In dealing with the relation of metaphysics to the formal sciences generally, the great difficulty which confronts us is that of determining exactly the boundaries which separate one from another. Among such pure sciences we have by universal admission to include at least two, pure formal logic and pure mathematics, as distinguished from the special applications of logic and mathematics to an empirical material. Whether we ought also to recognize ethics and æsthetics, in the sense of the general determination of the nature of the good and the beautiful, as non-empirical sciences, seems to be a more difficult question. It seems clear, for instance, that ethical discussions, such as bulk so largely in our contemporary literature, as to what is the right course of conduct under various conditions, are concerned throughout with an empirical material, namely, the existing peculiarities of human nature as we find it, and must therefore be regarded as capable only of an empirical and therefore problematic solution. Accordingly I was at one time myself tempted to regard ethics as a purely empirical science, and even published a lengthy treatise in defense of that point of view and in opposition to the whole Kantian conception of the possibility of a constructive Metaphysik der Sitten. It seems, however, possible to hold that in the question "What do we mean by good?" as distinguished from the question "What in particular is it right to do?" there is no more of a reference to the empirical facts of human psychology than in the question "What do we mean by truth?" and that there must therefore be a non-empirical answer to the problem. The same would of course hold equally true of the question "What is beauty?" If there are, however, such a pure science of ethics and again of æsthetics, it must at least be allowed that for the most part these sciences are still undiscovered, and that the ethical and æsthetical results hitherto established are in the main of an empirical nature, and this must be my excuse for confining the remarks of the next two paragraphs to the two great pure sciences of which the general principles may be taken to be now in large measure known.
That metaphysics and logic should sometimes have been absolutely identified, as for instance by Hegel, will not surprise us when we consider how hard it becomes on the view here defended to draw any hard and fast boundary line between them. For metaphysics, according to this conception of its scope, deals with the formulation of the self-evident principles implied, in there being such a thing as truth and the deductions which these principles warrant us in drawing. Thus it might be fairly said to be the supreme science of order, and it would not be hard to show that all the special questions commonly included in its range, as to the nature of space, time, causation, continuity, and so forth, are all branches of the general question, how many types of order among concepts are there, and what is their nature. A completed metaphysics would thus appear as the realization of Plato's splendid conception of dialectic as the ultimate reduction of the contents of knowledge to order by their continuous deduction from a supreme principle (or, we may add, principles). Now such a view seems to make it almost impossible to draw any ultimate distinction between logic and metaphysics. For logic is strictly the science of the mutual implication of propositions, as we see as soon as we carefully exclude from it all psychological accretions. In the question what are the conditions under which one proposition or group of propositions imply another, we exhaust the whole scope of logic pure and proper, as distinguished from its various empirical applications. This is the important point which is so commonly forgotten when logic is defined as being in some way a study of "psychical processes," or when the reference to the presence of "minds" in which propositions exist, is intended into logical science. We cannot too strongly insist that for logic the question so constantly raised in a multitude of text-books, what processes actually take place when we pass from the assertion of the premises to the assertion of the conclusion, is an irrelevant one, and that the only logical problem raised by inference is whether the assertion of the premises as true warrants the further assertion of the conclusion, supposing it to be made. (At the risk of a little digression I cannot help pointing out that the confusion between a logical and a psychological problem is committed whenever we attempt, as is so often done, to make the self-evidence of a principle identical with our psychological inability to believe the contradictory. From the strictly logical point of view, all that is to be said about the two sides of such an ultimate contradiction is that the one is true and the other is false. Whether it is or is not possible, as a matter of psychical fact for me to affirm with equal conviction, both sides of a contradiction, knowing that I am doing so, is a question of empirical psychology which is possibly insoluble, and at any rate seems not to have received from the psychologists the attention it deserves. But the logician, so far as I can see, has no interest as a logician in its solution. For him it would still be the case even though all mankind should actually and consciously affirm both sides of a given contradiction, that one of the affirmations would be true, and the other untrue.) Logic thus seems to become either the whole or an integral part of the science of order, and there remain only two possible ways of distinguishing it from metaphysics. It might be suggested that logical order, the order of implication between truths, is only one species of a wider genus, order in general by the side, for example, of spatial, temporal, and numerical order, and thus that logic is one subordinate branch of the wider science of metaphysics. Such a view, of course, implies that there are a plurality of ultimately independent forms of order irreducible to a single type. Whether this is the case, I must confess myself at present incompetent to decide, though the signal success with which the principles of number have already been deduced from the fundamental definitions and axioms of symbolic logic, and number itself defined, as by Mr. Russell, in terms of the purely logical concept of class-relation, seems to afford some presumption to the contrary. Or it may be held that the difference is purely one of the degree of completeness with which the inquiry into order is pursued. Thus the ordinary symbolic logic of what Schröder has called the "identical calculus," or "calculus of domains," consists of a series of deductions from the fundamental concepts of class and number, identical equality, totality or the "logical 1," zero or the null-class, and the three principles of identity, subsumption, and negation. The moment you cease to accept these data in their totality as the given material for your science, and to inquire into their mutual coherence, by asking for instance whether any one of them could be denied, and yet a body of consistent results deduced from the rest, your inquiry, it might be said, becomes metaphysics. So, again, the discussion of the well-known contradictions which arise when we try to apply these principles in their entirety and without modification to classes of classes instead of classes of individuals, or of the problem raised by Peano and Russell, whether the assertions "Socrates is a man" and "the Greeks are men" affirm the same or a different relation between their subject and predicate (which seems indeed to be the same question differently stated), would generally be allowed to be metaphysical. And the same thing seems to be equally true of the introduction of timerelations into the interpretation of our symbols for predication employed by Boole in his treatment of hypotheticals, and subsequently adopted by his successors as the foundation of the "calculus of equivalent statements."
