor, A. E., The Problem of Conduct.

WuNDT, W., Ethik.

AESTHETICS

CoHN, Allgemeine iEsthetik.

GuYAU, M. J., Les Problemes de I'esthetique contemporaine.

HiRN, Yrjo, The Origins of Art.

Lange, K., Das Wesen der Kunst.

Lipps, T., ^sthetik.

Puffer, E., Psychology of Beauty.

SouRiAu, P., La Beaute RationeUe.

VoLKELT, J., Sj'^stem der iEsthetik.

WiTASEK, S., Grundziige der allgemeinen aesthetik.



DEPARTMENT II — MATHEMATICS



DEPARTMENT II — MATHEMATICS



(Hall 7, September 20, 11.15 a. m.)

Chairman: Professor Henry S. White, Northwestern University.
Speakers: Professor Maxime Bocher, Harvard University.
Professor James P. Pierpont, Yale University.

The Chairman of the Department of Mathematics was Professor
Henry S. White, of Northwestern University. In opening the pro-
ceedings Professor White said:

" Influenced by patriotism and by pride in material progress, cities
and whole nations meet and celebrate the building of bridges, the
opening of long railways, the tunneling of difficult mountain passes,
the acquisition of new territories, or commemorate with festivity the
discovery of a continent. These things are real and significant to us
all.

" In the realm of ideas also there are events of no less moment,
discoveries and conquests that greatly enlarge the empire of human
reason. In the lapse of a century there may be many such notable
achievements, even in the domain of a single science.

" Mathematics is a science continually expanding; and its growth,
unlike some political and industrial events, is attended by universal
acclamation. We are wont to-day, as devotees of this noble and
useful science, to pass in review the newest phases of her expansion,
— I say newest, for in retrospect a century is but brief, — and to
rejoice in the deeds of the past. At the same time, also, we turn
an eye of aspiration and resolution towards the mountains, rivers,
deserts, and the obstructing seas that are to test the mathematicians
of the future."



THE FUNDAMENTAL CONCEPTIONS AND METHODS OF

MATHEMATICS

BY PEOFESSOE MAXIME BOCHEE

[Maxime Bocher, Professor of Mathematics, Harvard University, b. August 28,
1867, Boston, Mass. A.B. Harvard, 1888; Ph.D. Gottingen, 1891. In-
structor, Assistant Professor and Professor, Harvard University, 1891-.
Fellow of the American Academy. Author of Ueber die Reihenentwickel-
ungen der Potentialtheorie; and various papers on mathematics.]

I, Old and New Definitions of Mathematics

I AM going to ask you to spend a few minutes with me in consider-
ing the question: what is mathematics? In doing this I do not propose
to lay down dogmatically a precise definition; but rather, after hav-
ing pointed out the inadequacy of traditional views, to determine
what characteristics are common to the most varied parts of mathe-
matics but are not shared by other sciences, and to show how this
opens the way to two or three definitions of mathematics, any one of
which is fairly satisfactory. Although this is, after all, merely a dis-
cussion of the meaning to be attached to a name, 1 do not think that
it is unfruitful, since its aim is to bring unity into the fundamental
conceptions of the science with which we are concerned. If any of
you, however, should regard such a discussion of the meaning of words
as devoid of any deeper significance, I will ask you to regard this
question as merely a bond by means of which I have found it con-
venient to unite what I have to say on the fundamental conceptions
and methods of what, with or without definition, we all of us agree
to call mathematics.

