Despite all the foregoing considerations, however, we have still to face the objection that, even if these constructions be regarded as self-evident products of Thought, they, nevertheless, simply cannot be genuinely true of the final nature of Reality and must somehow be fallacious. For, from Mr. Bradley's side, it would be maintained that however inevitable the seeming of these endless processes, they become self-contradictory precisely when you take them to be real and yet endless. For who knows not the Aristotelian arguments, so often repeated in later thought, against the actual Infinite? Is not the complete Infinite the very type of a logical “monster?” Is not the very conception a self-contradiction? If thought, then, has to conceive Reality as infinite, so much the worse, one may say, for thought. The Real, whatever its appearance, cannot in itself be endless.

It is necessary to consider such arguments by themselves, for the moment, and apart from the foregoing considerations. Let us, then, briefly develope some of these often repeated reasons on account of which so many assert that Reality cannot be an infinite system at all.

One may begin with the case as Aristotle first stated it, in the Third Book of the Physics, and elsewhere. There can, indeed, exist a Reality that permits us, if we choose to number its parts, to distinguish within it what we call elements, in such wise that we can never end the process of numbering them. So space is for us capable of infinite, that is, of indefinite division, if you choose to try to take it to pieces. But such divisibility is a mere possibility. Space, if real, is not endlessly divided. It is only in potentia λείπεται οὐˆν δυνάμει εἰˆναι τὸ ἄπειρον divisible so far as you please to conceive its parts. The limitless exists, therefore, only in potentia. For were space actually either made up of endless parts, or in such wise real as to be infinitely great, there would result the contradiction of an actually infinite number as the number of the parts of a real collection. But a number actually infinite is contradictory; for it then could not be counted; it would have no determinate size; it would possess no totality; and it would so be formless and meaningless. Again, were any one portion of the world's material substance infinite, how could room be left for the other portions? Were the whole infinite, how could it be a whole at all? For any whole of reality is limited by its own form, and by the fact that, as an actual whole, it is perfectly determinate. The difficulty as to the infinite must be solved, then, by saying that what is real forms a definite and, for that reason, a finite totality; while within this totality there may be aspects which our thought discovers to be, in this or that respect, inexhaustible through any process of counting that follows some abstractly possible line of our own subjective distinctions or syntheses. We can say, of such aspects of the world, that you may go on as long as you please, in counting their special type of conceived complexities, without ever reaching the end. But this endlessness is potential only, and never actual.

These well-known Aristotelian considerations have formed the basis of every argument against the actual infinite in later thought. The special point of attack has, however, often shifted. In general, as the later arguments have repeatedly urged (quite in Aristotle's spirit), the infinitely complex, it real, must be knowable only through some finished synthesis of knowledge. But a finished synthesis is inconsistent (so one affirms) with the endlessness of the series of facts to be synthesized; and hence an infinite collection, if it existed, would be unknowable. On the other hand, an infinite collection, if real apart from knowledge, could be conceived to be altered by depriving it of some, or of a considerable fraction, of its constituent elements. The collection thus reduced (so one has often argued) would be at once finite (since it would have lost some of its members) and infinite, since no finite number would be equal to exhausting the remaining portion. Hence the reduced collection and, therefore, the original collection must be of a contradictory nature, and so impossible. In a variation of this argument often used, one employs, as an image, some such, instance as an inextensible rod, one end of which shall be in my hands, while I shall be supposed to believe that the rod, which stretches out of my sight into the heavens, is infinitely long, as well as quite incapable of being anywhere stretched. Suppose the rod hereupon drawn, or, if you please, anyway mysteriously moved, a foot towards me at this end. If I am to believe in the infinity and inextensibility of the rod, I shall believe that the whole of the rod, and every part thereof, is now a foot nearer to me than before. But in that case the furthest portion of the rod must also be a foot nearer than before, or must have been “drawn in out of the infinite,” as one writer has stated the case.1 It can therefore no longer be an infinite rod. Hence, it was not actually infinite before the drawing in of this end.

All such arguments insist, either upon the supposed fact that our own conception of an infinite series is necessarily a conception of an indefinite and, therefore, of an essentially incomplete sequence, or else upon the assertion that an infinite collection, if viewed as real, would prove to be in itself of a quantitatively indefinite and changeable character. In the one case, the argument continues by showing that an indefinite and incomplete sequence is incapable of being taken to be a finished reality beyond our thought. In the other case, one insists that the quantitatively indefinite collection, if viewed as real, would stand in conflict with the very notion of reality, since the real is, as such, the determinate. “The essence of number,” says Mr. Bosanquet,2 “is to construct a finite whole out of homogeneous units.” “An infinite number would be a number which is no particular number; for every particular number is finite.” “An infinite series3 … is not anything which we can represent in the form of number, and therefore cannot be, quâ infinite series, a fact in our world… . Our constructive judgment requires parts and a whole to give it meaning. Parts unrelated to any whole cannot be judged real by our thought. Their significance is gone and they are parts of nothing.”

More detailed, in the application of the general charge of indefiniteness thus made against the conception of the infinite collections, are the often used arguments such as exemplify how, if infinite collections are possible at all, one infinite must be greater than another, while yet, as infinite and determinate, all the boundless collections must (so one supposes) be equal. Or, again, in a similar spirit, one has pointed out that, by virtue of the properties which we have deliberately' attributed to the Ketten of the foregoing discussion, two infinite collections, if they existed, would be, in various senses of the term equal, at once equal and unequal to each other, or would contradict the axiom as to the whole and the part.4 These arguments can be illustrated by an endless list of examples, drawn from the realm of discrete collections of objects, as well as from cases where limitless extended lines, surfaces, or volumes are in question, and from cases where limitless divisibility is to be exemplified. The variety of the examples, however, need not confuse one as to the main issue. What is brought out, in every case, is that the infinite collections or multitudes, if real at all, must be in paradoxical contrast to all finite multitudes, and must also be in such contrast as to seem, at first sight, either quite indeterminate or else hopelessly incomplete, and, in either case, incapable of reality.

Upon a somewhat different basis rest a series of arguments which have more novelty, just because they are due to the experience of the modern exact sciences. In the seventeenth century one of the greatest methodical advances ever made in the history of descriptive science occurred, when the so-called Infinitesimal Calculus was invented. The Newtonian name, Fluxions, used for the objects to whose calculation the new science was devoted, indicated better than much of the more recent terminology, that one principal purpose of this advance in method, was to enable mathematical exactness to be used in the description of continuously varying quantities. But the generalization which was made when the Calculus appeared had been the outcome of a long series of studies of quantity, both temporal and spatial. And the Calculus brought under one method of treatment, not only the problems about continuous processes of actual change, such as motions, or other continuous physical alterations, but also problems regarding the properties, the relations, the lengths, and the areas of curves, and regarding the corresponding features of geometrical surfaces and solids. For, in all these objects alike, either continuous alterations, or else characters that, although matters of spatial coexistence, may be ideally expressed in terms of such continuous alterations, fell within the range of the methods of the Calculus.