However we may decide such questions, we seem at least driven by their existence to the recognition of two important conclusions. (1) The relation between logical and metaphysical problems is so close that you cannot in consistency deny the possibility of a science of metaphysics unless you are prepared with the absolute skeptic to go the length of denying the possibility of logic also, and reducing the first principles of inference to the level of formulae which have happened hitherto to prove useful but are, for all we know, just as likely to fail us in future application as not. (Any appeal to the doctrine of chances would be out of place here, as that doctrine is itself based on the very principles at stake.) (2) The existence of fundamental problems of this kind which remained almost or wholly unsuspected until revealed in our own time by the creation of a science of symbolic logic should console us if ever we are tempted to suspect that metaphysics is at any rate a science in which all the main constructive work has already been accomplished by the great thinkers of the past. To me it appears, on the contrary, that the recent enormous developments in the purely formal sciences of logic and mathematics, with the host of fundamental problems they open up, give promise of an approaching era of fresh speculative construction which bids fair to be no less rich in results than any of the great "golden" periods in the past history of our science. Indeed, but that I would avoid the slightest suspicion of a desire to advertise personal friends, I fancy I might even venture to name some of those to whom we may reasonably look for the work to be done.
Of the relation of metaphysics to pure mathematics it would be impertinent for any but a trained mathematician to say very much. I must therefore be content to point out that the same difficulty in drawing boundary lines meets us here as in the case of logic. Not so long ago this difficulty might have been ignored, as it still is by too many writers on the philosophy of science. Until recently mathematics would have been thought to be adequately defined as the science of numerical and quantitative relations, and adequately distinguished from metaphysics by the non-quantitative and non-numerical character of the latter, though it would probably have been admitted that the problem of the definition of quantity and number themselves is a metaphysical one. But in the present state of our knowledge such an account seems doubly unsatisfactory. On the one hand, we have to recognize the existence of branches of mathematics, such as the so-called descriptive geometry, which are neither quantitative nor numerical, and, on the other, quantity as distinct from number appears to play no part in mathematical science, while number itself, thanks to the labors of such men as Cantor and Dedekind, seems, as I have said before, to be known now to be only a special type of order in a series. Thus there appears to be ground for regarding serial order as the fundamental category of mathematics, and we are thrown back once more upon the difficult task of deciding how many ultimately irreducible types of order there may be before we can undertake any precise discrimination between mathematical and metaphysical science. However we may regard the problem, it is at least certain that the recent researches of mathematicians into the meaning of such concepts as continuity and infinity have, besides opening up new metaphysical problems, done much to transfigure the familiar ones, as all readers of Professor Royce must be aware. For instance I imagine all of us here present, even the youngest, were brought up on the Aristotelian doctrine that there is and can be no such thing as an actually existing infinite collection, but which of us would care to defend that time-honored position to-day? Similarly with continuity all of us were probably once on a time instructed that whereas "quantity" is continuous, number is essentially "discrete," and is indeed the typical instance of what we mean by the non-continuous. To-day we know that it is in the numberseries that we have our one certain and familiar instance of a perfect continuum. Still a third illustration of the transforming light which is thrown upon old standing metaphysical puzzles by the increasing formal development of mathematics may be found in the difficulties attendant upon the conception of the "infinitely little," once regarded as the logical foundation of the so-called Differential Calculus. With the demonstration, which maybe found in Mr. Russell's important work, that "infinitesimal," unlike "infinite," is a purely relative term, and that there are no infinitesimal real numbers, the supposed logical significance of the concept seems simply to disappear. Instances of this kind could easily be multiplied almost indefinitely, but those already cited should be sufficient to show how important are the metaphysical results which may be anticipated from contemporary mathematical research, and how grave a mistake it would be to regard existing metaphysical construction, e. g., that of the Hegelian system, as adequate in principle to the present state of our organized knowledge. In fact, all the materials for a new Kategorienlehre, which may be to the knowledge of our day what Hegel's Logic was to that of eighty years ago, appear to lie ready to hand when it may please Providence to send us the metaphysician who knows how to avail himself of them. The proof, given since this address was delivered, by E. Zermelo, that every assemblage can be well ordered, is an even more startling illustration of the remarks in the text.
