Leighton, J. A. “The Infinite New and Old.” The Philosophical Review 13, no. 5 (1904): 497. https://doi.org/10.2307/2176860
In the metaphysical application of the new concept of the infinite, great stress is laid on the peculiar relation of whole and part afforded by it. This relation is regarded as furnishing a key to the relation of the apparently finite human self to the Absolute. The whole is similar to the part, the part is equivalent to the whole, since every element in the whole corresponds to an element in the part. Professor Royce says the part equals the whole. But this is not equality in the ordinary quantitative sense. It is only a logical relation of one-one correspondence. The relationship may be called one of similarity, equivalence, one-one correspondence, etc., but it is certainly not equality in its ordinary meaning. This extraordinary sort of equality has no intelligible bearing on the relation between my will and an infinite will, between my struggling temporal life and this eternal and unvarying life, between my experience conditioned by change and error and an eternally complete and indivisible experience.
In short, these iterative processes of human thinking, defined by the new infinite, significant and suggestive of a timeless thinker though they be, neither prove the reality nor clearly illuminate the inward constitution of an absolute mind or self, which must somehow have a timeless, if perfect and indivisible, experience. Must not such a mind know all things in a radically different way from our minds? Must not even the infinite number of infinite series present themselves differently in an absolute mind, if they present themselves to it at all? And what can be the connection between an infinite mind, which occupies itself ever in thinking numerical and other forms of self-representative relations, and a supreme Self, regarded as sustaining human ideals, as making possible the fulfilment of specific human and practical purposes, and as conserving the complex and uniquely significant lives of human persons. The eternal play of an endless approximative asymptotic series of attempts at self-representation, or the notion of limitless serial orders, does not seem to be connected in any intelligible fashion with the existence of a multitude of imperfect and developing sentient beings. Such a play of purely abstract thought-relations scarcely affords a satisfactory foundation for human endeavor, or, indeed, for the growth of concrete knowledge. No positive relation has been shown to exist between the "new” infinite and the actual conditions of human action or common experience. Do we get from the " new "infinite any light on the place of our temporal activities in the universe? I fear not. If the notion of the infinite is to have any vital meaning at all we must approach it from some other quarter than that of abstract and symbolic logical operations developed in that department of science which is admittedly most remote from actual experience, and in which the very abstractness and aloofness from the conditions and structure of concrete experience make possible these new and beautiful formulas of serial order, etc. We are expressly informed, e. g., by Mr. Russell, that mathematical space can be constructed by an order of points, entirely without reference to the sensuous space-intuition of actual experience.
The notion of a perfect self or absolute mind, if it is to have any real meaning for us humans, must be determined by reference to the more significant aspects of human life. The infinite must be interpreted in terms of the fundamental activities and ideals of the concrete human self, and here at once we are faced by the antithesis between the temporal and the eternal, between the striving and growing and the perfect and complete. What is the relation of the human will to the Absolute as will? What is the relation of human deeds and sentiments and thoughts to the entire system of things? Here we face a central difficulty, and, indeed, I am disposed to think, the supreme problem of systematic philosophy. If we could determine the place of our temporal experiences and efforts in the ultimate reality, if we could in thought lay hold on the permanently significant in these experiences and efforts and see the ultimate goal and meaning of personal growth and of cosmic change, the problem of philosophy would be solved, and the "infinite" would cease to trouble us. But the new concept of the infinite does not advance us a single definitive step further towards the solution of these problems. We ask for bread and we are offered a stone.
After all these negations, I venture with hesitation to offer some positive considerations on the meaning of the infinite ; and, in so doing, I would remind the reader that the new concept of the infinite has already been recognized as a clear and beautiful illustration of the mind's power of self-transcendence. In the first place, we must distinguish carefully between the potential infinitude of human thought, which is but another name for the above-mentioned quality of self-transcendence, and existential or actual infinitude. The latter quality may belong to absolute being or ends prestissimo.
The human mind we know to be infinite only in promise and potency. We may perhaps assume that this promise and potency has somehow its roots in an actual infinite, that the capacity for transcending its existential conditions, for going beyond the data of experience and transforming the latter under the guidance of norms or ideal values which the human self-displays both in theoretical thinking and in practical endeavor and preeminently in the very discussion of its own final destiny, may entitle us to assume that these ideal values are evidences of the presence unawares of the actual angel of the infinite and perfect in the mind of man. But such considerations hardly furnish a gnostic insight into the synthesis of finite and infinite.
