# The World and the Individual, First Series/Supplementary Essay

GIFFORD LECTURES
FIRST SERIES
SUPPLEMENTARY ESSAY
THE ONE, THE MANY, AND THE INFINITE
Section I. Mr. Bradley’s Problem
The closing lecture of the foregoing series has begun the statement of the doctrine of the Individual. The reality of Many within One, and the necessity of the union of the One and the Many, have been maintained, side by side with some account of the nature that, as I also maintain, ought to be attributed to the Individual, whether you consider the Absolute Individual, or the Individuals of our finite world, — the men whose wills are expressed in our life. Now I should be glad to allow the general theory to stand, for the present, simply as stated; and to postpone altogether, until the second series of these lectures, the further defence of the doctrine, — were it not that the most thorough, and in very many respects by far the most important contribution to pure Metaphysics which has of late years appeared in England, has made known a Theory of Being with which, in some of its most significant theses, I heartily agree, while, nevertheless, this very Theory of Being, as it has been stated by its author, undertakes to render wholly impossible, for our human minds, as now we are constituted, any explicit and detailed reconciliation of the One and the Many, or any positive theory of how Individuals find their real place in the Absolute. Defining and defending a conception of the Absolute as “one system,” whose contents are “experience,” Mr. Bradley, to whose well-known book, ​Appearance and Reality, I am here referring, has, nevertheless, maintained that we are wholly unable to “construe” to ourselves the way in which the realm of Appearance finds its unity in the Absolute. He rejects, in consequence, every more detailed effort to interpret our own life in its relations to the Absolute, such as, in the foregoing discussions I have begun, and, in the second series of these lectures, hope to continue. The reason for this rejection, in Mr. Bradley’s case, is of the most fundamental kind. It is founded upon the most central theses of his Theory of Being. The proper place to discuss it is in close connection, therefore, with the general theory in question. I have stated my own case; but I feel obliged to try to do justice to Mr. Bradley’s interpretation. For if he is right, there is little hope for our further undertaking.
The task is no easy one. I myself owe a great debt to Mr. Bradley’s book, a debt manifest in my criticism of Realism in Lecture III, and in many other parts of my discussion. The book is itself a very elaborate argumentative structure. One ought not to make light of it by chance quotations. One cannot easily summarize its well-wrought reasonings in a few sentences. To discuss it carefully would have been wholly impossible in my general course of lectures. On the other hand, to sunder the discussion of it wholly from the present discourse, would have made such a critical enterprise as here follows, seem, for me, a thankless polemical task. For lengthy polemic regarding so serious a piece of work as Mr. Bradley’s is hardly to be tolerated apart from an attempt at construction. And so I have resolved to attempt the task in the form of an essay, supplementary to my own statement of a Theory of Being in these lectures, and preparatory to the discussion of Man and Nature in the next series.
Even here, however, I must attempt to construct as well as to object. And the effort will lead at once to problems which I had no time to discuss in the general lectures. Mr. Bradley, for instance, has shown that every effort to bring to unity the manifoldness of our world involves us in what he himself often calls an “infinite process.” In other words, if, in telling ​about the Absolute you try to show how the One and the Many are brought into unity, and how the Many develope out of the One, you find that, in attempting to define the Many at all, you have defined an actually infinite number. But an actually infinite multitude, according to Mr. Bradley, is a self-contradictory conception. The problem thus stated is an ancient phase of the general problem as to unity and plurality.
From the very outset of the philosophical study of the diversities of the universe, it has been noticed, that in many cases, where common sense is content to enumerate two, or three, or some other limited number of aspects or constituents of a supposed object, closer analysis shows that the variety contained in this object, if really existent at all, must be boundless, so that the dilemma: “Either no true variety of the supposed type is real, or else this variety involves an infinity of aspects,” has often been used as a critical test, to discredit some commonly received view as to the unity and variety of the universe or of some supposed portion thereof. Mr. Bradley has not been wanting in his appeal to this type of critical argument. But to give this argument its due weight, when it comes as a device for discrediting all efforts to define the nature of Individuals, requires one to attack the whole question of the actual Infinite, a question that recent discussions of the Philosophy of Mathematics have set in a decidedly new light, but that these discussions have also made more technical than ever. If I am to be just to this matter, I must therefore needs wander far afield. Nobody, I fear, except a decidedly technical reader, will care to follow. I have, therefore, hesitated long before venturing seriously to entertain the plan of saying, either here or elsewhere, anything about what seems to me the true, and, as I believe, the highly positive implication, of Mr. Bradley’s apparently most destructive arguments concerning Individual Being and concerning the meaning of the world of Appearance.
Yet the problem of the reality of infinite variety and multiplicity, — a problem thus made so prominent by Mr. Bradley’s whole method of procedure, — is one that no metaphysician ​can permanently evade. The doctrine that the conception of the actually infinite multitude is a self-contradictory conception is a familiar thesis ever since Aristotle. If this thesis is correct, as Mr. Bradley himself assumes, then Mr. Bradley’s results, as regards the limitations of our human knowledge of the Absolute, appear to be inevitable, and the effort of these present lectures to define the essential relations of the world and the individual must fail. On the other hand, however, if, as I believe, the very doctrine of the true nature of Individual Being, which these lectures defend, enables us, for the first time perhaps in the history of the discussion of the Infinite, to give a precise statement of the sense in which an Infinite Multitude can, without contradiction, be viewed as determinately real, — then a discussion of Mr. Bradley’s position, and of the whole problem of the One, the Many, and the Infinite, will prove an important supplement to our Theory of Being, and an essential basis for the vindication of our human knowledge of the general constitution of Reality. And so I must feel that, if the present task is extended and technical, the goal is nothing less than the defence of what I take to be a true theory of the whole meaning of life.
And so I am now minded to undertake the task of vindicating the concept of the actual Infinite against the charge of self-contradiction. I am minded, also, to attempt the closely related task of defending the concept of the Self against a like charge. In the same connection I shall undertake to show something of the true relations of the One and the Many in the real world. And in the course of this enterprise I shall found the positive discussion upon a criticism of Mr. Bradley’s position.
But now, at this point, let any weary reader whom my lectures may have already disheartened, but who nevertheless may kindly have proceeded so far, turn finally back. When you enter the realm of Mr. Bradley’s Absolute, it is much as it is at the close of Victor Hugo’s Toilers of the Sea, after the ship that carries away the lady has sunk below the horizon, and after the tide has just covered the rock where ​the desolate lover had been watching. “There was nothing,” says the poet, in his last words, “there was nothing now visible but the sea.” As for me, I love the sea, and am minded to find in it life, and individuality, and explicit law. And I go upon that quest. Whoever is not weary, and is not yet disheartened, and is fond of metaphysical technicality, is welcome to join the quest. But in the sea there are also, as Victor Hugo explained to us, very strange monsters. And Mr. Bradley, too, in his book, has had much to say of the “monsters,” philosophic and psychological, that the realm of Appearance contains, even in the immediate neighborhood of the Absolute. We shall meet some such reputed “monsters” in the course of this discussion. Let him who fears such trouble also turn back.
In this essay, I shall first try to state Mr. Bradley’s theses as to the problem of the One and the Many. Then I shall try to show how he himself seems to suggest a way by which, if we follow that way far enough, something may be done to solve what he leaves apparently hopeless. And, finally, I shall proceed upon the way thus opened until we have found whither it leads. We shall find it inevitably leading to the conception of the actually Infinite. We shall examine the known difficulties of that conception, and shall at last solve them by means of our own conception of the nature of determinateness and Individuality.
I. Mr. Bradley’s First Illustrations of His Problem
The general doctrine of the Absolute which Mr. Bradley maintains is the result of a critical analysis of a number of metaphysical conceptions which he opposes. Mr. Bradley’s work is divided into two books. The first book, entitled Appearance, has a mainly negative result. Beginning with the examination of the traditional distinction between primary and secondary qualities, Mr. Bradley shows that this distinction is incapable of furnishing a consistent account of the relation of the phenomenal to the real. The problem of inherence, ​attacked next in order, is declared to be, upon the basis of the ordinary conception of things and qualities, and of their relationship, insoluble. The reason given in this case is typical of Mr. Bradley’s position throughout the book, and, despite the general familiarity of the argument to readers of the Hegelian and Herbartian discussions of the concept of the thing, deserves special mention at this point.
A thing is somehow to be one, and “it has properties, adjectives which qualify it. We say that the thing is this or that, predicating of it the adjectives that express its qualities.” But it cannot be “all its properties if you take them each severally.” “Its reality lies somehow in its unity.” “But if, on the other hand, we inquire what there can be in the thing besides its several qualities, we are baffled once more. We can discover no real unity existing outside these qualities, or, again, existing within them.” To the hypothesis that the unity of the thing may be sufficiently expressed by asserting that “the qualities are, and are in relation,” Mr. Bradley replies that the meaning of is remains still doubtful when we say, “One quality, A is in relation with another quality, B” (p. 20). For still one does not, by here using is, intend to reduce A to simple identity with its relations to B, and so one is led to say, “The word to use, when we are pressed, should not be is, but only has.” But the has seems metaphorical. “And we seem unable to clear ourselves from the old dilemma, If you predicate what is different, you ascribe to the subject what it is not; and if you predicate what is not different you say nothing at all.” Nor does one better the case (p. 21) if one amends the phraseology here in question by asserting that the relation belongs equally to both A and B, instead of limiting the assertion in form to A alone. If the relation, however, be no mere attribute of A or of B, or of both of them, but a “more or less independent” fact, namely, the fact that “There is a relation C in which A and B stand,” then the problem of the unity of the thing becomes the problem as to ​the genuine tie that binds both A and B to their now relatively independent relation C. For C is now supposed to possess an existence which is not that of A or B, but something apart from either. This tie which unites A and B, in the thing, to C, hereupon appears as a new fact of relation, D, viz., the fact that A and B are so related to C that C becomes their relation to each other. “But such a make-shift at once leads to the infinite process. The new relation D can be predicated in no way of C, or of A and B; and hence we must have recourse to a fresh relation, E, which comes between D and whatever we had before. But this must lead to another, F; and so on indefinitely.” The consequence is that we are not aided by letting the “qualities and their relation fall entirely apart.” “There must be a whole embracing what is related, or there would be no differences, and no relation.” This remark applies not merely to things, and to the relations that are to bind into unity their qualities, but to space, and time, and to every case where varieties are in any way related. But although Mr. Bradley asserts thus early the general principle that variety must always find its basis in unity, he wholly denies that, in the present case, we have yet found or defined what the unity in question can be. He denies, namely, that the relational system offered to us so far by the qualities supposed to be inherent in the one thing, or to be related to one another, contains, or can be made to contain, any principle adequate to accomplish the required task, or to “justify the arrangement” that we try to make in conceiving the thing and its qualities as in relational unity.
II. The General Problem of “Relational Thought”
The defect in all these accounts of the nature of the thing is not due, according to Mr. Bradley’s view, to any accidental faults of definition. The defect depends upon a dilemma that first fully comes to light when the problem about relations and qualities is considered for itself, and apart from the special issue about the thing. The task of expounding this dilemma ​Mr. Bradley undertakes in Chapter III of his first book. Here his thesis is (p. 25), that “The arrangement of given facts into relations and qualities may be necessary in practice, but it is theoretically unintelligible.”
The true reason why the concept of the thing involved the foregoing paradoxes is now to become more obvious. It is set forth in three successive theses. First (p. 26): “Qualities are nothing without relations.” For qualities are different from one another. “Their plurality depends on relation, and without that relation they are not distinct” (p. 28). Even were qualities conceived as in themselves wholly separated from one another, and only for us related, still (p. 29) “Any separateness implies separation, and so relation, and is therefore, when made absolute, a self-discrepancy.” “If there is any difference, then that implies a relation.” Mr. Bradley enforces this assertion by a reference, made with characteristic skill, to the paradoxes of the Herbartian metaphysic of the einfache Qualitäten and the zufallige Ansichten (p. 30).
But if it is impossible to conceive qualities without relations, it is equally unintelligible to take qualities together with relations. For the qualities cannot be resolved into the relations. And, if taken with the relations, they “must be, and must also be related” (p. 31). But now afresh arises the problem as to how, in this instance, the variety involved in the also is reducible to the unity which each quality must by itself possess. For a quality, A, is made what it is both by its relations (since, as we have seen, these are essential to its being as a quality), and by something else, namely, by its own inner character. A has thus two aspects, both of which can be predicated of it. Yet “without the use of a relation it is impossible to predicate this variety of A,” just as it was impossible, except by the use of a relation, to predicate the various qualities of one thing. We have therefore to say that, within A, both its own inner character, as a quality, and its relatedness to other facts, are themselves, as varieties, facts; but such facts as constitute the being of A, so that they are united by a new relation, namely, by the very relation which ​makes them constitutive of A. Thus, however, “we are led by a principle of fission which conducts us to no end.” “The quality must exchange its unity for an internal relation.” This diversity “demands a new relation, and so on without limit.”
For similar reasons, a relation without terms being “mere verbiage” (p. 32), it follows that since the terms imply qualities, relation without qualities is nothing. But, on the other hand, if the relation stands related to the qualities, if it is anything to them, “we shall now require a new connecting relation.” But hereupon an endless process of the same kind as before is set up (p. 33). “The links are united by a link, and this bond of union is a link which has also two ends; and these require each a fresh link to connect them with the old.”
The importance for Mr. Bradley of the negative result thus reached lies in the great generality of the conceptions here in question, and in the consequent range covered by these fundamental considerations. “The conclusion,” says Mr. Bradley, “to which I am brought, is that a relational way of thought — any one that moves by the machinery of terms and relations — must give appearance and not truth. It is a make-shift, a device, a mere practical compromise, most necessary, but in the end most indefensible. We have to take reality as many, and to take it as one, and to avoid contradiction. We want to divide it, or to take it, when we please, as indivisible; to go as far as we desire in either of these directions, and to stop when that suits us. . . . But when these inconsistencies are forced together . . . the result is an open and staring inconsistency.”
In the subsequent chapters of Mr. Bradley’s first book, he himself sees, in a great measure, merely an application of the general principle just enunciated to such special problems as are exemplified by Space, by Time, by Causation, by Activity, and by the Self. For all these metaphysical conceptions are defined in terms of a “relational way” of thinking, and involve the problem of the One and the Many. To be sure, the discussion of the Self, in Chapters IX and X, brings the problem ​into decidedly new and important forms, but does not, in Mr. Bradley’s opinion, furnish any acceptable ground for its positive solution. “We have found,” he says, “puzzles in reality, besetting every way in which we have taken it.” The solution of these puzzles, if ever discovered, must be “a view not obnoxious to these mortal attacks, and combining differences in one so as to turn the edge of criticism” (p. 114). The mere appeal, however, to the fact of self-consciousness, does not furnish this needed explicit harmony of unity and variety. The Self does, indeed, unite diversity and unity in a profoundly important way; but the mere fact that this is somehow done does not show us how it is done.
III. The Problem of the One and the Many as Insoluble by Thought, yet solved by the Absolute
Despite this elaborate exposition of the apparent hopelessness of the problem as to the One and the Many, Mr. Bradley’s own theory of the Absolute, proposed in his second book, turns upon asserting that in Reality unity and diversity are positively reconciled, and reconciled, moreover, not by a simple abolition of either of the apparently opposed principles, but in a way that leaves to each its place. For first (p. 140), “Reality is one in this sense that it has a positive nature exclusive of discord. . . . Its diversity can be diverse only so far as not to clash.” Yet, on the other hand, “Appearance must belong to reality, and it must, therefore, be concordant and other than it seems. The bewildering mass of phenomenal diversity must hence somehow be at unity and self-consistent; for it cannot be elsewhere than in reality, and reality excludes discord. Or, again, we may put it so: The real is individual. It is one in the sense that its positive character embraces all differences in an inclusive harmony.” Further, “To be real . . . must be to fall within sentience” (p. 144). Or, again, to be real (p. 146) is “to be something which comes as a feature and aspect within one whole of feeling, something which, except as an integral element of such sentience, ​has no meaning at all.” In consequence, “The Absolute is one system,” and “its contents are . . . sentient experience.” “It will hence be a single and all-inclusive experience, which embraces every partial diversity in concord” (p. 147). It follows that, in the Absolute, none of the diversities which are to us so perplexing, and which, as exemplified by the cases of thing, quality, relation, Self, and the rest of the appearances, are so contradictory in their seeming, are wholly lost. For the Absolute, on the contrary, these diversities are all preserved; only they are “transmuted” into a whole, which is, in ways of which we have only a most imperfect knowledge, internally harmonious. As to the hints that we possess, regarding the nature of the Absolute, they are summarized as follows: “Immediate presentation” (p. 159) gives us the experience of a “whole” which “contains diversity,” but which is, nevertheless, “not parted by relations.” On the other hand, “relational form,” where known to us, points “everywhere to an unity,” — “a substantial totality, beyond relations and above them, a whole endeavoring without success to realize itself in their detail” (p. 160). Such facts and considerations give us “not an experience, but an abstract idea” of a “unity which transcends and yet contains every manifold appearance.” “We can form the general idea of an absolute experience in which phenomenal distinctions are merged, a whole becomes immediate at a higher stage without losing any richness.” But meanwhile we have “a complete inability to understand this concrete unity in detail.”
The ground of this, our inability, is the one already illustrated, namely, the necessary incapacity of a “relational way of thinking” to give us anything definite except Appearance, or to harmonize the One and the Many in concrete fashion, or to free our explicit accounts of the unity from the contradictions and infinite processes heretofore illustrated. A more precise exposition of the general defects of thought in question, Bradley undertakes to furnish in his fifteenth chapter, under the title Thought and Reality. Here the nature of relational thought, its inevitable sundering of the what and the ​that, and its inevitably infinite process in trying to unite them again, are two topics discussed, with the result, as Mr. Bradley states the case, that “Thought desires a consummation in which it is lost,” as “the river” runs “into the sea,” and "the self” loses itself “in love.” For every act of thought, in affirming its predicate of the subject, though all the while knowing that the quality or adjective is not the existent, explicitly faces its own Other, namely, precisely its object, the existent of which it thinks, the subject to which it applies its predicates. This existent, by virtue of its “sensuous infinitude,” or vaguely endless wealth of presented features, always defies our efforts exhaustively to define it in ideal terms (p. 176); and, by virtue of its “immediacy” (p. 177), possesses “the character of a single self-subsistent being,” — a character apparently inconsistent with the “sensuous infinitude.” Our thought, however, endeavoring to characterize this Other, seeks to make ideally explicit how, despite its endless wealth of presented features, it can be still a single individual, — a system of variety in unity. Attempting this task, thought is obliged to use the “relational form” in characterizing the subject; and this at once makes impossible the expression, in ideal terms, of either the self-dependence or the immediacy which the subject claims (p. 178). For, analyzing the subject, in order to define its wealth of content, thought, in the fashion before illustrated in the case of things, qualities, etc., is led to an infinite process, since every relation defined requires new relations to make it comprehensible. Both the internal and the external relations of the subject and of its contents, accordingly prove to be inexhaustible. Never, then, is thought’s ideal system of predicates adequate to the subject. The “sensuous infinitude” or undefined wealth that the subject at first presents, turns, while we think, into the explicitly infinite series of relational predicates. Moreover, even were thought's system ever completed, “that system would not be the subject.” For if it were, “it would wholly lose the relational form.”
The result is that thinking “desires to possess,” as its end ​and goal, a character of “immediate, self-dependent, all-inclusive individuality” (p. 179), while “individuality cannot be gained while we are confined to relations.” Thought, however, although not possessing the features of reality here in question, can recognize them as its own Other, can “desire them” (p. 180) “because its content has them already in an incomplete form. And in desire for the completion of what none has there is no contradiction.” “But, on the other hand (p. 181), such a completion would prove destructive; such an end would emphatically make an end of mere thought. It would bring the ideal content into a form which would be reality itself, and where mere truth and mere thought would certainly perish.” “It is this completion of thought beyond thought which remains forever an Other.” “Thought can understand that, to reach its goal, it must get beyond relations. Yet in its nature it can find no other working means of progress.”
Hence, “our Absolute,” once more, will include the differences of thought and reality, of “what” and “that.” “The self-consciousness of the part, its consciousness of itself even in opposition to the whole, — all will be contained within the one absorbing experience. For this will embrace all self-consciousness harmonized, though, as such, transmuted and suppressed.” But Mr. Bradley still insists that “we cannot possibly construe such an experience to ourselves.”
IV. Mr. Bradley’s Definition of “What would Satisfy the Intellect” as to the One and the Many
Mr. Bradley’s critics have very commonly expressed their disapproval of the extremely delicate position in which, by this theory, our finite thinking is left. We are obliged to define the Real as a system wherein unity and diversity are harmonized. We are to conceive this reality as a “sentient experience.” And in the Absolute Experience, nothing of our finite variety is to be wholly lost, but all is to be “transmuted.” Yet every instance, selected from our own human ​experience, where, through a process of thinking, or a type of mediated consciousness, we men seem to have won any sort of explicit synthesis and harmony of the One and the Many, is sternly rejected by Mr. Bradley, as furnishing no satisfactory guide to the final knowledge of the way in which, in the Absolute, unity and manifoldness are united. The critics have, accordingly, been sometimes disposed to accuse Mr. Bradley of seeking, in his Absolute, for bare identity without diversity; and sometimes tempted, on the other hand, to ask, complainingly, what sort of harmony would satisfy him, and why he supposes that any harmony of the One and the Many is attainable at all, even for the Absolute, when he himself rejects, as mere appearance, every proffered means, whereby harmony is to be defined.
In answer, Mr. Bradley has been led, in his second edition, to discuss, in an appendix, the problem of “Contradiction and the Contrary,” with special reference to its bearing upon the matter here at issue. The relation of the theory of the contrary to the problem of the relation of unity and diversity appears in the fundamental thesis of the discussion in question. This thesis is as follows (p. 562): “A thing cannot, without an internal distinction, be (or do) two different things; and differences cannot belong to the same thing, in the same point, unless in that point there is diversity. The appearance of such union may be fact, but is for thought a contradiction.” In expounding this statement of the principle of contradiction, Mr. Bradley first explains that the thesis “does not demand mere sameness,” which to thought “would be nothing.” A mere tautology “is not a truth in any way, in any sense, or at all.” The Law of Contradiction, then, does not forbid diversity. If it did, “it would forbid thinking altogether.” But the difficulty of the situation arises from the fact that, “Thought cannot do without differences; but, on the other hand, it cannot make them. And, as it cannot make them, so it cannot receive them from the outside, and ready-made.” ​Thought demands a reason and ground for diversity. It can neither pass from A to B without a reason, nor accept as final the fact that, external to thought’s process, A and B are found conjoined. If thought finds a diversity, it demands that this be “brought to unity” (p. 562). And so, if the mere fact of the conjunction of A and B appears, then thought must “either make or accept an arrangement which to it is wanton and without reason, — or, having no reason for anything else, attempt, against reason, to identify them simply” (p. 563). Nor can one meet this difficulty by merely asserting that there are certain ultimate complexes, given in experience, such that in them unity and variety are presented as obviously conjoined, while thought is to explain the “detail of the world” in terms of these fundamental complexes. No such “bare conjunction” is or possibly can be given; for when we find any kind of unity in diversity, that is, when we find diversities conjoined, we always also find a “background” (p. 564) which is a “condition of the conjunction’s existence” so that “the conjunction is not bare, but dependent,” and is presented to the intellect as “a connection, the bond of which is at present unknown.” “The intellect, therefore, while rejecting whatever is alien to itself, if offered as Absolute, can accept the inconsistent if taken as subject to conditions.”
