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

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of tlie categories has to-day become almost equally a problem for
the logicians of mathematics and for those students of philosophy
who take any serious interest in exactness of method in their own
branch of work. The result of this actual cooperation of men from
both sides is that, as I think, we are to-day, for the first time, in
sight of what is still, as I freely admit, a somewhat distant goal,
namely, the relatively complete rational analysis and tabulation of
the fundamental categories of human thought. That the student of
ethics is as much interested in such an investigation as is the meta-
physician, that the philosopher of religion needs a well-completed
table of categories quite as much as does the pure logician, every
competent student of such topics ought to admit. And that the
enterprise in question keenly interests the mathematicians is shown
by the prominent part which some of them have taken in the re-
searches in question. Here, then, is the type of recent scientific work
whose results most obviously bear upon the tasks of all of us alike.
A catalogue of the names of the workers in this wide field of
modern logic would be out of place here. Yet one must, indeed,
indicate what lines of research are especially in question. From the
purely mathematical side, the investigations of the type to which I
now refer may be viewed (somewhat arbitrarily) as beginning with
that famous examination into one of the postulates of Euclid's
geometry which gave rise to the so-called non-Euclidean geometry.
The question here originally at issue was one of a comparatively
limited scope, namely, the question whether Euclid's parallel-line
postulate was a logical consequence of the other geometrical prin-
ciples. But the investigation rapidly develops into a general study
of the foundations of geometry — a study to which contributions
are still almost constantly appearing. Somewhat independently
of this line of inquiry there grew up, during the latter half of the
nineteenth century, that reexamination of the bases of arithmetic
and analysis which is associated with the names of Dedekind, Weier-
strass, and George Cantor. At the present time, the labors of a num-
ber of other inquirers (amongst whom we may mention the school
of Peano and Fieri in Italy, and men such as Foincare and Couturat
in France, Hilbert in Germany, Bertrand Russell and Whitehead in
England, and an energetic group of our American mathematicians
— men such as Frofessor Moore, Frofessor Halsted, Dr. Hunting-
ton, Dr. Veblen, and a considerable number of others) have been
added to the earlier researches. The result is that we have recently
come for the first time to be able to see, with some completeness,
what the assumed first principles of pure mathematics actually are.
As was to be expected, these principles are capable of more than
one formulation, according as they are approached from one side or
from another. As was also to be expected, the entire edifice of pure


mathematics, so far as it has yet been erected, actually rests upon
a very few fundamental concepts and postulates, however you may
formulate them. What was not observed, however, by the earlier,
and especially by the philosophical, students of the categories, is
the form which these postulates tend to assume when they are
rigidly analyzed.

This form depends upon the precise definition and classification
of certain types of relations. The whole of geometry, for instance,
including metrical geometry, can be developed from a set of postu-
lates which demand the existence of points that stand in certain
ordinal relationships. The ordinal relationships can be reduced,
according as the series of points considered is open or closed, either
to the M^ell-known relationship in which three points stand when
one is between the other two upon a right line, or else to the ordinal
relationship in which four points stand when they are separated by
pairs; and these two ordinal relationships, by means of various log-
ical devices, can be regarded as variations of a single fundamental
form. Cayley and Klein founded the logical theory of geometry here
in question. Russell, and in another way Dr. Veblen, have given
it its most recent expressions. In the same way, the theory of whole
numbers can be reduced to sets of principles which demand the exist-
ence of certain ideal objects in certain simple ordinal relations. Dede-
kind and Peano have worked out such ordinal theories of the num-
ber concept. In another development of the theory of the cardinal
whole numbers, which Russell and Whitehead have worked out,
ordinal concepts are introduced only secondarily, and the theory
depends upon the fundamental relation of the equivalence or non-
equivalence of collections of objects. But here also a certain simple
tj^pe of relation determines the definitions and the development of
the whole theory.

Two results follow from such a fashion of logically analyzing the
first principles of mathematical science. In the first place, as just
pointed out, we learn how jew and simple are the conceptions and pos-
tulates upon which the actual edifice of exact science rests. Pure
mathematics, we have said, is free to assume what it chooses. Yet
the assumptions whose presence as the foundation principles of the
actually existent pure mathematics an exhaustive examination thus
reveals, show by their fewness that the ideal freedom of the mathe-
matician to assume and to construct what he pleases, is indeed, in.
practice, a very decidedly limited freedom. The limitation is, as we
have already seen, a limitation which has to do with the essential
significance of the fundamental concepts in question. And so the
result of this analysis of the bases of the actually developed and
significant branches of mathematics, constitutes a sort of empirical
revelation of what categories the exact sciences have practically


found to be of such significance as to be worthy of exhaustive treat-
ment. Thus the instinctive sense for significant truth, which has all
along been guiding the development of mathematics, comes at least
to a clear and philosophical consciousness. And meanwhile the es-
sential categories of thought are seen in a new light.

