Chapter XXXII: Section D: RELIGIOUS WORK. (Hall 1, September 24, 3 p. m.) (5)

Yet not even thus do we sufficiently state how closely related the two tasks are. For this very contrast, as we have also suggested, is, even within its own limits, no final or perfectly sharp contrast. There is a deep analogy between the two tasks. For the mathematician, as we have just seen, is not evenly interested in developing the consequences of any and every system of freely assumed postulates. He is no mere solver of arbitrary ideal puzzles in general. His systems of postulates are so chosen as to be not trivial, but significant. They are, therefore, in fact, but abstractly defined aspects of the very system of eternal truth whose expression is the universe. In this sense the mathematician is as genuinely interested as is the philosopher in the significant use of his scientific freedom. On the other hand, the philosopher, in reflecting upon the significance and the unity of fundamental ideas, can only do so with success in case he makes due inquiry into the logical consequences of given ideas. And this he can accomplish only if, upon occasion, he employs the exact methods of the mathematician, and develops his systems of ideal truth with the precision of which only mathematical research is capable. As a fact, then, the mathematician and the philosopher deal with ideal truth in ways which are not only contrasted, but profoundly interconnected. The mathematician, in so far as he consciously distinguishes significant from trivial problems, and ideal systems, is a philosopher. The philosopher, in so far as he seeks exactness of logical method, in his reflection, must meanwhile aim to be, within his own limits, a mathematician. He, indeed, will not in future, like Spinoza, seek to reduce philosophy to the mere development, in mathematical form, of the consequences of certain arbitrary hypotheses. He will distinguish between a reflection upon the unity of the system of truth and an abstract development of this or that selected aspect of the system. But he will see more and more that, in so far as he undertakes to be exact, he must aim to become, in his own way, and with due regard to his own purposes, mathematical; and thus the union of mathematical and philosophical inquiries, in the future, will tend to become closer and closer.

II

So far, then, I have dwelt upon extremely general considerations relating to the unity and the contrast of mathematical and philosophical inquiries. I can well conceive, however, that the individual worker in any one of the numerous branches of investigation which are represented by the body of students whom I am privileged to address, may at this point mentally interpose the objection that all these considerations are, indeed, far too general to be of practical interest to any of us. Of course, all we who study these so-called normative sciences are, indeed, interested in ideas, for their own sakes--in ideas so distinct from, although of course also somehow related to, phenomena. Of course, some of us are rather devoted to the development of the consequences of exactly stated ideal hypotheses, and others to reflecting as we can upon what certain ideas and ideals are good for, and upon what the unity is of all ideas and ideals. Of course, if we are wise enough to do so, we have much to learn from one another. But, you will say, the assertion of all these things is a commonplace. The expression of the desire for further mutual coöperation is a pious wish. You will insist upon asking further: "Is there just now any concrete instance in a modern type of research which furnishes results such as are of interest to all of us? Are we actually doing any productive work in common? Are the philosophers contributing anything to human knowledge which has a genuine bearing upon the interests of mathematical science? Are the mathematicians contributing anything to philosophy?"

These questions are perfectly fair. Moreover, as it happens, they can be distinctly answered in the affirmative. The present age is one of a rapid advance in the actual unification of the fields of investigation which are included within the scope of this present division. What little time remains to me must be devoted to indicating, as well as I can, in what sense this is true. I shall have still to deal in very broad generalities. I shall try to make these generalities definite enough to be not wholly unfruitful.