Meanwhile, the “mere conjunction,” if taken as such, is “for thought contradictory” (p. 565). For as soon as thought makes the conjunction its object, thought must “hold in unity” the elements of the conjunction. But finding these elements diverse, thought “can of itself supply no internal bond by which to hold them together, nor has it any internal diversity by which to maintain them apart.” If one replies that the elements are offered to thought “together and in conjunction,” Mr. Bradley retorts that the question is “how thought can think what is offered.” If thought were itself possessed of conjoining principles, of “a ‘together,’ a ‘between,’ and an ‘all at once,’” as its own internal principle, it could use them to explain the conjunction offered. But, as a fact (p. 566), “Thought cannot accept tautology, and yet demands unity in ​diversity. But your offered conjunctions, on the other side, are for it no connections or ways of union. They are themselves merely other external things to be connected.” It is, then, “idle from the outside to say to thought, ’Well, unite, but do not identify.’ How can thought unite except so far as in itself it has a mode of union? To unite without an internal ground of connection and distinction, is to strive to bring together barely in the same point, and that is self-contradiction.” Things, then, “are not contradictory because they are diverse,” but “just in so far as they appear as bare conjunctions.” Therefore it is that a mere together, “in space or time, is for thought unsatisfactory and, in the end, impossible.” But, on the other hand, every such untrue view must be transcended, and the Real is not self-contradictory, despite its diversities, since their real unity is, in the Absolute, present.
If one now asks what then “would satisfy the intellect, supposing it could be got” (p. 568), Mr. Bradley points out that if the ground of unity is “external to the elements into which the conjunction must be analyzed,” then the ground “becomes for the intellect a fresh element, and in itself calls for synthesis in afresh point of unity.” “But hereon,” he continues, “because in the intellect no intrinsic connections were found, ensues the infinite process.” This being the problem “The remedy might be here. If the diversities were complementary aspects of a process of connection and distinction, the process not being external to the elements, or, again, a foreign compulsion of the intellect, but itself the intellect’s own proprius motus, the case would be altered. Each aspect would of itself be a transition to the other aspect, a transition intrinsic and natural at once to itself, and to the intellect. And the Whole would be a self-evident analysis and synthesis of the intellect itself by itself. Synthesis here has ceased to be mere synthesis, and has become self-completion; and analysis, no longer mere analysis, is self-explication. And the question how or why the many are one and the one is many here loses its meaning. There is no why or how beside the self-evident process, and towards its own differ​ences this whole is at once their how and their why, their being, substance, and system, their reason, ground, and principle of diversity and unity” (id). Here, Mr. Bradley insists, the Law of Contradiction “has nothing to condemn.” Such an union or “identity of opposites” would not conflict with the Law of Contradiction, but would rather fulfil the law. If “all that we find were in the end such a self-evident and complete whole,” the end of the intellect, and so of philosophy, would have been won. But Mr. Bradley is (p. 569) “unable to verify a solution of this kind.” Hence, as he says, “Against my intellectual world the Law of Contradiction has claims nowhere satisfied in full.” Therefore “they are met in and by a whole beyond the mere intellect.” It is, however, no “abstract identity” that thus satisfies the demands of the intellect. “On the other hand, I cannot say that to me any principle or principles of diversity in unity are self-evident.” In consequence, while “self-existence and self-identity are to be found,” they are to be looked for neither in “bare identity,” nor in a relapse into a “stage before thinking begins,” but in “a whole beyond thought, a whole to which thought points and in which it is included.” Diversities exist. Therefore (p. 570) “they must somehow be true and real.” “Hence, they must be true and real in such a way that from A or B the intellect can pass to its further qualification without an external denomination of either. But this means that A and B are united, each from its own nature, in a whole which is the nature of both alike.” It is the failure of the intellect to define this whole positively and in detail, which is expressed in all the contradictions of the theory of appearance.
Section II. The One and the Many within the Realm of Thought or of Internal Meanings
So far, then, for a summary of Mr. Bradley’s general view regarding the mystery of unity in variety, and so much for the reasons which have led him, on the one hand, to maintain that real identity is never “simple,” or abstract, but involves ​real differences, and, on the other hand, to insist that the true ground of this union of identity and difference is always, to us, and to “thought,” something not manifest, but only presupposed as “beyond thought” What are we to hold of this doctrine?
I. Thought does Develope its own Varieties of Internal Meaning
Our first comment must repeat what several of Mr. Bradley’s critics have noticed. This is, that within at least one, perhaps limited, but still in any case for us mortals important region, Mr. Bradley himself finds and reports the working of a very “self-evident” principle of “diversity in unity.”
This is the region in which thought is itself the object whose process and movement, whose paradoxes and whose endless series of internal distinctions, we observe, or experience, while we read Mr. Bradley’s book, or any similarly deep examination of the realm of the “intellect.” In his Logic Mr. Bradley long since gave us a brilliant account of the movement of thought, — an account that he here lays at the basis of his discussion. The truth of a considerable portion of this earlier analysis of the thinking process, I should unhesitatingly accept. Now it may be indeed that the processes of thought, as Mr. Bradley examines them, constitute not only a relatively insignificant aspect of Reality, but also a portion to be labelled “Appearance.” Yet the point here in question is not, for the moment, the dignity or the extent of the thinking process in the life of the universe, but solely the exemplary value of the thinking process as an instance of a “self-evident,” even if extremely abstract union of unity and variety, of identity and diversity of aspects, in an objective realm. For thought, too, is a kind of life, and belongs to the realm of Reality, even if only as other appearances belong.
What we in general mean by this comment may first be very briefly developed. The special applications will indeed detain us longer. Mr. Bradley requires us to point out to him a case where diversities shall be “complementary aspects of a process ​of connection and distinction,” the process being no “foreign compulsion of the intellect, but itself the intellect’s own proprius motus . . . a self-evident analysis and synthesis of the intellect itself by itself.” He fails to find, as he looks through the World of Appearance, any case of the sort such as is sufficient to furnish any self-evident “principle or principles of diversity in unity.” Now we here desire to make a beginning in meeting his demand. We ask whether he has wholly taken account of the case that lies nearest of all to him in his research. This case is directly furnished by the intellect. Now the intellect may indeed not be all Reality. Thought may indeed, in the end, have to look “beyond itself” for its own “Other.” Yet Reality owns the intellect, too, along with the other Appearances. By Mr. Bradley’s hypothesis, Appearance is der Gottheit lebendiges Kleid, if by Gottheit we mean, for the moment, his Absolute. We have a right to use any rag torn by our own imperfect knowledge from this garment, to give us, if so may be, a hint of the weaving of the whole. The hint may prove poor. But only the trial can tell. And so, why not see how it is that the intellect, powerless though it be to make explicit the union of unity and diversity in the cases where experience furnishes from without “conjunctions” and their “background,” still manages to unite unity and diversity in its own internal processes? Might not this throw some light upon even our ultimate problem?
For the intellect, after all, has indeed its proprius motus. If it had not, how should we be thinking? And who has more often considered the proprius motus of the intellect, who has more frequently insisted that “thought involves analysis and synthesis,” than Mr. Bradley himself? Now the intellect, as Mr. Bradley observes, is discontent with its presented “external” object, the “conjunction” in space or in time, because of the uncomprehended unity in diversity of this presented object. The intellect seeks to define the ground of this unity, in case of the Thing, or of the world of Qualities and Relations, or of Space, or of Time, or in case of any of the other ​Appearances that seem external to thought. The intellect fails. Why? “Because it cannot do without differences, but, on the other hand, it cannot make them” (p. 562). But can Mr. Bradley wholly mean this assertion that the intellect cannot make differences? In the chapters upon the Thing, and upon the other objects presented, as from without, to the intellect, we are indeed shown, when Mr. Bradley’s argument is once accepted, that thought does not make, and does decline to receive ready made, the differences offered as real by these external objects, so long as they are taken in their abstraction.
But how is it possible for thought to discover the very fact that it cannot make, and that it declines to receive, certain differences, without itself making, of its own motion, certain other differences, whose internal unity it knows just in so far as it makes them? For when thought sets out to solve a problem, it has a purpose. This is its own purpose, and is, also, in so far an unity, not furnished as from without, but, in the course of the thinking process, developed as from within. When, after struggling to solve its problem, and to fulfil its purpose, thought finds itself in the presence of a puzzle that is so far ultimate, what, according to Mr. Bradley, does it see as the essence of this puzzle? It sees that a given hypothesis as to the unity of A and B (where A and B are the supposed “external” diversities, but where the hypothesis itself has been reflectively developed into its consequences through the inner movement of thought), — that this hypothesis, I say, either leads to various consequences which directly contradict one another, or else, by an internal and logical necessity, leads to an “infinite process,” — in other words, to an infinite variety of consequences. In either case, in addition to what thought so far finds puzzling about A and B, thought further sees a diversity, and a diversity that is now not the presented “conjunction” of A and B, but a necessary diversity constructively developed by thought's own movement. Thought learns that its own purpose developes this variety. For the hypothesis about A and B (viz., that they are “in relation” or are “substantive and adjective,” or whatever else the hypothe​sis may be) has developed, within itself, as thought has reflected upon it, a certain internal multiplicity of aspects. That the hypothesis developes these diversities, is a fact, — but a fact how discovered? The only answer is, by Reflection. Thought developes by its own processes the meaning, i.e. to use our own phraseology, the “internal meaning” of this hypothesis. The hypothesis perhaps leads to a self-contradiction concerning the nature of A and B. In that case, the hypothesis, taken apart from A and B themselves, as an object for reflection, is seen to imply that some account of A, or of B, or of A B, is both true and false. Now truth is diverse from falsity, and whoever observes that a given hypothesis implies, through the development of its “internal meaning,” the coëxistent truth and falsity of the same account of a supposed external fact, has observed a fact not now about A and B as such, but about this internal meaning of the hypothesis, taken by itself, — a fact lying within the circle of thought’s own movement. This fact is a diversity developed by thought’s proprius motus.
Or, again, the hypothesis leads to the “infinite process.” An “endless fission” is sometimes said to “break out” in the world of conceived relations and qualities. This “principle of endless fission” “conducts us to no end” (p. 31). “Within the relation” the plurality of the differences is said to “beget the infinite process” (p. 180). Now, when thought sees that all this must be, and is, the necessary outcome of “a relational way of thought” (p. 33), thought again sees a fact, but a fact now present in its own world of ideas, and as the “self-evident” outcome of its reflective effort to express its own purpose. But, as we insist, despite the diversity, thought’s purpose is, in each case of this type, consciously One. It is the purpose to find the ground for the conjunction of A and B. Reflection sees that this one purpose, left to its own development, becomes diverse, and expresses its own identity in a variety of aspects. When thought sees this result of its own efforts, and sees the result as necessary, as universal, as the consequence of a relational way of thinking, then I persist​ently ask, Does not thought here at least see in one instance, not only that identity and diversity are conjoined, but how they are this time connected, and how the one of them, here at least, expresses itself in the other?
May we not, then, for the moment, overlook our failures as to the understanding of the world external to thought, and turn to the consideration of our success in discovering something of the internal movement of thought. For, in our ignorance, our first interest is in observing not how little we know (since our ignorance itself is, indeed, brought home to us at every instant of our finitude), but in making a beginning at considering how much we can find out. We wanted to see how any unity could develope a plurality. We have already seen, if but dimly. Shall we not begin to use our insight?
I conclude, then, so far, that, if the argument of Mr. Bradley is sound, in the very sense in which I myself most accept its soundness, a “principle of diversity in unity,” in the case of the internal meaning of our ideas, is already, in several concrete cases, “self-evident.” It remains for us to become better acquainted with this principle. I must explicitly note that this union of One and Many in thought has to be a fact in the universe if it is self-evident, and has to be self-evident if Mr. Bradley’s argument is sound.
II. The Principle of Thought, which is responsible for the Infinite Processes. Definition of a Recurrent Operation of Thought
The principle in question can be made more manifest by a further reflection. The most important instances in Mr. Bradley’s argument are those wherein the “endless fission” appears; and what has led to this “endless fission” which so far forms our principal instance of the internal development of variety out of unity, appears, when reviewed, as in general, this: A certain “conjunction” was offered to us by sense. This “conjunction” thought undertook, by means of an hypothesis, to explain. The resulting process of “fission” had, however, wholly to do with the internal meaning of this ​hypothesis, and no longer with the original conjunction. It was a fact within the life of thought. The hypothesis ran thus: “The conjunction is to be explained as a relation, holding its own terms in unity.” Hereupon thought undertook so to think this hypothesis as to find its whole meaning. Thought hereupon reflectively observed, “But our relation, as soon as defined, becomes also a term of a new relation.” More in particular, the original question ran, "What is the unity of A and B?” The hypothesis said, “Their unity lies in their relation R; for the terms of a relationship are linked and unified by that relationship.” The reflective criticism runs, "But in creating R, as the ideal link between A and B, regarded now not as they were externally conjoined, but ideally as terms of a relationship, we have only recreated, in the supposed complex R A, or R B, or A R B, the type of situation originally presented. For A and B were to be objects of thought. They therefore needed a link. Therefore, as we said, they were to be viewed as terms linked by their relation. But the relation R, as soon as it is made an object of thought, becomes a term for the same reason which made us regard A and B as terms. For our implied principle was that objects of thought, if various, and yet united, are to be viewed as terms of a relationship. Our thinking process must therefore proceed to note, that if A and B are terms to be linked, R also, by the same right, is a term to be linked to A or to B, or to both, and so on ad infinitum.”
But the gist of this reflection may be better generalized thus: A thinking process of the type here in question recreates, although in a new instance, the very kind of ideal object that, by means of its process, it proposed to alter into some more acceptable form. The change of situation which it intended, leads, and must lead, to a reinstatement of essentially the same sort of situation as that which was to be changed. Or, again, The proposed solution reiterates the problem in a new shape. Therefore, the operation of thought here in question is what one may call, in the most general terms, an iterative, or, again, a recurrent, operation, — an operation whose result ​reinstates, in a new instance, the situation which gave rise to the operation, and to which the operation was applied.
Now, quite apart from the special circumstances of the problem about A and B, the observation that reflection makes upon the general nature of any iterative or recurrent process of thinking, becomes at once of great interest for the comprehension of the question about the One and the Many. We want to find some case of an unity which developes its own differences out of itself. Well, what more simple and obvious instance could we hope for than is furnished by an operation of thought, such that, when applied to a given situation, this operation necessarily, and in a way that we can directly follow, reinstates, in a new case, the very kind of situation to which it was applied? For this operation is a fact in the world. It begins in unity. It developes diversity. Let us, then, wholly drop, for the time, the problem about A and B, in so far as they were taken as facts of sense or of externality. Their “conjunction,” presented “from without,” we may leave in its mystery, until we are ready to return to the matter later. We have found something more obvious, viz., an iterative operation of thought, one which, when applied, is actually observed to develope out of one purpose many results, by recreating its own occasion for application. Now let us proceed with our generalization. Let there be found any such operation of thought, say C. C is to be one ideal operation of our thought just in so far as C expresses a single purpose. But let C be applied on occasion to some material, — no matter what. Let the material be M. Hereupon, as we reflect, let us be supposed to observe that the logical necessary result of applying C to M, the result of expressing the purpose in question in this material, or of ideally weaving the material M into harmony with the purpose C, is the appearance of a new material for thought, viz., M’ . Let us be supposed to observe, also, that M’ , taken as a content to be thought about, gives the same occasion for the application of C that M gave. Let the application of C to M’ be next observed to lead to M’’, in such wise that in M’’ there lies once more the occasion for the applica​tion of C. Let this series be observed to be endless, that is, to be such that, consistently with its nature, it can possess no last term. Then, as I assert, we shall see, in a special instance, how the endless series M, M’, M’’ . . . , just as a series of many ideally constructed facts, is developed by the one purpose, C, when once applied to any suitable material, M; and is developed, moreover, by internal necessity, as the very meaning of the objects M, M’, etc., and also as the meaning of the operation C itself, and not as a bare conjunction given from “without the intellect.” Now in such a case, I insist, we see how the One produces, out of itself, the Many.
Nor let one, objecting, interpose that since an “operation” is a case of activity, and since activity has been riddled by Mr. Bradley’s critical fire, the nature of every operation of thought must always remain mysterious. Let no one insist that since the supposed operation C is one fact, and its material M is another fact, in our world of ideal objects, the relation of C to M is as opaque as any other relation, so that we do not understand how C operates at all, nor yet how it changes M into M’, nor how the same operation C can persist, and be applied to M’ after it had been applied to M. Let no one further point out that since all the foregoing account of C, and of the endless series M, M’, M’’, involves Time as a factor in the “operation,” and since Time has been shown by Mr. Bradley to be a mysterious conjunction of infinite complexity, and so to be mere Appearance, therefore all the foregoing remains mysterious. For to all such objections I shall reply that I so far pretend to find “self-evident” about the iterative processes of thought, only so much as, in his own chosen instances, Mr. Bradley finds self-evident, namely, so much as constitutes the very meaning and ground of his condemnation of the mysterious and baffling Appearances. That the endless process is implied in a certain way of thinking, namely, in a “relational way,” Mr. Bradley reflectively observes. I accept the observation, so far as it goes, in the cases stated. But I ask why this is true. The answer lies in seeing that the endlessness of the ​process is due to the recurrent character of the operation of thought here in question. This relational way of thinking so operates as to reinstate, in a new case, the very type of situation that the explanation desired — the goal of the operation — was, in the former case, to reduce to some simple unity. The first complexity consequently survives the operation, unreduced to unity; while a new complexity, logically (not psychologically) due to the operation itself, appears as something necessarily implied. The reapplication of the same operation, if supposed accomplished, can but reinstate afresh the former type of situation. Hence the endless process. Now this process I consider not in so far as it is a mere temporal series of events, but in so far as it is the development, in a given case, of what a certain thought means. I do not assert the obvious existence of an Activity, but the logical necessity of a certain series of implications. The true meaning of the purpose C, expressed in the content M, logically gives rise to M’, which demands equally to be considered in the light of 'C, and thereupon implies M’’, and so on. Thus our argument does not depend upon a theory about how thought, as an “activity,” is a possible part of the world at all. I do not profess now to explain, say from a psychological point of view, the inmost nature of the operation in question, nor yet to find self-evident, in this place, the metaphysics of the time process. Mysteries still surround us; but we see what we see. And my point is that while we do not see all of what thought is, nor yet how it is able to weave its material into harmony with its purposes, nor yet what Time is, we do see that we think, and that this thought has, as it proceeds, its internal meaning, and that this meaning has, as its necessary and self-evident result, the reinstatement, in a new case, of the type of situation which the operation of the thought was intended to explain, or in some other wise to transform. When M is so altered by the operation C as to imply M’ , M’’, and so on, as the endless series of results of the iterative operation of thought, we see not only that this is so, but why this is so. And unless we see this, we see nothing whatever, whether in ​Appearance or in Reality. And here, then, the relation of Unity and Variety is clear to us.
Our generalization, however, of the process upon which Mr. Bradley insists, enables us to make more fruitful and positive our result. There are recurrent operations of thought. Whenever they act, they imply, upon their face, endless processes. Do such processes inevitably lead us to results wholly vain and negative? Is the union of One and Many which they make explicit an insignificant union? Or, on the other hand, is this union typical of the general constitution of Reality?
The first answer is that, at all events in the special science of mathematics, processes of this type are familiar, and lie at the basis of highly and very positively significant researches. If we merely name a few such instances of endless processes, we shall see that iterative thinking, if once made an ideal, — a method of procedure, — and not merely dreaded as a failure to reach finality, becomes a very important part of the life of the exact sciences, and developes results which have a very significant grade of Reality.
The classic instance of the recurrent or iterative operations of thought is furnished, in elementary mathematics, by the Number Series. A recurrent operation first developes the terms of this series; and thereby makes the counting of external objects, and all that, in our human science, follows therefrom, possible. A secondary recurrent operation, based upon the primary operation, appears in the laws governing the process called the “Addition” of whole numbers. A tertiary and once more recurrent operation appears in the laws governing Multiplication. In consequence of this recurrent nature of ​the thinking processes concerned, the number series itself is endless; the results of addition and multiplication, the sums and products of the various numbers, are not only endless, but capable of endless combinations; and, in general, the properties of numbers are themselves infinitely infinite in number. But in this case the mathematician does not mourn over the “endless fission” to which the number concepts are indeed due, but he regards the numbers as a storehouse of positive and often very beautiful novelties, which his science studies for their intrinsic interest.
If mathematical science thus begins, in the simplest construction, with the outcome of a recurrent process, it is no wonder that the later development of the science, as exemplified by the theories of negative and of fractional numbers, of irrational and of complex numbers, of infinite series and of infinite products, and of all that, in Analysis and in the Theory of Functions, depends upon these more elementary theories, is everywhere full of conceptions and methods that result from observing what happens when an operation of thought is recurrent, or is such as to reinstate, in its expressions, the occasion for new expressions. Without such recurrence, and without such infinite processes, mathematical science would be reduced to a very minute fraction of its present range and importance.
But we are here primarily concerned with the metaphysical aspect of the recurrent processes of thought. Important as are the countless mathematical instances of our type of operations, we must so deal with their general theory as to be able to identify the results of recurrent thinking whenever they occur, whether in mathematics or in other regions of our reflection.
I propose here, then, first to illustrate, and then to discuss theoretically, the nature and ideal outcome of any recurrent operation of thought, and to develope, in this connection, what one may call the positive nature of the concept of Infinite ​Multitude. We shall here see how there are cases, — and cases, too, of the most fundamental importance for the Theory of Being, where a single purpose, definable as One, demands for its realization a multitude of particulars which could not be a limited multitude without involving the direct defeat of the purpose itself. We shall in vain endeavor to escape from the consequences of this discovery by denouncing the purposes of the type in question as self-contradictory, or the Infinite in question as das Schlecht-Unendliche. On the contrary, we shall find these purposes to be the only ones in terms of which we can define any of the fundamental interests of man in the universe, and the only ones whose expression enables us to explain how unity and diversity are harmonized at all, or how Being gets its individuality and finality, or how anything whatever exists. Having made this clear, we shall endeavor to show, positively, that the concept of infinite variety in unity, to which these cases lead us, is consistent in itself, and is able to give our Theory of Being true definition.
Section III. Theory of the Sources and Consequences of any Recurrent Operation of Thought. The Nature of Self-Representative Systems
I shall begin the present section with illustrations. I shall make no preliminary assumption as to how our illustrations are related to the ultimate nature of things. For all that we at first know, we may be dealing, each time, with deceptive Appearance. We merely wish to illustrate, however, how a single purpose may be so defined, for thought, as to demand, for its full expression, an infinite multitude of cases, so that the alternative is, “Either this purpose fails to get expression, or the system of idealized facts in which it is expressed contains an infinite variety.” Whether or no the concept of such infinite variety is itself self-contradictory, remains to be considered later. ​
I. First Illustration of a Self-Representative System
The basis for the first illustration of the development of an Infinite Multitude out of the expression of a Single Purpose, ​which we shall here consider, may be taken, in a measure, from that world “external to thought” whose variety we still find a matter of “mere conjunction” and so opaque. For, despite the use of such a basis, our illustration will interest us not by reason of this aspect, but by reason of the opportunity thereby furnished for carrying out a certain recurrent process of thought, whose internal meaning we want to follow.
We are familiar with maps, and with similar constructions, such as representative diagrams, in which the elements of which a certain artificial or ideal object is composed, are intended to correspond, one to one, to certain elements in an external object. A map is usually intended to resemble the contour of the region mapped in ways which seem convenient, and which have a decidedly manifold sensuous interest to the user of the map; but, in the nature of the case, there is no limit to the outward diversity of form which would be consistent with a perfectly exact and mathematically definable correspondence between map and region mapped. If our power to draw map contours were conceived as perfectly exact, the ideal map, made in accordance with a given system of projection, could be defined as involving absolutely the aforesaid one to one correspondence, point for point, of the surface mapped and the representation. And even if one conceived space or matter as made up of indivisible parts, still an ideally perfect map upon some scale could be conceived, if one supposed it made up of ultimate space units, or of the ultimate material corpuscles, so arranged as to correspond, one by one, to the ultimate parts that a perfect observation would then distinguish in the surface mapped. In general, if A be the object mapped, and A’ be the map, the latter could be conceived as perfect if, while always possessing the desired degree of visible similarity of contours, it actually stood in such correspondence to A that for every elementary detail of A, namely, ɑ, b, c, d (be these details conceived as points or merely as physically smallest parts; as relations amongst the ​parts of a continuum, or as the relations amongst the units of a mere aggregate of particles), some corresponding detail, ɑ’, b’, c’, d’, could be identified in A’, in accordance with the system of projection used.
