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cases and in very many fields. This is the severest test to which any
theory can be put, and if it does not break down under it we may
feel the greatest confidence that, at least in cognate fields, it will
prove serviceable. But we can never be sure. The accepted modes
of exact reasoning may any day lead to a contradiction which would
show that what we regard as universally applicable principles are
in reality applicable only under certain restrictions.^
• To show that the danger which I here point out is not a purely

fanciful one, it is sufficient to refer to a very recent example. Inde-
pendently of one another, Frege and Russell have built up the theory
of arithmetic from its logical foundations. Each starts with a definite
list of apparently self-evident logical principles, and builds up a
seemingly flawless theory. Then Russell discovers that his logical
principles when applied to a very general kind of logical class lead
to an absurdity; and both Frege and Russell have to admit that
something is wrong with the foundations which looked so secure.
Now there is no doubt that these logical foundations will be somehow
recast to meet this difficulty, and that they will then be stronger
than ever before.^ But who shall say that the same thing will not
happen again?

It is commonly considered that mathematics owes its certainty
to its reliance on the immutable principles of formal logic. This,
as we have seen, is only half the truth imperfectly expressed. The
other half would be that the principles of formal logic owe such
degree of permanence as they have largely to the fact that they
have been tempered by long and varied use by mathematicians.
"A vicious circle!" you will perhaps say. I should rather describe
it as an example of the process known to mathematicians as the
method of successive approximations. Let us hope that in this
case it is really a convergent process, as it has every appearance of
being.

But to return to Peirce's definition. From what are these neces-

* If the v\ew which I here maintain is correct, it follows that if the tei-m " abso-
lute logical rigor " has a meaning, and if we should some time arrive at this abso-
lute standard, the only indication we should ever have of the fact would be that
for a long period, several thousand years let us say, the logical principles in ques-
tion had stood the test of use. But this state of affairs might equally well mean
that during that time the human mind had degenerated, at least with regard to
some of its functions. Consider, for instance, the twenty centuries following Euclid
when, without doubt, the high tide of exact thinking attained during Euclid's gen-
eration had receded.

* Cf. Poincar^'s view in La Science et I'Hi/pothese, p. 179, according to which
a theory never renders a greater service to science than when it breaks down.



CONCEPTIONS AND METHODS OF MATHEMATICS 461

sary conclusions to be drawn? The answer clearly implied is, from
any premises sufficiently precise to make it possible to draw neces-
sary conclusions from them. In geometry, for instance, we have a
large number of intuitions and fixed beliefs concerning the nature
of space: it is homogeneous and isotropic, infinite in extent in every
direction, etc.; but none of these ideas, however clearly defined
they may at first sight seem to be, gives any hold for exact reasoning.
This was clearly perceived by Euclid, who therefore proceeded to
lay down a list of axioms and postulates, that is, specific facts which
he assumes to be true, and from which it was his object to deduce all
geometric propositions. That his success here was not complete
is now well known, for he frequently assumes unconsciously further
data which he derives from intuition; but his attempt was a monu-
mental one.

III. The Abstract Nature of Mathematics

Now a further self-evident point, but one to which attention seems
to have been drawn only during the last few years, is this: since we
are to make no use of intuition, but only of a certain number of ^
explicitly stated premises, it is not necessary that we should have
any idea what the nature of the objects and relations involved in
these premises is.^ I will try to make this clear by a simple example.
In plane geometry we have to consider, among other things, points and
straight lines. A point may have a peculiar relation to a straight
line which we express by the words, the point hes on the line. Now
one of the fundamental facts of plane geometry is that two points
determine a line, that is, if two points are given, there exists one and
only one line on which both points lie. All the facts that I have just
stated correspond to clear intuitions. Let us, however, eliminate our
intuition of what is meant by a point, a line, a point lying on a line.
A slight change of language will make it easy for us to do this. In-
stead of points and lines, let us speak of two different kinds of objects,
say A-objects and 5-objects; and instead of saying that a point
lies on a line we will simply say that an A-object bears a certain
relation i^ to a 5-object. Then the fact that two points determine
a line will be expressed by saying: If any two yl -objects are given,
there exists one and only one 5-object to which they both bear the
relation R. This statement, while it does not force on us any specific
intuitions, will serve as a basis for mathematical reasoning ^ just as
well as the more familiar statement where the terms points and lines

