Mo.) Congress of Arts and Science (1904 : Saint Louis.

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This view, while not in accord with the prevailing ideas of mathe-
maticians, undoubtedly has its advantages as well as its dangers.
The non-deductive processes, of which I shall have more to say
presently, play too important a part in the life of mathematics to
be ignored, and the definition just given has the merit of not exclud-
ing them. It would seem, however, that the definition in the form
just given is too broad. It would include, for instance, the deter-
mination by experimental methods of what pairs of chemical com-
pounds of the known elements react on one another when mixed
under given conditions.

VI. Axioms and Postulates. Existence Theorems

If, however, we restrict ourselves to exact or deductive mathe-
matics, it will be seen that Kempe's definition becomes coextensive
with Peirce's. Here, in order to have a starting-point for deductive
reasoning, we must assume a certain number of facts or primitive
propositions concerning any mathematical system we wish to study,
of which all other propositions will be necessary consequences.*
We touch here on a subject whose origin goes back to Euclid and
which has of late years received great development, primarily at
the hands of Italian mathematicians.^

It is important for us to notice at this point that not merely these
primitive propositions but all the propositions of mathematics may
be divided into two great classes. On the one hand, we have pro-
-positions which state that certain specified objects satisfy certain
specified relations. On the other hand are the existence theorems,
which state that there exist objects satisfying, along with certain
specified objects, certain specified relations.^ These two classes of
propositions are well known to logicians and are designated by them

^ These primitive propositions may be spoken of as axioms or 'postulates, ac-
cording to^ the point of view we wish to take concerning their source, the word
axiom, which has been much misused of late, indicating an intuitional or empirical

^ Peano, Fieri, Eadoa, Burali-Forti. We may mention here also Hilbert, who,
apparently without knowing of the important work of his Italian predecessors,
has also done valuable work along these lines.

^ Or we might conceivably have existence theorems which state that there
exist relations which are satisfied by certain specified objects; or these two kinds
of existence theorems might be combined. If we take the point of view explained
in the second footnote on p. 467, all existence theorems will be of the type men-
tioned in the text.


universal and particular propositions respectively.^ It is only during
the last fifty years or so that mathematicians have become conscious
of the fundamental importance in their science of existence theorems,
which until then they had frequently assumed tacitly as they needed
them, without always being conscious of what they were doing.

It is sometimes held by non-mathematicians that if mathematics
were really a purely deductive science, it could not have gained
anything like the extent which it has without losing itself in trivial-
ities and becoming, as Poincare puts it, a vast tautology.^ This
view would doubtless be correct if all primitive propositions were
universal propositions. One of the most characteristic features of
mathematical reasoning, however, is the use which it makes of aux-
iliary elements. I refer to the auxiliary points and lines in proofs
by elementary geometry, the quantities formed by combining in
various ways the numbers which enter into the theorems to be
proved in algebra, etc. Without the use of such auxiliary elements
mathematicians would be incapable of advancing a step; and
whenever we make use of such an element in a proof, we are in reality
using an existence theorem.^ These existence theorems need not,
to be sure, be among the primitive propositions; but if not, they must
be deduced from primitive propositions some of which are existence
theorems, for it is clear that an existence theorem cannot be deduced
from universal propositions alone.* Thus it may fairly be said that
existence theorems form the vital principle of mathematics, but these
in turn, it must be remembered, would be impotent without the
material basis of universal propositions to work upon.

VII. Russell's Definition

We have so far arrived at the view that exact mathematics is
the study by deductive methods of what we have called a mathe-
matical system, that is, a class of objects and a class of relations
between them. If we elaborate this position in two directions we
shall reach the standpoint of Russell.^

In the first place Russell makes precise the term deductive method

* "All men are mortals" is a standard example of a universal proposition;
while as an illustration of a particular proposition is often given: "Some men are
Greeks. " That this is really an existence theorerti is seen more clearly when we
state it in the form: "There exists at least one man who is a Greek."