All this being understood, let us undertake to define a map that shall be in this sense perfect, but that shall be drawn subject to one special condition. It would seem as if, in case our map-drawing powers were perfect, we could draw our map wherever we chose to draw it. Let us, then, choose, for once, to draw it within and upon a part of the surface of the very region that is to be mapped. What would be the result of trying to carry out this one purpose? To fix our ideas, let us suppose, if you please, that a portion of the surface of England is very perfectly levelled and smoothed, and is then devoted to the production of our precise map of England. That in general, then, should be found upon the surface of England, map constructions which more or less roughly represent the whole of England, — all this has nothing puzzling about it. Any ordinary map of England spread out upon English ground would illustrate, in a way, such possession, by a part of the surface of England, of a resemblance to the whole. But now suppose that this our resemblance is to be made absolutely exact, in the sense previously defined. A map of England, contained within England, is to represent, down to the minutest detail, every contour and marking, natural or artificial, that occurs upon the surface of England. At once our imaginary case involves a new problem. This is now no longer the general problem of map making, but the nature of the internal meaning of our new purpose.
Absolute exactness of the representation of one object by another, with respect to contour, this, indeed, involves, as Mr. Bradley would say to us, the problem of identity in diversity; but it involves that problem only in a general way. Our map of England, contained in a portion of the surface of England, involves, however, a peculiar and infinite development of a special type of diversity within our map. For the map, in order to be complete, according to the rule given, will have to contain, as a part of itself, a representation of its own contour ​and contents. In order that this representation should be constructed, the representation itself will have to contain once more, as a part of itself, a representation of its own contour and contents; and this representation, in order to be exact, will have once more to contain an image of itself; and so on without limit. We should now, indeed, have to suppose the space occupied by our perfect map to be infinitely divisible, even if not a continuum.
One who, with absolute exactness of perception, looked down upon the ideal map thus supposed to be constructed, would see lying upon the surface of England, and at a definite place thereon, a representation of England on as large or small a scale as you please. This representation would agree in contour with the real England, but at a place within this map of England, there would appear, upon a smaller scale, a new representation of the contour of England. This representation, which would repeat in the outer portions the details of the former, but upon a smaller space, would be seen to contain yet another England, and this another, and so on without limit.
That such an endless variety of maps within maps could not physically be constructed by men, and that ideally such a map, if viewed as a finished construction, would involve us in all the problems about the infinite divisibility of matter and of space, I freely recognize. What I point out is that if my supposed exact observer, looking down upon the map, saw anywhere in the series of maps within maps, a last map, such that it contained within itself no further representation of the original object, he would know at once that the rule in question had not been carried out, that the resources of the map-maker had failed, and that the required map of England was imperfect. On the other hand, this endless variety of maps within maps, while its existence as a fact in the world might ​be as mysterious as you please, would, in one respect, present to an observer who understood the one purpose of the whole series, no mystery at all. For one who understood the purpose of the making within England a map of England, and the purpose of making this map absolutely accurate, would see precisely why the map must be contained within the map, and why, in the series of maps within maps, there could be no end consistently with the original requirement. Mathematically regarded, the endless series of maps within maps, if made according to such a projection as we have indicated, would cluster about a limiting point whose position could be exactly determined. Logically speaking, their variety would be a mere expression of the single plan, “Let us make within England, and upon the surface thereof, a precise map, with all the details of the contour of its surface.” Then the One and the Many would become, in one respect, clear as to their relations, even when all else was involved in mystery. We should see, namely, why the one purpose, if it could be carried out, would involve the endless series of maps.
But so far we have dealt with our illustration as involving a certain progressive process of map making, occurring in stages. We have seen that this process never could be ended without a confession that the original purpose had failed. But now suppose that we change our manner of speech. Whatever our theory of the meaning of the verb to be, suppose that some one, depending upon any authority you please, — say upon the authority of a revelation, — assured us of this as a truth about existence, viz., “Upon and within the surface of England there exists somehow (no matter how or when made) an absolutely perfect map of the whole of England.” Suppose that, for an instance, we had accepted this assertion as true. Suppose that we then attempted to discover the meaning implied in this one assertion. We should at once observe that in this one assertion, “A part of England perfectly maps all England, on a smaller scale,” there would be implied the assertion, not now of a process of trying to draw maps, but of the contemporaneous presence, in England, of an infinite number of ​maps, of the type just described. The whole infinite series, possessing no last member, would be asserted as a fact of existence. I need not observe that Mr. Bradley would at once reject such an assertion as a self-contradiction. It would be a typical instance of the sort of endlessness of structure that makes him reject Space, Time, and the rest, as mere Appearance. But I am still interested in pointing out that whether we continued faithful to our supposed revelation, or, upon second thought, followed Mr. Bradley in rejecting it as impossible, our faith, or our doubt, would equally involve seeing that the one plan of mapping in question necessarily implies just this infinite variety of internal constitution. We should, moreover, see how and why the one and the infinitely many are here, at least within thought’s realm, conceptually linked. Our map and England, taken as mere physical existences, would indeed belong to that realm of “bare external conjunctions.” Yet the one thing not externally given, but internally self-evident, would be that the one plan or purpose in question, namely, the plan fulfilled by the perfect map of England, drawn within the limits of England, and upon a part of its surface, would, if really expressed, involve, in its necessary structure, the series of maps within maps such that no one of the maps was the last in the series.
This way of viewing the case suggests that, as a mere matter of definition, we are not obliged to deal solely with processes of construction as successive, in order to define endless series. A recurrent operation of thought can be characterized as one that, if once finally expressed, would involve, in the region where it had received expression, an infinite variety of serially arranged facts, corresponding to the purpose in question. This consideration leads us back from our trivial illustration to the realm of general theory.
II. Definition of a Type of Self-Representative Systems
Let there be, then, any recurrent operation of thought, or any meaning in mind whose expression, if attempted, involves such a recurrent operation. That is, let there be ​any internal meaning such that, if you try to express it by means of a succession of acts, the ideal data which begin to express it demand, as a part of their own meaning, new data which, again, are new expressions of the same meaning, equally demanding further like expression. Then, if you endeavor to express this meaning in a series of successive acts, you get a series of results, M, M’, M’’, etc., which can never be finished unless the further expression of the purpose is somewhere abandoned. But such a successive series of attempts quickly gets associated in our minds with a sense of disappointment and fruitlessness, and perhaps this sense more or less blinds us to the true significance of the recurrent thinking processes. Let us try to avoid this mere feeling by dwelling upon the definition of the whole system of facts which, if present at once, would constitute the complete expression and embodiment of this one meaning. The general nature of the system in question is capable of a positive definition. Instead of saying, “The system, if gradually constructed by successive stages, has no last member,” we can say, in terms now wholly positive, (1) The system is such that to every ideal element in it, M, M’, or, in general, M$^{(r)}$, there corresponds one and only one other element of the system, ​which, taken in its order, is the next element of the system. This next element may be viewed, if we choose, as derived from its predecessor by means of the recurrent process. But it may also be viewed as in a relation to its predecessor, which is the same as the relation of a map to an object mapped. We shall accordingly call it, henceforth, the Image or Representation of this former element. (2) These images are all distinct, so that various elements always have various representatives. For the recurrent process is such that, in the system which should finally express it, one and only one element would be derived from any given element, or would be the next element in order after that given element. (3) At least one element, M, of the system, although imaged by another, is itself the image or representative of no other element, so that only a portion of the system is representative. A system thus defined we may call, for our present purposes, an instance of an internally Self-Representative System, or, more exactly, of a system precisely represented by a proper fraction or portion of itself. Of the whole system thus defined we can at once assert that if we take its elements in the order M, M’, M’’, etc., there is indeed no last member in the resulting series. The system is, therefore, defined as endless merely by being defined as thus self-representative. But since the self-representation of any system of facts is capable of definition, as a single internal purpose, in advance of the discovery that such purpose involves an endless series of constituents, we may, with Dedekind, use the generalized conception of a self-representation of the type here in question as a means of positively defining what we mean by an infinite system or multitude of elements. In thus proceeding, we further generalize the idea which the perfect map of England has already illustrated.
The positive definition of the concept of the Infinite thus resulting has no small speculative interest. Ordinarily one defines infinity merely by considering some indefinitely prolonged series of successive facts, by observing that the series in question does not, or at least, so far as one sees, need not, ​end at any given point, and by then saying, “A series taken thus as without end, may be called infinite.” We ourselves, so far in this discussion, have defined our infinite processes on the whole in a negative way. But the new definition of the infinity of our system uses positive rather than negative terms. The conception of a representation or of an imaging of one object by another, is wholly positive. This conception, if applied to the elements of a system A, with the proviso that A’, the image or the representation of A, shall form a constituent portion of A itself, remains still positive. But the system A, if defined as capable of this particular type of self-representation, proves, when examined, to contain, if it exists at all, an infinite number of elements. Whatever the metaphysical fate of the ideal object thus defined, the method of definition has a decided advantage over the older ones. It may be well at once to quote Dedekind’s original statement and illustration of the conception in question, in the passage cited in the note: —
“A System S is called ‘infinite’ when it is similar to a ​constituent (or proper) part of itself; in the contrary case S is called a ‘finite’ system.
“Theorem. — There exist infinite systems.
“Proof. — My own realm of thoughts (meine Gedankenwelt), i.e. the totality S, of all things that can be objects of my thought, is infinite. For if s is an element of S, it follows that the thought s’, viz., the thought, That s can be object of my thought, is itself an element of S. If one views s’ as the image (or representative) of the element s, the representation S’ of the system S, which is hereby defined, has the character that the representation S’ is a constituent portion (echter Theil) of S, since there are elements in S (for example, my own Ego) (?) which are different from every such thought s’, and which are, therefore, not contained in S’. Finally, it is plain that if ɑ and b are different elements of S, their images, ɑ’ and b’ are also different, so that the representation of S is distinct (deutlich) and similar. It follows that S [by definition], is infinite.”
Here, as we observe, the infinity of an ideal system is defined, and in a special case proved, without making any explicit reference to the number of its elements. That this number, negatively viewed, turns out to be no finite number, that is, to be that of a multitude with no last term, is for Dedekind a result to be later proved, — a secondary consequence of the infinity as first defined. The proof that my Gedankenwelt is infinite, is thus not my negative powerlessness to find the last term, but my positive power to image each of my thoughts s, by a new and reflective thought s’. It is the finite, and not the infinite, that here appears as the object negatively definable. For a finite system is one that cannot be adequately represented through a one-to-one correspondence with one of its own constituent parts. In any ​case, the infinite multitude of the elements of S developes, for thought, out of the single positive purpose stated so sharply in Dedekind’s definition.
III. Further Illustrations of Self-Representative Systems of the Type here Defined
This conception of a system that can be exactly represented or imaged, element for element, by one of its own constituent parts, has of course to meet the objection that such an idea appears, upon its face, paradoxical, even if it is not out and out self-contradictory. But before judging the conception, it is well to have in mind some illustrations of its range of application. A comparison of these will show that, if self-cor​respondent systems of the type here in question are mere Appearance, they are, at all events, Appearance worthy of study. A list of a few conceptions that are more or less obviously of the present type may make us pause before we lightly reject, as absurd, the offered definition.
First, then, the series of whole Numbers, as conceived objects, forms such a self-representative system. The same is true of all the secondary number-systems of higher arithmetic (the negative numbers, the rational numbers, the irrational numbers, the totality of the real numbers, the complex numbers) . And all continuous and discrete mathematical systems of any infinite type are similarly self-representative. But the mathematical objects are by no means the most philosophically interesting of the instances of our concept. For, next, we have the Self, the concept so elaborately studied by Mr. Bradley, and condemned by him as Appearance. And, indeed, if the Self is anything final at all, it is certainly in its complete expression (although of course not in our own psychological life from instant to instant) a self-representative system; and its metaphysical fate stands or falls with the possibility of such systems. Dedekind’s really very profound use of meine Gedankenwelt as his typical instance of the infinite, also suggests the interesting relation between the concept of the Self and that of the mere mathematical form called the number-series, a relation to which we shall soon return. Thirdly, the totality of Being, if conceived as in any way defined or characterized, or even as in any way even definable or characterizable, constitutes, in the present sense, a self-representative system. Obvious it is that our own Fourth Conception of Being defines the Absolute as a self-representative system. And, furthermore, despite his horror of the infinite, and despite his rejection of the Self as a final category, Mr. Bradley himself perforce has to describe his own Absolute as a self-representative system of our type, as we soon shall see. And if he attempted to view it otherwise, it would not be the Absolute or anything real at all. In brief, every system of which anybody can rationally assert anything is either a self-repre​sentative system, in the sense here in question, or else, being but a part of the real world, it is a more or less arbitrarily selected, or an empirically given portion or constituent of such a system, — a portion whose reality, apart from that of the whole system, is unintelligible.
Far from lacking totality, then, in the way in which the infinite, or rather the indefinite, multitude of such accounts as Mr. Bosanquet’s is said to lack totality, those genuinely self-representative systems, whose images are portions of their own objects, are the only ones which can be said to possess any totality whatever. It is they alone that are wholly positive in their definition. Finite systems are either capable only of negative definition, or, at all events, have positive characters only by virtue of their relation to their inclusive infinite, or, in our present sense, self-representative systems. Or, again, as we have already begun to see, only the processes of recurrent thought make explicit the true unity of the One and the Many. But these very processes express themselves in systems of the type now in question.
To make these matters clearer, it will be necessary to consider each of the just-mentioned illustrations more in detail. First, then, as to the simple case of the number-system, whose logical genesis we for the moment leave out of consideration, and whose general constitution we assume as known. The whole numbers first form what Cantor calls a wohl-definirte Menge, — or exactly defined multitude. That is, you can precisely distinguish between any conceived or presented object that is not a whole number (as, for example, one-half, or the moral law, or the odor of a rose), and an object that is a whole number, abstract or concrete (e.g. ten, or ten thousand, or the ​number of birds on yonder bough). Taking the whole numbers as the abstract numbers, i.e. as the members of a certain ideal series, arithmetically defined, the mathematician can, therefore, view them all as given by means of their universal definition, and their consequent clear distinction from all other objects of thought. Taking them thus as given, the numbers become entities of the type contemplated by our Third Conception of Being; and as such entities we can admit them here for the moment, not now asking whether or no they have, or can win, a reality of our Fourth type.
Now the numbers form, in infinitely numerous ways, a self-representative system of the type here in question. That is, as has repeatedly been remarked, by all the recent authors who have dealt with this aspect of the matter, the number-system, taken in its conceived totality, can be put in a one-to-one correspondence with one of its own constituent portions in any one of an endless number of ways. For the numbers, if once regarded as a given whole, form an endless ordered series, having a first term, a second term, and so on. But just so the even numbers, 2, 4, 6, etc., form an endless ordered series, having a first, second, third term, and so on. In the same way, too, the prime numbers form a demonstrably endless series, whereof there is a first member, a second member, and so forth. Or, again, the numbers that are perfect squares, those that are perfect cubes, and those, in general, that are of the form ɑ$^n$, where n is any one whole number, while a takes successively the value of every whole number, — all such derived systems of whole numbers, form similarly ordered series, wherein each member of each system has its determined place as first, second, third, or later member of its own system, while the system forms a series without end. Take, then, any ​whole number r, however large. Then, in the ideal class oF objects called whole numbers, there is a determinate even number which occupies the rth place in the series of even numbers, when the latter are arranged according to their sizes, beginning with 2. There is equally a prime number, occupying the rth place in a similarly ordered series of primes; and a square number occupying the rth place in a similarly ordered series of square numbers; and a cube occupying the rth place in a like arrangement of cubes; and an rth member in any particular series of numbers of the form ɑ$^n$, where n is any determinate whole number, and a is taken, in succession, as 1, 2, 3, etc. As all these things hold true for any r, however large, we can say, in general, that every whole number r has its correspondent rth member in any of the supposed series of systematically selected whole numbers, — even numbers, primes, square numbers, cubes, or what you will. But these various selected systems are such that each of them forms only a portion of the entire series of whole numbers. So that the whole series, taken as given, is in infinitely numerous ways capable of being put in a one-to-one relation to one of its own constituent parts.
I doubt not that this very fact might appear, at first blush, to bring out a manifest “contradiction” in the very conception of the “totality” of the whole numbers taken as “given.” But closer examination will show, as Couturat, Cantor, and the other authors here concerned (since Bolzano) have repeatedly pointed out, that the “contradiction” in question is really a contradiction only of the well-known nature of any finite collection. It was of such collections that the axiom, “The whole is greater than the part,” was first asserted. And of such collections alone is it with absolute generality true. Take any finite collection of whole numbers, however large; and then indeed the assertion of any of the foregoing one-to-one correspondences of the whole, with a mere part of itself, breaks down. But let us once see that taking any number r, however large, we can find the corresponding rth member in any of the ordered series of primes, squares, etc., and then we shall also ​see that the absolutely universal proposition, “Every whole has its single and separate correspondent member in any one of the various ordered series of selected whole numbers aforesaid,” is not only free from contradiction, but is easily demonstrable, and is a mere expression of the actual nature of the number-series, taken as an object of exact thought.
Highly important it is, however, to observe, that the property of the number-series here in question is most sharply conceived, not when one wearily tries, as Mr. Bosanquet has it, to count “without having anything in particular to count,” but when one rather tries to reflect, and then observes that the single feature about the number-system upon which all this conceivable complexity depends, is the simple and positive demand that is determined by the thought which conceives any order whatever. For order, as we shall soon more generally see, is comprehensible most of all in cases of self-representative systems of the present type. The numbers are simply a formally ordered collection of ideal objects. Whoever anywhere orders his own thoughts, either defines just such a self-representative system, or sets in order some empirically selected portion of a world that, in its totality, is such a system. And any system once self-representative, in this particular way, is infinitely self-representative. And if you will count its elements, you shall, then, always find that you can never finish the task.
Yet we are not yet done with showing, in this abstractly simple case of the numbers, what this type of self-representation implies. The numbers, namely, form a system not only self-representative in infinitely numerous ways, but also self-representative according to each of these ways, in a manner that can be doubly brought under our notice. Take, namely, the collection of series thus represented:— ​
| 1 2 4 8 | 2 4 8 16 | 3 6 12 24 | 4 8 16 32 | 5 10 20 40 | 6 12 24 48 | 7 14 28 56 | 8 16 32 64 | 9 18 36 72 | 10, etc. 20, etc. 40, etc. 80, etc. |
Each of these series, written in the horizontal rows, is ordered. Each is in such wise endless that to every number r, however large, there corresponds a determinate rth member of that particular series. And so each series illustrates the first point, namely, that the whole number-series may be put in a one-to-one correspondence with a part of itself. But each series is formed from the immediately preceding series by writing down, in order, the second, fourth, sixth, eighth member of that series, and so forth, as respectively the first, second, third, fourth member of the new series, and by proceeding, according to the same law, indefinitely. It is at once easy to illustrate a second principle regarding any such self-representative systems. To do this, let us observe that:—
First, Each new series is contained in the previous series as one of its constituent parts, so that each horizontal series is self-representative; while every one is a part of all of its predecessors.
Secondly, Each series is therefore to be derived from the former series in the same way in which the second series is derived from the first series.
Thirdly, The later series, therefore, bear to the earlier series, a relation parallel to that which characterized the members of the series of maps in our first illustration of the present type of self-representative systems.
For just as, in the former case, the one purpose to draw the exact map of England within England, gave rise to the endless series of maps within maps, just so, the one purpose, To represent the whole number-series (as to the order of its constituents) by a specially selected series of whole numbers, arranged in order as first, second, etc, — just so, I say, this one purpose involves of necessity the result that this second or representative series shall contain, as part of itself, an end​less series of parts within parts. Each of these contained parts represents a preceding part precisely in the way in which the first representative system represents the original system. The law of the process always is that in a self-representative system of the type here in question, if any part A’ can stand in a one-to-one relation to the elements of the whole system, A, then ipso facto there exists A’’ (a part of this part), such that A’’ is the image or representative in A’, of A’ as it was in A. A’’ stands, then, in the same relation to A’, as that in which A’ stands to A; and A’’ is also a part of A. To derive A’ from A, by any such process as the one just exemplified, is therefore at once to define, by recurrence, the derivation of A’’ from A’, or, if you please, the internal and representative presence of A’’ within A’, of A’’’ within A’’, and so on without end. Nor can any A’ be derived from A, in such wise as exactly to represent, while a part of A, the whole of A, without the consequent implied definition of the whole series, also endless, A, A’, A’’, A’’’, wherein each term is a representative of the former term. So that not only is A self-representative and endless, but each of the derived series is self-representative and endless, while the whole ordered system of series that one can write in the orderly sequence A, A’, A’’, A’’’ is again a self-representative sequence, and so on endlessly, — all this complexity resulting self-evidently from the expressions of a single purpose.
One sees, — self-representation of the present type remains persistently true to its tendency to develope types of variety out of unity. Trivial these types may indeed seem; yet the simplicity and the exactness of the derivation here in question will soon prove of use to us in a wholly different field. But it is now time to suggest, briefly, a still more general view of these self-representative systems.
IV. Remarks upon the Various Types of Self-Representative Systems
We have so far spoken, repeatedly, of the “present type” of self-representative systems, meaning the type that, in this ​paper, will especially interest us. In this type a system is capable of standing in an exact one-to-one correspondence with one of its own constituent portions. We are to be interested throughout this paper in cases of self-representation, such as Self-consciousness, and the relation between thought and Reality, and all the problems of Reflection, bring to our notice. And in all these cases, as we shall see, the system before us will combine the characters of selfhood and internal unity of nature, with the character of being also internally manifold, self-dirempted, Other than Self, and that in most complex and highly antithetic fashion. The relational systems of the type of the number-system especially exemplify — of course in a highly abstract fashion — the sort of unity in contrast, and of exact self-representation, which we are to learn to comprehend. Hence, the stress here to be laid upon one type of self-representative system.
Yet, mathematically regarded, this is indeed only one of several possible types of self-representation.
In the work by Dedekind already cited, the general name, Kette, is given to any self-representative system, whether of the present type or any other self-representative type. In the most general terms, a Kette is formed when a system is made to correspond, whether exactly, and element for element, or in any other way, either to the whole, or to a part of itself. The correspondence might be summary and inexact in type, if to many elements of the original system a single element of the representation or image were made to correspond, as, in a summary account or diagram, a single item or stroke can be made, at pleasure, to correspond to a whole series of facts in the original object which the account or the diagram represents. In this way, for instance, the one word prime can be made to correspond, in a given discussion, to all the prime numbers. If, in case of a Kette, the correspondence of the whole to the part is of this inexact type, the Kette need not be endless, but may even consist of the original object, and a single one of its constituent parts. Then all the later members of the Kette, the A’’, the A’’’, etc., of the previous account, ​fuse together in this one part, A’. If the map of England, before discussed, be an inexact and summary map, such as we actually always make, it need contain no part that visibly, or exactly, presents the place or the form of the map itself, as a part of the surface of England. But the Kette is constructed in such wise that the part is in exact correspondence to the whole when, as in Dedekind’s definition of the Infinite, the correspondence is ähnlich, so that any different elements in the object have different elements corresponding to them in the image, while every element has its own uniquely determined corresponding image. It will be observed that in case of inexact or dissimilar self-representation, we have a failure or external limitation of our self-representative purpose. Only exact self-representation is free from such external interference.