We have already emphasized one question which may be said to interest, in a very direct way, both the mathematician and the philosopher. The ideal postulates, whose consequences mathematical science undertakes to develop, must be, we have said, significant postulates, involving ideas whose exact definition and exposition repay the labor of scientific scrutiny. Number, space, continuity, functional correspondence or dependence, group-structure--these are examples of such significant ideas; the postulates or ideal assumptions upon which the theory of such ideas depends are significant postulates, and are not the mere conventions of an arbitrary game. But now what constitutes the significance of an idea, or of an abstract mathematical theory? What gives an idea a worthy place in the whole scheme of human ideas? Is it the possibility of finding a physical application for a mathematical theory which for us decides what is the value of the theory? No, the theory of functions, the theory of numbers, group theory, have a significance which no mathematician would consent to measure in terms of the present applicability or non-applicability of these theories in physical science? In vain, then, does one attempt to use the test of applied mathematics as the main criticism of the value of a theory of pure mathematics. The value of an idea, for the sciences which constitute our division, is dependent upon the place which this idea occupies in the whole organized scheme or system of human ideas. The idea of number, for instance, familiar as its applications are, does not derive its main value from the fact that eggs and dollars and star-clusters can be counted, but rather from the fact that the idea of numbers has those relations to other fundamental ideas which recent logical theory has made prominent--relations, for instance, to the concept of order, to the theory of classes or collections of objects viewed in general, and to the metaphysical concept of the self. Relations of this sort, which the discussions of the number concept by Dedekind, Cantor, Peano, and Russell have recently brought to light--such relations, I say, constitute what truly justified Gauss in calling the theory of numbers a "divine science." As against such deeper relations, the countless applications of the number concept in ordinary life, and in science, are, from the truly philosophical point of view, of comparatively small moment. What we want, in the work of our division of the sciences, is to bring to light the unity of truth, either, as in mathematics, by developing systems of truth which are significant by virtue of their actual relations to this unity, or, as in philosophy, by explicitly seeking the central idea about which all the many ideas cluster.

Now, an ancient and fundamental problem for the philosophers is that which has been called the problem of the categories. This problem of the categories is simply the more formal aspect of the whole philosophical problem just defined. The philosopher aims to comprehend the unity of the system of human ideas and ideals. Well, then, what are the primal ideas? Upon what group of concepts do the other concepts of human science logically depend? About what central interests is the system of human ideals clustered? In ancient thought Aristotle already approached this problem in one way. Kant, in the eighteenth century, dealt with it in another. We students of philosophy are accustomed to regret what we call the excessive formalism of Kant, to lament that Kant was so much the slave of his own relatively superficial and accidental table of categories, and that he made the treatment of every sort of philosophical problem turn upon his own schematism. Yet we cannot doubt that Kant was right in maintaining that philosophy needs, for the successful development of every one of its departments, a well-devised and substantially complete system of categories. Our objection to Kant's over-confidence in the virtues of his own schematism is due to the fact that we do not now accept his table of categories as an adequate view of the fundamental concepts. The efforts of philosophers since Kant have been repeatedly devoted to the task of replacing his scheme of categories by a more adequate one. I am far from regarding these purely philosophical efforts made since Kant as fruitless, but they have remained, so far, very incomplete, and they have been held back from their due fullness of success by the lack of a sufficiently careful survey and analysis of the processes of thought as these have come to be embodied in the living sciences. Such concepts as number, quantity, space, time, cause, continuity, have been dealt with by the pure philosophers far too summarily and superficially. A more thoroughgoing analysis has been needed. But now, in comparatively recent times, there has developed a region of inquiry which one may call by the general name of modern logic. To the constitution of this new region of inquiry men have principally contributed who began as mathematicians, but who, in the course of their work, have been led to become more and more philosophers. Of late, however, various philosophers, who were originally in no sense mathematicians, becoming aware of the importance of the new type of research, are in their turn attempting both to assimilate and to supplement the undertakings which were begun from the mathematical side. As a result, the logical problem of the 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 coöperation 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 metaphysician, 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 researches 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 principles. 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 reëxamination of the bases of arithmetic and analysis which is associated with the names of Dedekind, Weierstrass, and George Cantor. At the present time, the labors of a number of other inquirers (amongst whom we may mention the school of Peano and Pieri in Italy, and men such as Poincaré and Couturat in France, Hilbert in Germany, Bertrand Russell and Whitehead in England, and an energetic group of our American mathematicians--men such as Professor Moore, Professor Halsted, Dr. Huntington, 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 postulates 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 well-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 logical 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 existence of certain ideal objects in certain simple ordinal relations. Dedekind and Peano have worked out such ordinal theories of the number 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 nonequivalence of collections of objects. But here also a certain simple type 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 few and simple are the conceptions and postulates_ 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 mathematician 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 treatment. 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 essential 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 analysis 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, 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.

