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?

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 lie 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 processof 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 hypothesis 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 persistently 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.

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 resultreinstates, 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 to M′ be next observed to lead to M″, in such wise that M″ there lies once more the occasion for the application 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 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 otherwise 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.1 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.

• 1.

  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.