Yet even an exact self-correspondence can be brought to pass, within a system, by making it correspond not to a true portion of itself, but, member for member, to the whole of itself. Thus the system ɑbcd, consisting of the already distinguishable elements ɑ, b, etc., may be put in exact correspondence to itself by making b correspond to ɑ, and so represent ɑ, while, in similar fashion, c corresponds to b, d to c, and, finally, ɑ itself to d. In this case the system is, in a particular way, “transformed” into the image bcdɑ, in such wise as to be exactly self-representative. But the system ɑbcd might also be represented, element for element, by the system cbdɑ, where the order of the elements was again different, but where c now corresponded to the original ɑ, b to itself, d to c, and ɑ to d. Such “substitutions,” as they are called, give rise to self-representative systems of a type different from the one that we have heretofore had in mind. But in the general mathematical theory of “transformations,” and of “groups of operations,” self-representation of such types plays a great part. And in cases of such a type, to be sure, exact self-representation, and finitude of the system, are capable of perfect combination Such self-representations need not be endless, and can be exact. There are many remarkable instances known to ​descriptive physical science, where the correspondence used for scientific purposes is of this type. Such are the instances which occur in crystallography, where the symmetry of a physical object is studied by considering what group of rotations, or of internal reflections in one or in another plane, or of both combined, will bring any ideal crystal form to congruence with itself. All such operations as the rotations and reflections that leave the crystal form unaltered are, of course, operations which bring to light an essentially self-representative character in the crystal form, since by any one such operation the crystal form is made precisely to correspond with itself, while the operation can at once be followed by a new operation of the same type, which, again, leaves the form unaltered.
While, however, self-representative systems of ideal or of physical objects belonging to the later types play a great part in exact physical and in mathematical science, their study does not throw light upon the primal way in which the One and the Many, in the processes directly open to thought’s own internal observation, are genetically combined. For physical systems which permit these transformations of a whole into an exact image of itself are given as external “conjunctions,” such as crystal forms. We do not see them made. We find them. The ideal cases of the same type in pure mathematics have also a similar defect from the point of view of Bradley’s criticism. A system that is to be made self-representative through a “group of substitutions,” shows, therefore, the same diversities after we have operated upon it as before; and, furthermore, that congruence with itself which the system shows at the end of a self-representative operation of any type wherein all elements take the place of all, is not similar to what happens where, in our dealings with the universe, Thought and Reality, the Idea and its Other, Self and Not-Self, are brought into self-evident relations, and are at once contrasted with one another and unified in a single whole. Hence, we shall indeed continue to insist, in what follows, upon those self-representations wherein proper part ​and whole meet, and become in some wise precisely congruent, element for element. We mention the other types of self-representation only to eliminate them from the present discourse.
In case of these self-representative systems, of the type especially interesting to us, we have already illustrated how their particular kind of self-representation developes infinite variety out of unity in a peculiarly impressive way. The general law of the process in question may now be stated, in a still more precise and technical form.
We may once more use the thoroughly typical case of the number-system. We have seen, in general, the positive nature of its endlessness. We want now to define, in decidedly general terms, the infinite process whereby the numbers can be self-represented, in infinitely numerous ways, by a part of themselves, and to state, abstractly, the implications of any such process. Let, then, f(n) represent any “function” of a whole number, such that n is to take, successively, the value of any whole number from 1 onwards; while f(n) itself is, in value, always a determinate whole number. The values of f(n) shall never be repeated. They shall follow in endless succession, and, as we shall also here suppose, in the order of their magnitude from less to more. Not all the numbers shall appear amongst the values of f(n). In consequence, f(n), by means of its first, second, third values, etc., shall represent precisely the whole of the number-series, while forming only a part thereof. Otherwise let f(n) be an arbitrary function. Then it will always be true that f(n) will contain, as a part of itself, a series f$_1$(n), related to f(n) in precisely the same way in which f(n) is related to the original series of whole numbers. It ​will also be true that f$_1$(n) will contain a second series f$_2$(n), similarly related to f$_1$(n); and so on without end.
We have illustrated this truth. We now need to develope it for any and every series of f(n), however arbitrary. Consider, then, the values of f(n) as a part of the original number-series. These values of f(n) form an image or representative of the whole number-series in such wise that if r be a whole number appropriately chosen, some one value of f(n), say the value that corresponds to the number p in the original series, or, in other words, the pth value of f(n), is r. But since f(n) images the whole of the original number-series, it must contain, as a part of itself, a representation of its own self as it is in that number-series. In this representation, f$_1$(n), there is again a first member, a second member, and so on.
Now we can indeed speak of the series f$_1$(n) as “derived from” f(n) by a second and relatively new operation. But, as a fact, the very operation which defines the series f(n) already predetermines f$_1$(n), and no really second, or new operation is needed. For if every whole number has its correspondent, or “image,” in f(n), then, for that very reason, every separate “image,” being, by hypothesis, a whole number, has again, in f(n), its own image; and this image again its own image, and so on without end. Merely to observe these images of images, already present in f(n), is to observe, in succession, the various members of the series f$_1$(n). The law of the formation of f$_1$(n) is already determined, then, when f(n) is written, no matter how arbitrary f(n) itself may be.
In particular, let p be any whole number, and suppose that, according to the original self-representation of the numbers, f(p) = r. Then r also will have its image in the series f(n). Let that image be called f(r). Then f(r) = f(f(p)), is at once defined as f$_1$(p), that is, as that value which f$_1$(n) takes when n = p, or as the image of the image of p. It is easy to see that f$_1$(p) is the pth value, in serial order, of the series f$_1$(n). At the same time, since f$_1$(p) = f(r), and since f(r) occupies, ​in the series of values of f(n) the rth place, while f(p), or r, occupies the rth place in the original number-series, one can say, in general, that the successive values of f$_1$(n) are numbers which occupy in f(n) places precisely corresponding to the places which the successive values of f(n) themselves occupy in the original number-series. Thus the first member of f$_1$(n) is that one amongst the members of the series of values of f(n) whose place in that series of values corresponds to the place in the original series of whole numbers which was occupied by f(1). The second member of f$_1$(n) is, even so, that one amongst the series of values of f(n) which occupies the place in that series of values which f(2) occupies in the original number-series. And, in general, if, to the whole number p, in the original number-series, there corresponded the number r, as the image of that number in the series called f(n), then this pth member of the series called f(n) will have, as its image or representative in f$_1$(n), the number f(r), i.e. the value of f(n) when n = r. This number f(r) will constitute, of course, the pth member of f$_1$(n), and will occupy, in the series called f(n), the very same relative place which f(p) occupies in the original number-series.
Precisely so, f$_1$(n) contains, as a part of itself, its own image as it is in f(n) and also as it is in the original series. And this new image may be called f$_2$(n); and so on without end. Hence, one process of self-representation inevitably determines an endless Kette altogether parallel to our series of maps within maps of England. The general structure and development of any self-representative system of the present type have now been not only illustrated, but precisely defined and developed. Self-representation, of the type here in question, creates, at one stroke, an infinite chain of self-representations within self-representations. ​
V. The Self and the Relational System of the Ordinal Numbers. The Origin of Number; and the Meaning of Order
Having considered self-representation so much in the abstract, we may now approach nearer to the other illustrations of self-representative relational systems. To be sure, in beginning to do so, we shall, for the first time in this discussion, be able to state the precise logical source of the good order of the number-system, whose self-representative character, now so wearisomely illustrated, is simply due to the fact that the number-series is a purely abstract image, a bare, dried skeleton, as it were, of the relational system that must characterize an ideally completed Self. This observation, in the present form, cannot be said to be due to Hegel, although both his analysis and Fichte's account of the Self, imply a theory that apparently needs to be developed into this more modern form. But the contempt of the older Idealism for the careful analysis of mathematical forms, — its characteristic unwillingness to dwell upon the dry detail of the seemingly lifeless realm of the mathematically pure abstractions, is responsible for much of the imperfect development and relative vagueness of the idealistic Absolute. It is so easy for the philosopher to put on superior airs when he draws near to the realm of the mathematician. And Hegel, despite his laborious study of the conceptions of the Calculus, in his Logik, generally does so. The mathematician, one observes, is a mere “computer.” His barren Calcul, — what can it do for the deeper comprehension of truth? Truth is concrete. As a fact, however, these superior airs are usually the expression of an unwillingness even to spend as much time as one ought to spend over mathematical reading. And Hegel seems not to have solved the problem of the logic of mathematics. The truth is indeed concrete. But if alle Theorie is, after all, grau, and grün des Lebens Goldener Baum, the philosopher, as himself a thinker, merely shares with his colleague, the mathematician, the fate of having to deal with dead leaves and sections torn or cut from the tree of life, in his toilsome effort to make out what ​the life is. The mathematician’s interests are not the philosopher’s. But neither of the two has a monopoly of the abstractions; and in the end each of them — and certainly the philosopher — can learn from the other. The metaphysic of the future will take fresh account of mathematical research.
The foregoing observation as to the parallelism between the structure of the number-series and the bare skeleton of the ideal Self, is due, then, in its present form, rather to Dedekind than to the idealistic philosophers proper. It shall be briefly expounded in the form in which he has suggested it to me, although his discussion seems to have been written wholly without regard to any general philosophical consequences. And the present is the first attempt, so far as I know, to bring Dedekind’s research into its proper relation to general metaphysical inquiry.
The numbers have been so far taken as we find them. But how do we men come by our number-series? The usual answer is, by learning to count external objects. We see collections of objects, with distinguishable units, the “bare conjunctions” of Mr. Bradley once more. Their mysterious unity in diversity arouses our curiosity. We form the habit, however, ​of using certain familiar and easily observed collections (our fingers, for instance) as means for defining the nature of less familiar and more complex collections. The number-names, derived from these elementary processes of finger-counting, come to our aid in the further development of our thought about numbers. The decadic system makes possible, through a simple system of notation, the expression of numbers of any magnitude. And so the number-concept in its generality is born.
This usual summary view of the origin of the numbers has its obvious measure of historical and psychological truth. It leaves wholly unanswered, however, the most interesting problems as to the nature of the number-concept. For numbers have two characters. They are cardinal numbers, in so far as they give us an idea of how many constituents a given collection of objects contains. But they have also an ordinal character; for by using numbers, as the makers of watches, and bicycles, or as the printers of a series of banknotes, or of tickets, use them, we can give to any one object its place in a determinate series, as the first, the tenth, or the ten thousandth member of that series. Such ordinal use of numbers is a familiar device for identifying objects that, for any reason, we wish to view as individuals. Now, a very little consideration shows that the ordinal value of the numbers is of very fundamental importance for their use in giving us a notion of the cardinal numbers of multitudes of objects. For when we count objects by using either the fingers or the number-names, we always employ an already familiar ordered series of objects as the basis of our work. We put the members of this series in a “one-to-one” relation to the members of the collection of objects which we wish to count. We deal out our numbers, so to speak, in serial order, to the various objects to be counted. We thereby label the various objects as they are numbered, just as the makers of the banknotes stamp an ordinal number on each note of a given issue. Only when this process is completed do we recognize the cardinal number which tells us how many objects there are in the collection of the objects counted. ​And we recognize this result of counting by the simple device of giving to the whole collection counted a cardinal number corresponding to the last member of the ordinal number-series that we have thus dealt out. If, for instance, the last object labelled is the tenth in the series of objects set in order by the ordinal process of labelling, then the counted collection is said to contain ten objects.
Unless the numbers were, then, in our minds, already somehow a well-ordered series, they would help us no whit in counting objects. Nor does counting consist in the mere collection of acts of synthesis by which we each time add one more, in mind, to the collection of objects so far counted. For these acts of synthesis, however carefully performed, soon give us, if left to themselves, only the confused sense, “There is another object, — and another, — and another.” In such cases we soon “lose count.” We can “keep tally” of our objects only if we combine the successive series of acts of observing another, and yet another, object, in our collection of objects with the constant use of the already ordered series of number-names, whose value depends upon the fact that one of them comes first, another second, etc., and that we well know what this order means.
The ordinal character of the number-series is therefore its most important and fundamental character. But upon what mental process does the conception of any well-ordered series depend? The account of the origin of the number-series by the mere use of fingers or of names, does not yet tell us what we mean by any ordered series at all.
To this question, whose central significance, for the whole understanding of the number-concept, all the later discussions and the modern text-books recognize, various answers have been given. The order of a series of objects, presented or ​conceived, has been most frequently regarded, in the later discussions, either as a datum of sensuous experience, or else as an inexplicable and fundamental character of our process of conception. In either case the problem of the One and the Many is left unaualyzed. For an ordered series is a collection taken not only as One, but as a very special sort of unity, namely, as just this Order. That many things can be taken by us as in an ordered series, — this is true, but is once more the “bare conjunction” of Mr. Bradley’s discussion. We want to find out what act first brings to our consciousness that Many elements constitute One Order. Nearest to the foundation of the matter Dedekind seems to me to have come, when, without previously defining any number-series at all, he sets out with that definition of an infinite system of ideal objects which we have already stated, and then proceeds, substantially as follows, to show how this system can come to be viewed Whole.
Let there be a system N of objects, — a system defined as capable of the type of self-representation heretofore illustrated. That such a system is a valid object (of the type definable through our own Third Conception of Being), we have already seen by the one example of meine Gedankenwelt. For the ideally universal law of meine Gedankenwelt is that to every thought of mine, s, I can make correspond the thought, s’, viz., the thought, “This, s, is one of my thoughts.” Because of this single ideal law of the equally ideal Self here in question, the Gedankenwelt is already given as a conceptual system of many elements, — a system capable of exact representation by one of its own con​stituent portions. Now let us suppose our particular system N to be a system such as a particular portion, itself infinite, of the Gedankenwelt, would constitute. Namely, let us suppose our system N to be capable of a process of self-representation that first selects a single one of the elements of N (to be called One or element the first), and, that then represents the whole of N by that portion of N which is formed of all the elements of N except One. The result of this mode of self-representation is that N becomes, in the sense before defined, a Kette, represented by a part of itself, N’. This part, N’, by hypothesis, contains all of the N except the chosen first element named One. In consequence, and because of the very same sort of reasoning that we carried out in case of the map of England made within England, N’ will again contain, by virtue of the one principle of its constitution, a further part, N’’, which will be derived from N’ by leaving out a single element of N’, to be called Two, and defined as the second element of the system. Two will be, in fact, the name of that very element in N’ which, in the original mapping of N by N’, was the element that was made to represent, or to image, element One. But the process of expressing the meaning thus involved is now recurrent. For the one plan of representing N by N’, with the omission from N’ of the single element called One, has involved the representing of N’ by N’’, with the omission from N’’ of the single element now called Two, — an element which is merely the image in N’ of One in N. The same plan, however, not so much applied anew, as simply once fully expressed, implies that within N’’ there is an N’’’, an N$^{iv}$, and so on without end; just as the one plan of mapping England within England involved the endless series of maps. But each of the series of systems N’, N’’, N’’’, etc., differs from the previous one simply by the omission of a ​single element present in its predecessor. And the series of these successively omitted elements has an order absolutely predetermined by the one original plan. That order consists simply in the fact that each element omitted, when any of the new representations, N’’, N’’’, etc., is considered, is, upon each occasion, itself the Bild, the image or map or representative, of the very element that was previously omitted, when N’’, or N’’’, or other representation, was made. The endless series One, Two, Three, etc., is consequently the series of names of those objects whereof the first was omitted when the first representation or the mapping of N was made; while the second element represented, in the first map, N’, this first element of N. In the same way, in the second map, N’’, the element Three, the third element of the series, represented or pictured the second element, which latter, present in N’, had been omitted in N’’.
Thus the one plan of mapping or representing N by a part of itself, taken as a single act, accomplished at a stroke, logically involves what one can then express as an endless series of maps or images of the portion or element of N that is omitted from the first of the maps. And this endless ordered series of images of the omitted element of N, can be so carried out as to constitute a derived system that contains, in its turn, any member of N that you please, in a particular place, whose order in the series of successive images is absolutely predetermined by the one original plan. Hence, as Dedekind has it, “we say that the system N is, by this mode of representation, set in order (geordnet).” But, let us observe, this whole order, in all its infinite serial complexity, is logically accomplished by means of one act.
The series of images, or representations, of the element One, thus obtained, has of course, at first sight, a very artificial seeming. But a glance at the concrete case of the Gedankenwelt will show the sense of the process more directly. Let my Gedankenwelt be viewed in its totality, as a system self-represented in the way first defined. Then the one plan ​of representing any thought of mine, whether itself reflective or direct, by a reflective thought of the form, This is one of my thoughts, implies that about any primal thought of mine, say the thought, To-day is Tuesday, there ideally clusters an endless system, N, of thoughts whereof this thought, To-day is Tuesday, may be made the first member. These thoughts may follow one after another in time. But, logically, they are all determined at one stroke by the one purpose to reflect. The system N consists of the original thought, and then of the series of reflective thoughts of the form, This is one of my thoughts; — yes, and This last reflection is one of my thoughts; and This further reflection is one of my thoughts; and so on without end. Now the system N is known to be infinite, not by counting its members until you fail and give up the process in weariness, but by virtue of the universal plan that every one of its members shall have a corresponding reflective thought that shall itself belong to the system. Hereby already N is defined as infinite, before you have counted at all. But this very plan determines a fixed order of sequence, whether temporal or logical, amongst the constituent elements of N; because each new element, to be taken into account when you follow the order, is defined as that element whereby the last element is to be imaged, or reflectively represented. But this recurrent, or iterative, character of the operation of thought whereby you follow the series of elements, is really only the result of the single plan of self-representation whereby once for all the system N is ordered according to its defined first member. For the whole system N, once conceived as mapped, or represented by that portion of itself which does not include the element called One, is even thereby at one stroke defined as an ordered series of representations within representations, like our series of maps of England. This system of representations within representations of the whole of N, is given as a valid truth, totum simul, by the definition of the undertaking. The series of temporally successive reflective thoughts, however, is found to be ordered as a result of this constitution of the entire system; and therefore is its ​iterative meaning clear quite apart from any theory as to whether time and succession are appearance or reality.
Now the system N is, by definition, simply that system of thoughts which, if present at once, would express a complete self-consciousness as to the act of thinking that To-day is Tuesday. Were I just now not only to think this thought, but to think all that is directly implied in the mere fact that I think this thought, I should have present to me, at once, the whole system N as an ordered system of thoughts. Precisely so, the whole determined Gedankenwelt, if present at once, would be a Self, completely reflective regarding the fact that all of these thoughts were its own thoughts. But this complete reflection would, in all its portions, involve an ordered system of thoughts, whose purely abstract form, taken merely as an order, is everywhere precisely that of the number-system.
Self-representation, then, in the sense now so fully exemplified, is not merely, as it were, the property or accident of the number-system; but is, logically speaking, its genetic principle. When order is not a mere “external conjunction,” when we know not merely that facts seem in order, but what the order is, and how it is one order through all of its manifold expressions, we do so by virtue of comprehending the internal meaning of a plan whereby a system of conceived objects comes to be represented through a portion of itself. Dedekind has shown that this view is adequate to the logical development of the various properties of the number-system. What we here observe is that the consequent constitution of the number-system is explicitly defined as, of course in the barest and most abstract outline, the form of a completed Self. Here, then, the Intellect, “of its own movement,” “itself by itself,” defines what, in our temporal experience, whether sensuous or thoughtful, it of course nowhere finds given, namely, a self-representative system of objects, parallel in structure to what the structure of a Gedankenwelt would be if it were the Welt of a completely self-conscious Thought, none of whose acts failed to be its own intellectual objects. This ​concept comes to us as positive, and wholly in advance of counting. It involves, first, the general definition of a Kette, of the type here in question, whose properties, taken in their abstraction, are as exactly definable as those of a triangle. Not every such Kette is a Self, or a Gedankenwelt; for of course the general concept of a system possessing some sort of one-to-one correspondence, can be applied in any region, however abstract; and a Kette may therefore be defined where the objects in question are taken to be either dead matter or else mere fiction. Consequently the mathematical world is simply full of Ketten of the present and of other types. But the notable facts are, first, that the present type of Kette becomes the very model of an ordered system, and, secondly, that it becomes this by virtue of the fact that in structure it is precisely parallel to the structure of an ideal Self. Herein the intellect does indeed, of itself, comprehend its own work, even though this work be but an ideal creation.
But all order in the world of space, of time, of quantity, or of morals, however rich its wealth of life, of meaning, or of beauty may be, is order because it presents to us systems of facts that may be viewed as having a first, a second, a third constituent, or some higher form of order; while the rank, dignity, worth, magnitude, proportion, structure, description, explanation, law, or other reasonableness of any of these objects in our world depends, for us, upon our power to recognize in them what, for a given purpose, comes first, what second, and so on, amongst their elements or their higher constituents. The absolutely universal application of the concept of order wherever the intellect recognizes in any sense its own, in heaven or upon earth, shows us the interest of considering even these barest abstractions regarding simple order. The number-series is indeed the absolutely abstract, but also the absolutely universal and inclusive type of all order, — the one thing that every rational being, however much he may differ in constitution from us men, must, in some shape, possess, just in so far as he knows any complete order or system at all, divine or diabolical, moral or physical, æsthetic or social, ​formal or concrete. For the deepest essence of the number-series lies not in its power to aid us in finding how many units there are in this or that collection, but in its expression of the notion that something is first, and something next, in any type of orderly connection that we may be capable of knowing. It is the relational system of the numbers, taken in their wholeness as one act, which here interests us. Those degrade arithmetical truth who conceive it merely as the means for estimating the cardinal numbers of collections of objects. The science of arithmetic is rather the abstract science of ordered collections. But all collections, if they have any rational meaning, are ordered and orderly. Hence, it is indeed worth while to know where it is that we first clearly learn what order means.
Now it is not very hard to see, and to say, that I first recognize order as a form of unity in multiplicity when I learn, of myself, to put something first, and something next, and self-consciously to know that I do so. That counting my fingers, or learning the names of the numbers, first sets me upon the way to attain this degree of self-consciousness, is true enough. But our question is what the concept of order, as the one transparent form of unity in manifoldness, directly implies. In following the analysis of the number-concept, we have been led to the point where this becomes an answerable question. Given, as “bare conjunction,” is what you will. The intellect, however, as Mr. Bradley well says, accepts only what it can make for itself. The first object that it can make for itself, however, is seen, as Mr. Bradley also says, to involve the seeming of an endless process. The single purpose of the intellect, in any effort at self-comprehension, proves to be recurrent precisely when it is most obvious and necessary. The infinite task looms up before us; and, in impatient weariness, we talk of “endless fission” breaking out everywhere, and are fain to give up the task; failing, however, to observe that just hereby we have already seen how the One must express itself, by the very self-movement of the intellect, as the Many. If we reflect afresh, however, we observe that what we have seen ​is due to the fact that the only systems of ideal objects which the intellect can define without taking account of “bare external conjunctions,” are systems such that to whatever object we have presupposed, another object, expressing the same intellectual purpose, must correspond, as the next object in question. This fact, however, is due to the simple necessity of the reflective process in which we are involved.
Our thought seeks its own work as its object. That is of the very essence of this effort to let the intellect express its self-movement. But making its own work its object, observing afresh what it has done, is merely reinstating, as a fact yet to be known, the very process whose first result is observed when the intellect contemplates its own just accomplished deed. Reflection, then, implies, to be sure, what, in time, must appear to us as an endless process. We are not interested, however, in the mere feeling of weariness which this endless process (in consequence of still another “bare conjunction,” of a psychological nature) involves to one of us mortals when he first observes its necessity. What interests us is the positive structure of the whole intellectual world. We have found that structure. It is the structure of a self-representative system of the type that we now have in mind. We frankly define all such systems as endless, so far as concerns the variety of their elements. But hereupon we indeed observe that, as self-representative, they are, in a perfectly transparent way, self-ordered. The trivial illustration of the map within the country mapped, has been followed by the more exact illustrations of the self-representative character of the complete number-system when once its traditional structure is accepted as something given and present in totality. With these examples of self-ordered unity in the midst of infinite diversity, we have returned to the question of the logical genesis of the very conception of order of which the number-system is the first example. We have found the answer to our question in the assertion that since a self-representative system, of the type here in question, once assumed as an ideal object, determines its own order, and assigns to its constitu​ents their place as first, next, and so on, and since only such self-representative systems result from the undisturbed expression of the intellect's internal meanings, therefore, an order that shall be transparent to the intellect, or that shall appear to it as its own deed, must be of the type exemplified in Dedekind’s analysis.