The new method, however, seemed to involve, at first, the conception both of “infinitely small” quantities, and of devices whereby an “infinite number” of such quantities could be summed together, or otherwise submitted to computation. The science of the continuous, in the realm of geometrical forms, as well as in the realm of physical changes, thus seemed to depend upon the conception both of the infinitely small and of the infinitely great; and the successful application of the results of such science in the realm of physics, was sometimes used as a proof that nature contains actually infinite and actually infinitesimal collections or magnitudes. But the early methods of the Infinitesimal Calculus were not free from inexactness, and led, upon occasion, to actually false conclusions. Hence, the paradoxes apparently involved in the logical bases of the science attracted more and more critical attention, as time went on; and, as a consequence, within the present century, the whole method of the Calculus has been repeatedly and carefully revised,—with the result, to be sure, that the conceptions of the actually infinite, in the sense here in question, and the actually infinitesimal (in the older sense of the term), have been banished from the principal modern test-books of both the Differential and the Integral Calculus. The terms, “Infinite ”and “Infinitesimal,” have been, indeed, very generally retained in such text-books for the sake of conciseness of expression; but with a definition that wholly avoids all the problems which our foregoing discussion has raised. The infinite and the infinitesimal of the Calculus can, therefore, no longer be cited in favor of a theory of the “actually Infinite.”

In the world of varying quantities, namely, it often happens that, by the terms of definition of a given problem, you have upon your hands a varying quantity (call it X) which, consistently with these terms, you are able to make, or to assume, as large as you please. In such cases, if some one else is supposed to have predesignated, as the value of X, any definite magnitude that he pleases, say X₁, then you are at liberty, under the conditions of the problem, to assume the value of X as larger still, i.e. as greater than any such previously assigned definite value X₁. Now, whenever the variable X has this character, in a given problem, then, according to the fashion of speech used in the Calculus, you may define X either simply as infinite, or as capable of being increased to infinity; and in the Calculus you are indeed often enough interested in learning what happens to some quantity whose value depends upon X, when X thus increases without limit, or, as they briefly say, becomes infinite. But in all such cases the term infinite, as used in the modern text-books of the Calculus, is, by definition, simply an abbreviation for the whole conception just defined. The variable X need not even be, at any moment, actually at all large in order to be, in this sense, infinite. It only so varies that, consistently with the conditions of the problem, it can be made larger than a predesignated value, whatever that value may be. And the Calculus is simply often interested in computing the consequences of such a manner of variation on the part of X.

Now, unquestionably a quantity that is called infinite in this sense is not the actually infinite against which Aristotle argued. It is merely the limitlessly increasing variable or the potentially infinite magnitude which he willingly admitted as a valid conception. A parallel definition of the infinitesimal is even more frequently employed in the modern text-books of the Calculus, just because the infinitesimal is mentioned more frequently than the infinite. In this sense, a variable magnitude is infinitesimal merely when it can be made and kept as small as we will, consistently with the conditions of the problem in which it appears. Thus neither the infinite nor the infinitesimal of the modern treatment of the Calculus has any fixed character, as a finished or finally given quantity, nor any character which could be defined as a determinately real somewhat, apart from our defining thought, and apart from the conditions of a given problem. The Calculus is deeply interested in computing results of such variation without limit; but as a branch of mathematics, it is, in fact, not at all directly interested in our present problem about the actually infinite.5

Now, this result of the whole experience of the students of the Calculus with the logic of their own science,—this outcome of the modern critical restudy of the bases of the science of the continuously variable quantities,—tends of itself to indicate (as one may say, and as objectors to the actually Infinite have often said) that the conception of the actually Infinite, formerly confounded with the conceptions lying at the bases of the Calculus, is, as a fact, not only in this region, but everywhere, scientifically superfluous; while the conception of the Infinite merely in potentia, originally defended by Aristotle, thus triumphs in the very realm where, for a time, its rival seemed to have found a firm foothold.6

Yet it has indeed to be observed that, from the mathematical point of view, not the questions of the Calculus, but certain decidedly special problems of the Theory of Numbers, and of the modern Theory of Functions, have given the mathematical basis for these newer efforts towards an exact and positive definition of the Infinite. As a fact, in our foregoing statement of the merely prima facie case for the recent definition of the positively Infinite, we have deliberately refrained from making any mention of the special problems about continuity, or of the conceptions of the Calculus. And it has also been noted that Cantor, who has done so much to make specific the positive concept of das Eigentlich-Unendliche, and who has also given us one of the very first of the exact definitions of continuous quantity ever discovered,—himself rejects the actually infinitesimal quantities as quite impossible; and does so quite as vigorously as he accepts and defends the actually infinite quantities; so that he fully agrees that the infinitesimal must remain where the Calculus leaves it, namely, simply the variable small at will.7 It must therefore be distinctly understood that, in the discussion of the reality of the infinite quantities and multitudes, appeal need no longer be made to the conceptions of quantity peculiar to the Calculus; while, in general, the majority of those concerned in this inquiry expressly admit that the logic of the Calculus is quite independent of the present issue, and that the infinite of the Calculus is simply the variable large at will, which therefore need not be at any moment, even notably large at all.8

And now, finally, there is also urged against any conception of the actually Infinite the well-known consideration that the conception of such infinity involves an empty and worthless repetition of the same, over and over,—a mere “counting when there is nothing to count,” or, in the realm of explicit reflection, a vain observationthat I am I, and that I am I, again, even in saying that I am I,—or an equally inane insistence that I know, and know that I know, and so on. The non-mathematical often dislike numbers, especially the large ones, and therefore easily make light of a wisdom that seems only to count, in monotonous inefficacy. Even the more reflective thinkers often believe, with Spinoza, that knowing that I know can imply nothing essentially new, at all events after the reflection has been two or three times repeated. The Hindoo imagination, with its love for large numbers, often strikes the Western mind as childish. And in all such cases, since mere size, as such, rightly seems unworthy of the admiration that it has excited in untrained minds, it has appeared to many to be the more rational thing to say that wisdom involves rather Hegel's Rückkehr aus der unendlichen Flucht than any acceptance of the notion that infinite magnitudes or multitudes can be real.