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 classes of relations, namely, the class represented, 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 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, constitutes 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 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 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 classes 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 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 quantity 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, 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, 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 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 decisions.

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 entities; and so all that is true of the relations of logical classes may consequently be stated merely by ascribing certain perfectly symmetrical 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 logician, 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 conception 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 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 classes or of statements_. The formal identity 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 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 relation 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 description 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 logician 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_ 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 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 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 æsthetics, 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 analysis 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 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 Kantian 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 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, æsthetics. Dr. Emch, in an important paper in the _Monist_, called attention, some time since, to the symmetry groups to which certain æsthetically pleasing forms belong, and endeavored to point out the empirical relations between these groups and the æsthetic effects in question. The grounds for such a connection between the groups in question and the observed æsthetic 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 æsthetic 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 æsthetic 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 æsthetic analysis 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 coöperation, 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 coöperation of the mathematicians and of the philosophers.

_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
passed, 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 passed
away. Philosophical criticism began its work and the naïve
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 fictitious 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 partition 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, identity, 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 identity, or identity 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 space 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 superstition 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.
Headmaster High School, Salem, Mass., 1862-64; Assistant
Professor of Mathematics, Washington University, St. Louis,
1864-66; Tileston Professor of Political Economy, _ibid._
1866-69; Professor of Logic and the Philosophy of Science,
Massachusetts Institute of Technology, 1871-79; Lecturer on
Ethics, Harvard University, 1879-80; Lecturer on Logic and
Speculative Philosophy, University of Michigan, 1883-84.
Member and vice-president St. Louis Philosophical Society;
member California Historical Society; American Historical
Association; American Association for the Advancement of
Science; National Geographic Society, etc. Author of
_Treatise on Analytic Geometry_, 1869; _The Limits of
Evolution_, 1901, 2d edition, 1904; joint author and editor
of _The Conception of God_, 1897, etc. Editor Philosophical
Publications of University of California; American Editorial
Representative _Hibbert Journal_, London.]

The duty has been assigned me, honored colleagues, of addressing you on the Fundamental Conceptions and the Methods of our common pursuit--philosophy. In endeavoring to deal with the subject in a way not unworthy of its depth and its extent, I have found it impossible to bring the essential material within less compass than would occupy, in reading, at least four times the period granted by our programme. I have therefore complied with the rule of the Congress which directs that, if a more extended writing be left with the authorities for publication, the reading must be restricted to such a portion of it as will not exceed the allotted time. I will accordingly read to you, first, a brief summary of my entire discussion, by way of introduction, and then an excerpt from the larger document, which may serve for a _specimen_, as our scholastic predecessors used to say, of the whole inquiry I have carried out. The impression will, of course, be fragmentary, and I must ask beforehand for your most benevolent allowances, to prevent a judgment too unfavorable.

The discussion naturally falls into two main parts: the first dealing with the Fundamental Conceptions; and the second, with the Methods.