And so, as far as we have gone, the circle of our investigation is provisionally completed. The intellect has been studying itself, and, as the abstract and merely formal expression of the orderly aspect of its own ideally conceived complete Self, and of any ideal system that it is to view as its own deed, the intellect finds precisely the Number System, — not, indeed, primarily the cardinal numbers, but the ordinal numbers. Their formal order of first, second, and, in general, of next, is an image of the life of sustained, or, in the last analysis, of complete Reflection. Therefore, this order is the natural expression of any recurrent process of thinking, and, above all, is due to the essential nature of the Self when viewed as a totality. Here, then, although we are still merely in the world of forms, we know something about the One and the Many.
VI. On the Realm of Reality as a Self-Representative System
We must now proceed to apply our previous considerations to the question of the constitution of any realm of Being, or of any universe.
Suppose, in the first place, for a moment, that one is to conceive the universe in realistic terms, as a realm whose existence is supposed to be independent of the mere accident that any one does or does not know or conceive it. Suppose such a world to be once for all there. Then it is possible to show that this supposed universe has the character of a self-representative system, and that, too, even if you try to define its ultimate constitution as unknowable.
For, in the first place, at the moment when you suppose that any fact exists, independently of whether you know it or not, it is obvious that you must in reality be making, or at ​least, by hypothesis, trying to make, this supposition. For unless the supposition is really attempted, there is no conception of F in question at all. But if the supposition is itself a fact, then, at that instant, when the supposition is made, the world of Being contains at least two facts, namely, F, and your supposition about F. Call the supposition f; and symbolize the universe by U. Then the least possible universe that can exist, at the moment when your hypothesis is made, will be such that U = F + f.
Having proceeded so far, however, we cannot stop. As we saw in analyzing the realistic concept, Realism hopelessly endeavors to assert that, although what we now call F and f are alike real, they have no essential relations to each other. For our present purpose, however, we need only note that whether or no the relations of F and f are in the least essential to the being of either F or f, taken in themselves, still, when F and f are once together and related, the relations are at least as real as their terms. Or, even if we confine ourselves strictly to our symbols, it remains obviously true that in order merely to report the supposed facts, we had to write, as the actual constitution of our universe, at least F + f. Now this universe, as thus symbolized, has not merely a twofold, but a threefold constitution. It consists of F, and of f, and of their +, i.e. of the relation, as real as both of them, which we try to regard as non-essential to the Being of either of them, but which, for that very reason, has to be something wholly other than themselves, just as they are supposed to be different from each other. A system such as Herbart’s depends, indeed, upon trying to reduce this + to a Zufällige Ansicht, which is supposed, for that reason, to be no part of the realm of the “reals.” But, in answer to any such effort, we must stubbornly insist (and here in entire agreement with Mr. Bradley) upon declaring that either this Zufällige Ansicht stands for a real fact, for something which is, or else the whole hypothesis falls to the ground. For the essence of the hypothesis is that, f rightly supposes F to exist, or, in other words, that the relation between F and f is one of genuine reference, ​assertion, or truth on the part of f, and of actual expression of the truth of this assertion by the very existence of F. Therefore, the relation between F and f is supposed to be a real fact. Since, by hypothesis, it is independent of the mere existence of F and of f, or since, if you please, F, by hypothesis, might have been real without f, and f, if false, might have existed, as a mere opinion, in the absence of any F, the relation which we have expressed by + has its own place in Being, and is a third and, by the realistic hypothesis, a separate fact; so that now U contains at least three facts, all different from one another.
Hereupon, of course, Mr. Bradley’s now familiar form of argument enters with its full rights. Unquestionably a world with three facts in it, — facts such that, by definition, either f or F might have existed wholly alone, and apart from the third fact, is a world where legitimate questions can be raised about the ties that bind the third fact to the other two. These ties are themselves facts. The + is linked to f and to F, and the “endless fission” unquestionably “breaks out.” The relation itself is seen entering into what seem new relations. The reason why this fission breaks out is now more obvious to us. It lies not in the impotence of our intellect, impotent as our poor human wits no doubt are, but in the self-representative character of any relational system. In our realistic world the system is such that, to any object, there corresponds, as another object (belonging to the same system), the relation between this first object and the rest of the universe. Or, in general, if in the world there is an object, F, then there is that relation, R, whereby F is linked to the rest of the world. But to R, as itself an object, there therefore corresponds, at the very least, R’, its own relation to the rest of the world; and the whole system F + R + R’ is as self-representative, and therefore as endless, as the number-system, and for precisely the same reason: viz., because it images, and, by hypothesis, expresses, in the abstract form of a supposed “independent Being,” the very process of the Self which undertakes to say, “F exists.” ​
Now, it would be wholly useless for a realist to attempt to escape from this consequence by persistently talking, as some realists do, about the defective nature of our poor human thought, and about the Unknowability of the Real. For the question is not as to what we do not know, but merely as to what we do know, about the supposed Independent Beings. And what we do know is, that by definition they form a Kette of the type now in question. They cannot escape from this consequence of their own definition by declaring their true Being to be unknowable. For if they attempt thus to escape, we shall very simply point out that, as unknowable, and as thus different from our definition of their Being, they, the realities, have now merely a twofold form of Being, namely, their Unknowable and their Knowable form. For, after all, we are supposed to know that they are, and that they appear to us in the form of a Kette. The problem of the “two natures” in one being, is, then, upon the hands of any realist who, like Mr. Spencer, thus divides his world; and this relation, whether knowable or unknowable, between the Knowable and the Unknowable aspects, or regions of Reality, will become something different from either of the two; and the new system will once more be a Kette, precisely like its predecessor, and for the same reason.
But, finally, one may attempt to escape from the entire situation by declaring that F, in the foregoing account, is, by hypothesis, a fact that does not need f, since f is, by supposition, a conscious process, — an idea, — and F is F whether or no anybody supposes it to exist, or knows it in any way. “Suppose now,” a realist may say, “that there were no knowledge or ideas at all, but only the facts independent of all minds, and totally separate from one another. Then the realistic world would not be an endless Kette.” Therefore it only becomes one, per accidens, when known.
In reply, I should point out, that if the world that contains F contains also any other facts, any diversity whatever, Mr. Bradley’s repeated analysis of the “endless fission” will at once apply, and the world will become a self-representative ​system in the former sense. But F, if supposed to be wholly alone, and to be the only Being, and absolutely simple, is still not exempt from the universal self-diremption. When you think of it, — now, for instance, it is not alone. It is, by hypothesis, just now in the same world with the thoughts that define it. “But it is such that it need not be together with the thoughts that think it. It could exist independently.” Yes, but to exist alone, and to exist in company with another, are not the same thing. F, then, has two aspects, or potencies: the aspect that enables it to exist independently of f, or of any thought, and its power to exist in relation to, and along with f, and with the rest of the Kette determined by the presence of f. F, the same F, has these two states of being, — its existence alone, and what Herbart called its Zusammen. Now just as the Zusammen is, by hypothesis, a fact, which nobody gets rid of by calling it a Zufällige Ansicht, so to be in Zusammen is to be in a state very different from the “Being, alone and without a Second,” which F has before f comes. Call F, when taken as alone, F$_1$ and F, when taken as in company, F$_2$. Then the problem, How are F$_1$ and F$_2$ related? gives rise to the same sort of Kette with which Mr. Bradley has made us so familiar.
I agree, then, wholly with Mr. Bradley, that every form of realistic Being involves such endless or self-representative constitution. And I agree with him that, in particular, realistic Being breaks down upon the contradictions resulting from this constitution. I do not, however, accept the view that to be self-representative is, as such, to be self-contradictory. But I hold that any world of self-representative Being must be of such nature as to partake of the constitution of a Self, either because it is a Self, or because it is dependent for its form upon the Self whose work or image it is. But the realistic world is not able to accept this constitution. In case of the realistic type of Being, then, the endless fission proves to be an endless corruption and destruction of whatever had appeared to be the fact. Why? For the reason pointed out, but without any mention of the mere infinity of the relational ​process, in our third lecture. You want from a realist the facts, and all the facts, which are essential to his scheme. He names you the facts. You point out that since he inevitably names you a variety of facts, he must also admit that the connections or relations of these facts are real. And then you rightly add that the system in question must be self-representative and endless. But hereupon first appears the contradiction of Realism, viz., when you see that none of these endlessly numerous connections actually connect, because they are to be connections amongst beings that, by definition, are independent of knowledge, and therefore, as we saw, of one another, in such wise that their ties and links, if ever these ties seem to exist at all, must, upon examination, be found to be other real beings, as independent of the facts that they were to link as these, in their first essence, were of one another. The endlessly many elements of this world turn out, then, to be endlessly sundered. The Kette of the realist is a chain of hopelessly parted links. It is this aspect of the matter which gives their true cogency to the arguments of Mr. Bradley’s first book. We do not see, then, how the real that is in any final sense independent of knowledge can be either One or Many or both One and Many. And we do not see this because we can see and define nothing but what is linked with knowledge. But within knowledge itself we do, indeed, still find the self-representative system.
So much for the realistic conception of Being. But if we turn to another conception of the nature of reality, namely, to our Third Conception of Being, then we once more find that this conception, too, involves a self-representative system of the type here in question. For this result has been already illustrated by the number-system, by the Gedankenwelt of Dedekind, and by the other mathematical instances cited; since all of these objects, when mathematically defined, appear primarily as beings of the third type of our list. Whether they possess any deeper form of Being, we have yet to see. In general, however, it is interesting to note that, in the proof of the mathematical possibility or validity of infinite systems given ​by Bolzano, in the passage of his Paradoxien des Unendlichen, already cited, the typical instance chosen to exemplify the infinite is that system of truth, or of wahre Sätze, whose validity follows from any primary Satz, or from any collection of such Sätze. If the proposition A is true, it follows, as Bolzano points out, that the proposition which asserts that “A is true,” is also true. Call this proposition A’. Then the proposition “A’ is true,” is also true; and so on endlessly. While Bolzano has not Dedekind’s exact conception of the nature of a Kette, and does not expressly use Dedekind’s positive definition of the infinite, his example of the series of true propositions, A, A’, A’’, etc., — each of which is different from its predecessor, since it makes its predecessor the subject of which it asserts the predicate true, — is an example chosen wholly in the spirit of Dedekind’s later selection of the Gedankenwelt, and is an extremely simple instance of a self-representative system.
Realism, and the Third Conception of Being in our list, share alike, then, whatever difficulties may cluster about the conception of an infinitely self-representative system. What conception of Being can escape from this fate? Our own Fourth Conception?
No, as we must now expressly point out, our own conception of what it is to be makes the Real a Kette of the present type. ​For from our point of view, to be, or to be real, means to express, in final and determinate form, the whole meaning and purpose of a system of ideas. But the fact that a given experience anywhere fulfils a particular purpose, implies that this purpose itself is, in some wise, a fact, and has its place in reality. But if this purpose is real, it must, by our hypothesis, be real as a fulfilment of a purpose not absolutely and simply identical with itself. And so any particular purpose of the Absolute is itself such as it is, because it fulfils a particular purpose other than itself. Hence, for us, the Absolute must be a self-representative ordered system, or Kette, of purposes fulfilled; and the ordered system in question must be infinite. I accept this consequence. The Absolute must have the form of a Self. This I have repeatedly maintained in former discussions. Despite that horror of the infinite which Mr. Bradley’s counsel would tend to keep alive in me, I still insist upon the necessity of the consequence. But I also insist upon several important aspects of the Kette in terms of which the Absolute is for me defined. And these aspects enable me to conceive the Absolute not only as infinite, but also as determinate, and not only as a form, but as a life.
First, the implied internal variety is subject to, and is merely expressive of, the perfectly precise and determinate unity of the single plan whereby, at one stroke, the Absolute is defined, or rather defines itself, as a self-representative system. Secondly, because of the now so wearisomely analyzed character of a Kette of the type here in question, the self-possession or self-consciousness of the Absolute does not imply any simple identity of subject and object in the absolute Self. The map of England (the subjective aspect in our original illustration) is not identical with the whole of England. Yet, in the supposed Kette of maps, once taken as real, the whole of England is mapped within itself. Order primarily implies a first that is represented by the second, third, and later members of the order, but that, as first, is itself representative of nothing else. The Absolute, in my conception, has this first aspect, which is essential at once to the immediacy of its ​experience, and to the individuality which, in my agreement with Mr. Bradley, I attribute to the whole. But this first aspect of Being must needs be represented, within itself, by the second, third, and other aspects. In other words, a full possession of the fulfilment of purpose, in final and determinate form, involves, as the first element in the conception of Absolute Being, the fact that purpose is fulfilled. But this fact is experienced, is known, is present, is seen. Otherwise it is no fact, and the world has no Being. But the fact that this first fact is known, or experienced, is itself a fact, a second fact. This, too, is known; and so on without end.
Thirdly, as I conceive, this whole series without end — a series which can equally well be expressed in terms of knowledge and in terms of purpose — is for the final view, and in the Absolute, no series of sundered successive states of temporal experience, but a totum simul, a single, endlessly wealthy experience. And, fourthly, by the very nature of the type of self-representation here in question, no one fashion of self-representation is required as the only one in such a realm of Being. As the England of our illustration could be self-mapped, if at all, then by countless series of various maps, not found in the same part of England and not in the least inconsistent with one another; and as the number-series, — that abstract image of the bare form of every self-representative system of the type here in question, — can be self-represented in endlessly various ways, — so, too, the self-representation of the Absolute permitted by our view is confined to no one necessary case; but is capable of embodiment in as many and various cases of self-representation, in as many different forms of selfhood, each individual, as the nature of the absolute plan involves. So that our view of the Selfhood of the Absolute, if possible at all, leaves room for various forms of individuality within the one Absolute; and we have a new opening for a possible Many in One, — an opening whose value we shall have to test in another way in our second series of lectures.
Our own view, then, also implies that the Absolute is a ​Kette of the type now in question. But if one insists that such a doctrine is inevitably self-contradictory and vain, — where shall one still look for escape from this fate which besets, so far, all of the views as to the Real?
Shall one turn to Mysticism? Mysticism, viewed in its philosophical aspect, as we have viewed it in these lectures, knows of a One that is to be in no sense really Many. Every Kette must, then, for the mystic, prove an illusion. But, unfortunately for the mystic, the inevitableness of an infinite process is nowhere more manifest than in the movement of his own thought while, weary of finitude, this thought indulges endlessly its sad luxury of a troubled contemplation of its own defects. For this thought, as finite, is, by hypothesis, nothing real at all. Yet it reveals, in its own negative way, the road to absolute peace and truth. This road, however, is a path in the essentially pathless wilderness. This revelation is explicitly an absolute darkness. While you think, you have not won the truth; for thought is illusion. But if you merely cease to think, you have thereby won nothing at all. The Absolute is really known as such by contrast with your illusion. It is so far just the Other. You seek it in thought, and find it not. But perhaps the ineffable experience comes. Ich bin Gott geworden, says the Schwester Katrei of the tract usually (and, as the critics now tell us, wrongly) attributed to Meister Eckhart. This experience, whenever it comes, — why is it said to be an experience of Being? Viewed from without, it seems a mere transient state of feeling in somebody's mind. But no; it shall be no mere feeling, for it reveals all that thought had ever sought. The peace that passeth understanding fulfils all the needs of understanding. Hence, in this peace thought finds itself satisfied, and ceases. Therefore is Being here attained. Yet if this be the mystical insight, — what has been gained? Thought the deceiver, thought the illusory, bears witness to its own refutation and to its own fulfilment in the peace of the Absolute; for only when this evidence is given of the final satisfaction of all thought’s demands is the truth known. And thus the sole ​testimony that Being is what the mystic declares it to be, is a witness borne by this self-detected and hopeless liar, thought, — whose words are the speech of one who exists not at all, but only falsely pretends to exist, and whose ideas are merely lies. This liar, at the moment of the mystical vision, declares that he rests content; and therefore we know, forsooth, that we have come upon “that which is,” and have caught the “deep pulsations of the world.” We accept, then, the last testimony of the wholly hardened and hopeless deceiver; and this dying word of false thought is our sole proof of the Absolute Truth.
Can this be really the mystic’s ultimate wisdom? No; the unconscious silence in which he ought forever to dwell, once broken by his first utterance when he teaches his doctrine, leads him to endless speech, — but to speech all of the same infinitely self-denying kind. The ineffable is ineffable. Therefore it is indeed “hard to frame, in matter-moulded forms of speech,” the meaning of what has been won at the instant of the mystical vision. This difficult task is, in fact, a self-representative and infinite task. For it is the task of endless denial even of every previous act of denial. The only word as to the Absolute must be Neti, Neti, — It is not so, not so. But this only word needs endless repetition in new forms. The Absolute, if you will, was not well reported when we just gave, as the reason for the truth of the mystical insight, the fact that thought found itself at rest in the presence of God. For the thought really finds not itself, at all. It finds, as the truth, only its own Other. But in what way does it find its Other as the truth? Answer, By seeing, in the endless process of its own failure, the necessity of its own defeat, — the need of Another. So then — as we afresh observe — thought does know itself as a failure. It does represent to itself its own defeat. It does, then, learn, by a dialectic process, to comprehend its own lying nature. But herewith we return to our starting-point, and can only continue the same process without end.
In brief, mysticism turns upon a recognition of the failure of all thinking to grasp Reality. But this recognition is itself ​thought’s own work. Thought is, so far, a system which represents to itself its own nature, — as a nature doomed to failure. If you try to express this recognition, however, not as thought's work, but as a direct revelation, in a merely immediate experience, of a final fact, you at once rediscover that this fact is final only if it is known, as in contrast to the failure of thought. The failure of thought must, therefore, once more be known to thought. But such self-knowledge on thought’s part can only be won through the ineffable experience; and so you proceed back and forth without end. The reason for this particular endless chain is that mysticism turns upon a process whereby something, namely, thought, is to represent to itself its own negation and defeat. The consequence is a self-representative system of failure, in which every new attempt, based upon the failure of the former attempts to win the truth, itself involves the process of transcending the former failure by means of the very principle whose failure is to be observed.
And now, at last, let us ask, Does Mr. Bradley’s Absolute escape the common fate of all of our conceptions of Being? Is Mr. Bradley’s Absolute alone exempt from being a self-representative system of the type here in question?
I am obliged to answer this question in the negative. Mr. Bradley’s account of the Absolute often comes near to the use of mystical formulations, but Mr. Bradley is of course no mystic; and nobody knows better than he the self-contradictions inherent in the effort to view the real as a simple unity, without real internal multiplicity. As we have seen, Mr. Bradley’s Absolute is One, and yet does possess, as its own, all the manifoldness of the world of Appearance. The central difficulty of metaphysics, for Mr. Bradley, lies in the fact that we do not know how, in the Absolute, the One and the Many are reconciled. But that they both are in the Real is certain. Reality is explicitly called by Mr. Bradley a System. “We insist that all Reality must keep a certain character. The whole of its contents must be experience; they must come together into one system, and this unity itself must be expe​rience. It must include and must harmonize every possible fragment of appearance” (op. cit., p. 548). “Reality is one experience, self-pervading, and superior to mere relations” (p. 552). Now that Reality, while a “system,” is to be viewed as experience, this assertion is due to Mr. Bradley's definition of what it is to be real. “I mean that to be real is to be indissolubly one with sentience. It is to be something which comes as a feature and aspect within one whole of feeling, something which, except as an integral aspect of such sentience, has no meaning at all” (p. 146). “You cannot find fact unless in unity with sentience, and one cannot in the end be divided from the other, either actually or in idea.”
Now this account of the Absolute must of course be taken literally. It is not a speech about an Unknowable. It is, indeed, not an effort to tell how the unity is accomplished in detail. But it is a general, and by hypothesis a true account, of what the final unity must accomplish. We have therefore a right to observe that Mr. Bradley’s Absolute, however much above our poor relational way of thinking its unity may be, really has two aspects that, although inseparable, are still distinguishable. The varieties of the world are somehow “absorbed,” or “rearranged,” in the unity of the Absolute Experience. This is one aspect. But the other aspect is that, since this absorption itself is real, — is a fact, — and since to be real is to be one with sentience, the fact that the absorption occurs, that the One and the Many are harmonized, and that the Absolute is what it is, is also a fact presented within the sentient experience of the Absolute. It is not, then, that the rivers of Appearance merely flow into the silent sea of Reality, and are there lost. No; this sentient Absolute, by hypothesis, feels, experiences, is aware, that it thus absorbs its differences. In general, whatever the Absolute is, its experience must make manifest to itself. For either this is true, or else Mr. Bradley’s definition of Reality is meaningless. Let A be any character of the Absolute. Then the fact that A is a character of the Absolute, as such, and not of the mere appearances, is also a genuine fact. As such, it is a fact experienced. ​
The Absolute therefore must not merely be A, but experience itself, as possessing the character of A. It is, for instance, “above relations.” If this is a fact, and if this statement is true of the Absolute, then the Absolute must experience that it is above relations. For Mr. Bradley’s definition of Reality requires this consequence. The Absolute of Mr. Bradley must not, like the mystical Absolute, merely ignore the relations as illusion. It must experience their “transformation” as a fact, — and as its own fact. Or, again, the Absolute is that in which thought has been “taken up” and “transformed,” so that it is no longer “mere thought.” Well, this too is to be a fact. In consequence of Mr. Bradley’s definition of what he means by the word “real,” this fact must take its place amongst the totality of fact that is in its wholeness experienced. The Absolute, then, experiences itself as the absorber and transmuter of thought. Or, yet again, the Absolute is so much above “personality” that Mr. Bradley (p. 532) finds “intellectually dishonest” “most of those” who insist upon regarding the Absolute as personal. Well, this transcendence of personality is a fact. But “Reality must be one experience; and to doubt this conclusion is impossible.” “Show me your idea of an Other, not a part of experience, and I will show you at once that it is, throughout and wholly, nothing else at all.” Hence, the fact that the Absolute transcends personality is a fact that the Absolute itself experiences as its own fact, and is “nothing else at all” except such a fact.
As we have before learned, the category of the Self is far too base, in Mr. Bradley’s opinion, to be Reality, and must be mere appearance. The Absolute, then, is above the Self, and above any form of mere selfhood. The fact that it is thus above selfhood is something “not other than experience”; but is wholly experience, and is the Absolute Experience itself. In fine, then, the Absolute, in Mr. Bradley’s view, knows itself so well, — experiences so fully its own nature, — that it sees itself to be no Self, but to be a self-absorber, “self-pervading” to be sure (p. 552), and “self-existent,” but aware ​of itself, in the end, as something in which there is no real Self to be aware of. Or, in other words, the Absolute is really aware of itself as being not Reality, but Appearance, just in so far as it is a Self. Meanwhile, of course, this Absolute experiences, also, the fact that it is an “individual”; that it is a “system”; that it “holds all content in an individual experience”; that “no feeling or thought of any kind can fall outside its limits” (p. 147); that it “stands above and not below its internal distinctions” (p. 533); that “it is not the indifference, but the concrete identity of all extremes.” For all these statements are said by Mr. Bradley, in various places, to be accounts of what the Absolute really is. But if the Absolute is all these things, it can be so only in case it experiences itself as the possessor of these characters. Yet all the concrete self-possession of the Absolute remains something above Self; and apparently the Absolute thus knows itself to be, as a Self, quite out of its own sight!
Now in vain does one endeavor to assert all this, and yet to add that we know not how, in detail, all this can be true of the Absolute. We know, at all events, that apart from what is flatly self-contradictory in the foregoing expressions, Mr. Bradley’s Absolute is a self-representative system, which views itself as the possessor of what, through all the unity, remains still in one aspect another than itself, namely, the whole world of Appearance. And we know, therefore, that the Absolute, despite all Mr. Bradley’s objections to the Self, escapes from selfhood and from all that selfhood implies, or even transcends selfhood, only by remaining to the end a Self. In other words, it really escapes from selfhood in no genuine fashion whatever. For it can escape from selfhood only by experiencing, as its own, this, its own escape. This consequence is clear. Whatever is in the Absolute is experienced doubly. Namely, what is there is experienced, and that this content is experienced by the Absolute itself, — this final fact is also experienced. Hence, the whole Absolute must be infinite in precisely Dedekind’s positive sense of the term. Mr. Bradley’s Absolute is a Kette in the same sense as every other ​fundamental metaphysical conception. For it is a self-experiencing and, therefore, self-representative system.
