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.1

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.2 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, a, 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, a′, 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.3

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.4 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.5 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 similar6 to a constituent (or proper) part of itself; in the contrary case S is called a ’finite“ system.

“Theorem.—There exist infinite systems.7

“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 a and b are different elements of S, their images, a′ 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.8 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-correspondent 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-representative 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,9 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.10 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.11 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 aⁿ, 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 aⁿ, 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,”12 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	3	4	5	6	7	8	9	10,	etc.
2	4	6	8	10	12	14	16	18	20,	etc.
4	8	12	16	20	24	28	32	36	40,	etc.
8	16	24	32	40	48	56	64	72	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 endless 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 abcd, consisting of the already distinguishable elements a, b, etc., may be put in exact correspondence to itself by making b correspond to a, and so represent a, while, in similar fashion, c corresponds to b, d to c, and, finally, a itself to d. In this case the system is, in a particular way, “transformed” into the image bcda, in such wise as to be exactly self-representative. But the system abcd might also be represented, element for element, by the system cbda, where the order of the elements was again different, but where c now corresponded to the original a, b to itself, d to c, and a 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 partand whole meet, and become in some wise precisely congruent, element for element. 13 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 bind 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 f1(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 f₁(n) will contain a second series f₂(n), similarly related to f₁(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, f₁(n), there is again a first member, a second member, and so on.

Now we can indeed speak of the series f₁(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 f₁(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 f₁(n). The law of the formation of f₁(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 f₁(p), that is, as that value which f₁(n) takes when n = p, or as the image of the image of p. It is easy to see that f₁(p) is the pth value, in serial order, of the series f₁(n). At the same time, since f₁(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 f₁(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 f₁(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 f₁(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 f₁(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 f₁(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, f₁(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 f2(n); and so on without end.14 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.15 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 (out 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.16 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 unanalyzed. 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 constituent 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.17 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).”18 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 itsiterative 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 Gedatikenwelt, 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. Thisconcept 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 selfconsciously 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-conscious ness, 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 constituents 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 he 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, 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 tins 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-representativesystem 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, F₁, and F, when taken as in company, F₂. Then the problem, How are F₁ and F₂ 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 givenby 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.19

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 imago 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 experience. 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 off 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,”20 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 a part 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.21 No philosophy that wholly ignores this elementary 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.

1.
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 Philosophie 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. 481; Bd. 49, p. 207. Some of Cantor's results are now the common property of the later test-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, Leçmons 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.
2.
Compare the general discussion of “Correspondence” in the course of Lecture VII.
3.
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.
4.
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?
5.
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.
6.
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.
7.
Es giebt unendliche Systems. 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.
8.
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 (Tübingen, 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. Garolinischen Akad. d. Natur-forscher for 1898, entitled Ueber Zwei Deftnitionem 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 yon 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.
9.
See Bosanquet's Logic, loc. cit. et sq.
10.
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 un avoidable, internal meaning. The infinity then follows of itself.
11.
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, tom. 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.
12.
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.
13.
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.
14.
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. 345-404. Compare Borel, op. cit., pp. 104-106.
15.
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. III, 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.
16.
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 textbooks, 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.
17.
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 individualiting interest or Will which selects. But this will is here presupposed only in the abstract.
18.
Op. cit., § 6, 71, p. 20.
19.
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.
20.
“Our standard is Reality in the form of self-existence” (p. 375).
21.
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 here 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, hut is the consequence of a purely abstract account of the formal Logic of the concept of Reality in any of its forms.

The Gifford Lectures

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SECTION III. THEORY OF THE SOURCES AND CONSEQUENCES OF ANY RECURRENT OPERATION OF THOUGHT. THE NATURE OF SELF-REPRESENTATIVE SYSTEMS

I. First Illustration of a Self-Representative System

II. Definition of a Type of Self-Representative Systems

III. Further Illustrations of Self-Representative Systems of the Type here Defined

IV. Remarks upon the Various Types of Self-Representative Systems

V. The Self and the Relational System of the Ordinal Numbers. The Origin of Number; and the Meaning of Order

VI. On the Realm of Reality as a Self-Representative System

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