* This was essentially, Kempe's point of view in the papers to be referred to
presently. In the geoHiiS+ric example which follows it was clearly brought out
by H. Wiener: Jahresbericfd d. deutschen Maihematiker-Vereinigung, vol. i (1891),
p. 45. _

^ In conjunction, of course, with further postulates with which we need not
here concern ourselves.



462 MATHEMATICS

are used. But more than this. Our A-objects, our J5-objects, and our
relation R may be given an interpretation, if we choose, very different
from that we had at first intended.

We may, for instance, regard the Ji-objects as the straight Hnes in
a plane, the 5-objects as the points in the same plane (either finite
or at infinity), and when an A-object stands in the relation i^ to a
5-object, this may be taken to mean that the line passes through the
point. Our statement would then become : Any two lines being given,
there exists one and only one point through which they both pass.
Or we may regard the A-objects as the men in a certain community,
the 5-objects as the women, and the relation of an ^-object to a
-B-object as friendship. Then our statement would be: In this com-
munity any two men have one, and only one, woman friend in com-
mon.

These examples are, I think, sufficient to show what is meant
when I say that we are not concerned in mathematics with the
nature of the objects and relations involved in our premises, except
in so far as their nature is exhibited in the premises themselves.
Accordingly mathematicians of a critical turn of mind, during the
last few years, have adopted more and more a purely nominalistic
attitude towards the objects and relations involved in mathematical
investigation. This is, of course, not the crude mixture of nominalism
and empiricism of the philosopher Hobbes, whose claim to mathe-
matical fame, it may be said in passing, is that of a circle-squarer.*
The nominalism of the present-day mathematician consists in treating
the objects of his investigation and the relations between them as
mere symbols. He then states his propositions, in effect, in the fol-
lowing form: If there exist any objects in the physical or mental
world with relations among themselves which satisfy the conditions
which I have laid down for my symbols, then such and such facts
will be true concerniag them.

It will be seen that, according to Peirce's view, the mathematician
as such is in no wise concerned with the source of his premises or with
their harmony or lack of harmony with any part of the external
world. He does not even assert that any objects really exist which
correspond to his symbols. Mathematics may therefore be truly
said to be the most abstract of all sciences, since it does not deal
directly with reality.^

ThiS; then, is Peirce's definition of mathematics. Its advantages
in the direction of unifying our conception of mathematics and of
assigning to it a definite place among the other sciences are clear.

^ Hobbes practically obtains as the ratio of a circir'rerence to its diameter
the value \/lO. Cf. for instance Molesworth's edition of JHobbes's English Works,
vol. VII, p. 431.

' Cf. the very interesting remarks along this line of C. S. Peircain The Monist,
vol, vii, pp. 23-24.



CONCEPTIONS AND METHODS OF MATHEMATICS 463

What are its disadvantages? I can see only two. First that, as has
been already remarked, the idea of drawing necessary conclusions
is a slightly vague and shifting one. Secondly, that it lays exclusive
stress on the rigorous logical element in mathematics and ignores
the intuitional and other non-rigorous tendencies which form an
important element in the great bulk of mathematical work concern-
ing which I shall speak in greater detail later.