I conclude, then, so far, that by no device can we avoid conceiving the realm of Being as infinite in precisely the positive sense, now so fully illustrated. The Universe, as Subject-Object, contains a complete and perfect image, or view of itself. Hence it is, in structure, at once One, as a single system, and also an endless Kette. Its form is that of a Self. To observe this fact is simply to reflect upon the most elementary and fundamental implications of the concept of Being. The Logic of Being has, as a central theorem, the assertion, Whatever is, is apart of a self-imaged system, of the type herein discussed. This truth is common property for all, whether realists or idealists, whether sceptics or dogmatists. And hence our trivial illustration of the ideally perfect map of England within England, turns out to be, after all, a type and image of the universal constitution of things. I am obliged to regard this result as of the greatest weight for any metaphysical enterprise. No philosophy that wholly ignores this ele​mentary fact can be called rational. And hereby we have indeed found a sense in which the “endless fission” of Mr. Bradley’s analysis expresses not mere Appearance but Being. Here is a law not only of Thought but also of Reality. Here is the true union of the One and the Many. Here is a multiplicity that is not “absorbed” or “transmuted,” but retained by the Absolute. And it is a multiplicity of Individual facts that are still One in the Absolute.
Section IV. Infinity, Determinateness, and Individuality
Despite all the foregoing considerations, however, we have still to face the objection that, even if these constructions be regarded as self-evident products of Thought, they, nevertheless, simply cannot be genuinely true of the final nature of Reality and must somehow be fallacious. For, from Mr. Bradley’s side, it would be maintained that however inevitable the seeming of these endless processes, they become self-contradictory precisely when you take them to be real and yet endless. For who knows not the Aristotelian arguments, so often repeated in later thought, against the actual Infinite? Is not the complete Infinite the very type of a logical “monster?” Is not the very conception a self-contradiction? If thought, then, has to conceive Reality as infinite, so much the worse, one may say, for thought. The Real, whatever its appearance, cannot in itself be endless.
I. The Objections to the Actually Infinite
It is necessary to consider such arguments by themselves, for the moment, and apart from the foregoing considerations. Let us, then, briefly develope some of these often repeated reasons on account of which so many assert that Reality cannot be an infinite system at all. ​
One may begin with the case as Aristotle first stated it, in the Third Book of the Physics, and elsewhere. There can, indeed, exist a Reality that permits us, if we choose to number its parts, to distinguish within it what we call elements, in such wise that we can never end the process of numbering them. So space is for us capable of infinite, that is, of indefinite division, if you choose to try to take it to pieces. But such divisibility is a mere possibility. Space, if real, is not endlessly divided. It is only in potentia divisible so far as you please to conceive its parts. The limitless exists, therefore, only in potentia; λείπεται οὖν δυνάμει εἶναι τὸ ἄπειρον. For were space actually either made up of endless parts, or in such wise real as to be infinitely great, there would result the contradiction of an actually infinite number as the number of the parts of a real collection. But a number actually infinite is contradictory; for it then could not be counted; it would have no determinate size; it would possess no totality; and it would so be formless and meaningless. Again, were any one portion of the world’s material substance infinite, how could room be left for the other portions? Were the whole infinite, how could it be a whole at all? For any whole of reality is limited by its own form, and by the fact that, as an actual whole, it is perfectly determinate. The difficulty as to the infinite must be solved, then, by saying that what is real forms a definite and, for that reason, a finite totality; while within this totality there may be aspects which our thought discovers to be, in this or that respect, inexhaustible through any process of counting that follows some abstractly possible line of our own subjective distinctions or syntheses. We can say, of such aspects of the world, that you may go on as long as you please, in counting their special type of conceived complexities, without ever reaching the end. But this endlessness is potential only, and never actual.
These well-known Aristotelian considerations have formed the basis of every argument against the actual infinite in later thought. The special point of attack has, however, often shifted. In general, as the later arguments have repeatedly ​urged (quite in Aristotle’s spirit), the infinitely complex, it real, must be knowable only through some finished synthesis of knowledge. But a finished synthesis is inconsistent (so one affirms) with the endlessness of the series of facts to be synthesized; and hence an infinite collection, if it existed, would be unknowable. On the other hand, an infinite collection, if real apart from knowledge, could be conceived to be altered by depriving it of some, or of a considerable fraction, of its constituent elements. The collection thus reduced (so one has often argued) would be at once finite (since it would have lost some of its members) and infinite, since no finite number would be equal to exhausting the remaining portion. Hence the reduced collection and, therefore, the original collection must be of a contradictory nature, and so impossible. In a variation of this argument often used, one employs, as an image, some such instance as an inextensible rod, one end of which shall be in my hands, while I shall be supposed to believe that the rod, which stretches out of my sight into the heavens, is infinitely long, as well as quite incapable of being anywhere stretched. Suppose the rod hereupon drawn, or, if you please, anyway mysteriously moved, a foot towards me at this end. If I am to believe in the infinity and inextensibility of the rod, I shall believe that the whole of the rod, and every part thereof, is now a foot nearer to me than before. But in that case the furthest portion of the rod must also be a foot nearer than before, or must have been “drawn in out of the infinite,” as one writer has stated the case. It can therefore no longer be an infinite rod. Hence, it was not actually infinite before the drawing in of this end.
All such arguments insist, either upon the supposed fact ​that our own conception of an infinite series is necessarily a conception of an indefinite and, therefore, of an essentially incomplete sequence, or else upon the assertion that an infinite collection, if viewed as real, would prove to be in itself of a quantitatively indefinite and changeable character. In the one case, the argument continues by showing that an indefinite and incomplete sequence is incapable of being taken to be a finished reality beyond our thought. In the other case, one insists that the quantitatively indefinite collection, if viewed as real, would stand in conflict with the very notion of reality, since the real is, as such, the determinate. “The essence of number,” says Mr. Bosanquet, “is to construct a finite whole out of homogeneous units.” “An infinite number would be a number which is no particular number; for every particular number is finite.” “An infinite series . . . is not anything which we can represent in the form of number, and therefore cannot be, quâ infinite series, a fact in our world. . . . Our constructive judgment requires parts and a whole to give it meaning. Parts unrelated to any whole cannot be judged real by our thought. Their significance is gone and they are parts of nothing.”
More detailed, in the application of the general charge of indefiniteness thus made against the conception of the infinite collections, are the often used arguments such as exemplify how, if infinite collections are possible at all, one infinite must be greater than another, while yet, as infinite and determinate, all the boundless collections must (so one supposes) be equal. Or, again, in a similar spirit, one has pointed out that, by virtue of the properties which we have deliberately attributed to the Ketten of the foregoing discussion, two infinite collections, if they existed, would be, in various senses of the term equal, at once equal and unequal to each other, or would ​contradict the axiom as to the whole and the part. These arguments can be illustrated by an endless list of examples, drawn from the realm of discrete collections of objects, as well as from cases where limitless extended lines, surfaces, or volumes are in question, and from cases where limitless divisibility is to be exemplified. The variety of the examples, however, need not confuse one as to the main issue. What is brought out, in every case, is that the infinite collections or multitudes, if real at all, must be in paradoxical contrast to all finite multitudes, and must also be in such contrast as to seem, at first sight, either quite indeterminate or else hopelessly incomplete, and, in either case, incapable of reality.
Upon a somewhat different basis rest a series of arguments which have more novelty, just because they are due to the experience of the modern exact sciences. In the seventeenth century one of the greatest methodical advances ever made in the history of descriptive science occurred, when the so-called Infinitesimal Calculus was invented. The Newtonian name, Fluxions, used for the objects to whose calculation the new science was devoted, indicated better than much of the more recent terminology, that one principal purpose of this advance in method, was to enable mathematical exactness to be used in the description of continuously varying quantities. But the generalization which was made when the Calculus appeared had been the outcome of a long series of studies of quantity, both temporal and spatial. And the Calculus brought under one method of treatment, not only the problems about continuous processes of actual change, such as motions, or other continuous physical alterations, but also problems regarding the properties, the relations, the lengths, and the areas of curves, and ​regarding the corresponding features of geometrical surfaces and solids. For, in all these objects alike, either continuous alterations, or else characters that, although matters of spatial coexistence, may be ideally expressed in terms of such continuous alterations, fell within the range of the methods of the Calculus.
The new method, however, seemed to involve, at first, the conception both of “infinitely small” quantities, and of devices whereby an “infinite number” of such quantities could be summed together, or otherwise submitted to computation. The science of the continuous, in the realm of geometrical forms, as well as in the realm of physical changes, thus seemed to depend upon the conception both of the infinitely small and of the infinitely great; and the successful application of the results of such science in the realm of physics, was sometimes used as a proof that nature contains actually infinite and actually infinitesimal collections or magnitudes. But the early methods of the Infinitesimal Calculus were not free from inexactness, and led, upon occasion, to actually false conclusions. Hence, the paradoxes apparently involved in the logical bases of the science attracted more and more critical attention, as time went on; and, as a consequence, within the present century, the whole method of the Calculus has been repeatedly and carefully revised, — with the result, to be sure, that the conceptions of the actually infinite, in the sense here in question, and the actually infinitesimal (in the older sense of the term), have been banished from the principal modern text-books of both the Differential and the Integral Calculus. The terms, “Infinite” and “Infinitesimal,” have been, indeed, very generally retained in such text-books for the sake of conciseness of expression; but with a definition that wholly avoids all the problems which our foregoing discussion has raised. The infinite and the infinitesimal of the Calculus can, therefore, no longer be cited in favor of a theory of the “actually Infinite.”
In the world of varying quantities, namely, it often happens that, by the terms of definition of a given problem, ​you have upon your hands a varying quantity (call it X) which, consistently with these terms, you are able to make, or to assume, as large as you please. In such cases, if some one else is supposed to have predesignated, as the value of X, any definite magnitude that he pleases, say X$_1$ then you are at liberty, under the conditions of the problem, to assume the value of X as larger still, i.e. as greater than any such previously assigned definite value X$_1$. Now, whenever the variable X has this character, in a given problem, then, according to the fashion of speech used in the Calculus, you may define X either simply as infinite, or as capable of being increased to infinity; and in the Calculus you are indeed often enough interested in learning what happens to some quantity whose value depends upon X, when X thus increases without limit, or, as they briefly say, becomes infinite. But in all such cases the term infinite, as used in the modern text-books of the Calculus, is, by definition, simply an abbreviation for the whole conception just defined. The variable X need not even be, at any moment, actually at all large in order to be, in this sense, infinite. It only so varies that, consistently with the conditions of the problem, it can be made larger than a predesignated value, whatever that value may be. And the Calculus is simply often interested in computing the consequences of such a manner of variation on the part of X.
Now, unquestionably a quantity that is called infinite in this sense is not the actually infinite against which Aristotle argued. It is merely the limitlessly increasing variable or the potentially infinite magnitude which he willingly admitted as a valid conception. A parallel definition of the infinitesimal is even more frequently employed in the modern text-books of the Calculus, just because the infinitesimal is mentioned more frequently than the infinite. In this sense, a variable magnitude is infinitesimal merely when it can be made and kept as small as we will, consistently with the conditions of the problem in which it appears. Thus neither the infinite nor the infinitesimal of the modern treatment of the Calculus has any ​fixed character, as a finished or finally given quantity, nor any character which could be defined as a determinately real somewhat, apart from our defining thought, and apart from the conditions of a given problem. The Calculus is deeply interested in computing results of such variation without limit; but as a branch of mathematics, it is, in fact, not at all directly interested in our present problem about the actually infinite.
Now, this result of the whole experience of the students of the Calculus with the logic of their own science, — this outcome of the modern critical restudy of the bases of the science of the continuously variable quantities, — tends of itself to indicate (as one may say, and as objectors to the actually Infinite have often said) that the conception of the actually Infinite, formerly confounded with the conceptions lying at the bases of the Calculus, is, as a fact, not only in this region, but everywhere, scientifically superfluous; while the conception of the Infinite merely in potentia, originally defended by Aristotle, thus triumphs in the very realm where, for a time, its rival seemed to have found a firm foothold.
Yet it has indeed to be observed that, from the mathematical point of view, not the questions of the Calculus, but certain decidedly special problems of the Theory of Numbers, and of the modern Theory of Functions, have given the mathematical basis for these newer efforts towards an exact and positive definition of the Infinite. As a fact, in our foregoing statement of the merely prima facie case for the recent definition of the positively Infinite, we have deliberately re​framed from making any mention of the special problems about continuity, or of the conceptions of the Calculus. And it has also been noted that Cantor, who has done so much to make specific the positive concept of das Eigentlich-Unendliche, and who has also given us one of the very first of the exact definitions of continuous quantity ever discovered, himself rejects the actually infinitesimal quantities as quite impossible; and does so quite as vigorously as he accepts and defends the actually infinite quantities; so that he fully agrees that the infinitesimal must remain where the Calculus leaves it, namely, simply the variable small at will. It must therefore be distinctly understood that, in the discussion of the reality of the infinite quantities and multitudes, appeal need no longer be made to the conceptions of quantity peculiar to the Calculus; while, in general, the majority of those concerned in this inquiry expressly admit that the logic of the Calculus is quite independent of the present issue, and that the infinite of the Calculus is simply the variable large at will, which therefore need not be at any moment, even notably large at all.
And now, finally, there is also urged against any conception of the actually Infinite the well-known consideration that the conception of such infinity involves an empty and worthless repetition of the same, over and over, — a mere “counting when there is nothing to count,” or, in the realm of explicit reflection, a vain observation that I am I, and that I am I, again, even in saying that I am I, — or an equally inane insistence ​that I know, and know that I know, and so on. The non-mathematical often dislike numbers, especially the large ones, and therefore easily make light of a wisdom that seems only to count, in monotonous inefficacy. Even the more reflective thinkers often believe, with Spinoza, that knowing that I know can imply nothing essentially new, at all events after the reflection has been two or three times repeated. The Hindoo imagination, with its love for large numbers, often strikes the Western mind as childish. And in all such cases, since mere size, as such, rightly seems unworthy of the admiration that it has excited in untrained minds, it has appeared to many to be the more rational thing to say that wisdom involves rather Hegel’s Rückkehr aus der unendlichen Flucht than any acceptance of the notion that infinite magnitudes or multitudes can be real.
II. The Infinite as One Aspect only of Being
All the foregoing objections to the conception of the actually infinite rest, in large measure, upon a true and perfectly relevant principle. As a fact, what is real is ipso facto determinate and individual. It is this for the reasons pointed out in the closing lectures of the present series. It is this because it is such that No Other can take its place. The Real is the final, the determinate, the totality. And now, not only is this principle valid, but it is indeed supreme in every metaphysical inquiry. And therefore we shall, to be sure, find it true that in case, despite all the foregoing highly important objections, we succeed in reconciling infinity with determinateness, we shall still be unable to assert that the Reality is anything merely infinite. For infinity, as such, is at best a character, — a feature having the value of an universal. If the Absolute is in any sense an infinite system, it is certainly also an unique and individual system; and its uniqueness involves something very clearly distinguishable from its mere infinity. The Absolute is, in its determinate Reality, certainly exclusive of an infinity of mere possibilities. In this respect I shall here simply repeat the position taken in the discussion supple​mentary to the book called the Conception of God. It is, then, perfectly true, for me, as for the opponents of the actual Infinite, that much must be viewed as, in the abstract, “possible,” which is nowhere determinately presented in any final experience of the fulfilment of truth. The special illustration used, in my former book, to exemplify this fact, namely, the illustration of the points on the continuous line, points which are “possible” in an infinitely infinite collection of ways, but which, however presented, cannot exhaustively constitute the determinate continuity of the line, — this, I say, is an illustration involving other problems besides those of the actual Infinite. The existence of the line, taken as a geometrical fact, contains more than the possible multitudes of multitudes of the points on the line can ever express. And this more includes, also, a something more determinate than the multitudes of the points can conceivably present. Hence, as I argued in my former book, and as I still deliberately maintain, the Absolute cannot experience the nature of the line by merely exhausting any infinitude of the points. But to this illustration I can here devote no further space, since the discussion of continuity, and especially of the geometrical continuum, lies outside of the scope of this paper. It is quite consistent, however, to hold, as I do, that while the Absolute indeed, by reason of its determinateness, excludes and must exclude infinitely infinite “bare possibilities,” known to mere thought, from presentation in any individual way, except as ideas of excluded objects, the Absolute still finds present, in the individual whole of its Selfhood, an actually infinite, because self-representative, system of experienced fact. The points on the line, then, if my former illustration is indeed well chosen, are not exhaustively presented, as constituting the whole line, in any experience, whatever, Absolute or relative. But this, as we now have to see, is not because the actually Infinite is, to the Absolute, something unrepresented, but because the determinate geometrical continuity of the individual line is something more, and more determinate, than any infinitude of ​points can express. And this individuality of the line I can and do express by saying that, even to a final view, the essence of the individual continuity of any one line involves the “bare possibility” of systems of ideal points over and above any that are found present in this final experience of the line. Even if the Absolute, then, observes infinitely infinite collections of points, it sees that the individual continuity of the line is more than they present. This I still assert.
In general, as we shall see, by virtue of what here follows, a fair account of the completeness of the Absolute must be just to two aspects. They are the ultimate aspects of Reality. Their union constitutes, once more, the world-knot. And the reason of their union is the one made explicit in our seventh Lecture. The Real is determinate and individual; and the Real is expressive of all that universal ideas, taken in their wholeness, actually demand, or mean, as their absolutely satisfactory fulfilment. In this twofold thesis, as I understand, I am wholly in agreement with Mr. Bradley. But I differ from him by maintaining that we know more than he admits concerning how the Real combines these two aspects. I maintain, then, with a full consciousness of the paradoxes involved, that the Reality is indeed a Self, whatever else it is or is not. For the Absolute, as I insist, would have to be not apparently, but really a Self, even in order to be (as Mr. Bradley seems to imagine his Absolute) a sort of self-absorbing sponge, that endlessly sucked in, and “transformed,” its own selfhood, until nothing was left of itself but the mere empty spaces where the absorbent Self had been. For the category of Self is indeed immortal. Deny it, and, in denying, you affirm it. As a fact, however, the Absolute is no sponge. It is not a cryptic or self-ashamed, but an absolutely self-expressive self. And to see how it can be so without contradiction, is simply to see how the concept of the actually Infinite, despite all the foregoing objections, is not self-contradictory, is not indeterminate, is not merely based upon wearisome reflections of the same; but is a positive and concrete conception, quite capable of individual embodiment. This is what we shall see in what ​here follows. The concept of the actually Infinite once in general vindicated from the charge of self-contradiction, all objection to conceiving the Absolute as a Self will vanish; and the transparent union of the One and the Many, which reflective thought has already shown us within its own realm, will become the universal law of Being.
But, on the other hand, if the Absolute is a Self, and, as such, an Infinite, this does not mean that it is anything yon please, or that it is at once all possible things, or that it views its realm of fact as having all possible characters at once, and hence as having no character in particular. This Self, and no Other, this world and no Other, this totality of experience, and nothing else, — such is what has to be presented when the Real is known as the real. The Infinite will have to be also a determinate Infinite, a self-selected case of its type. For the world as merely thought, or as merely defined in idea, is the world viewed with an abstract or bare universality, and as that which still demands its Other, and which refers to that Other as valid and possible. The world of thought is, as such, an effort to characterize this Other, to imitate it, to correspond to it, and, of course, if so may be, to find it. Hence the world of mere thought has, as its very life, a principle of dissatisfaction; and when it conceives its object as the Truth, it defines, in the object, only the sense in which there is to be agreement or correspondence between the object and the thought. Consequently, an idea taken merely as an imitation of another, or taken as having an external meaning, expresses the Truth only as a barely universal validity. And one who merely takes thought as thought conceives the shadow land which shall, nevertheless, somehow have the value of a standard. In that realm, — the realm of mere validity, — all is mere character, and type, and possibility. And thought is the endlessly restless definition of another, and yet another. And this is true even when thought conceives an Infinite. Hence, infinity, as merely conceived, is indeed not yet Reality as Reality.
Now, the opponents of the actual Infinite, ever since Aristotle, have always seen, and rightly seen, that, as defined by ​mere thinking about external meanings, the world is not finally defined. The restlessly infinite, as such, they have condemned as in so far unreal. For whoever sees reality, sees that which has no Other like itself, which seeks no Other to define its being, which is itself no mere correspondence between one object and another, and, despite its unquestionable character as the fulfilment of thought, no mere agreement between a thought and a fact. The Heal, then, has not the character which bare thought, as such, emphasizes, — the character of being essentially incomplete. It has wholeness. Its meaning is internal and not external. Therefore, it is indeed a finished fact. It cannot, then, be infinite if infinity implies incompleteness.
But, once more, is the Real for that reason finite? Because it excludes the search for another beyond itself, does it therefore contain no infinite wealth of presented content within itself? This is precisely the question. In emphasizing the exclusiveness of the Real we must be just to the fact that, whatever it excludes, it cannot, from our point of view, be poorer, less wealthy, less manifold in genuine meaning, than the false Other, which its reality reduces to a bare and unrealized possibility of thought. That the world is what it determinately is, means, from our point of view, that its being excludes an infinitely complex system of “barely possible” other contents, which, just because they are excluded from Reality, are conceived by a thought such that not all of its “barely possible” ideal objects could conceivably be actualized at once. In this sense, for us, just as for the partisans of the barely possible and unactualized infinite, there are indeed ideas of infinitely numerous facts which remain, from an Absolute point of view, hypotheses contrary to fact. We agree, moreover, with our opponents, that no process expresses reality in so far as this process merely seeks, without end, for another and another object or fact. Hence, for us, as for our opponents, the Infinite, when taken merely as an endless ​process, is falsely taken. As merely that which you cannot exhaust by counting, the Infinite is, by the hypothesis, never found, presented or completed, so long as you simply count. Hence we wholly agree that the Infinite, just in so far as it is viewed as indeterminate, incomplete, or merely endless, is not rightly viewed; and that in so far it is indeed unreal. We also fully agree that Absolute knowledge unquestionably recognizes, as an object for its own relatively abstract thought, a distinctly unreal Infinite, namely, the Infinite of the excluded ideal “bare possibilities” aforesaid. In all this we quite agree with our opponents, and prize their insistence upon the determinateness of the final truth.
Nevertheless, we shall perforce insist upon these theses: —
(1) The true Infinite, both in multitude and in organization, although in one sense endless, and so incapable in that sense of being completely grasped, is in another and precise sense something perfectly determinate. Nor is it a mere monotonous repetition of the same, over and over. Each of its determinations has individuality, uniqueness, and novelty about its own nature.
(2) This determinateness is a character which, indeed, includes and involves the endlessness of an infinite series; but the mere endlessness of the series is not its primary character, being simply a negatively stated result of the self-representative character of the whole system.
(3) The endlessness of the series means that by no merely successive process of counting, in God or in man, is its wholeness ever exhausted.
(4) In consequence, the whole endless series, in so far as it is a reality, must be present, as a determinate order, but also all at once, to the Absolute Experience. It is the process of successive counting, as such, that remains, to the end, incomplete, so as to imply that its own possibilities are not yet realized. Hence, the recurrent processes of thought reveal eternal truth about the infinite constitution of real Being, — their everlasting pursued Other; but themselves, — as mere processes in time, — they are not that Other. Their true Other ​is, therefore, that self-representative System of which they are at once portions, imitations, and expressions.