All the foregoing objections to the conception of the actually infinite rest, in large measure, upon a true and perfectly relevant principle. As a fact, what is real is ipso facto determinate and individual. It is this for the reasons pointed out in the closing lectures of the present series. It is this because it is such that No Other can take its place. The Real is the final, the determinate, the totality. And now, not only is this principle valid, but it is indeed supreme in every metaphysical inquiry. And therefore we shall, to be sure, find it true that in case, despite all the foregoing highly important objections, we succeed in reconciling infinity with determinateness, we shall still be unable to assert that the Reality is anything merely infinite. For infinity, as such, is at best a character,—a feature having the value of an universal. If the Absolute is in any sense an infinite system, it is certainly also an unique and individual system; and its uniqueness involves something very clearly distinguishable from its mere infinity. The Absolute is, in its determinate Reality, certainly exclusive of an infinity of mere possibilities. In this respect I shall here simply repeat the position taken in the discussion supplementary to the book called the Conception of God.9 It is, then, perfectly true, for me, as for the opponents of the actual Infinite, that much must be viewed as, in the abstract, “possible,” which is nowhere determinately presented in any final experience of the fulfilment of truth. The special illustration used, in my former book, to exemplify this fact, namely, the illustration of the points on the continuous line,—points which are “possible” in an infinitely infinite collection of ways, but which, however presented, cannot exhaustively constitute the determinate continuity of the line,—this, I say, is an illustration involving other problems besides those of the actual Infinite. The existence of the line, taken as a geometrical fact, contains more than the possible multitudes of multitudes of the points on the line can ever express. And this more includes, also, a something more determinate than the multitudes of the points can conceivably present. Hence, as I argued in my former book, and as I still deliberately maintain, the Absolute cannot experience the nature of the line by merely exhausting any infinitude of the points. But to this illustration I can here devote no further space, since the discussion of continuity, and especially of the geometrical continuum, lies outside of the scope of this paper. It is quite consistent, however, to hold, as I do, that while the Absolute indeed, by reason of its determinateness, excludes and must exclude infinitely infinite “bare possibilities,” known to mere thought, from presentation in any individual way, except as ideas of excluded objects, the Absolute still finds present, in the individual whole of its Selfhood, an actually infinite, because self-representative, system of experienced fact. The points on the line, then, if my former illustration is indeed well chosen, are not exhaustively presented, as constituting the whole line, in any experience, whatever, Absolute or relative. But this, as we now have to see, is not because the actually Infinite is, to the Absolute, something unrepresented, but because the determinate geometrical continuity of the individual line is something more, and more determinate, than any infinitude of points can express. And this individuality of the line I can and do express by saying that, even to a final view, the essence of the individual continuity of any one line involves the “bare possibility” of systems of ideal points over and above any that are found present in this final experience of the line. Even if the Absolute, then, observes infinitely infinite collections of points, it sees that the individual continuity of the line is more than they present. This I still assert.

In general, as we shall see, by virtue of what here follows, a fair account of the completeness of the Absolute must be just to two aspects. They are the ultimate aspects of Reality. Their union constitutes, once more, the world-knot. And the reason of their union is the one made explicit in our seventh Lecture. The Real is determinate and individual; and the Real is expressive of all that universal ideas, taken in their wholeness, actually demand, or mean, as their absolutely satisfactory fulfilment. In this twofold thesis, as I understand, I am wholly in agreement with Mr. Bradley. But I differ from him by maintaining that we know more than he admits concerning how the Real combines these two aspects. I maintain, then, with a full consciousness of the paradoxes involved, that the Reality is indeed a Self, whatever else it is or is not. For the Absolute, as I insist, would have to be not apparently, but really a Self, even in order to be (as Mr. Bradley seems to imagine his Absolute) a sort of self-absorbing sponge, that endlessly sucked in, and “transformed,” its own selfhood, until nothing was left of itself but the mere empty spaces where the absorbent Self had been. For the category of Self is indeed immortal. Deny it, and, in denying, you affirm it. As a fact, however, the Absolute is no sponge. It is not a cryptic or self-ashamed, but an absolutely self-expressive self. And to see how it can be so without contradiction, is simply to see how the concept of the actually Infinite, despite all the foregoing objections, is not self-contradictory, is not indeterminate, is not merely based upon wearisome reflections of the same; but is a positive and concrete conception, quite capable of individual embodiment. This is what we shall see in what here follows. The concept of the actually Infinite once in general vindicated from the charge of self-contradiction, all objection to conceiving the Absolute as a Self will vanish; and the transparent union of the One and the Many, which reflective thought has already shown us within its own realm, will become the universal law of Being.

But, on the other hand, if the Absolute is a Self, and, as such, an Infinite, this does not mean that it is anything you please, or that it is at once all possible things, or that it views its realm of fact as having all possible characters at once, and hence as having no character in particular. This Self, and no Other, this world and no Other, this totality of experience, and nothing else,— such is what has to be presented when the Real is known as the real. The Infinite will have to be also a determinate Infinite, a self-selected case of its type. For the world as merely thought, or as merely defined in idea, is the world viewed with an abstract or bare universality, and as that which still demands its Other, and which refers to that Other as valid and possible. The world of thought is, as such, an effort to characterize this Other, to imitate it, to correspond to it, and, of course, if so may be, to find it. Hence the world of mere thought has, as its very life, a principle of dissatisfaction; and when it conceives its object as the Truth, it defines, in the object, only the sense in which there is to be agreement or correspondence between the object and the thought. Consequently, an idea taken merely as an imitation of another, or taken as having an external meaning, expresses the Truth only as a barely universal validity. And one who merely takes thought as thought conceives the shadow land which shall, nevertheless, somehow have the value of a standard. In that realm,—the realm of mere validity,—all is mere character, and type, and possibility. And thought is the endlessly restless definition of another, and yet another. And this is true even when thought conceives an Infinite. Hence, infinity, as merely conceived, is indeed not yet Reality as Reality.

Now, the opponents of the actual Infinite, ever since Aristotle, have always seen, and rightly seen, that, as defined by mere thinking about external meanings, the world is not finally defined. The restlessly infinite, as such, they have condemned as in so far unreal. For whoever sees reality, sees that which has no Other like itself, which seeks no Other to define its being, which is itself no mere correspondence between one object and another, and, despite its unquestionable character as the fulfilment of thought, no mere agreement between a thought and a fact. The Real, then, has not the character which bare thought, as such, emphasizes,—the character of being essentially incomplete. It has wholeness. Its meaning is internal and not external. Therefore, it is indeed a finished fact. It cannot, then, be infinite if infinity implies incompleteness.

But, once more, is the Real for that reason finite? Because it excludes the search for another beyond itself, does it therefore contain no infinite wealth of presented content within itself? This is precisely the question. In emphasizing the exclusiveness of the Heal we must be just to the fact that, whatever it excludes, it cannot, from our point of view, be poorer, less wealthy, less manifold in genuine meaning, than the false Other, which its reality reduces to a bare and unrealized possibility of thought. That the world is what it determinately is, means, from our point of view, that its being excludes an infinitely complex system of “barely possible” other contents, which, just because they are excluded from Reality, are conceived by a thought such that not all of its “barely possible” ideal objects could conceivably be actualized at once. In this sense, for us, just as for the partisans of the barely possible and unactualized infinite, there are indeed ideas of infinitely numerous facts which remain, from an Absolute point of view, hypotheses contrary to fact.10 We agree, moreover, with our opponents, that no process expresses reality in so far as this process merely seeks, without end, for another and another object or fact. Hence, for us, as for our opponents, the Infinite, when taken merely as an endless process, is falsely taken. As merely that which you cannot exhaust by counting, the Infinite is, by the hypothesis, never found, presented or completed, so long as you simply count. Hence we wholly agree that the Infinite, just in so far as it is viewed as indeterminate, incomplete, or merely endless, is not rightly viewed; and that in so far it is indeed unreal. We also fully agree that Absolute knowledge unquestionably recognizes, as an object for its own relatively abstract thought, a distinctly unreal Infinite, namely, the Infinite of the excluded ideal “bare possibilities” aforesaid. In all this we quite agree with our opponents, and prize their insistence upon the determinateness of the final truth.

Nevertheless, we shall perforce insist upon these theses:—

(1) The true Infinite, both in multitude and in organization, although in one sense endless, and so incapable in that sense of being completely grasped, is in another and precise sense something perfectly determinate. Nor is it a mere monotonous repetition of the same, over and over. Each of its determinations has individuality, uniqueness, and novelty about its own nature.