IV. Geometry an Experimental Science

Some of you will also regard it as an objection that there are
subjects which have almost universally been regarded as branches
of mathematics but are excluded by this definition. A striking
example of this is geometry, I mean the science of the actual space
we live in; for though geometry is, according to Peirce's definition,
preeminently a mathematical science, it is not exclusively so. Until
a system of axioms is established mathematics cannot begin its work.
Moreover, the actual perception of spatial relations, not merely
in simple cases but in the appreciation of complicated theorems, is
an essential element in geometry which has no relation to mathe-
matics as Peirce understands the term. The same is true, to a con-
siderable extent, of such subjects as mechanical drawing and model-
making, which involve, besides small amounts of physics and math-
ematics, mainly non-mathematical geometry. Moreover, although the
mathematical method is the traditional one for arriving at the truth
concerning geometric facts, it is not the only one. Direct appeal to
the intuition is often a short and fairly safe cut to geometric results;
and on the other hand experiments may be used in geometry, just
as they are used every day in physics, to test the truth of a proposi-
tion or to determine the value of some geometric magnitude.^

We must, then, admit, if we hold to Peirce's view, that there is
an independent science of geometry just as there is an independent
science of physics, and that either of these may be treated by math-
ematical methods. Thus geometry becomes the simplest of the
natural sciences, and its axioms are of the nature of physical laws,
to be tested by experience and to be regarded as true only within
the limits of error of observation. This view, while it has not yet
gained universal recognition, should, I believe, prevail, and geo-
metry be recognized as a science independent of mathematics, just
as psychology is gradually being recognized as an independent
science and not as a branch of philosophy.

The view here set forth, according to which geometry is an ex-
perimental science like physics or chemistry, has been held ever

1 I am thinking of measurements and observations made on accurately con-
structed drawings and modeL. A famous example is Galileo's determination of
the area of a cycloid by cutting out a cycloid from a metallic sheet and weighing it-



464 MATHEMATICS

since Gauss's time by almost all the leading mathematicians who
have been conversant with non-Euclidean geometry.^ Recently,
however, Poincare has thrown the weight of his great authority
against this view,^ claiming that the experiments by which it is
sought to test the truth of geometric axioms are really not geometrical
experiments at all but physical ones, and that any failure of these
experiments to agree with the ordinary geometrical axioms could
be explained by the inaccuracy of the 'physical laws ordinarily as-
sumed. There is undoubtedly an important element of truth here.
Every experiment depends for its results not merely on the law we
wish to test, but also on other laws which for the moment we assume
to be true. But, if we prefer, we may, in many cases, assume as
true the law w^e were before testing and our experiment will then
serve to test some of the remaining laws. If, then, we choose to stick
to the ordinary Euclidean axioms of geometry in spite of what any
future experiments may possibly show, we can do so, but at the cost,
perhaps, of our present simple physical laws, not merely in one
branch of physics but in several. Poincare 's view ^ is that it will
always be expedient to preserve simple geometric laws at all costs,
an opinion for which I fail to see sufficient reason.

V. Kempe's Definition

Let us now turn from Peirce's method of defining mathematics to
Kempe's, which, however, I shall present to you in a somewhat
modified form.* The point of view adopted here is to try to define
mathematics, as other sciences are defined, by describing the objects
with which it deals. The diversity of the objects with which mathe-
matics is ordinarily supposed to deal being so great, the first step
must be to divest them of what is unessential for the mathematical
treatment, and to try in this way to discover their common and
characteristic element.

The first point on which Kempe insists is that the objects of mathe-
matical discussion, whether they be the points and lines of geometry,
the numbers real or complex of algebra or analysis, the elements of
groups or anything else, are always individuals, infinite in number
perhaps, but still distinct individuals. In a particular mathematical
investigation we ma}^ and usually do, have several different kinds of
individuals; as for instance, in elementary plane geometry, points,
straight lines, and circles. Furthermore, we have to deal with certain
relations of these objects to one another. For instance, in the example

' Gauss, Riemaim, Helmholtz are the names which -will carry perhaps the
greatest weight.

^ Cf. La Science et VHypoihese. Paris, 1903.

^ L. c, chapter v. In particular, p. 93.

* Kempe has set forth his ideas in rather popular form in the Proceedings of
the London Mathematical Society, vol. xxvi (1894], p. 5; and in Nature, vol. xliii
1890), p. 156, where references to his more technical ^Titings wiU be found.



