International Congress of Arts and Science Part 31

The second result still more directly concerns a philosophical logic. It is this: Since the few types of relations which this sort of a.n.a.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?_ 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 number 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 countries--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 generation, and, in our time, to Charles Peirce in this country, very important stages in the growth of these problems are due. Russell, in his work on the _Principles of Mathematics_ has very lately undertaken 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, ent.i.tled, respectively, _The Theory of Mathematical Form_, and _The a.n.a.logy between the Logical Theory of Cla.s.ses 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.

III

The two most obviously and universally important kinds of relations 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 cla.s.ses of relations, namely, the cla.s.s represented, although by no means exhausted, by the various relations actually called, in different branches of science by the one name equality, this cla.s.s 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 cla.s.ses of relations, namely, those of the type of before and after, or greater and less--this cla.s.s 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 relation 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, const.i.tutes a one-dimensional 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 cla.s.ses 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 cla.s.ses 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 cla.s.ses 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 leveling relations are symmetrical 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 cla.s.ses of objects on the same level, by means of the use of symmetrical 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 arranging 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 cla.s.sic 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 quant.i.ty 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 themselves, 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 symmetrical. 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 substantial agreement. Russell's brilliant book is a development of the logic of mathematics very largely in terms of the two types of relations which, in my own way, I have just characterized; although Russell gives due regard, of course, to certain other types of relations.

But hereupon the question arises, "Are these two types of relations what Russell holds them to be, namely, ultimate and irreducible logical facts, una.n.a.lyzable 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 cla.s.ses_, 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, equivalent 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 cla.s.ses 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 cla.s.ses, 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 cla.s.ses, of statements, or of decisions.

It follows that, in describing the logician's world of possible cla.s.ses 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 ent.i.ties; and so all that is true of the relations of logical cla.s.ses may consequently be stated merely by ascribing certain perfectly symmetrical and h.o.m.ogeneous 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 logician, simply vanishes. The categories of the logician's world of cla.s.ses, of statements, or of decisions, are marvelously simple. All the relations present may be viewed as variations of the mere conception of opposition as distinct from non-opposition.

All this holds, of course, so far, merely for the logician's world of cla.s.ses 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 contribution 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 pairs or triads of cla.s.ses or of statements_. The formal ident.i.ty of the geometrical relation called "between" with a purely logical relation which one can define as existing or as not existing amongst the members of a given triad of logical cla.s.ses, or of logical statements, is shown by Kempe in a fas.h.i.+on 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 relation is precisely identical, in all its properties, with the fundamental ordinal relation of geometry.

Thus the order-systems of geometry and a.n.a.lysis appear simply as special cases of the more general order-system of pure logic. The whole, both of a.n.a.lysis and of geometry, can be regarded as a description of certain selected groups of ent.i.ties, 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 logician finds various statements, or various decisions inevitably standing_. "Let me," says in substance Kempe, "choose from the logician's ideal world of cla.s.ses or decisions, what ent.i.ties 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 s.p.a.ce of _n_ dimensions." In other words, all of the geometer's figures and relations can be precisely pictured by the relational structure of a selected system of cla.s.ses 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 relations 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 categories--then I reply that we are already not without evidence that such general researches, abstract though they may seem, are bearing fruits which have much more than a merely special interest. Apart from its most general problems, that a.n.a.lysis of mathematical 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 general metaphysical bearings. I refer to the logical ident.i.ty 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 ident.i.ty 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 Kantian mathematical antinomies needs to be explicitly and totally revised in the light of Cantor's modern theory of infinite collections.

To pa.s.s at once to another, and a very different instance: The modern 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, however, there are signs that group theory will soon prove to be of importance for the definition of some of the fundamental concepts of that most refractory branch of philosophical inquiry, aesthetics. Dr. Emch, in an important paper in the _Monist_, called attention, some time since, to the symmetry groups to which certain aesthetically 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 largely 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 the unexpected, and in case of Miss Puffer's research, quite unintended, appearance of group theory in recent aesthetic a.n.a.lysis is to me an impressive instance of the use of relatively new mathematical conceptions in philosophical regions which _seem_, at first sight, very remote from mathematics.

That both the group concept and the concept of the self just suggested are sure to have also a wide application in the ethics of the future, I am myself well convinced. In fact, no branch of philosophy is without close relations to all such studies of fundamental categories.

These are but hints and examples. They suffice, I hope, to show that the workers in this division have deep common interests, and will do well, in future, to study the arts of cooperation, and to regard one another's progress with a watchful and cordial sympathy. In a word: Our common problem is the theory of the categories. That problem can be solved only by the cooperation of the mathematicians and of the philosophers.

[Ill.u.s.tration: THE UNIVERSITY OF PARIS IN THE THIRTEENTH CENTURY

_Hand-painted Photogravure from a Painting by Otto Knille. Reproduced from a Photograph of the Painting by permission of the Berlin Photograph Co._

This famous painting is now in the University of Berlin. Thomas Aquinas, one of the greatest of the scholastic philosophers, surnamed the "Angelic Doctor," is delivering a learned discourse before King Louis IX. To the right of the King stands Joinville, the French chronicler.

The Dominican monk with his hand to his face is Guillaume de Saint Amour, and Vincent de Beauvais, and another Dominican are seated with their backs to the platform desk from which Thomas Aquinas is making his animated address. The picture is thoroughly characteristic of a University disputation at the close of the Middle Ages.]