(2) This determinateness is a character which, indeed, includes and involves the endlessness of an infinite series; but the mere endlessness of the series is not its primary character, being simply a negatively stated result of the self-representative character of the whole system.

(3) The endlessness of the series means that by no merely successive process of counting, in God or in man, is its whole ness ever exhausted.

(4) In consequence, the whole endless series, in so far as it is a reality, must be present, as a determinate order, but also all at once, to the Absolute Experience. It is the process of successive counting, as such, that remains, to the end, incomplete, so as to imply that its own possibilities are not yet realized. Hence, the recurrent processes of thought reveal eternal truth about the infinite constitution of real Being,—their everlasting pursued Other; but themselves,—as mere processes in time,—they are not that Other. Their true Otheris, therefore, that self-representative System of which they are at once portions, imitations, and expressions.

(5) The Reality is such a self-represented and infinite system. And therein lies the basis of its very union, within itself of the One and the Many. For the one purpose of self-representation demands an infinite multiplicity to express it; while no multiplicity is reducible to unity except through processes involving self-representation.

(6) And, nevertheless, the Real is exclusive as well as inclusive. On the side of its thought the Absolute does conceive a barely possible infinity, other than the real infinity,—a possible world, whose characters, as universal characters, are present to the Absolute, and are known by virtue of the fact that the Absolute also thinks. But these possibilities are excluded by reason of their conflict with the Absolute Will.

(7) Yet, in meaning, the infinite Reality, as present, is richer than the infinity of bare possibilities that are excluded. But for that very reason the Reality presented, in the final and determinate experience of the Absolute, cannot be less than infinitely wealthy, both in its content and in its order. Its unity in its wholeness, and its infinite variety in expression, are both of an individual character. The constituent individuals are not “absorbed” or “transmuted” in the whole. The whole is One Self; but therefore is all its own constitution equally necessary to its Selfhood. Hence it is an Individual of Individuals.

With less of complexity and, if you please, with less of paradox, no theory of Being can be rendered coherent. Our present purpose is to bring these various aspects of the twofold nature of Being, as Infinite Being and as Determinate Being, to light and to definition.

We shall return, therefore, to the consideration of the main points made by our objectors, and, as we meet them shall even thereby justify, without needing formally to repeat, our various theses.

The principal one amongst all the traditional objections to the Infinite is, as we have seen, the thought that the Infinite, as such, is merely an endlessly sought or an endlessly incomplete somewhat; while the real, as such, is very rightly to be viewed as the determinate. Hence, the actually Infinite, one insists, would be at once determinate and indeterminate, and so would be contradictory.

Now, whatever may be said about the actually Infinite, we have already seen that the infinite of the merely conceptual but valid type, the infinite of the realm of mathematical possibilities, is certainly as determinate a conception as any merely universal idea can ever be, and, as thus determinate, involves no contradiction whatever. Cling to our Third Conception of Reality; and then, indeed, there can be no doubt whatever that the Infinite is real. For there is no contradiction, there is only a necessarily valid truth involved in saying that to any whole number r, however large, there inevitably does correspond one number, and only one, which stands amongst all numbers as the rth member in the ordered series of whole numbers that are squares, or in the ordered series of the cubes, or in the ordered series, if you please, of numbers of the form of a¹⁰⁰ or a¹⁰⁰⁰, where the exponent is fixed, but where the number that is to be raised to the power indicated takes successively the series of values, 1, 2, 3, … r. The inevitable result is that to every whole number r, without a possible exception, there corresponds, in the realm of validity, and corresponds uniquely, just that particular whole number which you get if you raise r to the second, third, or hundredth, or thousandth power. Moreover, this ideal ordering of all the whole numbers, without exception, in a one-to-one relation (let us say) to their own thousandth powers, is in such wise predetermined by the very nature of number that, if you undertake to calculate the thousandth power (let us say) of the number 80,000,000, your result is in no wise left to you, as a bare possibility that your private will can capriciously decide.The result is lawfully fixed beforehand by the very essence of mathematical validity, i.e. by the very expression of your own final Will in its wholeness. Your calculation can only bring this result to light in your own private experience of numbers. It is an arithmetically true result quite apart from your instantaneous observation. Its triviality, as a mere matter for computation, is not now in question. Its eternal validity, however, interests us. Every number, then, speaking in terms of mathematical validity, already has its own thousandth power, whether you chance to have observed or to have computed that thousandth power or not. Yet, in any finite collection of whole numbers, those which are the thousandth powers of the whole numbers constitute at most an incomparably minute part of the whole collection. But, on the other hand, viewed with reference to the logically valid truth about all the numbers, these powers, as a mere part of the whole series of whole numbers, still occupy such a logically predetermined place that they are set, by their values, in a one-to-one relation to the members of the whole series; so that not a small portion, but absolutely all of the whole numbers, have their correspondents among the thousandth powers. Now, all these are facts of thought, just as valid as any conceptual constructions, however simple, and just as true as that 2 + 2 = 4. And by themselves these truths, trivial if you please, are, in all their wearisome-ness, not “monstrous” at all, but simply the necessary consequences of an exact conception of the nature of number.

“Monstrous, however,” so one may reply, “would be the assertion that in any real world there could be determinate facts corresponding to all this merely ideal complexity.” On the contrary, as we might at once retort, it would be monstrous if all these truths were merely “valid,” in a purely formal way, without any correspondent facts whatever in the real world. Can mere validity hang in the void? Must it not possess a determinate basis?

The issue, then, is at once the issue about the Third Conception of Being in our list. Either the truth, the world of mere forms, can indeed hang in the void, valid, but nowhere concrete, or else, just because the infinite is valid, it has its place, as fact, in the determinate experience of the Absolute. At all events, the Infinite, in such cases as have just been cited, is something quite as determinately valid as any barely universal conception can be. And unless it is true that two and two would make four in a world where no experience ever observed the fact, it is true that the infinitely numerous properties of the numbers need some concrete representation.

I grant, however, that these are but preliminary considerations. Every validity, as a bare universal, must be a reflectively abstract expression of a fact that ultimately exists in individual embodiment in the Absolute. Yet, on the other hand, you cannot predetermine the nature of this individual expression merely by pointing out that the possibilities in question appear to us endless. For the endlessness might be one of those matters of bare external conjunction of which Mr. Bradley so often speaks. Thus space appears to us endless. I fully grant that we are not warranted in making any one assertion about the Absolute view of the meaning of our spatial experience, by virtue of the mere fact that going on and on endlessly in space appears to us possible, and that, consequently, we can define propositions that would be valid if this possibility is endlessly realized by the Absolute. In passing from the Third to the Fourth Conception of Being, what we did was to see that nothing can be valid unless a determinate individual experience has present to it all that gives warrant for this validity. Because our fleeting experience never gives such final warrant, we are forced to seek for the ground and the basis of any valid truth once recognized by us, and to seek this basis in a realm that is Other than our own experience as it comes to us. This Other is, finally, the Absolute in its wholeness. But we do not assert that the Absolute realizes our validity merely as we happen to think it.