It remains to say something of the relation of metaphysical speculation to the various sciences which make use of empirical premises. On this topic I maybe allowed to be all the more brief, as I have quite recently expressed my views at fair length in an extended treatise (Elements of Metaphysics, Bks. 3 and 4), and have nothing of consequence to add to what has been there said. The empirical sciences, as previously defined, appear to fall into two main classes, distinguished by a difference which corresponds to that often taken in the past as the criterion by which science is to be separated from philosophy. We may study the facts of temporal sequence either with a view to the actual control of future sequences or with a view to detecting under the sequence some coherent purpose. It is in the former way that we deal with facts in mechanics, for instance, or in chemistry, in the latter that we treat them when we study history for the purpose of gaining insight into national aims and character. We may, if we please, with Professor Royce, distinguish the two attitudes toward fact as the attitude respectively of description and of appreciation or evaluation. Now as regards the descriptive sciences, the position to which, as I believe, metaphysicians are more and more tending is that here metaphysics has, strictly speaking, no right at all to interfere. Just because of the absence from metaphysics itself of all empirical premises, it can be no business of the metaphysician to determine what the course of events will be or to prescribe to the sciences what methods and hypotheses they shall employ in the work of such determination. Within these sciences any and every hypothesis is sufficiently justified, whatever its nature, so long as it enables us more efficiently than any other to perform the actual task of calculation and prediction. And it was owing to neglect of this caution that the Naturphilosophie of the early nineteenth century speedily fell into a disrepute fully merited by its ignorant presumption. As regards the physical sciences, the metaphysician has indeed by this time probably learned his lesson. We are not likely to-day to repeat the mistake of supposing that it is for us as metaphysicians to dictate what shall be the physicist's or chemist's definition of matter or mass or elementary substance or energy, or how he shall formulate the laws of motion or of chemical composition. Here, at any rate, we can see that the metaphysician's work is done when his analysis has made it clear that we are dealing with no self-evident truths such as the laws of number, but with inductive, and therefore problematic and provisional results of empirical assumptions as to the course of facts, assumptions made not because of their inherent necessity, but because of their practical utility for the special task of calculation. It is only when such empirical assumptions are treated as self-evident axioms, in fact when mechanical science gives itself out as a mechanistic philosophy, that the metaphysician obtains a right to speak, and then only for the purpose of showing by analysis that the presence of the empirical postulates which is characteristic of the natural sciences of itself excludes their erection into a philosophy of first principles.
What is important in this connection is that we should recognize quite clearly that psychology stands in this respect on precisely the same logical footing as physics or chemistry. It is tempting to suppose that in psychology, at any rate, we are dealing throughout with absolute certainties, realities which "consciousness" apprehends just as they are without any of that artificial selection and construction which, as we are beginning to see, is imposed upon the study of physical nature by the limitations of our purpose of submitting the course of events to calculation and manipulation. And it is a natural consequence of this point of view to infer that since psychology deals directly with realities, it must be taken as the foundation of the metaphysical constructions which aim at understanding the general character of the real as such. The consequence, indeed, disappears at once if the views maintained in this address as to the intimate relation of metaphysics and logic, and the radical expulsion from logic of all discussion of mental processes as such, be admitted. But it is still important to note that the premises from which the conclusion in question was drawn are themselves false. We must never allow ourselves to forget that, as the ever-increasing domination of psychology by the highly artificial methods of observation and experiment introduced by Fechner and Wundt is daily making more apparent, psychology itself, like physics, deals not directly with the concrete realities of individual experience, but with an abstract selected from that experience, or rather a set of artificial symbols only partially corresponding with the realities symbolized, and devised for the special object of submitting the realm of mental sequences to mathematical calculation. We might, in fact, have based this inference upon the single reflection that every psychological "law" is obtained, like physical laws, by the statistical method of elimination of individual peculiarities, and the taking of an average from an extended series of measurements. For this very reason, no psychological law can possibly describe the unique realities of individual experience. We have in psychology, as in the physical sciences, the duty of suspecting exact correspondence between the single case and the general "law" to be of itself proof of error somewhere in the course of our computation. These views, which I suppose I learned in the first instance from Mr. F. H. Bradley's paper called A Defence of Phenomenalism in Psychology, may now, I think, be taken as finally established beyond doubt by the exhaustive analysis of Professor Münsterberg's Grundzüge der Psychologie. They possess the double advantage of freeing the psychologist once for all from any interference by the metaphysician in the prosecution of his proper study, and delivering metaphysics from the danger of having assumptions whose sole justification lies in their utility for the purpose of statistical computation thrust upon it as self-evident principles. For their full discussion I may perhaps be allowed to refer to the first three chapters of the concluding book of my Elements of Metaphysics.