^ Cf. La Science et VHypothese, p. 10.

^ Even when in algebra we consider the sum of two numbers a + 6, we are using
the existence theorem which says that, any two numbers a and b being given,
there exists a number c which stands to them in the relation wliich we indicate in
ordinary language by saying that c is the sum of a and b.

* The power which resides in the method of mathematical induction, so called,
comes from the fact that this method depends on an existence theorem. It is,
however, not the only fertile principle in mathematics as Poincare would have
us believe (cf. La Science et VHypothese). In fact there are great branches of
mathematics, like elementary geometry, in which it takes little or no part.

^ The Principles of Mathematics, Cambridge, England, 1903.


by laying down explicitly a list of logical conceptions and prin-
ciples which alone are to be used; and, secondly, he insists,^ on the
contrary, that no mathematical system, to use again the technical
term introduced above, be studied in pure mathematics whose exist-
ence cannot be established solely from the logical principles on which
all mathematics is based . Inasmuch as the development of mathemat-
ics during the last fifty years has shown that the existence of most,
if not all the mathematical systems which have proved to be im-
portant can be deduced when once the existence of positive integers
is granted, the point about which interest must centre here is the
proof, which Russell attempts, of the existence of this latter sys-
tem. ^ This proof will necessarily require that, among the logical
principles assumed, existence theorems be found. Such theorems
do not seem to be explicitly stated by Russell, the existence theorems
which make their appearance further on being evolved out of some-
what vague philosophical reasoning. There are also other reasons,
into which I cannot enter here, why I am not able to regard the
attempt made in this direction by Russell as completely successful.'
Nevertheless, in view of the fact that the system of finite positive
integers is necessary in almost all branches of mathematics (we
cannot speak of a triangle or a hexagon without having the numbers
three and six at our disposal), it seems extremely desirable that the
system of logical principles which we lay at the foundation of all
mathematics be assumed, if possible, broad enough so that the
existence of positive integers — at least finite integers — follows from
it; and there seems little doubt that this can be done in a satisfactory
manner. "When this has been done we shall perhaps be able to regard,
with Russell, pure mathematics as consisting exclusively of deduc-
tions "by logical principles from logical principles."

VIII. The Non-Deductive Elements in Mathematics

I fear that many of you will think that what I have been saying
is of an extremely one-sided character, for I have insisted merely on
the rigidly deductive form of reasoning used and the purely abstract
character of the objects considered in mathematics. These, to the
great majority of mathematicians, are only the dry bones of the
science. Or, to change the simile, it may perhaps be said that instead
of inviting you to a feast I have merely shown you the empty dishes

' In the formal definition of mathematics at the beginning of the book this is
not stated or in any way impUed; and yet it comes out so clearly thronghoiit
the book that this is a point of view which the author regards as essential, that
I have not hesitated to include it as a part of his definition.

^ Cf. also Burali-Forti, Congres internationale de philosophie. Paris, a'oI. hi,
p. 280.

^ Russell's unequivocal repudiation of nominalism in mathematics seems to
me a serious if not an insurmountable barrier to progress.