When we regard any valid truth as implying a variety of valid assertions, all for us matters of conceived possible experience, we often take the Many, thus conceived by us, as a mere fact, an uncomprehended “conjunction.” I agree altogether with Mr. Bradley that such varieties might seem, to a higher experience, artificial, and that, as such, they might be “transmuted” even in coming to their unity in the higher view. For in such cases we never experience that these varieties are self-evidently what they seem to us. And our conception that they are many is associated with a confession of ignorance as to what they are. A good example of all this is furnished by our conception of what our own lives, or the course of human history, would have been, if certain critical events had never taken place.11 What, in such instances, we have on our hands is an ignorance as to the whole ground and meaning of the critical events themselves. A fuller knowedge of what they meant might render much of our speech about the “possibilities” in question obviously vain.

Determinate decisions of the will involve rendering invalid countless possibilities that, but for this choice, might have been entertained as valid. In such cases the nature of the rejected possibilities is sufficiently expressed, in concrete form, by the will that decides, if only it knows itself as deciding, and is fully conscious of how and why it decides. That Absolute insight would mean absolute decision, and so a refusal to get presented in experience endlessly numerous contents that, but for the decision, would have been possible,—this I maintain as a necessary aspect of the whole conception of individuality. Whoever knows not decisions that exclude, knows not Being. For apart from such exclusion of possibilities, one would face barely abstract universals, and would, therefore, still seek for Another. Our whole conception of Being agrees, then, with Mr. Bradley's in insisting that the bare what, the idea as a mere thought, still pursuing, and imitatively characterizing its Other, not only does not face Being as Being, but can never, of itself, decide what its own final expression shall be. Thought must win satisfaction not as mere Thought, but also as decisive Will, determining itself to final expression in a way that the abstract universals of mere thinking can characterize, but never exhaust. Thus, and thus only, can be found that which admits of no Other. So far, then, it is indeed true that nothing is proved real merely by proving its abstract consistency as a mere idea taken apart from the rest of the world.

Or, again, the realm of validity is not exhausted by presented fact in the way suggested by one of Amadeus Hoffman's most horrible fancies (I believe in the Elixiere des Teufels), according to which a hero, persistently beset by a double, always finds that, whenever he, in his relative strength, resists a great temptation, and avoids a crime, this miserable double, whom he all the while vaguely takes to be in a way himself, appears,—pale, wretched, fate-driven,—and does, or at least attempts, in very fact, the deed that the hero had rejected. No; whoever knows Being, finds himself satisfied in the presence of a will fulfilled, and needs no fate-driven other Self, no outcast double, to realize for him the possibilities whose validity he rejects. For in rejecting, he wins. And Being is a destruction as well as an accomplishment of Experience.

Upon all this I have elsewhere insisted. That the very essence of individuality is a Will that permits no Other to take the place of this fulfilment,—a Love that finds in this wholeness of life its own,—I have pointed out in an argument that the Tenth Lecture of the present course has merely summarized.12 And therefore I am perfectly prepared to admit that when we define as valid, in the realm of mathematical truth, an infinite wealth of ideal forms, we need not, on that account alone, and apart from other reasons, declare that the Absolute Life realizes these forms in their variety as defined by us. Their true meaning it must somehow get present to itself,—otherwise it would face Another of which it was essentially ignorant. But its realization of their meaning may well imply an exclusion of their variety, just in so far as that variety, when conceived by us, expresses our ignorance of what principle of multiplicity is here at work, of how the One and the Many here concerned are related, and of what decision of Will would give these forms a concrete meaning in the universal life.

It remains, then, returning to the typical case of the numbers, to see in what sense a determinate expression of their whole meaning can be found in the life of a Will that fulfils itself through exclusive decisions, but that does not ignore any genuinely significant aspect of the truth. For our Absolute is not in such wise exclusive of content as to impoverish its wealth of ideal characters; and, on the other hand, it is not in such wise inclusive of bare possibilities as to oppose to whatever fact it chooses as its own, the fatal Other deed of Amadeus Hoffman's double-willed and distracted hero.

And here, of course, an opponent of the actual Infinite will be ready with the very common observation that the numbers are indeed, apart from the concrete objects numbered, of a trivial validity. “In a life,” he may say, “in a world of decisions and of concrete values, a barren contemplation of the properties of the numbers can have but a narrow place. Hence, no fulfilment of the hopeless task of wandering from number to number need be expected as a part of the Absolute life.”

Moreover, such an objector will insist that all these Ketten involve mere repetition of the same sort of experience over and over. “To carry such repetition to the infinite end,—what purpose,” he will say, “can such an ideal fulfil?” The individual fulfilment of the meaning of the number-series, in the final view, may well, then, take the form of knowing that there are indeed numbers, that they are made in a certain way, that the plan of their order has a particular type, and that this type is exemplified thus and thus by a comparatively few concretely presented ideas of whole numbers. Otherwise, the numbers may be left as unrealized as are those other excluded possibilities of the Will exemplified.

But against this view one has next to point out that, observed a little more closely, even the numbers have characters not reducible to any limited collection of universal types. They do not prove to be a monotonous series of contents, involving mere repetition of the same ideas. On the contrary, to know them at all well, is to land in them properties involving the most varied and novel features, as you pass from number to number, or bring into synthesis various selected groups of numbers. Consider, for instance, the prime numbers. Distributed through the number-series in ways that are indeed capable of partial definition through, general formulas, they still conform to no single known principle that enables us to determine, a priori, and in merely universal terms, exactly what and where each prime shall be. They have been discovered by an essentially empirical process which has now been extended, by the tabulators of the prime numbers, far into the millions. Yet the process much resembles any other empirical process. Its results are reported by the tabulators as the astronomers catalogue the stars. The primes have, as it were, relatively individual characters,13 which cannot be reduced to any barren repetition of the same thing over and over. One may call them uninteresting. But one must not judge the truth by one's private dislike of mathematics, just as, of course, one must not exaggerate the importance of mere forms. Here, then, is one instance of endless novelty within the number-series.

But the real question is, How shall the genuine meaning of all this series of truths be in any way grasped, unless the insight which grasps is adequate to the endless wealth of novel, and relatively individual truth that the various numbers present as one passes on in the series? For the will cannot consciously decide against the further realization of certain types of possibility, unless it clearly knows their value. And this it must know in exhaustive, even if ideal and abstractly universal terms. Nobody can fairly tell what value in life numerical truth may possess, unless he first knows that truth. And the numbers whose ordered rationality is, for us men, the very basis of our exact science, show a wealth of truth

These, then, are mere hints of the inexhaustible properties of the number-series. I speak still as layman j but I am convinced that these significant properties are quite as inexhaustible as the number-series itself. Now, the value of such properties you can never tell until you see what they are. Their meaning in the life of reason can only be estimated when they are present. Hence, you can never wisely decide not to know them until you have first known them. But they are not to be known merely as the endless repetitions of the same over and over. Hence it is wholly vain to say, “Numbers come from counting, and counting is vain repetition of the same over and over.” Whoever views the numbers merely thus, knows not whereof he speaks. It is not “counting, with nothing to count”; it is finding what Order means, that is the task of a true Theory of Numbers.

As a fact, then, the number-series in its wholeness seems to be a realm not only of inexhaustible truth, but of a truth that possesses an everywhere relatively individual type. And its validity has relations that we, at present, but imperfectly know, and a rational value that appears to be fundamental in every orderly inquiry.