The second result still more directly concerns a philosophical logic.
It is this: Since the few types of relations which this sort of ana-
lysis reveals as the fundamental ones in exact science are of such
importance, the logic of the present day is especially required to face
the questions : What is the nature of our concept of relations f What
are the various possible types of relations? Upon what does the
variety of these types depend? What unity lies beneath the variety?

As a fact, logic, in its modern forms, namely, first that symbolic
logic which Boole first formulated, which Mr. Charles S. Peirce and
his pupils have in this country already so highly developed, and
which Schroeder in Germany, Peano's school in Italy, and a num-
ber of recent English writers have so effectively furthered — and
secondly, the logic of scientific method, which is now so actively
pursued, in France, in Germany, and in the English-speaking coun-
tries — this whole movement in modern logic, as I hold, is rapidly
approaching new solutions of the problem of the fundamental nature
and the logic of relations. The problem is one in which we are all
equally interested. To De Morgan in England, in an earlier genera-
tion, and, in our time, to Charles Peirce in this country, very im-
portant stages in the growth of these problems are due. Russell, in
his work on the Principles of Mathematics has very lately under-
taken to sum up the results of the logic of relations, as thus far
developed, and to add his own interpretations. Yet I think that
Russell has failed to get as near to the foundations of the theory
of relations as the present state of the discussion permits. For
Russell has failed to take account of what I hold to be the most
fundamentally important generalization yet reached in the general
theory of relations. This is the generalization set forth as early as
1890, by Mr. A. B. Kempe, of London, in a pair of wonderful but
too much neglected, papers, entitled, respectively. The Theory of
Mathematical Form, and The Analogy between the Logical Theory
of Classes and the Geometrical Theory of Points. A mere hint first
as to the more precise formulation of the problem at issue, and then
later as to Kempe's special contribution to that problem, may be in
order here, despite the impossibility of any adequate statement.


The two most obviously and universally important kinds of rela-
tions known to the exact sciences, as these sciences at present exist,
are: (1) The relations of the type of equality or equivalence; and


(2) the relations of the type of before and after, or greater and less.
The first of these two classes of relations, namely, the class repre-
sented, although by no means exhausted, by the various relations
actually called, in different branches of science by the one name
equality, this class I say, might well be named, as I myself have
proposed, the leveling relations. A collection of objects between
any two of which some one relation of this type holds, may be said
to be a collection whose members, in some defined sense or other,
are on the same level. The second of these two classes of relations,
namely, those of the type of before and after, or greater and less
— this class of relations, I say, consists of what are nowadays often
called the serial relations. And a collection of objects such that, if
any pair of these objects be chosen, a determinate one of this pair
stands to the other one of the same pair in some determinate rela-
tion of this second type, and in a relation which remains constant
for all the pairs that can be thus formed out of the members of this
collection — any such collection, I say, constitutes a one-dimen-
sional open series. Thus, in case of a file of men, if you choose any
pair of men belonging to the file, a determinate one of them is, in the
file, before the other. In the number series, of any two numbers,
a determinate one is greater than the other. Wherever such a state
of affairs exists, one has a series.

Now these two classes of relations, the leveling relations and the
serial relations, agree with one another, and differ from one another
in very momentous ways. They agree with one another in that both
the leveling and the serial relations are what is technically called
transitive; that is, both classes conform to what Professor James
has called the law of "skipped intermediaries." Thus, if A is equal
to B, and B is equal to C, it follows that A is equal to C. If A is
before B, and B is before C, then A is before C. And this property,
which enables you in your reasonings about these relations to skip
middle terms, and so to perform some operation of elimination, is
the property which is meant when one calls relations of this type
transitive. But, on the other hand, these two classes of relations
differ from each other in that the leveling relations are, while the
serial relations are not, symmetrical or reciprocal. Thus, if A is equal
to B, B is equal to A. But if X is greater than Y, then Y is not
greater than X, but less than X. So the levehng relations are sym-
metrical transitive relations. But the serial relations are transitive
relations which are not symmetrical.

All this is now well known. It is notable, however, that nearly
all the processes of our exact sciences, as at present developed,
can be said to be essentially such as lead either to the placing of sets
or classes of objects on the same level, by means of the use of sym-
metrical transitive relations, or else to the arranging of objects in


orderly rows or serieS; by means of the use of transitive relations
which are not symmetrical. This holds also of all the applications
of the exact sciences. Whatever else you do in science (or, for that
matter, in art), you always lead, in the end, either to the arrang-
ing of objects, or of ideas, or of acts, or of movements, in rows or
series, or else to the placing of objects or ideas of some sort on the
same level, by virtue of some equivalence, or of some invariant
character. Thus numbers, functions, lines in geometry, give you
examples of serial relations. Equations in mathematics are classic
instances of leveling relations. So, of course, are invariants. Thus,
again, the whole modern theory of energy consists of two parts,
one of which has to do with levels of energy, in so far as the quan-
tity of energy of a closed system remains invariant through all the
transformations of the system, while the other part has to do with
the irreversible serial order of the transformations of energy them-
selves, which follow- a set of unsymmetrical relations, in so far as
energy tends to fall from higher to lower levels of intensity within
the same system.