DEPARTMENT I--PHILOSOPHY

DEPARTMENT I--PHILOSOPHY

(_Hall 6, September 20, 11.15 a. m._)

CHAIRMAN: PROFESSOR BORDEN P. BOWNE, Boston University.

SPEAKERS: PROFESSOR GEORGE H. HOWISON, University of California.

PROFESSOR GEORGE T. LADD, Yale University.

In opening the Department of Philosophy, the Chairman, Professor Borden P. Bowne, LL.D., of Boston University, made an interesting address on the Philosophical Outlook. Professor Bowne said in part:--

I congratulate the members of the Philosophical Section on the improved outlook in philosophy. In the generation just pa.s.sed, philosophy was somewhat at a discount. The great and rapid development of physical science and invention, together with the profound changes in biological thought, produced for a time a kind of chaos. New facts were showered upon us in great abundance, and we had no adequate philosophical preparation for dealing with them. Such a condition is always disturbing. The old mental equilibrium is overthrown and readjustment is a slow process. Besides, the shallow sense philosophy of that time readily lent itself to mechanical and materialistic interpretations, and for a while it seemed as if all the higher faiths of humanity were permanently discredited. All this has pa.s.sed away. Philosophical criticism began its work and the nave dogmatism of materialistic naturalism was soon disposed of.

It quickly appeared that our trouble was not due to the new facts, but to the superficial philosophy by which they had been interpreted. Now that we have a better philosophy, we have come to live in perfect peace with the facts once thought disturbing, and even to welcome them as valuable additions to knowledge....

The brief naturalistic episode was not without instruction for us. It showed conclusively the great practical importance of philosophy. Had we had thirty years ago the current philosophical insight, the great development of the physical and biological sciences would have made no disturbance whatever. But being interpreted by a crude scheme of thought, it produced somewhat of a storm.

Philosophy may not contribute much of positive value, but it certainly has an important negative function in the way of suppressing pretentious dogmatism and fict.i.tious knowledge, which often lead men astray. It is these things which produce conflicts of science and religion or which find in evolution the solvent of all mysteries and the source of all knowledge.

Concerning the part.i.tion of territory between science and philosophy, there are two distinct questions respecting the facts of experience. First, we need to know the facts in their temporal and spatial order, and the way they hang together in a system of law. To get this knowledge is the function of science, and in this work science has inalienable rights and a most important practical function.

This work cannot be done by speculation nor interfered with by authority of any kind. It is not surprising, then, that scientists in their sense of contact with reality should be indignant with, or feel contempt for, any who seek to limit or proscribe their research. But supposing this work all done, there remains another question respecting the causality and interpretation of the facts. This question belongs to philosophy. Science describes and registers the facts with their temporal and spatial laws; philosophy studies their causality and significance. And while the scientist justly ignores the philosopher who interferes with his inquiries, so the philosopher may justly reproach the scientist who fails to see that the scientific question does not touch the philosophic one....

In the field of metaphysics proper I note a strong tendency toward personal idealism, or as it might be called, Personalism; that is, the doctrine that substantial reality can be conceived only under the personal form and that all else is phenomenal. This is quite distinct from the traditional idealisms of mere conceptionism. It holds the essential fact to be a community of persons with a Supreme Person at their head while the phenomenal world is only expression and means of communication. And to this view we are led by the failure of philosophizing on the impersonal plane, which is sure to lose itself in contradiction and impossibility. Under the form of mechanical naturalism, with its tendencies to materialism and atheism, impersonalism has once more been judged and found wanting. We are not likely to have a recurrence of this view unless there be a return to philosophical barbarism. But impersonalism at the opposite pole in the form of abstract categories of being, causality, unity, ident.i.ty, continuity, sufficient reason, etc., is equally untenable. Criticism shows that these categories when abstractly and impersonally taken cancel themselves. On the impersonal plane we can never reach unity from plurality, or plurality from unity; and we can never find change in ident.i.ty, or ident.i.ty in change. Continuity in time becomes mere succession without the notion of potentiality, and this in turn is empty. Existence itself is dispersed into nothingness through the infinite divisibility of s.p.a.ce and time, while the law of the sufficient reason loses itself in barren tautology and the infinite regress.

The necessary logical equivalence of cause and effect in any impersonal scheme makes all real explanation and progress impossible, and shuts us up to an unintelligible oscillation between potentiality and actuality, to which there is no corresponding thought....

Philosophy is still militant and has much work before it, but the omens are auspicious, the problems are better understood, and we are coming to a synthesis of the results of past generations of thinking which will be a very distinct progress. Philosophy has already done good service, and never better than in recent times, by destroying pretended knowledge and making room for the higher faiths of humanity. It has also done good service in helping these faiths to better rational form, and thus securing them against the defilements of superst.i.tion and the cavilings of hostile critics. With all its aberrations and shortcomings, philosophy deserves well of humanity.

PHILOSOPHY: ITS FUNDAMENTAL CONCEPTIONS AND ITS METHODS

BY GEORGE HOLMES HOWISON

[George Holmes Howison, Mills Professor of Intellectual and Moral Philosophy and Civil Polity, University of California.

b. Montgomery County, Maryland, 1834. A.B. Marietta College, 1852; M.A. 1855; LL.D. _ibid._ 1883. Post-graduate, Lane Theological Seminary, University of Berlin, and Oxford.