We can, then, neither assert that to all the varieties which our thought may chance to conceive as possible, there correspond just as many final facts for an Absolute Experience; nor yet can we, on the other hand, exclude from concrete presentation, as final facts, such wholes as include an infinite series, merely because, for us, if we do not take due account of mathematical truth, the series seems to involve the empty repetition of “one more” and “one more.” For, as Poincaré has so finely pointed out, in the article before cited, it is precisely the “reasoning by recurrence “which is, in mathematics, the endless source of new results. Hereby, in the combination of his previous results for the sake of new insight, the mathematician is preserved from mere “identities,” and gets novelties. The “reasoning by recurrence,” however, is that form of reasoning whereby one shows that if a given truth holds in n cases, it holds for the n + 1st case, and so for all cases. Such processes of passing to “one more” instance of a given type, are processes not of barren repetition, but of genuine progress to higher stages of knowledge.

Precisely so it is, too, if one takes account of that other aspect of ordered series which it has been one principal purpose of this paper to emphasize. The numbers have interested us, not from any Pythagorean bias, but because their Order is the expression, not only of a profoundly significant aspect of all law in the world, but of the very essence of Selfhood, when formally viewed. Now reflective selfhood, taken merely as the abstract series, I know, and I know that I know, etc., appears to be a vain repetition of the same over and over. But this it appears merely if you neglect the concrete content which every new reflection, when taken in synthesis with previous reflections, inevitably implies in case of every living subject-matter. A life that knows not itself differs from the same life conscious of itself, by lacking precisely the feature that distinguishes rational morality alike from innocence and from brutish naïveté. A knowledge that is self-possessed differs from an unreflective type of consciousness by having all the marks that separate insight from blind faith.

“Thus we see,” says Spinoza, in a most critical passage of his Ethics,14 “that the infinite essence and the eternity of God are known to all… . That men have not an equally clear cognition of God as they have of ordinary abstract ideas, is due to the fact that God cannot be imagined, as bodies are imagined, and that they have associated the name of God with the images of things that they are accustomed to see.” All the ignorance and unwisdom whose consequences Spinoza sets forth in the Third and Fourth Parts of his Ethics, are thus declared, in this passage, to be due to the failure of the ordinary human mind to reflect upon, and to observe, an idea of the truth, i.e. of God, which it still always possesses, and which not the least of minds can really be without. For God's essence is “equally in the part and in the whole.” Thus vast, then, is the difference in our whole view of ourselves and of the universe which is to be the outcome of mere self-consciousness. Yet the same Spinoza, in a passage not long since cited in our notes, can assert that whoever has a true idea knows that he has it, and in a parallel passage can even make light of all reflective insight, as a useless addition to one's true ideas.

This really marvellous vacillation of Spinoza, as regards the central importance of self-consciousness in the whole life of man and of the universe, is full of lessons as to the fallacy of ignoring the positive meaning of reflective insight. This positive meaning once admitted, it is impossible to assert that any limited series of reflective acts can exhaust the self-representative significance of any concrete life. The properties of the number-series, the inexhaustible wealth of the concept of Order, and the fecundity of the mathematical “conclusion from n to n + 1,” are mere hints of what a reflective series implies, and of the infinity of every genuine reflective series. For, on the one hand, we have now sufficiently seen that the fecundity in question is due to the essentially reflective character of the process whereby the conclusion from n to n+1 is justified.15 On the other hand, our argument as to the universal fecundity of reflective processes, as merely illustrated by the wealth of the number-forms, is an argument a fortiori.

It is easy, as we have seen, to make light of mere numbers because they are so formal, and because one wearies of mathematics. But our present case is simply this: Of course the numbers, taken in abstract divorce from life, are mere forms. But if in the bare skeleton of selfhood, if in the dry bones of that museum of mere orderliness, the arithmetical series,—if, even here, we find such an endless wealth of relatively unique results of each new act of reflection, in case that act is taken in synthesis with the foregoing acts,—what may not be, what must be, the wealth of meaning involved in a reflective series whose basis is a concrete life, whose reflections give this life at each stage new insight into itself, and whose syntheses with all foregoing acts of reflection are themselves, if temporally viewed, as it were, new acts in the drama of this life? If such a life is to be present totum simul to the Absolute, how shall not the results of endless acts of reflection, each of an individual meaning, but all given, at one stroke, as an expression of the single purpose to reflect and to be self-possessed,—how shall all these facts not appear as elements in the unity of the whole, elements neither “transmuted” nor “suppressed,” but comprehended in their organic unity?

Unless the Absolute is a Self, and that concretely and explicitly, it is no Absolute at all. And unless it exhausts an infinity, in its presentations, it cannot be a Self. That even in thus exhausting it also excludes from itself the infinity that it wills to exclude, I equally insist. But I also maintain that this exclusion can only be based upon insight, and that, unless the positive infinity is present, as the self-represented whole that is accepted, the exclusion is blind, and our conception of Being lapses into mere Realism. But even Realism, as we have seen, is equally committed to the actually infinite.16

And yet one will persistently retort, “Your idea of the complete exhaustion of what you all the while declare to be, as infinite, an inexhaustible series, is still a plain contradiction.”

I reply that I am anxious to report the facts, as one finds them whenever one has to deal with any endless Kette. The facts are these: (1) This series, if real, is inexhaustible by any process of successive procedure, whereby one passes from one member to the next. It is then expressly a series with no last term. Try to go through it from, first to last, and the process can never be completed. Now this negative character of the series, if it is real, is as true for the Absolute as for a boy at school. In this sense, namely, viewed as a succession, since the series has no last term, its last term cannot be found by God or man, and does not exist. In this sense, too, any effort to complete the series will fail. In this sense, therefore, the series indeed has no “totality,” because it needs none. In this sense, finally, it would indeed be contradictory to speak of it as a totality. And all this is admitted, and need not be further illustrated.

(2) The sense in which the series is a totality is, however, if the series is real, not at all the sense in which it merely has no last member. The series is not to be exhausted in the sense in which it is indeed inexhaustible. But you may and must take it otherwise. The sense in which it is a totality expressly depends upon that concept of totum simul which I have everywhere in this discussion emphasized. To grasp this aspect of the case, you must view it in two stages. Take the series then first as a purely conceptual entity, as a mere idea, or “bare possibility.” The one purpose of the perfect internal self-representation of any system of elements in the fashion, and according to the type of self-representation, here in question, defines, for any Kette formed upon the basis of that purpose, all of the ideal objects that are to belong to the Kette. And this purpose defines them all at once, as we saw in dealing with f1 (n), and the rest of those series that are involved in any Kette. Now this endless wealth of detail is defined at one stroke, so that it is henceforth eternally predetermined, as a valid truth, precisely what does and what does not belong to that Kette. And the various series and this Kette are here one and the same thing. To find whether this or that element belongs to the Kette, may or may not involve, for you, a long time. It will involve for you succession, processes of counting, and much more of the sort indefinitely. This, however, is due to your fortune as a human observer. But the definition of the series has predetermined at one stroke all the results that you thus, taking them in succession, can never exhaust, and has predetermined these results as a fixed Order, wherein every element has its precise place, next after a previous element, next before a subsequent one. As for the before and after, in this Order, they, too, are ideally predetermined, not as themselves successions, but as valid and simultaneous relations. That a come first, b second, etc., is determined by the definition, all at once. The definition of the Kette does not, however, like your acts in counting, first determine a and afterwards b. In the truly valid series it is the a and b that are simultaneously first and next. You must not confuse then the eternally valid and simultaneously predetermined aspects of this order with the temporal succession of your verifications of the order.