When we turn to the sciences which aim at the appreciation or evaluation of empirical fact, the case seems rather different. It may fairly be regarded as incumbent on the metaphysician to consider how far the general conception he has formed of the character of reality can be substantiated and filled in by our empirical knowledge of the actual course of temporal sequence. And thus the way seems to lie open to the construction of what may fairly be called a Philosophy of Nature and History. For instance, a metaphysician who has rightly or wrongly convinced himself that the universe can only be coherently conceived as a society of souls or wills may reasonably go on to ask what views seem best in accord with our knowledge of human character and animal intelligence as to the varying degrees of organized intelligence manifested by the members of such a hierarchy of souls, and the nature and amount of mutual intercourse between them. And again, he may fairly ask what general way of conceiving what we loosely call the inanimate world would at once be true to fundamental metaphysical principles and free from disagreement with the actual state of our physical hypotheses. Only he will need to bear in mind that since conclusions on these points involve appeal to the present results of the inductive sciences, and thus to purely empirical postulates, any views he may adopt must of necessity share in the problematic and provisional character of the empirical sciences themselves, and can have no claim to be regarded as definitely demonstrated in respect of their details. I will here only indicate very briefly two lines of inquiry to which these reflections appear applicable. The growth of evolutionary science, with the new light it has thrown upon the processes by which useful variations may be established without the need for presupposing conscious preëxisting design, naturally gives rise to the question whether such unconscious factors are of themselves sufficient to account for the actual course of development so far as it can be traced, or whether the actual history of the world offers instances of results which, so far as we can see, can only have issued from deliberate design. And thus we seem justified in regarding the problem of the presence of ends in Nature as an intelligible and legitimate one for the philosophy of the future. I would only suggest that such an inquiry must be prosecuted throughout by the same empirical methods, and with the same consciousness of the provisional character of any conclusions we may reach which would be recognized as in place if we were called on to decide whether some peculiar characteristic of an animal group or some singular social practice in a recently discovered tribe does or does not indicate definite purpose on the part of breeders or legislators.
The same remarks, in my opinion, apply to the familiar problems of Natural Theology relative to the existence and activity of such non-human intelligences as are commonly understood by the names "God" or "gods." Hume and Kant, as it seems to me, have definitely shown between them that the old-fashioned attempts to demonstrate from self-evident principles the existence of a supreme personal intelligence as a condition of the very being of truth all involve unavoidable logical paralogisms. I should myself, indeed, be prepared to go further, and to say that the conception of a single personality as the ground of truth and reality can be demonstrated to involve contradiction, but this I know is a question upon which some philosophers for whom I entertain the profoundest respect hold a contrary opinion. The more modest question, however, whether the actual course of human history affords probable ground for believing in the activity of one or more non-human personalities as agents in the development of our species I cannot but think a perfectly proper subject for empirical investigation, if only it be borne in mind that any conclusion upon such a point is inevitably affected by the provisional character of our information as to empirical facts themselves, and can claim in consequence nothing more than a certain grade of probability. With this proviso, I cannot but regard the question as to the existence of a God or of gods as one upon which we may reasonably hope for greater certainty as our knowledge of the empirical facts of the world's history increases. And I should be inclined only to object to any attempt to foreclose examination by forcing a conclusion either in the theistic or in the atheistic sense on alleged grounds of a priori metaphysics. In a word, I would maintain not only with Kant that the "physico-theological" argument is specially deserving of our regard, but with Boole that it is with it that Natural Theology must stand or fall.
NOTE ON EXTENSION AND INTENSION OF TERMS
Among the numerous difficulties which beset the teaching of the elements of formal logic to beginners, one of the earliest is that of deciding whether all names shall be considered to have meaning both in extension and intension. As we all know, the problem arises in connection with two classes of names, (1) proper names of individuals, (2) abstract terms. I should like to indicate what seems to me the true solution of the difficulty, though I do not remember to have seen it advocated anywhere in just the form I should prefer.