The old idea that mathematics is the science of quantity, or that
it is the science of space and number, or indeed that it can be charac-
terized by any enumeration of several more or less heterogeneous
objects of study, has pretty well passed away among those mathe-
maticians who have given any thought to the question of what
mathematics really is. Such definitions, which might have been
i^itelligently defended at the beginning of the nineteenth century,
became obviously inadequate as subjects like projective geometry,
the algebra of logic, and the theory of abstract groups were de-
veloped; for none of these has any necessary relation to quantity
(at least in any ordinary understanding of that word), and the last
two have no relation to space. It is true that such examples have
had little effect on the more or less orthodox followers of Kant,
who regard mathematics as concerned with those conceptions which



CONCEPTIONS AND METHODS OF MATHEMATICS 457

are obtained by direct intuition of time and space without the aid of
empirical observation. This view seems to have been held by such
eminent mathematicians as Hamilton and DeMorgan; and it is a
very difficult position to refute, resting as it does on a purely meta- 1
physical foundation which regards it as certain that we can evolve !
out of our inner consciousness the properties of time and space,;
According to this view the idea of quantity is to be deduced from
these intuitions; but one of the facts most \'ividly brought home to
pure mathematicians during the last half-century is the fatal weak-
ness of intuition when taken as the logical source of our knowledge
of number and quantity.^

The objects of mathematical study, even when we confine our
attention to what is ordinarily regarded as 'pure mathematics are,
then, of the most varied description; so that, in order to reach a
satisfactory conclusion as to what really characterizes mathematics,
one of tv/o methods is open to us. On the one hand w-e may seek
some hidden resemblance in the various objects of mathematical
investigation, and having found an aspect common to them all we
may fix on this as the one true object of mathematical study. Or,
on the other hand, we may abandon the attempt to characterize
mathematics by means of its objects of study, and seek in its methods
its distinguishing characteristic. Finally, there is the possibility of
our combining these two points of view. The first of these methods is
that of Kempe, the second will lead us to the definition of Benjamin
Peirce, while the third has recently been elaborated at great length
by Russell. Other mathematicians have naturally followed out more
or less consistently the same ideas, but I shall nevertheless take the
liberty of using the names Kempe, Peirce, and Russell as convenient
designations for these three points of view. These different methods
of approaching the question lead finally to results which, without
being identical, still stand in the most intimate relation to one an-
other, as we shall now see. Let us begin with the second method.

II. Peirce' s Definition

More than a third of a century ago Benjamin Peirce wrote: ^
Mathematics is the science which draws necessary conclusions. Accord-
ing to this view there is a mathematical element involved in every
inquiry in which exact reasoning is used. Thus, for instance,^ a
jury listening to the attempt of the counsel for the prisoner to prove
an alibi in a criminal case might reason as follows : " If the witnesses

^_ I refer here to such facts as that there exist continuous functions mthout
derivatives, whereas the direct untutored intuition of space would lead any one
to believe that every continuous curve has tangents.

^ Linear Associative Algebra. Ijitbographed 1870. Reprinted in the American
Journal of Mathematics, vol. iv.

^ This illustration was suggested by the remarks by J. Richard, Sur la philoso-
phie des mathmeatiques. Paris, Gauth'ier-Villars, 1903!^ p. 50.



458 MATHEMATICS

are telling the truth when they say that the prisoner was in St. Louis
at the moment the crime was committed in Chicago, and if it is
true that a person cannot be in two places at the same time, it follows
that the prisoner was not in Chicago when the crime was committed."
This, according to Peirce, is a bit of mathematics; while the further
reasoning by which the jury would decide whether or not to believe
the witnesses, and the reasoning (if they thought any necessary)
by which they would satisfy themselves that a person cannot be
in two places at once, would be inductive reasoning, which can give
merely a high degree of probability to the conclusion, but never
certainty. This mathematical element may be, as the example
just given shows, so slight as not to be worth noticing from a prac-
tical point of view. This is almost always the case in the transac-
tions of daily life and in the observational sciences. If, however, we
turn to such subjects as chemistry and mineralogy, we find the
mathematical element of considerable importance, though still
subordinate. In physics and astronomy its importance is much
greater. Finally in geometry, to mention only one other science, the
mathematical element predominates to such an extent that this
science has been commonly rated a branch of pure mathematics,
whereas, according to Peirce, it is as much a branch of applied
mathematics as is, for instance, mathematical physics.