CONCEPTIONS AND METHODS OF MATHEMATICS 465

Just cited, a given point may or may not lie on a given line; a given
line may or may not touch a given circle; three or more points may
or may not be collinear, etc. This example shows how in a single
mathematical problem a large number of relations may be involved,
relations some of which connect two objects, others three, etc.
Moreover these relations may connect like or they may connect
unlike objects; and finally the order in which the objects are taken
is not by any means immaterial in general, as is shown by the relation
between three points which states that the third is coUinear with and
lies between the first two.

But even this is not all; for, besides these objects and relations
of various kinds, we often have operations by which objects can be
combined to yield another object, as, for instance, addition or multi-
plication of numbers. Here the objects combined and the resulting
object are all of the same kind, but this is by no means necessary.
We may, for instance, consider the operation of combining two
points and getting the perpendicular bisector of the line connecting
them; or we may combine a point and a line and get the perpen-
dicular dropped from the point on the line.

These few examples show how diverse the relations and operations, |
as well as the objects of mathematics, seem at first sight to be. Out \
of this apparent diversity it is not difficult to obtain a very great j
uniformity by simply restating the facts in a little different language, i
We shall find it convenient to indicate that the objects a,b, c, . . . ,
taken in the order named, satisfy a relation R by simply writing
R(a. h, c, . . .), where it should be understood that among the
objects a, h. c, , . . the same object may occur a number of times.
On the other hand, if two objects a and b are combined to yield
a third object c, we may wi'ite a o h=c,^ Vvhere the symbol o is
characteristic of the special operation with which we are concerned.

Let us first notice that the equation aoh=c denotes merely
that the three objects a, b, c bear a certain relation to one another,
say R{a, b, c). In other words the idea of an operation or law of
combination between the objects we deal with, however convenient
and useful it may be as a matter of notation, is essentially merely
a way of expressing the fact that the objects combined bear a certain
relation to the object resulting from their combination. Accordingly,
in a purely abstract discussion like the present, where questions of
practical convenience are not involved, we need not consider such
rules of combination.^

' I speak here merely of dyadic opCTafcions, — i. e., of operations by which
two objects are combined to yield a third, — these being by far the most import-
ant as well as the simplest. What is said, however, obviously applies to opera-
tions by which any number of objects are combined.

^ ^ Even from the point of view of the technical mathematician it may some-
times be desirable to adopt the point of view of a relation rather than that of an
operation. This is seen, for instance, in laying down a system of postulates for the



466 MATHEMATICS

Furthermore, it is easy to see that when we speak of objects of
different kinds, as, for instance, the points and Hnes of geometry, we
are introducing a notion which can very readily be expressed in our
relational notation. For this purpose we need merely to introduce
a further relation which is satisfied by two or more objects when and
only when they are of the same "kind."

Let us turn finally to the relations themselves. It is customary
to distinguish here between dyadic relations, triadic relations, etc.,
according as the relation in question connects two objects, three
objects, etc. There are, however, relations which may connect any
number of objects, as, for instance, the relation of collinearity which
may hold between any number of points. Any relation holds for
certain ordered groups of objects but not for others, and' it is in no
way necessary for us to fix our attention on the fact, if it be true,
that the number of objects in all the groups for which a particular
relation holds is the same. This is the point of view we shall adopt,
and we shall relegate the property that a relation is dyadic, triadic,
etc., to the background along with the various other properties
relations may have,^ all of which must be taken account of in the
proper place.

We are thus concerned in any mathematical investigation, from
our present point of view, with just two conceptions: first a set, or
as the logicians say, a class of objects a, b, c, . . .; and secondly a
class of relations R, S, T, . . . . We may suppose these objects
divested of any qualitative, quantitative, spatial, or other attributes
which they may have had, and regard them merely as satisfying or not
satisfying the relations in question, where, again, we are wholly
indifferent to the nature which these relations originally had. And
now we are in a position to state what I conceive to be really the
essential point in Kempe's definition of mathematics; although I
have omitted one of the points on which he insists most strongly,^
by saying:

If we have a certain class of objects and a certain class of relations,
and if the only questions which we investigate are whether ordered
groups of these objects do or do not satisfy the relations, the results
of the investigation are called mathematics.

theory of abstract groups (cf., for example, Huntington, Bulletin of the Ameri-
can Mathematical Society, June, 1902), where the postulate:
If a and b belong to the class, a o b belongs to the class,
•which in this form looks indecomposable, immediately breaks up, when stated in
the relational form, into the following two:

1 . If a and 6 belong to the class, there exists an element c of the class such that
R{a, b, c).