The entire conceivable universe then, and all of our present exact
science, can be viewed, if you choose, as a collection of objects or
of ideas that, whatever other types of relations may exist, are at
least largely characterized either by the leveling relations, or by
the serial relations, or by complexes of both sorts of relations. Here,
then, we are plainly dealing with very fundamental categories.
The "between" relations of geometry can of course be defined, if
you choose, in terms of transitive relations that are not symmet-
rical. There are, to be sure, some other relations present in exact
science, but the two types, the serial and leveling relations, are
especially notable.

So far the modern logicians have for some time been in substan-
tial agreement. Russell's brilliant book is a development of the
logic of mathematics very largely in terms of the tv\ro types of rela-
tions which, in my own way, I have just characterized; although
Russell gives due regard, of course, to certain other types of rela-

But hereupon the question arises, "Are these two types of rela-
tions what Russell holds them to be, namely, ultimate and irre-
ducible logical facts, unanalyzable categories — mere data for the
thinker? Or can we reduce them still further, and thus simplify
yet again our view of the categories?

Here is where Kempe's generalization begins to come into sight.
These two categories, in at least one very fundamental realm of
exact thought, can be reduced to one. There is, namely, a world
of ideal objects which especially interest the logician. It is the
world of a totality of possible logical classes, or again, it is the ideal


world, equivalent in formal structure to the foregoing, but composed
of a totality of 'possible statements, or thirdly, it is the world, equiva-
lent once more, in formal structure, to the foregoing, but consisting
of a totality of possible acts of will, of possible decisions. When we
proceed to consider the relational structure of such a world, taken
merely in the abstract as such a structure, a relation comes into
sight which at once appears to be peculiarly general in its nature.
It is the so-called illative relation, the relation which obtains between
two classes when one is subsumed under the other, or between two
statements, or two decisions, when one implies or entails the other.
This relation is transitive, but may be either symmetrical or not
symmetrical; so that, according as it is symmetrical or not, it may
be used either to establish levels or to generate series. In the order
system of the logician's world, the relational structure is thus, in
any case, a highly general and fundamental one.

But this is not all. In this the logician's world of classes, or of
statements, or of decisions, there is also another relation observable.
This is the relation of exclusion or mutual opposition. This is a
purely symmetrical or reciprocal relation. It has two forms —
obverse or contradictory opposition, that is, negation proper, and
contrary opposition. But both these forms are purely symmetrical.
And by proper devices each of them can be stated in terms of the
other, or reduced to the other. And further, as Kempe incidentally
shows, and as Mrs. Ladd Franklin has also substantially shown in
her important theory of the syllogism, it is possible to state every
proposition, or complex of propositions involving the illative relation,
in terms of this purely symmetrical relation of opposition. Hence,
so far as mere relational form is concerned, the illative relation itself
may be wholly reduced to the symmetrical relation of opposition.
This is our first result as to the relational structure of the realm of
pure logic, that is, the realm of classes, of statements, or of deci-

It follows that, in describing the logician's world of possible classes
or of possible decisions, all unsymmetrical, and so all serial, relations
can be stated solely in terms of symmetrical relations, and can be entirely
reduced to such relations. Moreover, as Kempe has also very prettily
shown, the relation of opposition, in its two forms, just mentioned,
need not be interpreted as obtaining merely between pairs of objects.
It may and does obtain between triads, tetrads, n-ads of logical en-
tities; and so all that is true of the relations of logical classes may
consequently be stated merely by ascribing certain perfectly sym-
metrical and homogeneous predicates to pairs, triads, tetrads, n-ads
of logical objects. The essential contrast between symmetrical
and unsymmetrical relations thus, in this ideal realm of the logi-
cian, simply vanishes. The categories of the logician's world of


classes, of statements, or of decisions, are marvelously simple. All
the relations present may be viewed as variations of the mere con-
ception of opposition as distinct from non-opposition.