It is clear from what has just been said that, from Peirce's point
of view, mathematics does not necessarily concern itself with quanti-
tative relations, and that any subject becomes capable of mathe-
matical treatment as soon as it has secured data from which import-
ant consequences can be drawn by exact reasoning. Thus, for
example, even though psychologists be right when they assure us
that sensations and the other objects with which they have to deal
cannot be measured, we need still not necessarily despair of one day
seeing a mathematical psychology, just as we already have a math-
ematical logic.

I have said enough, I think, to show what relation Peirce's con-
ception of mathematics has to the applications. Let us then turn
to the definition itself and examine it a little more closely. You
have doubtless already noticed that the phrase, " the science which
draws necessary conclusions, " contains a word which is very much
in need of elucidation. What is a necessary conclusion? Some of
you wall perhaps think that the conception here involved is one
about which, in a concrete case at least, there can be no practical
diversity of opinion among men with well- trained minds; and in
fact when I spoke a few minutes ago about the reasoning of the
jurymen when listening to the lawyer trying to prove an alibi, I
assumed tacitly that this is so. If this really were the case, no further
discussion would be necessary, for it is not my purpose to enter into



CONCEPTIONS AND METHODS OF MATHEMATICS 459

any purely philosophical speculations. But unfortunately we can-
not dismiss the matter in this way; for it has happened not infre-
quently that the most eminent men, including mathematicians,
have differed as to whether a given piece of reasoning was exact or
not; and, what is worse, modes of reasoning which seem absolutely
conclusive to one generation no longer satisfy the next, as is shown.
by the way in which the greatest mathematicians of the eighteenth
century used geometric intuition as a means of drawing what they|
regarded as necessary conclusions/

I do not wish here to raise the question whether there is such a
thing as absolute logical rigor, or whether this whole conception of
logical rigor is a purely psychological one bound to change with
changes in the human mind. I content myself with expressing the
belief, which I will try to justify a little more fully in a moment,
that as we never have found an immutable standard of logical rigor
in the past, so we are not likely to find it in the future. However
this may be, so much we can say with tolerable confidence, as past
experience shows, that no reasoning which claims to be exact can
make any use of intuition, but that it must proceed from definitely
and completely stated premises according to certain principles of
formal logic. It is right here that modern mathematicians break
sharply with the tradition of a 'priori synthetic judgments (that is,
conclusions drawn from intuition) which, according to Kant, form an
essential part of mathematical reasoning.

If then we agree that "necessary conclusions" must, in the present
state of human knowledge, mean conclusions drawn according to
certain logical principles from definitely and completely stated
premises, we must face the question as to what these principles
shall be. Here, fortunately, the mathematical logicians from Boole
down to C. S. Peirce, Schroder, and Peano have prepared the field
so well that of late years Peano and his followers ^ have been able
to make a rather short list of logical conceptions and principles upon
which it would seem that all exact reasoning depends.^ We must
remember, however, when we are tempted to put implicit confidence
in certain fundamental logical principles, that, owing to their extreme
generality and abstractness, no very great weight can be attached
to the mere fact that these principles appeal to us as obviously

* All writers on elementary geometry from Euclid down almost to the close
of the nineteenth century use intuition freely, though usually unconsciously, in
obtaining results which they are unable to deduce from their axioms. The first
few demonstrations of Euclid are criticised from this point of view by Russell in
his Principles of Mathematics, vol. i, 404-407. Gauss's first proof (1799) that 5ss
every algebraic equation has a root gives a striking example of the use of intuition ^}
in what was intended as an absolutely rigorous proof by one of the greatest and at
the same time most critica,} mathematical minds the world has ever seen.

' And, independently, Frege.

8 It is not intended to assert that a single list has been fixed upon. Different
writers naturally use different lists. . .



460 MATHEMATICS

K j true; for, as I have said, other modes of reasoning which are now
"^ / universally recognized as faulty have appealed in just this way to
the greatest minds of the past. Such confidence as we feel must,
I think, come from the fact that those modes of reasoning which
we trust have withstood the test of use in an immense number of



Online LibraryMo.) Congress of Arts and Science (1904 : Saint LouisCongress of Arts and Science : universal exposition, St. Louis, 1904 → online text (page 50 of 68)