and explained how the feast would be served if only the dishes were
filled.^ I fully agree with this opinion, and can only plead in excuse
that my subject was the jundamental conceptions and methods of
mathematics, not the infinite variety of detail and application
which give our science its real vitality. In fact I should like to
subscribe most heartily to the view that in mathematics, as else-
where, the discussion of such fundamental matters derives its interest
mainly from the importance of the theory of which they are the
so-called foundations.^ I like to look at mathematics almost more
as an art than as a science; for the activity of the mathematician,
constantly creating as he is, guided though not controlled by the
external world of the senses, bears a resemblance, not fanciful I
believe but real, to the activity of an artist, of a painter let us say.
Rigorous deductive reasoning on the part of the mathematician
may be likened here to technical skill in drawing on the part of the
painter. Just as no one can become a good painter without a certain
amount of this skill, so no one can become a mathematician without
the power to reason accurately up to a certain point. Yet these
qualities, fundamental though they are, do not make a painter or
a mathematician worthy of the name, nor indeed are they the most
important factors in the case. Other qualities of a far more subtle
sort, chief among which in both cases is imagination, go to the
making of the good artist or good mathematician. I must content
myself merely by recalling to you this somewhat vague and difficult
though interesting field of speculation which arises when we attempt
to attach value to mathematical work, a field which is familiar
enough to us all in the analogous case of artistic or literary criticism.
We are in the habit of speaking of logical rigor and the considera-
tion of axioms and postulates as the foundations on which the superb
structure of modern mathematics rests; and it is often a matter of
wonder how such a great edifice can rest securely on such a small
foundation. Moreover, these foundations have not always seemed so
secure as they do at present. During the first half of the nineteenth
century certain mathematicians of a critical turn of mind — Cauchy,
Abel, Weierstrass, to mention the greatest of them — perceived to
their dismay that these foundations were not sound, and some of the
best efforts of their lives were devoted to strengthening and improv-
ing them. And yet I doubt whether the great results of mathematics

^ Notice that just as the empty dishes could be filled by a great variety of
viands, so the empty symbols of mathematics can be given meanings of the most
varied sorts.

^ Cf. the following remark by Study, Jahresbericht der deutschcn Mathematiker-
Vereinigung, vol. xi (1902), p. 313:

" So wertvoU auch Untersuchungen uber die systematische Stellung der math-
ematischen Grundbegriffe sind . . . wertvoller ist doch noch der materielle Inhalt
der einzelnen Disciplinen, um dessentwillen aUein ja derartige Untersuchungen
i'lberhaupt Zweck haben. ..."


seemed less certain to any of them because of the weakness they
perceived in the foundations on which these results are built up.
The fact is that what we call mathematical rigor is merely one of
the foundation stones of the science; an important ai;id essential
one surely, yet not the only thing upon which we can rely. A science
which has developed along such broad lines as mathematics, with
such numerous relations of its parts both to one another and to other
sciences, could not long contain serious error without detection.
This explains how, again and again, it has come about, that the
most important mathematical developments have taken place by
methods which cannot be wholly justified by our present canons of
\ mathematical rigor, the logical "foundation" having been supplied
! only long after the superstructure had been raised. A discussion
■ and analysis of the non-deductive methods which the creative
mathematician really uses would be both interesting and instructive.
Here I must content myself with the enumeration of a few of them.
First and foremost there is the use of intuition, whether geometrical,
mechanical, or physical. The great service which this method has
rendered and is still rendering to mathematics both pure and applied
is so well known that a mere mention is sufficient.

Then there is the method of experiment; not merely the physical
experiments of the laboratory or the geometrical experiments I
had occasion to speak of a few minutes ago, but also arithmetical
experiments, numerous examples of which are found in the theory
of numbers and in analysis. The mathematicians of the past fre-
quently used this method in their printed works. That this is now
seldom done must not be taken to indicate that the method itself is
not used as much as ever.

Closely allied to this method of experiment is the method of
analogy, which assumes that something true of a considerable num-
ber of cases will probably be true in analogous cases. This is, of
course, nothing but the ordinary method of induction. But in mathe-
matics induction may be employed not merely in connection wdth
the experimental method, but also to extend results won by deduct-
ive methods to other analogous cases. This use of induction has
often been unconscious and sometimes overbold, as, for instance,
when the operations of ordinary algebra were extended without
scruple to infinite series.