So far, then, you have taken the series as a valid Order, whose ideal totality lies in the singleness of a plan that it is supposed to express. And now comes the second stage of the process of defining our Kette as real. Here is indeed the decisive step. All the members of the series are at once validly predetermined. That we have seen. Whatever can be precisely defined, however, can be supposed immediately given. So now simply suppose that the members are all seen, experienced, presented, not as they follow one after another, in your successive apperception of a few of them, but precisely as the definition predetermines them, namely, all at once. Hereupon you define the series as a fact, not merely valid, but presented. And so to define it is to define it as actually infinite.

And now I challenge you: “Where is the contradiction in this conception of the presented infinite totality?” Try to point out the precise place of the contradictory element in the system as defined.

You may reply: “The contradiction lies here: That the series has no last term is admitted; yet if all its terms are present, the series must be completely presented. But a completed and ordered series must have a last term. How otherwise should it be completed?”

I rejoin: There is finality and finality, completion and completion. The sort of finality possessed by the series is expressly of one sort, and not of another. By hypothesis the series is not in such wise completely presented that its last term is seen. For it has indeed no last term. But it is, by hypothesis, so presented that all the terms, precisely as the single purpose of the definition demands them, are present. The definition was not self-contradictory in demanding them as its ideal fulfilment. How should the presentation become contradictory by merely showing what the consistent definition had called for? And now in no other sense is the series, as presented, complete, than in the one sense of showing, in the supposed experience, all of its own ideally defined members. It is not complete in having any closing term.

Your reply to this statement will doubtless at last appeal to the decisive consideration regarding the nature of any individual fact of Being. You will say: “But the determinate presentation of a series of facts involves precisely that sort of completion of the series which makes it possess a last member. For the series, if given, is an Individual Whole, presented as such a complex individual in experience; and as an individual, the series needs precise limits. As it has a first, so then, if completely individuated, it must be finished by a last member. Otherwise it would lack the determination necessary to distinguish an Individual Being from a general idea.”17

If the objection be thus stated, it raises afresh the whole question: What is an individual fact of experience? What is an individual whole in experience? Now I have set forth in the foregoing lectures (see Lectures VII and X), and have still more minutely developed elsewhere,18 a thesis about individuality whose relative novelty in the discussion of that topic, and whose special importance with regard to the issue about the determinateness of the Infinite, I must here insist upon. That every individual Being is determinate, I fully maintain. But how and upon what basis does such determination rest? When, and upon what ground, could one say: I have seen an individual whole? Never, I must insist, upon the ground that one has seen a group of facts with a sharply marked boundary, or with a definite localization in space or in time,or with, any temporal or spatial terminus. 19 A finished series of data simply does not constitute an individual whole merely by becoming finished. It is perfectly true that such a finished whole, with its boundary, its last term, or what limit you will, may be viewed and rightly viewed, as an individual; but only for reasons which lie far deeper than its mere possession of limits, and which, in their turn, might be present if such limits were quite undiscoverable. If you insist that only such limited wholes are ever viewed by us men as individual wholes, I retort that we men have never experienced the direct presence of any individual whole whatever. For us, individuals are primarily the objects presupposed, but never directly observed, by love and by its related passions,—in brief, by the exclusive affections which give life all its truest interests. As we associate these affections with those contents of experience whose empirical limits we also experience as essential to their form, the spatially or numerically boundless comes to seem (as it especially seemed to the Greek), the essentially formless, and hence unindividuated realm, where chaos reigns.

But such mere prejudices of our ordinary apprehension vanish, if we look more closely at what individual wholeness means. Never presented in our human experience, individuality is the most characteristic feature of Being. Its true definition, however, implies three features, no one of which has any necessary connection with last terms, or with ends, or with any other such accidents of ordinary sense perception, and of the temporal enumeration of details. These three features are as follows: First, an individual whole must conform to an ideal definition, which is precise, and free from ambiguity, so that if you know this individual type, you know in advance precisely what kind of fact belongs to the defined whole, and in what way. Secondly, the individual whole must embody this type in the form of immediate experience. And thirdly, the individual whole must so embody the type that no other embodiment would meet precisely the purpose, the Will, fulfilled by this embodiment. It is the third of these features that is the really decisive one, The satisfied Will, as such, is the sole Principle of Individuation. This is our theory of individuality. Here it conies to our aid.

For wherever in the universe these three conditions are together fulfilled, determinate individual wholeness gets presented. In our human experience their union, as a fact, is only postulated, and never found present, in the objects which constitute our empirical world. Hence in vain do you choose empirical series such as have last terms, and say, “Lo! these are typical individual wholes. If the Absolute sees individuality, in any collection of facts, he sees it as of this determinate type.” On the contrary, as we men observe these things, they appear to us to be individuals, solely because we presuppose our own individuality as Selves, and then, in the light of this presupposition, regard these serial acts of ours as individual wholes, merely because in them we have found a relative satisfaction of a purpose.

That finite series are individual wholes at all, is therefore itself a presupposition—never a datum. I take myself to be an individual Self, whose acts, as my own, are unique with the assumed uniqueness of my own purposes. Any one of the various series of my acts which attains, for the moment, its relative goal, is thereby the more marked as my own, and as one. But it is not directly experienced as any individual fact of Being at all, and that for the reason set forth in our seventh lecture. That we are individuals is true, and that our finite series of acts have their own place in Being is also true. But their finitude has only accidental relations to their individuality.

But now, in case of such a Kette as we are supposing real, what is lacking to constitute it a determinate whole? It has ideal totality. For a single ideal purpose defines the type of all facts that shall belong to it? and distinguishes them from facts of all other types, and predetermines their order, assigning to every element its ideal place. We suppose now an experience embodying all these elements in such wise that immediacy and idea completely fuse, so that what is hereconceived is also given. We finally suppose this to be such an experience, for the Self whose Kette this is, that in possessing this series he views himself as this Being and no other. Now this last feature of itself constitutes determinateness. To demand that the series should have its end, temporal or spatial, is to mistake wholly the nature of individuality; is to overlook the primacy of the decisive Will as the sole begetter of individuality; and is to apply to the Absolute a character derived from certain experiences of ours which we merely view as individual experiences in the light of a postulate, while, for this very postulate, only the Absolute itself can furnish the adequate warrant and realization.

Our own definition of individuality then, by freeing us from bondage to mere temporal and spatial limits, leaves us free to regard as determinate and as real an experience that contains, and that does not merely “absorb” a wealth of detail which in itself is endless. In so far as this wealth is endless, it does indeed force every process of successive synthesis to remain unfinished; and therefore, in so far as you merely count the successive steps, you shall never find what makes the whole determinate. There is indeed no infinite number belonging to, or terminating, the series of whole numbers. All whole numbers are finite. It is the totality of the whole numbers that constitutes an infinite multitude. But the determinate-ness of this infinite whole is given, not when the last whole number is counted (for that indeed would be self-contradictory), but when the completely conscious Self knows itself as this Being, and no other. And this it knows not when it performs its last act, but when it views its whole wealth of life as the determinate satisfaction of its Will.