(5) The Reality is such a self-represented and infinite system. And therein lies the basis of its very union, within itself of the One and the Many. For the one purpose of self-representation demands an infinite multiplicity to express it; while no multiplicity is reducible to unity except through processes involving self-representation.
(6) And, nevertheless, the Real is exclusive as well as inclusive. On the side of its thought the Absolute does conceive a barely possible infinity, other than the real infinity, — a possible world, whose characters, as universal characters, are present to the Absolute, and are known by virtue of the fact that the Absolute also thinks. But these possibilities are excluded by reason of their conflict with the Absolute Will.
(7) Yet, in meaning, the infinite Reality, as present, is richer than the infinity of bare possibilities that are excluded. But for that very reason the Reality presented, in the final and determinate experience of the Absolute, cannot be less than infinitely wealthy, both in its content and in its order. Its unity in its wholeness, and its infinite variety in expression, are both of an individual character. The constituent individuals are not “absorbed” or “transmuted” in the whole. The whole is One Self; but therefore is all its own constitution equally necessary to its Selfhood. Hence it is an Individual of Individuals.
With less of complexity and, if you please, with less of paradox, no theory of Being can be rendered coherent. Our present purpose is to bring these various aspects of the twofold nature of Being, as Infinite Being and as Determinate Being, to light and to definition.
We shall return, therefore, to the consideration of the main points made by our objectors, and, as we meet them shall even thereby justify, without needing formally to repeat, our various theses. ​
III. The Infinite as Determinate
The principal one amongst all the traditional objections to the Infinite is, as we have seen, the thought that the Infinite, as such, is merely an endlessly sought or an endlessly incomplete somewhat; while the real, as such, is very rightly to be viewed as the determinate. Hence, the actually Infinite, one insists, would be at once determinate and indeterminate, and so would be contradictory.
Now, whatever may be said about the actually Infinite, we have already seen that the infinite of the merely conceptual but valid type, the infinite of the realm of mathematical possibilities, is certainly as determinate a conception as any merely universal idea can ever be, and, as thus determinate, involves no contradiction whatever. Cling to our Third Conception of Reality; and then, indeed, there can be no doubt whatever that the Infinite is real. For there is no contradiction, there is only a necessarily valid truth involved in saying that to any whole number r, however large, there inevitably does correspond one number, and only one, which stands amongst all numbers as the rth member in the ordered series of whole numbers that are squares, or in the ordered series of the cubes, or in the ordered series, if you please, of numbers of the form of ɑ$^{100}$ or ɑ$^{1000}$ , where the exponent is fixed, but where the number that is to be raised to the power indicated takes successively the series of values, 1, 2, 3, . . . r. The inevitable result is that to every whole number r, without a possible exception, there corresponds, in the realm of validity, and corresponds uniquely, just that particular whole number which you get if you raise r to the second, third, or hundredth, or thousandth power. Moreover, this ideal ordering of all the whole numbers, without exception, in a one-to-one relation (let us say) to their own thousandth powers, is in such wise predetermined by the very nature of number that, if you undertake to calculate the thousandth power (let us say) of the number 80,000,000, your result is in no wise left to you, as a bare possibility that your private will can capriciously decide. ​The result is lawfully fixed beforehand by the very essence of mathematical validity, i.e. by the very expression of your own final Will in its wholeness. Your calculation can only bring this result to light in your own private experience of numbers. It is an arithmetically true result quite apart from your instantaneous observation. Its triviality, as a mere matter for computation, is not now in question. Its eternal validity, however, interests us. Every number, then, speaking in terms of mathematical validity, already has its own thousandth power, whether you chance to have observed or to have computed that thousandth power or not. Yet, in any finite collection of whole numbers, those which are the thousandth powers of the whole numbers constitute at most an incomparably minute part of the whole collection. But, on the other hand, viewed with reference to the logically valid truth about all the numbers, these powers, as a mere part of the whole series of whole numbers, still occupy such a logically predetermined place that they are set, by their values, in a one-to-one relation to the members of the whole series; so that not a small portion, but absolutely all of the whole numbers, have their correspondents among the thousandth powers. Now, all these are facts of thought, just as valid as any conceptual constructions, however simple, and just as true as that 2 + 2 = 4. And by themselves these truths, trivial if you please, are, in all their wearisomeness, not "monstrous" at all, but simply the necessary consequences of an exact conception of the nature of number.
“Monstrous, however,” so one may reply, “would be the assertion that in any real world there could be determinate facts corresponding to all this merely ideal complexity.” On the contrary, as we might at once retort, it would be monstrous if all these truths were merely “valid,” in a purely formal way, without any correspondent facts whatever in the real world. Can mere validity hang in the void? Must it not possess a determinate basis?
The issue, then, is at once the issue about the Third Conception of Being in our list. Either the truth, the world of mere forms, can indeed hang in the void, valid, but nowhere ​concrete, or else, just because the infinite is valid, it has its place, as fact, in the determinate experience of the Absolute. At all events, the Infinite, in such cases as have just been cited, is something quite as determinately valid as any barely universal conception can be. And unless it is true that two and two would make four in a world where no experience ever observed the fact, it is true that the infinitely numerous properties of the numbers need some concrete representation.
I grant, however, that these are but preliminary considerations. Every validity, as a bare universal, must be a reflectively abstract expression of a fact that ultimately exists in individual embodiment in the Absolute. Yet, on the other hand, you cannot predetermine the nature of this individual expression merely by pointing out that the possibilities in question appear to us endless. For the endlessness might be one of those matters of bare external conjunction of which Mr. Bradley so often speaks. Thus space appears to us endless. I fully grant that we are not warranted in making any one assertion about the Absolute view of the meaning of our spatial experience, by virtue of the mere fact that going on and on endlessly in space appears to us possible, and that, consequently, we can define propositions that would be valid if this possibility is endlessly realized by the Absolute. In passing from the Third to the Fourth Conception of Being, what we did was to see that nothing can be valid unless a determinate individual experience has present to it all that gives warrant for this validity. Because our fleeting experience never gives such final warrant, we are forced to seek for the ground and the basis of any valid truth once recognized by us, and to seek this basis in a realm that is Other than our own experience as it comes to us. This Other is, finally, the Absolute in its wholeness. But we do not assert that the Absolute realizes our validity merely as we happen to think it.
When we regard any valid truth as implying a variety of valid assertions, all for us matters of conceived possible experience, we often take the Many, thus conceived by us, as a mere fact, an uncomprehended “conjunction.” I agree altogether with ​Mr. Bradley that such varieties might seem, to a higher experience, artificial, and that, as such, they might be “transmuted” even in coming to their unity in the higher view. For in such cases we never experience that these varieties are self-evidently what they seem to us. And our conception that they are many is associated with a confession of ignorance as to what they are. A good example of all this is furnished by our conception of what our own lives, or the course of human history, would have been, if certain critical events had never taken place. What, in such instances, we have on our hands is an ignorance as to the whole ground and meaning of the critical events themselves. A fuller knowledge of what they meant might render much of our speech about the “possibilities” in question obviously vain.
Determinate decisions of the will involve rendering invalid countless possibilities that, but for this choice, might have been entertained as valid. In such cases the nature of the rejected possibilities is sufficiently expressed, in concrete form, by the will that decides, if only it knows itself as deciding, and is fully conscious of how and why it decides. That Absolute insight would mean absolute decision, and so a refusal to get presented in experience endlessly numerous contents that, but for the decision, would have been possible, — this I maintain as a necessary aspect of the whole conception of individuality. Whoever knows not decisions that exclude, knows not Being. For apart from such exclusion of possibilities, one would face barely abstract universals, and would, therefore, still seek for Another. Our whole conception of Being agrees, then, with Mr. Bradley’s in insisting that the bare what, the idea as a mere thought, still pursuing, and imitatively characterizing its Other, not only does not face Being as Being, but can never, of itself, decide what its own final expression shall be. Thought must win satisfaction not as mere Thought, but also as decisive Will, determining itself to final expression in a way that the abstract universals of ​mere thinking can characterize, but never exhaust. Thus, and thus only, can be found that which admits of no Other. So far, then, it is indeed true that nothing is proved real merely by proving its abstract consistency as a mere idea taken apart from the rest of the world.
Or, again, the realm of validity is not exhausted by presented fact in the way suggested by one of Amadeus Hoffman’s most horrible fancies (I believe in the Elixiere des Teufels), according to which a hero, persistently beset by a double, always finds that, whenever he, in his relative strength, resists a great temptation, and avoids a crime, this miserable double, whom he all the while vaguely takes to be in a way himself, appears, — pale, wretched, fate-driven, — and does, or at least attempts, in very fact, the deed that the hero had rejected. No; whoever knows Being, finds himself satisfied in the presence of a will fulfilled, and needs no fate-driven other Self, no outcast double, to realize for him the possibilities whose validity he rejects. For in rejecting, he wins. And Being is a destruction as well as an accomplishment of Experience.
Upon all this I have elsewhere insisted. That the very essence of individuality is a Will that permits no Other to take the place of this fulfilment, — a Love that finds in this wholeness of life its own, — I have pointed out in an argument that the Tenth Lecture of the present course has merely summarized. And therefore I am perfectly prepared to admit that when we define as valid, in the realm of mathematical truth, an infinite wealth of ideal forms, we need not, on that account alone, and apart from other reasons, declare that the Absolute Life realizes these forms in their variety as defined by us. Their true meaning it must somehow get present to itself, — otherwise it would face Another of which it was essentially ignorant. But its realization of their meaning may well imply an exclusion of their variety, just in so far as that variety, when conceived by us, expresses our ignorance of ​what principle of multiplicity is here at work, of how the One and the Many here concerned are related, and of what decision of Will would give these forms a concrete meaning in the universal life.
It remains, then, returning to the typical case of the numbers, to see in what sense a determinate expression of their whole meaning can be found in the life of a Will that fulfils itself through exclusive decisions, but that does not ignore any genuinely significant aspect of the truth. For our Absolute is not in such wise exclusive of content as to impoverish its wealth of ideal characters; and, on the other hand, it is not in such wise inclusive of bare possibilities as to oppose to whatever fact it chooses as its own, the fatal Other deed of Amadeus Hoffman’s double-willed and distracted hero.
And here, of course, an opponent of the actual Infinite will be ready with the very common observation that the numbers are indeed, apart from the concrete objects numbered, of a trivial validity. “In a life,” he may say, “in a world of decisions and of concrete values, a barren contemplation of the properties of the numbers can have but a narrow place. Hence, no fulfilment of the hopeless task of wandering from number to number need be expected as a part of the Absolute life.”
Moreover, such an objector will insist that all these Ketten involve mere repetition of the same sort of experience over and over. “To carry such repetition to the infinite end, what purpose,” he will say, “can such an ideal fulfil?” The individual fulfilment of the meaning of the number-series, in the final view, may well, then, take the form of knowing that there are indeed numbers, that they are made in a certain way, that the plan of their order has a particular type, and that this type is exemplified thus and thus by a comparatively few concretely presented ideas of whole numbers. Otherwise, the numbers may be left as unrealized as are those other excluded possibilities of the Will exemplified.
But against this view one has next to point out that, observed a little more closely, even the numbers have characters not reducible to any limited collection of universal types. ​They do not prove to be a monotonous series of contents, involving mere repetition of the same ideas. On the contrary, to know them at all well, is to find in them properties involving the most varied and novel features, as you pass from number to number, or bring into synthesis various selected groups of numbers. Consider, for instance, the prime numbers. Distributed through the number-series in ways that are indeed capable of partial definition through general formulas, they still conform to no single known principle that enables us to determine, a priori, and in merely universal terms, exactly what and where each prime shall be. They have been discovered by an essentially empirical process which has now been extended, by the tabulators of the prime numbers, far into the millions. Yet the process much resembles any other empirical process. Its results are reported by the tabulators as the astronomers catalogue the stars. The primes have, as it were, relatively individual characters, which cannot be reduced to any barren repetition of the same thing over and over. One may call them uninteresting. But one must not judge the truth by one’s private dislike of mathematics, just as, of course, one must not exaggerate the importance of mere forms. Here, then, is one instance of endless novelty within the number-series.
But the real question is, How shall the genuine meaning of all this series of truths be in any way grasped, unless the insight which grasps is adequate to the endless wealth of novel, and relatively individual truth that the various numbers present as one passes on in the series? For the will cannot consciously decide against the further realization of certain types of possibility, unless it clearly knows their value. And this it must know in exhaustive, even if ideal and abstractly universal terms. Nobody can fairly tell what value in life numerical truth may possess, unless he first knows that truth. And the numbers whose ordered rationality is, for us men, the very basis of our exact science, show a wealth of truth ​that we find more and more baffling the further we go. The “perfect numbers” form a series that may be as full of interest, for all that I know, as the primes. The properties of the “Arithmetical Triangle” are linked in the most unexpected fashion with the laws of our statistical science, and with the nature of certain orderly combinations of vast importance in other branches of mathematical inquiry. Countless other combinations of numbers form topics, not only of numerous well-known plays and puzzles, but of scientific investigations whose character is actually adventurous, — so arduous is their course, and so full of unexpected bearings upon other branches of knowledge has been their outcome. Nobody amongst us can pretend to fathom the value for concrete science, and for life, that has yet to be derived from advances in the Theory of Numbers.
These, then, are mere hints of the inexhaustible properties of the number-series. I speak still as layman; but I am convinced that these significant properties are quite as inexhaustible as the number-series itself. Now, the value of such properties you can never tell until you see what they are. Their meaning in the life of reason can only be estimated when they are present. Hence, you can never wisely decide not to know them until you have first known them. But they are not to be known merely as the endless repetitions of the same over and over. Hence it is wholly vain to say, “Numbers come from counting, and counting is vain repetition of the same over and over.” Whoever views the numbers merely thus, knows not whereof he speaks. It is not “counting, with nothing to count”; it is finding what Order means, that is the task of a true Theory of Numbers.
As a fact, then, the number-series in its wholeness seems to be a realm not only of inexhaustible truth, but of a truth that possesses an everywhere relatively individual type. And its validity has relations that we, at present, but imperfectly know, and a rational value that appears to be fundamental in every orderly inquiry.
We can, then, neither assert that to all the varieties which ​our thought may chance to conceive as possible, there correspond just as many final facts for an Absolute Experience; nor yet can we, on the other hand, exclude from concrete presentation, as final facts, such wholes as include an infinite series, merely because, for us, if we do not take due account of mathematical truth, the series seems to involve the empty repetition of “one more” and “one more.” For, as Poincaré has so finely pointed out, in the article before cited, it is precisely the “reasoning by recurrence” which is, in mathematics, the endless source of new results. Hereby, in the combination of his previous results for the sake of new insight, the mathematician is preserved from mere “identities,” and gets novelties. The “reasoning by recurrence,” however, is that form of reasoning whereby one shows that if a given truth holds in n cases, it holds for the n + 1st case, and so for all cases. Such processes of passing to " one more " instance of a given type, are processes not of barren repetition, but of genuine progress to higher stages of knowledge.
Precisely so it is, too, if one takes account of that other aspect of ordered series which it has been one principal purpose of this paper to emphasize. The numbers have interested us, not from any Pythagorean bias, but because their Order is the expression, not only of a profoundly significant aspect of all law in the world, but of the very essence of Selfhood, when formally viewed. Now reflective selfhood, taken merely as the abstract series, I know, and I know that I know, etc., appears to be a vain repetition of the same over and over. But this it appears merely if you neglect the concrete content which every new reflection, when taken in synthesis with previous reflections, inevitably implies in case of every living subject-matter. A life that knows not itself differs from the same life conscious of itself, by lacking precisely the feature that distinguishes rational morality alike from innocence and from brutish naïveté. A knowledge that is self-possessed differs from an unreflective type of consciousness by having all the marks that separate insight from blind faith.
“Thus we see,” says Spinoza, in a most critical passage of ​his Ethics, “that the infinite essence and the eternity of God are known to all. . . . That men have not an equally clear cognition of God as they have of ordinary abstract ideas, is due to the fact that God cannot be imagined, as bodies are imagined, and that they have associated the name of God with the images of things that they are accustomed to see.” All the ignorance and unwisdom whose consequences Spinoza sets forth in the Third and Fourth Parts of his Ethics, are thus declared, in this passage, to be due to the failure of the ordinary human mind to reflect upon, and to observe, an idea of the truth, i.e. of God, which it still always possesses, and which not the least of minds can really be without. For God’s essence is “equally in the part and in the whole.” Thus vast, then, is the difference in our whole view of ourselves and of the universe which is to be the outcome of mere self-consciousness. Yet the same Spinoza, in a passage not long since cited in our notes, can assert that whoever has a true idea knows that he has it, and in a parallel passage can even make light of all reflective insight, as a useless addition to one's true ideas.
This really marvellous vacillation of Spinoza, as regards the central importance of self-consciousness in the whole life of man and of the universe, is full of lessons as to the fallacy of ignoring the positive meaning of reflective insight. This positive meaning once admitted, it is impossible to assert that any limited series of reflective acts can exhaust the self-representative significance of any concrete life. The properties of the number-series, the inexhaustible wealth of the concept of Order, and the fecundity of the mathematical “conclusion from n to n + 1,” are mere hints of what a reflective series implies, and of the infinity of every genuine reflective series. For, on the one hand, we have now sufficiently seen that the fecundity in question is due to the essentially reflective character of the process whereby the conclusion from n to n + 1 is justified. On the other hand, our argument as to the ​universal fecundity of reflective processes, as merely illustrated by the wealth of the number-forms, is an argument a fortiori.
It is easy, as we have seen, to make light of mere numbers because they are so formal, and because one wearies of mathematics. But our present case is simply this: Of course the numbers, taken in abstract divorce from life, are mere forms. But if in the bare skeleton of selfhood, if in the dry bones of that museum of mere orderliness, the arithmetical series, — if, even here, we find such an endless wealth of relatively unique results of each new act of reflection, in case that act is taken in synthesis with the foregoing acts, — what may not be, what must be, the wealth of meaning involved in a reflective series whose basis is a concrete life, whose reflections give this life at each stage new insight into itself, and whose syntheses with all foregoing acts of reflection are themselves, if temporally viewed, as it were, new acts in the drama of this life? If such a life is to be present totum simul to the Absolute, how shall not the results of endless acts of reflection, each of an individual meaning, but all given, at one stroke, as an expression of the single purpose to reflect and to be self-possessed, — how shall all these facts not appear as elements in the unity of the whole, elements neither “transmuted” nor “suppressed,” but comprehended in their organic unity?
Unless the Absolute is a Self, and that concretely and explicitly, it is no Absolute at all. And unless it exhausts an infinity, in its presentations, it cannot be a Self. That even in thus exhausting it also excludes from itself the infinity that it wills to exclude, I equally insist. But I also maintain that this exclusion can only be based upon insight, and that, unless the positive infinity is present, as the self-represented whole that is accepted, the exclusion is blind, and our conception of Being lapses into mere Realism. But even Real​ism, as we have seen, is equally committed to the actually infinite.
IV. The Infinite as a Totality
And yet one will persistently retort, “Your idea of the complete exhaustion of what you all the while declare to be, as infinite, an inexhaustible series, is still a plain contradiction.”
I reply that I am anxious to report the facts, as one finds them whenever one has to deal with any endless Kette. The facts are these: (1) This series, if real, is inexhaustible by any process of successive procedure, whereby one passes from one member to the next. It is then expressly a series with no last term. Try to go through it from first to last, and the process can never be completed. Now this negative character of the series, if it is real, is as true for the Absolute as for a boy at school. In this sense, namely, viewed as a succession, since the series has no last term, its last term cannot be found by God or man, and does not exist. In this sense, too, any effort to ​complete the series will fail. In this sense, therefore, the series indeed has no “totality,” because it needs none. In this sense, finally, it would indeed be contradictory to speak of it as a totality. And all this is admitted, and need not be further illustrated.
(2) The sense in which the series is a totality is, however, if the series is real, not at all the sense in which it merely has no last member. The series is not to be exhausted in the sense in which it is indeed inexhaustible. But you may and must take it otherwise. The sense in which it is a totality expressly depends upon that concept of totum simul which I have everywhere in this discussion emphasized. To grasp this aspect of the case, you must view it in two stages. Take the series then first as a purely conceptual entity, as a mere idea, or “bare possibility.” The one purpose of the perfect internal self-representation of any system of elements in the fashion, and according to the type of self-representation, here in question, defines, for any Kette formed upon the basis of that purpose, all of the ideal objects that are to belong to the Kette. And this purpose defines them all at once, as we saw in dealing with f$_1$ (n), and the rest of those series that are involved in any Kette. Now this endless wealth of detail is defined at one stroke, so that it is henceforth eternally predetermined, as a valid truth, precisely what does and what does not belong to that Kette. And the various series and this Kette are here one and the same thing. To find whether this or that element belongs to the Kette, may or may not involve, for you, a long time. It will involve for you succession, processes of counting, and much more of the sort indefinitely. This, however, is due to your fortune as a human observer. But the definition of the series has predetermined at one stroke all the results that you thus, taking them in succession, can never exhaust, and has predetermined these results as a fixed Order, wherein every element has its precise place, next after a previous element, next before a subsequent one. As for the before and after, in this Order, they, too, are ideally predetermined, not as themselves successions, but as valid and simultaneous relations. That ɑ ​come first, b second, etc., is determined by the definition, all at once. The definition of the Kette does not, however, like your acts in counting, first determine ɑ and afterwards b. In the truly valid series it is the ɑ and b that are simultaneously first and next. You must not confuse then the eternally valid and simultaneously predetermined aspects of this order with the temporal succession of your verifications of the order.
So far, then, you have taken the series as a valid Order, whose ideal totality lies in the singleness of a plan that it is supposed to express. And now comes the second stage of the process of defining our Kette as real. Here is indeed the decisive step. All the members of the series are at once validly predetermined. That we have seen. Whatever can be precisely defined, however, can be supposed immediately given. So now simply suppose that the members are all seen, experienced, presented, not as they follow one after another, in your successive apperception of a few of them, but precisely as the definition predetermines them, namely, all at once. Hereupon you define the series as a fact, not merely valid, but presented. And so to define it is to define it as actually infinite.
And now I challenge you: “Where is the contradiction in this conception of the presented infinite totality?” Try to point out the precise place of the contradictory element in the system as defined.
You may reply: “The contradiction lies here: That the series has no last term is admitted; yet if all its terms are present, the series must be completely presented. But a completed and ordered series must have a last term. How otherwise should it be completed?”
I rejoin: There is finality and finality, completion and completion. The sort of finality possessed by the series is expressly of one sort, and not of another. By hypothesis the series is not in such wise completely presented that its last term is seen. For it has indeed no last term. But it is, by hypothesis, so presented that all the terms, precisely as the single purpose of the definition demands them, are present. The definition was not self-contradictory in demanding them ​as its ideal fulfilment. How should the presentation become contradictory by merely showing what the consistent definition had called for? And now in no other sense is the series, as presented, complete, than in the one sense of showing, in the supposed experience, all of its own ideally defined members. It is not complete in having any closing term.
Your reply to this statement will doubtless at last appeal to the decisive consideration regarding the nature of any individual fact of Being. You will say: “But the determinate presentation of a series of facts involves precisely that sort of completion of the series which makes it possess a last member. For the series, if given, is an Individual Whole, presented as such a complex individual in experience; and as an individual, the series needs precise limits. As it has a first, so then, if completely individuated, it must be finished by a last member. Otherwise it would lack the determination necessary to distinguish an Individual Being from a general idea.”