Finally there is what may perhaps be called the method of optim-
ism, which leads us either willfully or instinctn^ely to shut our eyes
to the possibility of evil. Thus the optimist wjo treats a problem in
algebra or analytic geometry will say, if he stops to reflect on what
he is doing: "1 know that I have no right to divide by zero; but
there are so many other values whicTi the expression by which I am
dividing might have that I will assume that the Evil One has not


thrown a zero in my denominator this time." This method, if a pro-
ceeding often unconscious can be called a method, has been of great
service in the rapid development of many branches of mathematics,
though it may well be doubted whether in a subject as highly devel-
oped as is ordinary algebra it has not now survived its usefulness.^
While no one of these methods can in aiiy way compare with
that of rigorous deductive reasoning as a method upon which to
base mathematical results, it would be merely shutting one's eyes
to the facts to deny them their place in the life of the mathematical
world, not merely of the past but of to-day. There is now, and there
always will be room in the world for good mathematicians of every
grade of logical precision. It is almost equally important that the
small band whose chief interest hes in accuracy and rigor should
not make the mistake of despising the broader though less accurate
work of the great mass of their colleagues; as that the latter should
not attempt to shake themselves wholly free from the restraint the
former would put upon them. The union of these two tendencies
in the same individuals, as it was found, for instance, in Gauss and
Cauchy, seems the only sure way of avoiding complete estrangement
between mathematicians of these two types.

' Cf. the very suggestive remarks by Study, Jahresbericht d. Deutschen Math-
ematiker-Vereinigung, vol. xi (1902), p. 100, footnote, in which it is pointed out
how rigor, in cases of this sort, may not merely serve to increase the correctness of
the result, but actually to suggest new fields for mathematical investigation.




The extraordinary development of mathematics in the last century
is quite unparalleled in the long history of this most ancient of
sciences. Not only have those branches of mathematics which were
taken over from the eighteenth century steadily grown, but entirely
new ones have sprung up in almost bewildering profusion, and
many of these have promptly assumed proportions of vast extent.

As it is obviously impossible to trace in the short time allotted to
me the history of mathematics in the nineteenth century even in
merest outline, I shall restrict myself to the consideration of some
of its leading theories.

Theory of Functions of a Complex Variable

Without doubt one of the most characteristic features of mathe-
matics in the last century is the systematic and universal use of the
complex variable. Most of its great theories received invaluable aid
from it, and many owe their very existence to it. What would the
theory of differential equations or elliptic functions be to-day without
it, and is it probable that Poncelet, Steiner, Chasles, and von Staudt
would have developed synthetic geometry with such elegance and
perfection without its powerful stimulus?

The necessities of elementary algebra kept complex numbers
persistently before the eyes of every mathematician. In the eight-
eenth century the more daring, as Euler and Lagrange, used them
sparingly; in general one avoided them when possible. Three events,
however, early in the nineteenth century changed the attitude of
mathematicians toward this mysterious guest. In 1813 Argand
published his geometric interpretation of complex numbers. In
1824 came the discovery by Abel of the imaginary period of the
elliptic function. Finally Gauss in his second memoir on biquadratic
residues (1832) proclaims them a legitimate and necessary element
of analysis.

The theory of function of a complex variable may be. said to have
had its birth when Cauchy discovered his integral theorem

ff{x)dx =

published in 1825. In a long series of publications beginning with
the Cours d' Analyse (1821), Cauchy gradually developed his theory
of functions and applied it to problems of the most diverse nature;


for example, existence theorems for implicit functions and the solu-
tions of certain differential equations, the development of functions
in infinite series and products, and the periods of integrals of one
and many valued functions.

Meanwhile Germany is not idle; Weierstrass and Riemann de-
velop Cauchy's theory along two distinct and original paths. Weier-
strass starts with an explicit analytical expression, a power series,
and defines his function as the totality of its analytical continua-
tions. No appeal is made to geometric intuition, his entire theory
is strictly arithmetical. Riemann growing up under Gauss and
Dirichlet not only relies largely on geometric intuition, but" he also
does not hesitate to impress mathematical physics into his service.
Two noteworthy features of his theory are the many leaved surfaces
named after him, and the extensive use of conformal representation.