And thus, having vindicated the conception of the really Infinite, we are free, upon the basis of the general argument of these lectures, to assert that the Absolute is no absorber and transmuter, but an explicit possessor and knower of an infinite wealth of organized individual facts,—the facts, namely, of the Absolute Life and Selfhood. How these facts are One and also Many, we now in general know, precisely in so far as wereflectively grasp the true nature of Thought. For the Other which Thought restlessly seeks is simply itself in individual expression,—or, in other words, its own purpose in a determinate and conscious embodiment. Since this embodiment has to assume the form of Selfhood, its detail must be infinite. The world is-an endless Kette, whatever else it is. Yet this infinite wealth of detail is not opposed to, but is the very expression of the internal meaning of the purpose to be and to comprehend the Self. The infinite wealth is determinate because it fulfils a precisely definable purpose in an unique way, that permits no other to take its place as the embodiment of the Absolute Will. And the One and the Many are so reconciled, in this account, that the Absolute Self; even in order to be a Self at all, has to express itself in an endless series of individual acts, so that it is explicitly an Individual Whole of Individual Elements. And this is the result of considering Individuality, and consequently Being, as above all an expression of Will, and of a Will in which both Thought and Experience reach determinateness of expression.

• 1.

  Constantin Gutberlet, Zeitsehrift für Philosophie (Ulrici-Falckenberg), Bd. 92, Hft. II, p. 199. The wording of the example is a little different in the text cited. The force of the argument no longer exists for one who approaches the concept of the Infinite through that of the Kette. Cantor observes as much in his answer to Gutberlet in the same journal. The puzzle turns upon falsely identifying the properties of finite and infinite quantities.

• 2.
  Logic, I, p. 175. We have already seen how imperfect this view of the number-series is, since the number-series, as a product of thought, is primarily ordinal, and its essence is to express, very abstractly, the orderly development of a reflective purpose.
• 3.
  Loc. cit., p. 177.
• 4.

  Couturat, in his dialectical discussion between the “finitist” and the “infinitist,” in L'Infini Mathematique, p. 443 sqq., gives full room to a statement of these arguments of his opponents. Our account of the Ketten has discounted them in advance. Dedekind's Definition of the Infinite deliberately makes naught of them. If infinite multitudes corresponding to his definition can be proved real, these paradoxes will be simply obvious properties of such multitudes.

• 5.

  All this is not only admitted, but insisted upon by Cantor himself, as a preliminary to his own discussion of das Eigentlich-Unendliche, which he sharply distinguishes from such Uneiyentlicher concept of the Infinite as has to be used in the Calculus. See his separately published Grundzüge einer allgemeinen Mannigfaltigkeitslehre (Leipzig, 1883), p. 1, sqq. Compare the statement in Professor Franz Meyer's lecture, before cited, to the same effect.

• 6.

  This line of argument against the Infinite has often been used,—most recently perhaps by F. Evellin, in his two articles directed against the metaphysical use of Cantor's theories, in the Revue Philosophique for February and November, 1898.

• 7.

  See Cantor's statement In the Zeitschr f. Philos., Bd. 88, p. 230; and in the same journal, Bd. 91, p. 112, in a passage there quoted from a letter addressed by Cantor to Weierstrass. I am unable to understand how Mr. Charles Peirce, in his paper in the Monist (1892, p. 537 of Vol. 2) is led to attribute to Cantor his own opinion as to the infinitesimals.

• 8.

  Mr. Charles Peirce, as I understand his statements in the Monist (loc. cit.), appears to stand almost alone amongst recent mathematical logicians outside of Italy, in still regarding the Calculus as properly to be founded upon the conception of the actually infinite and infinitesimal. In Italy, Veronese has used in his Geometry the concept of the actually infinitesimal.

• 9.

  New York, 1897, p. 194, sqq.

• 10.

  See Conception of God, pp. 196, 198, 201, 213–214. See also the concluding lecture of the present series.

• 11.

  On such possibilities, “counter to fact,” see again the discussion in the Conception of God, loc. cit., and in later passages of the same essay.

• 12.

  See the Conception of God, Supplementary Essay, Part III, especially pp. 247–270. Compare Part IV, pp. 303–315.

• 13.

  Of course they are in no sense true individuals, but taken as members of their series, they have relatively unique features that we find more and more baffling the further we go. The “perfect numbers” form a series that may be as full of interest, for all that I know, as the primes. The properties of the “Arithmetical Triangle” are linked in the most unexpected fashion with the laws of our statistical science, and with the nature of certain orderly combinations of vast importance in other branches of mathematical inquiry. Countless other combinations of numbers form topics, not only of numerous well-known plays and puzzles, but of scientific investigations whose character is actually adventurous,—so arduous is their course, and so full of unexpected bearings upon other branches of knowledge has been their outcome. Nobody amongst us can pretend to fathom the value for concrete science, and for life, that has yet to be derived from advances in the Theory of Numbers.

• 14.

  Part II, Prop. 47, Scholium.

• 15.

  Dedekind, op. cit., p. 15, § 4, 59, has given a formal proof of the validity of the “conclusion from n to n + 1.” His proof, an extraordinarily brilliant feat of logical analysis, has been exhaustively analyzed, by Schroeder, in the passage before cited. It involves a peculiarly subtle reflection upon what the process of self-representation implies,—a reflection as easy to ignore as it is important to bring to clear light.

• 16.

  As for my reasons for speaking of an Absolute Will at all, despite Mr. Bradley's repeated objections, I must insist that we have precisely the same reasons for attributing a generalized type of Will to the Absolute that we have for attributing to it Experience. And the grounds for this conclusion have been stated at length in Lecture VII of the foregoing series. My insistence means mere report of the facts, in the best accessible language. To say that the Absolute has or is Will, is simply to say that it knows its object, namely itself in its wholeness, as this and no other, despite the fact that the “mere” Thought, which it also possesses, consists, as abstract thought, in defining such an Other, and because of the fact that this and no other satisfies or fulfils the complete internal meaning of the Absolute itself. That Thought, Will, and Experience are not “transmuted” but concretely present from the Absolute point of view, is a thesis merely equivalent to saying that the Absolute consciously views itself as the immediately given fulfilment of purpose in this and no other life. As immediately given fact, the life is Experience. In so far as the purpose is distinguished from its fulfilment, one has an Idea seeking its Other. And this is Thought. In so far as this and no other life fulfils purpose, we have Will. All these are concretely distinguished aspects of the fact, if the Absolute is a Self, and views itself as such. If this is not true, the Absolute is less than nothing.

• 17.

  Here, as I believe, is the deepest ground for that Aristotelian objection to the Infinite as “no totality,” which we have now so often met. The whole question, then, is as to the true essence of Individuality.

• 18.
  Conception of God, Part III of the Supplementary Essay of that work. See also ibid, p. 331: “Chasms do not individuate.”
• 19.

  See, as against the theory of space and time as principles of individuation, the Conception of God, p, 260, sqq.