All this holds, of course, so far, merely for the logician's world of
classes or of decisions. There, at least, all serial order can actually
be derived from wholly symmetrical relations. But Kempe now
very beautifully shows (and here lies his great and original contri-
bution to our topic) — he shows, I say, that the ordinal relations
of geometry, as well as of the number-system, can all be regarded
as indistinguishable from mere variations of those relations which,
in pure logic, one finds to be the symmetrical relations obtaining within
fairs or triads of classes or of statements. The formal identity of the
geometrical relation called "between" with a purely logical relation
M^hich one can define as existing or as not existing amongst the mem-
bers of a given triad of logical classes, or of logical statements, is
shown by Kempe in a fashion that I cannot here attempt to expound.
But Kempe's result thus enables one, as I believe, to simplify the
theory of relations far beyond the point which Russell in his brilliant
book has reached. For Kempe's triadic relation in question can be
stated, in what he calls its obverse form, in perfectly symmetrical
terms. And he proves very exactly that the resulting logical rela-
tion is precisely identical, in all its properties, with the fundamental
ordinal relation of geometry.

Thus the order-systems of geometry and analysis appear simply
as special cases of the more general order-system of pure logic. The
whole, both of analysis and of geometry, can be regarded as a de-
scription of certain selected groups of entities, which are chosen,
according to special rules, from a single ideal world. This general
and inclusive ideal world consists simply of all the objects which can
stand to one another in those symmetrical relations wherein the pure lo-
gician finds various statements, or various decisions inevitably standing.
" Let me," says in substance Kempe, " choose from the logician's
ideal world of classes or decisions, what entities I will; and I will
show you a collection of objects that are in their relational structure,
precisely identical with the points of a geometer's space of n dimen-
sions." In other words, all of the geometer's figures and relations can
be precisely pictured by the relational structure of a selected system
of classes or of statements, whose relations are wholly and explicitly
logical relations, such as opposition, and whose relations may all
be regarded, accordingly, as reducible to a single type of purely
symmetrical relation.

Thus, for all exact science, and not merely for the logician's special
realm, the contrast between symmetrical and unsymmetrical rela-
tions proves to be, after all, superficial and derived. The purely
logical categories, such as opposition, and such as hold within the


calculus of statements, are, apparently, the basal categories of all
the exact science that has yet been developed. Series and levels are
relational structures that, sharply as they are contrasted, can be
derived from a single root.

I have restated Kempe's generalization in my own way. I think
it the most promising step towards new light as to the categories
that we have made for some generations.

In the field of modern logic, I say, then, work is doing which is
rapidly tending towards the unification of the tasks of our entire
division. For this problem of the categories, in all its abstractness,
is still a common problem for all of us. Do you ask, however, what
such researches can do to furnish more special aid to the workers
in metaphysics, in the philosophy of religion, in ethics, or in aesthetics,
beyond merely helping towards the formulation of a table of cate-
gories — then I reply that we are already not without evidence that
such general researches, abstract though they may seem, are bear-
ing fruits which have much more than a merely special interest.
Apart from its most general problems, that analysis of mathemat-
ical concepts to which I have referred has in any case revealed
numerous unexpected connections between departments of thought
which had seemed to be very widely sundered. One instance of such
a connection I myself have elsewhere discussed at length, in its gen-
eral metaphysical bearings. I refer to the logical identity which
Dedekind first pointed out between the mathematical concept of
the ordinal number of series and the philosophical concept of the
formal structure of an ideally completed self. I have maintained
that this formal identity throws light upon problems which have as
genuine an interest for the student of the philosophy of religion as
for the logician of arithmetic. In the same connection it may be
remarked that, as Couturat and Russell, amongst other writers,
have very clearly and beautifully shown, the argument of the Kant-
ian mathematical antinomies needs to be explicitly and totally
revised in the light of Cantor's modern theory of infinite collections.
To pass at once to another, and a very different instance : The mod-
ern mathematical conceptions of what is called group theory have
already received very wide and significant applications, and promise
to bring into unity regions of research which, until recently, appeared
to have little or nothing to do with one another. Quite lately, how-
ever, there are signs that group theory will soon prove to be of im-
portance for the definition of some of the fundamental concepts of
that most refractory branch of philosophical inquiry, sesthetics. Dr.
Emch, in an important paper in the Monist, called attention, some
time since, to the symmetry groups to which certain sestheticallj''
pleasing forms belong, and endeavored to point out the empirical
relations between these groups and the aesthetic effects in question.


The grounds for such a connection between the groups in question
and the observed aesthetic effects, seemed, in the paper of Dr. Emch
to be left largelj'^ in the dark. But certain papers recently published
in the country by Miss Ethel Puffer, bearing upon the psychology
of the beautiful (although the author has approached the subject
without being in the least consciously influenced, as I understand,
by the conceptions of the mathematical group theory), still actually
lead, if I correctly grasp the writer's meaning, to the doctrine that
the aesthetic object, viewed as a psychological whole, must possess
a structure closely, if not precisely, equivalent to the ideal structure
of what the mathematician calls a group. I myself have no authority
regarding aesthetic concepts, and speak subject to correction. But

Online LibraryMo.) Congress of Arts and Science (1904 : Saint LouisCongress of Arts and Science : universal exposition, St. Louis, 1904 → online text (page 18 of 68)