The history of functions as first developed is largely a theory of
algebraic functions and their integrals. A general theory of func-
tions is only slowly evolved. For a long time the methods of Cauchy,
Riemann, and Weierstrass were cultivated along distinct lines by
their respective pupils. The schools of Cauchy and Riemann were
the first to coalesce. The entire rigor which has recently been im-
parted to their methods has removed all reason for founding, as
Weierstrass and his school have urged, the theory of functions on
a single algorithm, namely, the power series. We may therefore say
that at the close of the century there is only one theor}^ of functions
in which the ideas of its three great creators are harmoniously united.

Let us note briefly some of its lines of advance. Weierstrass early
observed that an analytic expression might represent different
analytic functions in different regions. Associated with this is the
phenomenon of natural boundaries. The question therefore arose.
What is the most general domain of definition of an analytic function?
Runge has shown that any connected region may serve this purpose.
An important line of investigation relates to the analytic expression
of a function by means of infinite series, products, and fractions.
Here may be mentioned Weierstrass 's discovery of prime factors;
the theorems of Mittag-Leffler and Hilbert; Poincare's uniform-
ization of algebraic and analytic functions by means of a third
variable, and the work of Stieljes, Pade, and Van Vleck on infinite
fractions. Since an analytic function is determined by a single
power series, which in general has a finite circle of convergence, two
problems present themselves: determine, first, the singular points of
the analytic function so defined, and, second, an analytic expression
valid for its whole domain of definition. The celebrated memoir of
Hadamard inaugurated a long series of investigations on the first
problem; while Mittag-Leffier's star theorem is the most important
result yet obtained relating to the second.


Another line of investigation relates to the work of Poincar^,
Borel, Fade, et al., on divergent series. It is, indeed, a strange vicissi-
tude of our science that these series which early in the century
were supposed to be banished once and for all from rigorous mathe-
matics should at its close be knocking at the door for readmission.

Let us finally note an important series of memoirs on integral
transcendental functions, beginning with Weierstrass, Laguerre, and

Algebraic Functions and their Integrals

A branch of the theory of functions has been developed to such
an extent that it may be regarded as an independent theory; we
mean the theory of algebraic functions and their integrals. The
brilliant discoveries of Abel and Jacobi in the elliptic functions from
1824 to 1829 prepared the way for a similar treatment of the hyper-
elliptic case. Here a difficulty of gravest nature was met. The cor-
responding integrals have 2p linearly independent periods; but as
Jacobi had shown, a one valued function having more than two
periods admits a period as small as we choose. It therefore looked
as if the elliptic functions admitted no further generalization.
Guided by Abel's theorem, Jacobi at last discovered the solution to
the difficulty (1832); to get functions analogous to the elliptic func-
tions we must consider functions not of one but of p independent
variables, namely, the p independent integrals of the first species.
The great problem now before mathematicians, known as Jacobi's
Problem of Inversion, was to extend this apergu to the case of any
algebraic configuration and develop the consequences. The first to
take up this immense task were Weierstrass and Riemann, whose
results belong to the most brilliant achievements of the century.
Among the important notions hereby introduced we note the fol-
lowing: the birational transformation, rank of an algebraic con-
figuration, class invariants, prime functions, the theta and multiply
periodic functions in several variables. Of great importance is
Riemann's method of proving existence theorems, as also his repre-
sentation of algebraic functions by means of integrals of the second

A new direction was given to research in this field by Clebsch, who
considered the fundamental algebraic configuration as defining a
curve. His aim was to bring about a union of Riemann's ideas and
the theory of algebraic curves for their mutual benefit. Clebsch's
labors were continued by Brill and Nother; in their work the tran-
scendental methods of Riemann are placed qaitt in the background.
More recently Klein and his school have sought to unite the tran-
scendental methods of Riemann with the geometric direction begun
by Clebsch, making systematic use of homogeneous coordinates and

Online LibraryMo.) Congress of Arts and Science (1904 : Saint LouisCongress of Arts and Science : universal exposition, St. Louis, 1904 → online text (page 52 of 68)