Positively regarded, the actual or existential infinite is a limiting notion like the square root of 2. We indefinitely approximate it in our thinking and doing, but under present conditions we do not actually comprehend it or attain unto it. We may conceive this existential infinite as the ideal limit of thought and volition. It does not present to our minds as boundless in space or endless in time, but rather as the complete and perfect, transcending space and time. The infinite, then, in this sense, is the goal of thinking and of practical endeavor. It is really the limiting notion of the indefinite series of thoughts, aspirations, and deeds in which we strive to approach and realize the ideally perfect or Absolute. This series seems to us now, as we look before and after, to be endless. And just as a life is presented in the successive steps of its development, and a supreme end is unfolded in the successive steps towards its fulfilment, we may presuppose the actual infinite to be inherently involved in our approximations towards it. But when we think of the goal or end as a reality now, the actual infinite becomes the limit of our apparent infinitude of thought, feeling, and action. And our apparent infinitude is the possibility of indefinite continuance in thought, deed, etc.
In knowing a limit, we transcend it and set it further on. This self-transcendence, whether it be in acquiring knowledge or in the deeds which go to make character, is at once a negation and an affirmation. We negate that which is for us now, as attained, in seeking to transcend it. We affirm that which is not but is to be. In setting forward the limit or goal, we at once confess the present unreality for us of that which we seek, and we postulate its reality as that unto which we may attain. There is here a dialectic which involves the mutual implication of the finite and the infinite. The existentially finite human spirit is potentially infinite. Butut cannot be even potentially infinite unless its repeated self-transcendence is grounded on a reality which is the common basis of finite and infinite. The infinite as actual now appears beyond the finite self. It is at once the goal and the presupposition of the incessant, self-transcending efforts of the human spirit in thought and deed, i. e., in the very concrete pulse and movement of life itself.
If we should come to possess the infinite in every truth, if we should, by the falling away of the veil of time, apprehend as it really is that which we now call the infinite, it would no doubt at once seem both strange and familiar. We should long feel our own finitude; but, on the other hand, the merely infinite would no longer mean anything to us. As the attained goal of hitherto indefinite endeavor, the infinite would be transformed into a more positive and satisfying reality. In truth the goal is not infinite. It is more concrete and individual. It must be a reality which transcends the opposition of finite and infinite engendered by the temporal character of our present activities. Now it appears to us as a terminus or limit, just the square root of 2, although not in itself infinite, is a limit which is approached by an infinite series of numbers. This is the paradox of the infinite, viz., that the fruition of our experiences and the fulfilment of our purposes, in other words, the actual attainment or possession of infinitude, would mean the complete evanescence of the notion of an infinite. In so far as we attain to, or apprehend, perfection and the completed reality in any fundamental activity of life, for the nonce at least, the contrast between our existential finitude and the hitherto indefinitely distant goal or limit of our striving falls away. We feel the presence of an Absolute, and the infinite is lost in being attained, since our state of being then seems wholly throbbing with the positive and the actual. Hence the very notion of an infinite springs out of a present consciousness of impermanence and imperfection which seeks ever the permanent and perfect.
The notion of the infinite has for life and religion the significance of a limiting concept. In this respect, it is akin to the notion of God; and, like the latter, it represents in religious feeling and metaphysical speculation the craving for completeness, i. e., timeless perfection. Therefore, the positive content of our notion of the infinite is to be derived from the chief or fundamental directions or tendencies in which feeling and thought seek completeness with reference to life as a totality. The infinite is the limiting notion or point of fulfilment for certain fundamental tendencies of the human spirit in relation to the conditions of its life and activity. I shall endeavor in the space left to indicate very summarily the meaning of the infinite in the chiefs of these relations. We are dealing here simply with tendencies of the life-process in the human self.
The infinite, in relation to existence in time, is not the endless but the timeless, i.e., its being and life are not in any sense episodes in time, are neither increased nor diminished, nor in anyway realized in subjection to temporal conditions ; and yet, of course, since the infinite is a limiting concept standing in relation to our finite lives, the temporal life of man and the course of history must have positive significance in relation to the timeless infinite, and be somehow taken up into the thought and vitally connected with the activity of the latter. But this starts a very difficult problem, perhaps insoluble, and I cannot attempt even to deal with it here. In relation to space, the true infinite is not the indefinitely boundless but that which is limited to no space and is indeed the ultimate limit of space-conditions of existence. Here again, of course, conditions of finite existence must have some positive significance for the infinite.