2. If a, b, c, d belong to the class, and if R{a, b, c) and R{a, b, d), then c = d.

^ For instance, the property of symmetry. A relation is said to be symmetrical
if it holds or fails to hold independently of the order in which the objects are taken.

^ Namely, that the only relation that need be considered is that of being "in-
distinguishable," i. e., a symmetrical and transitive relation between two groups
of objects.



CONCEPTIONS AND METHODS OF MATHEMATICS 467

It is convenient to have a term to designate a class of objects
associated with a class of relations between these objects. Such an
aggregate we will speak of as a mathematical system. If now we have
two different mathematical systems, and if a one-to-one correspond-
ence can be set up between the two classes of objects, and also
between the two classes of relations in such a way that whenever
a certain ordered set of objects of the first system satisfies a relation
of that system, the set consisting of the corresponding objects of the
second system satisfies the corresponding relation of that system,
and vice versa, then it is clear that the two systems are, from our
present point of view, mathematically equivalent, however different
the nature of the objects and relations may be in the two cases. ^ To
use a techtiical term, the two systems are simply isomorphic.^

It will be noticed that in the definition of mathematics just given
nothing is said as to the method by which we are to ascertain whether
or not a given relation holds between the objects of a given set. The
method used may be a purely empirical one, or it may be partly or
wholly deductive. Thus, to take a very simple case, suppose our class
of objects to consist of a large number of points in a plane and sup-
pose the only relation between them with which we are concerned
is that of collinearity. Then, if the points are given us by being
marked in ink on a piece of white paper, we can begin by taking three
pins, sticking them into the paper at three of the points; then, by
sighting along them, we can determine whether or not these points
are collinear. We can do the same with other groups of three
points, then with all groups of four points, etc. The same result
can be obtained with much less labor if we make use of certain
simple properties which the relation of collinearity satisfies, pro-
perties which are expressed by such propositions as:

R(a, h, c) implies R(b, a, c),

R{a, h, c, d) implies R(a, h, c),

R{a, b, c) and R{a, h, d) together imply R{a, h, c, d), etc.

By means of a small number of propositions of this sort it is easy
to show that no empirical observations as to the collinearity of
groups of more than three points need be made, and that it may
not be necessary to examine even all groups of three points. Having

^ The point of view here brought out, including the term isomorphism, was
first developed in a special case, — the theory of groups.

^ Inasmuch as the relations in a mathematical system are themselves objects,
we may, if we choose, take our class of objects so as to include these relations as
well as what we called objects before, some of which, we may remark in passing,
may themselves be relations. Looked at from this point of _ view, we need one
additional relation which is now the only one which we explicitly call a relation.
If we denote this relation by inclosing the objects which satisfy it in parentheses,
then if the relation denoted before by R{a, h) is satisfied, we should now write
{R, a, b), whereas we should not have (a, R, b) {S, R, a, b), etc. Thus we see that
any mathematical system may be regarded as consisting of a class of objects and
a single relation between them.



468 " MATHEMATICS

made this relatively small number of observations, the remaining
results would be obtained deductively. Finally, we may suppose
the points given by their coordinates, in which case the complete
answer to our question may be obtained by the purely deductive
method of analytic geometry.

According to the modified form of Kempe's definition which I
have just stated, mathematics is not necessarily a deductive science.



Online LibraryMo.) Congress of Arts and Science (1904 : Saint LouisCongress of Arts and Science : universal exposition, St. Louis, 1904 → online text (page 51 of 68)