In the former, after presenting the conception of philosophy itself, as _the consideration of things in the light of the whole_, I take up the involved Fundamental Concepts in the following order:--

I. Whole and Part;

II. Subject and Object (Knowing and Being, Mind and Matter; Dualism,
Materialism, Idealism);

III. Reality and Appearance (Noumenon and Phenomenon);

IV. Cause and Effect (Ground and Consequence; Causal System);

V. One and Many (Number System; Monism and Pluralism);

VI. Time and Space (their relation to Number; their Origin and
Real Meaning);

VII. Unconditioned and Conditioned (Soul, World, God; their
Reinterpretation in terms of Pluralism);

VIII. The True, the Beautiful, the Good (their relation to the
question between Monism and Pluralism).

These are successively dealt with as they rise one out of the other in the process of interpreting them and applying them in the actual creation of philosophy, as this goes on in the historic schools. The theoretic progress of philosophy is in this way explained by them, in its movement from natural dualism, or realism, through the successive forms of monism, materialistic, agnostic, and idealistic, until it reaches the issue, now coming so strongly forward within the school of idealism, between the adherents of monism and those of pluralism.

The importance of the Fundamental Concepts is shown to increase as we pass along the list, till on reaching Cause and Effect, and entering upon its full interpretation into the complete System of Causes, we arrive at the very significant conception of the RECIPROCITY OF FIRST CAUSES, and through it come to the PRIMACY OF FINAL CAUSE, and the derivative position of the other forms of cause, Material, Formal, Efficient. The philosophic strength of idealism, but especially of idealistic pluralism, comes into clear light as the result of this stage of the inquiry. But it appears yet more decidedly when One and Many, Time and Space, and their interrelations, are subjected to analysis. So the discussion next passes to the higher conceptions, Soul, World, God, by the pathway of the correlation Unconditioned and Conditioned, and its kindred contrasts Absolute and Relative, Necessary and Contingent, Infinite and Finite, corroborating and reinforcing the import of idealism, and, still more decidedly, that of its plural form. Finally, the strong and favorable bearing of this last on the dissolution of agnosticism and the habilitation of the ideals, the True, the Beautiful, and the Good, in a heightened meaning, is brought out.

This carries the inquiry to the second part of it, that of the Philosophical Methods. Here I recount these in a series of six: the Dogmatic, the Skeptical, the Critical, the Pragmatic, the Genetic, the Dialectic. These, I show, in spite of the tendency of the earlier members in the series to over-emphasis, all have their place and function in the development of a complete philosophy, and in fact form an ascending series in methodic effectiveness, all that precede the last being taken up into the comprehensive Critical Rationalism of the last. Methodology thus passes upward, over the ascending and widening roadways of (1) Intuition and Deduction; (2) Experience and Induction; (3) Intuition and Experience adjusted by Critical Limits; (4) Skepticism reinforced and made _quasi_-affirmative by Desire and Will; (5) Empiricism enlarged by substitution of cosmic and psychic history for subjective consciousness; (6) Enlightened return to a Rationalism critically established by the inclusion of the preceding elements, and by the sifting and the grading of the Fundamental Concepts through their behavior when tested by the effort to make them universal. In this way, the methods fall into a System, the organic principle of which is this principle of Dialectic, which proves itself alone able to establish _necessary_ truths; that is, _truths indeed_,--judgments that are seen to exclude their opposites, because, in the attempt to substitute the opposite, the place of it is still filled by the judgment which it aims to dislodge.

And now, with your favoring leave, I will read the excerpt from my larger text.

The task to which, in an especial sense, the cultivators of philosophy are summoned by the plans of the present Congress of Arts and Science, is certainly such as to stir an ambition to achieve it. At the same time, it tempers eagerness by its vast difficulty, and the apprehension lest this may prove insuperable. The task, the officers of the Congress tell us, is no less than to promote the unification of all human knowledge. It requires, then, the reduction of the enormous detail in our present miscellany of sciences and arts, which to a general glance, or even to a more intimate view, presents a confusion of differences that seems overwhelming, to a system nevertheless clearly harmonious,--founded, that is to say, upon universal principles which control all differences by explaining them, and which therefore, in the last resort, themselves flow lucidly from a single supreme principle. Simply to state this meaning of the task set us, is enough to awaken the doubt of its practicability.