In relation to knowledge, an infinite consciousness means, primarily, not the capacity to think in serial order, but to penetrate directly and immediately the obstinate facts of experience which are to us opaque, and into which we gain insight only slowly and by constant effort. An infinite intellect must be intuitive, i.e., the contents of its own immediate consciousness and malforms and sorts of existence must be present to its thought luminously, instantaneously, and continuously. To such an intellect, all objects of thought are as clear in every relation as indirectly created by itself. But we do not need to assume that it has no objects of thought or experience that are not directly created by itself. We need not assert, in order to admit the reality of an absolute self, that there is in the universe only one thinker order. Of course, we do not understand from our own experience the inner constitution of such an infinite intuitive intellect. But, as I have maintained, the infinite is a limiting concept, we must be satisfied to negatively determine its meaning in this relation, I. e., as the limiting condition of thought and knowledge in us.
In relation to goodness, an infinite will must be devoid of all inherent temptation or struggle. There can be in such a will no gap between purpose and achievement, no interval between wildland deed, and no conflict of desires. In other words, a goodness positively infinite transcends the human moral struggle. The infinitely good is the limiting notion of the humanly good. The latter approaches the former as goodness becomes second nature, as it passes from self-conscious struggle and choice into moral habitude, and good conduct becomes the spontaneous expression of 'good feeling.' The opposing concepts of duty and inclination, then, have no direct application to the action of an infinitely good will. The infinite or final limit of our consciously sought moral goodness is a state of volition other and higher than itself. This other seems to be what the Christian means by infinite love. The attainment of an infinite goodness would be its transformation into a higher and spontaneous state of action in harmony with reality. The infinitely good is that goal of our moral endeavor which sets the limit to our struggles. But here again we know not how many efforts lie between us and the goal. Perfect goodness, being indefinitely removed from our present attainments, we call infinite.
In relation to power, the infinite must be wholly self-active. Its action can in no way be originated or called forth by any power hostile to or underived from itself. This action can be permanently obstructed by no obstacle which it does not itself set up or allow (the human will, for example, might conceivably obstruct the Divine will, but if the latter were infinite in power, we should have to assume that, from the depths of its ethical nature as love, the Divine will consented to this obstruction as a condition of human moral freedom). No sort of being could be said to possess infinite power unless it were the creative source of all power. But an infinite ethical power might give relatively independent power to created or finite wills. Indeed, unless we admit in the infinite power or will the reality of self limitation, it follows that there is only one truly active being in the universe, and that we finite doers are absorbed in the infinite doer. This conception would make the realization of the infinite the absolute negation of the finite. The synthesis of finite and infinite would be that of the lamb and the wolf. But if one starts from the assumption of a reality in the finite and individual, the notion of infinite power must be subordinated to that of infinite love or ethical will. Otherwise, the ground is cut from under one's feet, the potential infinitude of the human self is denied, and we are plunged into the inane. There is a dialectic here which can only be overcome by recognizing that omnipotence Isa notion to be transcended, and that it merely represents for us the limit beyond our indefinite consciousness of power in ourselves and the world.
I have here tried to indicate very briefly the meanings of a notion which has its deepest roots in the moral and religious life and in the accompanying metaphysical craving, rather than in pure thought. The new infinite of symbolic logic and mathematics illustrates clearly from the side of pure thought the mind’s self-transcendence of the actual; and to this extent it shows pure thought to be in harmony with ethical and religious feeling and speculation. But whether such notions are more than perennial illusions of the human mind, whether reality ultimately meets these demands of feeling and action as well as of thought, must be decided on other and more fundamental considerations. The problem of the place of the developing human self and of change in general in the universe, still remains the central problem to which the notion of the infinite is auxiliary and supplemental.
However, one may try to answer this metaphysical question, I venture to assert that the most positive, comprehensive, and fruitful notion of the infinite is that of the ideal limit of actual human thoughts, feelings, and deeds. But this invites the further and paradoxical conclusion that the infinite has significance for us only so long as we fall short of perfection, and that perfection once achieved, the notion of the infinite must vanish from thought.
Here, on the threshold of metaphysics, the present discussion must end, and I will only say in conclusion that if the term infinities to continue to be used in philosophical and theological discussion, a sharp distinction must be made between the potential and the actual infinites, i. e., between the infinite as the law or principle of serial order, etc., in human thinking, and the infinite as the absolute limit or fruition of human striving. This distinction is the same as that expressed in Cantor's terms, the" transfinite" and the "absolutely infinite." The new notion of the infinite in its application to metaphysics seems to fluctuate between these two meanings.