If the objection be thus stated, it raises afresh the whole question: What is an individual fact of experience? What is an individual whole in experience? Now I have set forth in the foregoing lectures (see Lectures VII and X), and have still more minutely developed elsewhere, a thesis about individuality whose relative novelty in the discussion of that topic, and whose special importance with regard to the issue about the determinateness of the Infinite, I must here insist upon. That every individual Being is determinate, I fully maintain. But how and upon what basis does such determination rest? When, and upon what ground, could one say: I have seen an individual whole? Never, I must insist, upon the ground that one has seen a group of facts with a sharply marked boundary, or with a definite localization in space or in time, ​or with any temporal or spatial terminus. A finished series of data simply does not constitute an individual whole merely by becoming finished. It is perfectly true that such a finished whole, with its boundary, its last term, or what limit you will, may be viewed and rightly viewed, as an individual; but only for reasons which lie far deeper than its mere possession of limits, and which, in their turn, might be present if such limits were quite undiscoverable. If you insist that only such limited wholes are ever viewed by us men as individual wholes, I retort that we men have never experienced the direct presence of any individual whole whatever. For us, individuals are primarily the objects presupposed, but never directly observed, by love and by its related passions, — in brief, by the exclusive affections which give life all its truest interests. As we associate these affections with those contents of experience whose empirical limits we also experience as essential to their form, the spatially or numerically boundless comes to seem (as it especially seemed to the Greek), the essentially formless, and hence unindividuated realm, where chaos reigns.
But such mere prejudices of our ordinary apprehension vanish, if we look more closely at what individual wholeness means. Never presented in our human experience, individuality is the most characteristic feature of Being. Its true definition, however, implies three features, no one of which has any necessary connection with last terms, or with ends, or with any other such accidents of ordinary sense perception, and of the temporal enumeration of details. These three features are as follows: First, an individual whole must conform to an ideal definition, which is precise, and free from ambiguity, so that if you know this individual type, you know in advance precisely what kind of fact belongs to the defined whole, and in what way. Secondly, the individual whole must embody this type in the form of immediate experience. And thirdly, the individual whole must so embody the type that no other embodiment would meet precisely the purpose, the Will, fulfilled by this embodiment. ​It is the third of these features that is the really decisive one. The satisfied Will, as such, is the sole Principle of Individuation. This is our theory of individuality. Here it comes to our aid.
For wherever in the universe these three conditions are together fulfilled, determinate individual wholeness gets presented. In our human experience their union, as a fact, is only postulated, and never found present, in the objects which constitute our empirical world. Hence in vain do you choose empirical series such as have last terms, and say, “Lo! these are typical individual wholes. If the Absolute sees individuality, in any collection of facts, he sees it as of this determinate type.” On the contrary, as we men observe these things, they appear to us to be individuals, solely because we presuppose our own individuality as Selves, and then, in the light of this presupposition, regard these serial acts of ours as individual wholes, merely because in them we have found a relative satisfaction of a purpose.
That finite series are individual wholes at all, is therefore itself a presupposition — never a datum. I take myself to be an individual Self, whose acts, as my own, are unique with the assumed uniqueness of my own purposes. Any one of the various series of my acts which attains, for the moment, its relative goal, is thereby the more marked as my own, and as one. But it is not directly experienced as any individual fact of Being at all, and that for the reason set forth in our seventh lecture. That we are individuals is true, and that our finite series of acts have their own place in Being is also true. But their finitude has only accidental relations to their individuality.
But now, in case of such a Kette as we are supposing real, what is lacking to constitute it a determinate whole? It has ideal totality. For a single ideal purpose defines the type of all facts that shall belong to it, and distinguishes them from facts of all other types, and predetermines their order, assigning to every element its ideal place. We suppose now an experience embodying all these elements in such wise that immediacy and idea completely fuse, so that what is here ​conceived is also given. We finally suppose this to be such an experience, for the Self whose Kette this is, that in possessing this series he views himself as this Being and no other. Now this last feature of itself constitutes determinateness. To demand that the series should have its end, temporal or spatial, is to mistake wholly the nature of individuality; is to overlook the primacy of the decisive Will as the sole begetter of individuality; and is to apply to the Absolute a character derived from certain experiences of ours which we merely view as individual experiences in the light of a postulate, while, for this very postulate, only the Absolute itself can furnish the adequate warrant and realization.
Our own definition of individuality then, by freeing us from bondage to mere temporal and spatial limits, leaves us free to regard as determinate and as real an experience that contains, and that does not merely “absorb” a wealth of detail which in itself is endless. In so far as this wealth is endless, it does indeed force every process of successive synthesis to remain unfinished; and therefore, in so far as you merely count the successive steps, you shall never find what makes the whole determinate. There is indeed no infinite number belonging to, or terminating, the series of whole numbers. All whole numbers are finite. It is the totality of the whole numbers that constitutes an infinite multitude. But the determinateness of this infinite whole is given, not when the last whole number is counted (for that indeed would be self-contradictory), but when the completely conscious Self knows itself as this Being, and no other. And this it knows not when it performs its last act, but when it views its whole wealth of life as the determinate satisfaction of its Will.
And thus, having vindicated the conception of the really Infinite, we are free, upon the basis of the general argument of these lectures, to assert that the Absolute is no absorber and transmuter, but an explicit possessor and knower of an infinite wealth of organized individual facts, — the facts, namely, of the Absolute Life and Selfhood. How these facts are One and also Many, we now in general know, precisely in so far as we ​reflectively grasp the true nature of Thought. For the Other which Thought restlessly seeks is simply itself in individual expression, — or, in other words, its own purpose in a determinate and conscious embodiment. Since this embodiment has to assume the form of Selfhood, its detail must be infinite. The world is an endless Kette, whatever else it is. Yet this infinite wealth of detail is not opposed to, but is the very expression of the internal meaning of the purpose to be and to comprehend the Self. The infinite wealth is determinate because it fulfils a precisely definable purpose in an unique way, that permits no other to take its place as the embodiment of the Absolute Will. And the One and the Many are so reconciled, in this account, that the Absolute Self, even in order to be a Self at all, has to express itself in an endless series of individual acts, so that it is explicitly an Individual Whole of Individual Elements. And this is the result of considering Individuality, and consequently Being, as above all an expression of Will, and of a Will in which both Thought and Experience reach determinateness of expression.
* ↑ Page 19. I cite throughout from the second edition of Appearance and Reality.
* ↑ Note A of the second edition, pp. 562, sqq. — a paper reprinted from Mind with omission.
* ↑ The precise sense in which the Number Series itself is the outcome of a recurrent operation of thought will be explained, in general accord with Dedekind’s theory, further on. Addition and Multiplication, in any particular instance, as in the adding or in the multiplying of 7 and 5, are of course operations terminated by the finding of the particular sum or product, and in so far they are finite and non-recurrent. But the laws of Addition and Multiplication (e.g., the Associative law), and the relation of both these operations to one another and to the number system, are dependent, in part, upon the fact that the result of every addition or multiplication of whole numbers is itself a whole number, uniquely determined, and, as a number, capable of entering into the formation of new sums and products.
* ↑ The discussion of the instances and conceptions of Multitude and Infinity, contained in what follows, is largely dependent upon various recent contributions to the literature of the subject. Prominent among the later authors who have dealt with our problem from the mathematical side, is George Cantor. For his now famous theory of the Mächtigkeiten or grades of infinite multitude, and for his discussions of the purely mathematical aspects of his problem, one may consult his earlier papers, as collected in the Acta Mathematica, Vol. II. With this theory of the Mächtigkeiten I shall have no space to deal in this paper, but it is of great importance for forming the conception of the determinate Infinite. Upon the more philosophical aspects of the same researches, Cantor wrote a brief series of difficult and fragmentary, but fascinating discussions in the Zeitschrift für Philosophic und Philosophische Kritik: Bd. 88, p. 224; Bd. 91, p. 81; Bd. 92, p. 240. In recent years (1895-97) Cantor has begun a systematic restatement of his mathematical theories in the Mathematische Annalen: Bd. 48, p. 48; Bd. 49, p. 207. Some of Cantor’s results are now the common property of the later text-books, such as Dini’s Theory of Functions, and Weber’s Algebra. Upon Cantor’s investigations is also based the remarkable and too much neglected posthumous philosophical essay of Benno Kerry: System einer Theorie der Grenzbegriffe (Leipzig, 1890) — a fragment, but full of ingenious observations. The general results of Cantor are summarized in a supplementary note to Couturat’s L’Infini Mathematique (Paris, 1896), on pp. 602-655 of that work. Couturat’s is itself the most important recent general treatment of the philosophical problem of the Infinite; and the Third Book of his Second Part (p. 441, sqq.) ought to be carefully pondered by all who wish fairly to estimate the “contradictions” usually attributed to the concept of the Infinite Multitude. A further exposition of Cantor’s most definite results is given, in a highly attractive form, by Borel, Leçons sur la Théorie des Fonctions, Paris, 1898. Side by side with Cantor, in the analysis of the fundamental problem regarding number, and multitude, stands Dedekind, upon whose now famous essay, Was Sind und Was Sollen die Zahlen? (2te Auflage, Braunschweig, 1893), some of the most important of the recent discussions of the nature of self-representative systems are founded. See also the valuable discussion of the iterative processes of thought by G. F. Lipps, in Wundt’s Studien (Bd. XIV, Hft. 2, for 1898); and the extremely significant remarks of Poincaré on the nature of mathematical reasoning in the Revue de Metaphysique et de Morale for 1894, p. 370. Other references are given later in this discussion.
* ↑ Compare the general discussion of “Correspondence” in the course of Lecture VII.
* ↑ In the older discussions of continuity, this concept was very generally confounded with that of infinite divisibility. The confusion is no longer made by mathematicians. Continuity implies infinite divisibility. The converse does not hold true.
* ↑ Leere Wiederholung is one of Hegel’s often repeated expressions in regard to such series. There is a certain question-begging involved in condemning a process because of one’s subjective sense of fatigue. Yet Bosanquet, in his Logic (Vol. I, p. 173), begins his subtle discussion of infinite number and series with an instance intended to illustrate the merely wearisome vanity of search that seems to be involved in a case of endless looking beyond for our goal. I wholly agree with Bosanquet when he demands that the “element of totality” (p. 173) must be present in the work of our thought, that is, as the ultimate test of its truth. Wholeness and finality our object must have, before we can properly rest in the contemplation of its real nature. But as we shall soon see, the question is whether a real and objective totality, — a full expression of meaning, — cannot, at the same time, be the explicit expression of such an internal meaning as can permit no last term in any series of successive operations whereby we may try to express this meaning. We tire soon of such “tasks without end.” But does the totum simul of Reality fail to express, in detail, the whole of what such processes mean?
* ↑ More or less vaguely this positive property of infinite multitudes was observed as a paradox whenever the necessity of conceiving “one infinite as greater than another,” or as containing another as a part of itself, was recognized. The paradox was in this sense felt already by Aristotle in the third Book of the Physics, ch. 5 (cf. Spinoza’s Ethics, Part I, Prop. XV. Scholium, where the well-known solution is that the true infinite is essentially indivisible, having no parts and no multitude). Explicitly the property of infinite multitudes here in question was insisted upon by Bolzano in his Paradoxien des Unendlichen (1851). Cantor, and, in America, Mr. Charles Peirce, have since made this aspect of the infinite multitudes prominent. Most explicitly, however, Dedekind has built up his entire theory of the number concept upon defining the infinite multitude or system simply in these positive terms, without previous definition of any numbers at all. See his op. cit., §5, 64, p. 17.
* ↑ In previous definitions, in Dedekind’s text, two systems have been defined as similar (ähnlich), when one of them can be made to correspond, element for element, with the other, any two different elements having different representations. And a proper part (echter Theil), or constituent portion, of a system, has been defined as one produced by leaving out some elements of the whole.
* ↑ Es giebt unendliche Systeme. Es giebt, is of course here used to express existence within the realm of consistent mathematical definitions. The conception of Being in question is the Third Conception of our own list.
* ↑ That the finite and infinite here quite change places is pointed out in an interesting way by Professor Franz Meyer, in his Antrittsrede at Tübingen entitled Zur Lehre vom Unendlichen (Tubingen, 1889). The same observation is made by Kerry in his comments upon Dedekind (in Kerry’s before-cited Theorie der Grenzbegriffe, p. 49). Bolzano, who, in his Paradoxien des Unendlichen had much earlier reached a position in many ways near to that of Dedekind, proves the existence of the infinite in a closely similar, but less exact way. Schroeder, in his very elaborate essay in the Abhandlungen der Leopold. Carolinischen Akad. d. Naturforscher for 1898, entitled Ueber Zwei Definitionem der Endlichkeit, insists indeed that this whole distinction between positive and negative definitions is, from the point of view of formal Logic, vain, and that Mr. Charles Peirce’s definition of finite systems, given in the American Journal of Mathematics, Vol. 7, p. 202, while it is the polar opposite of Dedekind’s definition of the Infinite, is, logically speaking, at once equivalent to Dedekind’s definition, and yet as positive as the latter, although Mr. Peirce, in the passage in question, starts from the finite, and not from the infinite. Schroeder seems to me quite right in regarding the distinction between essentially positive and essentially negative definitions as one for which a purely formal Logic has no place. But as a fact, the distinction in question, between what is positive and what is negative, has an import wholly metaphysical. Our interest in it here lies in the fact that if you begin, in Dedekind’s way, with the positive concept of the Infinite, you need not presuppose the “externally given” Many, but may develope the multitude out of the internal meaning of a single purpose. Mr. Charles Peirce, in his parallel definition of finite systems, has first to presuppose them as given facts of experience. We, however, are seeking to develope the Many out of the One.
* ↑ See Bosanquet’s Logic, loc. cit. et sq.
* ↑ Mr. Charles Peirce, as noted above, has indeed given a perfectly positive and exact definition of a finite system; but in order to set that definition to work you have first to suppose your Many externally given, while, in order to define the Gedankenwelt, or the Self, or, as we shall later see, the Real World, you have only to presuppose a single, and unavoidable, internal meaning. The infinity then follows of itself.
* ↑ How they are to be defined is of course itself a significant logical problem, whereof we shall soon hear more. Cantor’s account of the well-defined multitude, Menge, or ensemble, is found in French translation in the Acta Mathematica, torn. II, p. 363. On the general sense in which any multitude can be viewed as given for purposes of mathematical discussion, see Borel’s Leçons (cited above), p. 2.
* ↑ Logic, Vol. I, p. 175. In the Theory of Numbers, the properties of the whole numbers are indeed interesting for themselves “without anything in particular to count,” just because they form an ordered series, whose properties are the properties of all ordered systems.
* ↑ Upon the various types of Ketten, finite and infinite, “cyclical” and “open,” see the very minute analysis given by Bettazzi, in his papers entitled Sulla Catena di un Ente in un gruppo, and Gruppi finiti ed infiniti di Enti, in the Atti of the Turin Academy of Sciences (for 1895-96), Vol. 31, pp. 447 and 506. Bettazzi, in the second of these papers, expresses some dissatisfaction with Dedekind’s definition of the Infinite, but withdraws his objections in a later paper, Atti, Vol. 32, p. 353.
* ↑ On the properties of a Kette, see further in addition to Dedekind, Schroeder, in the latter’s Algebra der Relative, in the 3d Vol. of his Logik, pp. 346-404. Compare Borel, op. cit., pp. 104-106.
* ↑ Hegel indeed defines the positive Infinite as das Fürsichseiende, and sets it in opposition to the merely negative Infinitive, or das Schlecht-Unendliche. See the well-known discussion in the Logik, Werke, 2te Auflage, Bd. Ill, p. 148, sqq. Dr. W. T. Harris, in his Hegel (Chicago, 1890), and in other discussions, has ably defended and illustrated the Hegelian statements. They are applied to the problem of the quantitative Infinite by Hegel in the Logik, in the volume cited, p. 272 sqq. But near as Hegel thus comes to the full definition of the Infinite, his statement of the matter remains rather a postulate that the self-representative system shall be found, than a demonstration and exact explanation of its reality. The well-known Hegelian assertions that the only true image of the Infinite is the closed cycle (Logik, loc. cit., p. 156), that the quantitative infinite is a return to quality (loc. cit., p. 271), and that the rational fraction, taken as the equivalent of the endless decimal, is the one typical example of the completed quantitatively infinite process, — these, all of them valuable as emphasizing various aspects of the concept of the infinite, appear in the present day wholly inadequate to the complexity of our problem, and rather hinder than aid its final expression.
* ↑ Couturat, in the work cited, gives an admirable summary of the present phases of the discussion; only that he fails, I think, to appreciate the importance and originality of Dedekind’s method of deducing the ordinal concept. The views of Helmholtz and Kronecker are discussed with especial care by Couturat. Veronese, in the introduction to his Principles of Geometry (known to me in the German translation, Grundzüge der Geometrie, übers v. Schepp, Leipzig, 1894) gives a very elaborate development of the number-concept upon the basis of the view that the order of a series of conceived objects is an ultimate fact or absolute datum for thought (op. cit., § 3, 14-28, 46-50). Amongst the recent text-books, Fine’s Number-System of Arithmetic and Algebra holds an important place. See also the opening chapter of Harkness and Morley’s Introduction to the Theory of Analytic Functions.
* ↑ In order to accomplish this selection, the concept of an individual content, distinguished, within the system, as this and no other, must of course be presupposed as valid. Such a concept already implies an individuating interest or Will which selects. But this will is here presupposed only in the abstract.
* ↑ Op. cit., § 6, 71, p. 20.
* ↑ The parallel Kette of knowledge was observed by Spinoza, Ethics, P. II, Prop. 43. In the tract, De Intell. Emendat., however, Spinoza tries to explain away the significance of the endlessness of the resulting series. In the Ethics he says that whoever knows, knows that he knows, so that to an adequate idea, an adequate idea of this idea is necessarily joined by God and man. But in the Tractatus he asserts that the idea of the idea is not a necessary accompaniment of the adequate idea, but merely may follow upon the adequate idea if we choose. The contrast of expression in the two passages is remarkable; and the question is of the most critical importance for the whole system of Spinoza. For if the idea, when adequate, is actually self-representative, the form of parallelism between extension and thought, asserted by Spinoza, finally breaks down, since, to avoid the troubles about the infinite, Spinoza expressly makes extended substance indivisible, so as to avoid making it a self-representative system. Furthermore, in any case, no precisely parallel process to the idea of the idea is to be found in extended substance.
* ↑ “Our standard is Reality in the form of self-existence” (p. 375).
* ↑ I was years ago much struck by the remarkable proof, in the first volume of Schroeder’s Algebra der Logik, of the purely formal proposition that no simply constituted Universe of Discourse could be defined, in terms of the Algebra of Logic, as the absolute whole of Being, without an immediately stateable self-contradiction, resulting from the mere definition of the symbols used in that Algebra. See Schroeder, Vol. I, p. 245. The metaphysical interest of this purely symbolic result is not mentioned by Schroeder himself. The proof given by him turns, however, upon showing that if you regard provisionally, as the “whole of the universe,” or as “all that is,” any simply defined universe of classes of objects, you are confronted by contradictions as soon as you reflect that the “totality of what is” also contains a realm of secondary objects that you may define by reflecting upon the classes contained in the first universe, and by classifying these classes themselves from new points of view. This realm of secondary objects, however, does not consistently belong to the primary universe that in a purely formal way you first defined. The true totality of Being can therefore only be defined by an endless process, or is an endless reflective system. This proof of Schroeder’s first brought home to me the fact that the necessity for defining reality in self-reflecting or endless terms is not dependent upon any one metaphysical interpretation of the world, whether realistic or idealistic, but is the consequence of a purely abstract account of the formal Logic of the concept of Reality in any of its forms.
* ↑ Constantin Gutberlet, Zeitschrift für Philosophie (Ulrici-Falckenberg), Bd. 92, Hft. II, p. 199. The wording of the example is a little different in the text cited. The force of the argument no longer exists for one who approaches the concept of the Infinite through that of the Kette. Cantor observes as much in his answer to Gutberlet in the same journal. The puzzle turns upon falsely identifying the properties of finite and infinite quantities.
* ↑ Logic, I, p. 175. We have already seen how imperfect this view of the number-series is, since the number-series, as a product of thought, is primarily ordinal, and its essence is to express, very abstractly, the orderly development of a reflective purpose.
* ↑ Loc. cit., p. 177.
* ↑ Couturat, in his dialectical discussion between the “finitist” and the “infinitist,” in L’Infini Mathematique, p. 443 sqq., gives full room to a statement of these arguments of his opponents. Our account of the Ketten has discounted them in advance. Dedekind’s Definition of the Infinite deliberately makes naught of them. If infinite multitudes corresponding to his definition can be proved real, these paradoxes will be simply obvious properties of such multitudes.
* ↑ All this is not only admitted, but insisted upon by Cantor himself, as a preliminary to his own discussion of das Eigentlich-Unendliche, which he sharply distinguishes from such Uneigentlicher concept of the Infinite as has to be used in the Calculus. See his separately published Grundzüge einer allgemeinen Mannigfaltigkeitslehre (Leipzig, 1883), p. 1, sqq. Compare the statement in Professor Franz Meyer’s lecture, before cited, to the same effect.
* ↑ This line of argument against the Infinite has often been used, — most recently perhaps by F. Evellin, in his two articles directed against the metaphysical use of Cantor’s theories, in the Revue Philosophique for February and November, 1898.
* ↑ See Cantor’s statement in the Zeitschr f. Philos., Bd. 88, p. 230; and in the same journal, Bd. 91, p. 112, in a passage there quoted from a letter addressed by Cantor to Weierstrass. I am unable to understand how Mr. Charles Peirce, in his paper in the Monist (1892, p. 537 of Vol. 2) is led to attribute to Cantor his own opinion as to the infinitesimals.
* ↑ Mr. Charles Peirce, as I understand his statements in the Monist (loc. cit.), appears to stand almost alone amongst recent mathematical logicians outside of Italy, in still regarding the Calculus as properly to be founded upon the conception of the actually infinite and infinitesimal. In Italy, Veronese has used in his Geometry the concept of the actually infinitesimal.
* ↑ New York, 1897, p. 194, sqq.
* ↑ See Conception of God, pp. 196, 198, 201, 213-214. See also the concluding lecture of the present series.
* ↑ On such possibilities, “counter to fact,” see again the discussion in the Conception of God, loc. cit., and in later passages of the same essay.
* ↑ See the Conception of God, Supplementary Essay, Part III, especially pp. 247-270. Compare Part IV, pp. 303-315.
* ↑ Of course they are in no sense true individuals, but taken as members of their series, they have relatively unique features.
* ↑ Part II, Prop. 47, Scholium.
* ↑ Dedekind, op. cit., p. 15, §§ 4, 69, has given a formal proof of the validity of the “conclusion from n to n + 1.” His proof, an extraordinarily brilliant feat of logical analysis, has been exhaustively analyzed, by Schroeder, in the passage before cited. It involves a peculiarly subtle reflection upon what the process of self-representation implies, — a reflection as easy to ignore as it is important to bring to clear light.
* ↑ As for my reasons for speaking of an Absolute Will at all, despite Mr. Bradley’s repeated objections, I must insist that we have precisely the same reasons for attributing a generalized type of Will to the Absolute that we have for attributing to it Experience. And the grounds for this conclusion have been stated at length in Lecture VII of the foregoing series. My insistence means mere report of the facts, in the best accessible language. To say that the Absolute has or is Will, is simply to say that it knows its object, namely itself in its wholeness, as this and no other, despite the fact that the “mere” Thought, which it also possesses, consists, as abstract thought, in defining such an Other, and because of the fact that this and no other satisfies or fulfils the complete internal meaning of the Absolute itself. That Thought, Will, and Experience are not “transmuted” but concretely present from the Absolute point of view, is a thesis merely equivalent to saying that the Absolute consciously views itself as the immediately given fulfilment of purpose in this and no other life. As immediately given fact, the life is Experience. In so far as the purpose is distinguished from its fulfilment, one has an Idea seeking its Other. And this is Thought. In so far as this and no other life fulfils purpose, we have Will. All these are concretely distinguished aspects of the fact, if the Absolute is a Self, and views itself as such. If this is not true, the Absolute is less than nothing.
* ↑ Here, as I believe, is the deepest ground for that Aristotelian objection to the Infinite as “no totality,” which we have now so often met. The whole question, then, is as to the true essence of Individuality.
* ↑ Conception of God, Part III of the Supplementary Essay of that work. See also ibid, p. 331: “Chasms do not individuate.”
* ↑ See, as against the theory of space and time as principles of individuation, the Conception of God, p. 260, sqq.