This doubt, we are bound to confess, has more and more impressed itself upon the general mind, the farther this has advanced in the experience of scientific discovery. The very increase in the multiplicity and complexity of facts and their causal groupings increases the feeling that at the root of things there is "a final inexplicability"--total reality seems, more and more, too vast, too profound, for us to grasp or to fathom. And yet, strangely enough, this increasing sense of mysterious vastness has not in the least prevented the modern mind from more and more asserting, with a steadily increasing insistence, the essential and unchangeable unity of that whole of things which to our ordinary experience, and even to all our sciences, appears such an endless and impenetrable complex of differences,--yes, of contradictions. In fact, this assertion of the unity of all things, under the favorite name of the Unity of Nature, is the pet dogma of modern science; or, rather, to speak with right accuracy, it is the stock-in-trade of a _philosophy_ of science, current among many of the leaders of modern science; for every such assertion, covering, as it tacitly and unavoidably does, a view about the absolute whole, is an assertion belonging to the province of philosophy, before whose tribunal it must come for the assessment of its value. The presuppositions of all the special sciences, and, above all, this presupposition of the Unity and Uniformity of Nature, common to all of them, must thus come back for justification and requisite definition to philosophy--that uppermost and all-inclusive form of cognition which addresses itself to the whole as whole. In their common assertion of the Unity of Nature, the exponents of modern science come unawares out of their own province into quite another and a higher; and in doing so they show how unawares they come, by presenting in most instances the curious spectacle of proclaiming at once their increasing belief in the unity of things, and their increasing disbelief in its penetrability by our intelligence:--

_In's Innere der Natur,
Dringt kein erschaffner Geist,_

is their chosen poet's expression of their philosophic mood. Curious we have the right to call this state of the scientific mind, because it is to critical reflection so certainly self-contradictory. How can there be a real unity belonging to what is inscrutable?--what evidence of unity can there be, except in intelligible and explanatory continuity?

But, at all events, this very mood of agnostic self-contradiction, into which the development of the sciences casts such a multitude of minds, brings them,--brings all of us,--as already indicated, into that court of philosophy where alone such issues lawfully belong, and where alone they can be adjudicated. If the unification of the sciences can be made out to be real by making out its sole sufficient condition, namely, that there is a genuine, and not a merely nominal, unity in the whole of reality itself,--a unity that explains because it is itself, not simply intelligible, but the only completely intelligible of things,--this desirable result must be the work of philosophy. However difficult the task may be, it is rightly put upon us who belong to the Department listed first among the twenty-four in the programme of this representative Congress.

I cannot but express my own satisfaction, as a member of this Department, nor fail to extend my congratulations to you who are my colleagues in it, that the Congress, in its programme, takes openly the affirmative on this question of the possible unification of knowledge. The Congress has thus declared beforehand for the practicability of the task it sets. It has even declared for its not distant accomplishment; indeed, not impossibly, its accomplishment through the transactions of the Congress itself; and it indicates, by no uncertain signs, the leading, the determining part that philosophy must have in the achievement. In fact, the authorities of the Congress themselves suggest a solution of their own for their problem. In their programme we see a renewed Hierarchy of the Sciences, and at the summit of this appears now again, after so long a period of humiliating obscuration, the figure of Philosophy, raised anew to that supremacy, as Queen of the Sciences, which had been hers from the days of Plato to those of Copernicus, but which she began to lose when modern physical and historical research entered upon its course of sudden development, and which, until recently, she has continued more and more to lose as the sciences have advanced in their career of discoveries,--ever more unexpected, more astonishing, yet more convincing and more helpful to the welfare of mankind. May this sign of her recovered empire not fail! If we rejoice at the token, the Congress has made it our part to see that the title is vindicated. It is ours to show this normative function of philosophy, this power to reign as the unifying discipline in the entire realm of our possible knowledge; to show it by showing that the very nature of philosophy--its elemental concepts and its directing ideals, its methods taken in their systematic succession--is such as must result in a view of universal reality that will supply the principle at once giving rise to all the sciences and connecting them all into one harmonious whole.