But this criticism misses Zeno's point entirely. Zeno would have been perfectly willing to grant that if the tortoise can be overtaken at all, he can be overtaken in (say) twenty seconds, but he would still have insisted that he can't be overtaken at all. Leave Achilles and the tortoise out of the account altogether, he would have said—they complicate the case unnecessarily. Take any single process of change whatever, take the twenty seconds themselves elapsing. If time be infinitely divisible, and it must be so on intellectualist principles, they simply cannot elapse, their end cannot be reached; for no matter how much of them has already elapsed, before the remainder, however minute, can have wholly elapsed, the earlier half of it must first have elapsed. And this ever re-arising need of making the earlier half elapse first leaves time with always something to do before the last thing is done, so that the last thing never gets done. Expressed in bare numbers, it is like the convergent series 1/2 plus 1/4 plus 1/8…, of which the limit is one. But this limit, simply because it is a limit, stands outside the series, the value of which approaches it indefinitely but never touches it. If in the natural world there were no other way of getting things save by such successive addition of their logically involved fractions, no complete units or whole things would ever come into being, for the fractions' sum would always leave a remainder. But in point of fact nature doesn't make eggs by making first half an egg, then a quarter, then an eighth, etc., and adding them together. She either makes a whole egg at once or none at all, and so of all her other units. It is only in the sphere of change, then, where one phase of a thing must needs come into being before another phase can come that Zeno's paradox gives trouble.
And it gives trouble then only if the succession of steps of change be infinitely divisible. If a bottle had to be emptied by an infinite number of successive decrements, it is mathematically impossible that the emptying should ever positively terminate. In point of fact, however, bottles and coffee-pots empty themselves by a finite number of decrements, each of definite amount. Either a whole drop emerges or nothing emerges from the spout. If all change went thus drop-wise, so to speak, if real time sprouted or grew by units of duration of determinate amount, just as our perceptions of it grow by pulses, there would be no zenonian paradoxes or kantian antinomies to trouble us. All our sensible experiences, as we get them immediately, do thus change by discrete pulses of perception, each of which keeps us saying 'more, more, more,' or 'less, less, less,' as the definite increments or diminutions make themselves felt. The discreteness is still more obvious when, instead of old things changing, they cease, or when altogether new things come. Fechner's term of the 'threshold,' which has played such a part in the psychology of perception, is only one way of naming the quantitative discreteness in the change of all our sensible experiences. They come to us in drops. Time itself comes in drops.
Our ideal decomposition of the drops which are all that we feel into still finer fractions is but an incident in that great transformation of the perceptual order into a conceptual order of which I spoke in my last lecture. It is made in the interest of our rationalizing intellect solely. The times directly felt in the experiences of living subjects have originally no common measure. Let a lump of sugar melt in a glass, to use one of M. Bergson's instances. We feel the time to be long while waiting for the process to end, but who knows how long or how short it feels to the sugar? All felt times coexist and overlap or compenetrate each other thus vaguely, but the artifice of plotting them on a common scale helps us to reduce their aboriginal confusion, and it helps us still more to plot, against the same scale, the successive possible steps into which nature's various changes may be resolved, either sensibly or conceivably. We thus straighten out the aboriginal privacy and vagueness, and can date things publicly, as it were, and by each other. The notion of one objective and 'evenly flowing' time, cut into numbered instants, applies itself as a common measure to all the steps and phases, no matter how many, into which we cut the processes of nature. They are now definitely contemporary, or later or earlier one than another, and we can handle them mathematically, as we say, and far better, practically as well as theoretically, for having thus correlated them one to one with each other on the common schematic or conceptual time-scale.
Motion, to take a good example, is originally a turbid sensation, of which the native shape is perhaps best preserved in the phenomenon of vertigo. In vertigo we feel that movement is, and is more or less violent or rapid, more or less in this direction or that, more or less alarming or sickening. But a man subject to vertigo may gradually learn to co-ordinate his felt motion with his real position and that of other things, and intellectualize it enough to succeed at last in walking without staggering. The mathematical mind similarly organizes motion in its way, putting it into a logical definition: motion is now conceived as 'the occupancy of serially successive points of space at serially successive instants of time.' With such a definition we escape wholly from the turbid privacy of sense. But do we not also escape from sense-reality altogether? Whatever motion really may be, it surely is not static; but the definition we have gained is of the absolutely static. It gives a set of one-to-one relations between space-points and time-points, which relations themselves are as fixed as the points are. It gives positions assignable ad infinitum, but how the body gets from one position to another it omits to mention. The body gets there by moving, of course; but the conceived positions, however numerously multiplied, contain no element of movement, so Zeno, using nothing but them in his discussion, has no alternative but to say that our intellect repudiates motion as a non-reality. Intellectualism here does what I said it does—it makes experience less instead of more intelligible.
We of course need a stable scheme of concepts, stably related with one another, to lay hold of our experiences and to co-ordinate them withal. When an experience comes with sufficient saliency to stand out, we keep the thought of it for future use, and store it in our conceptual system. What does not of itself stand out, we learn to cut out; so the system grows completer, and new reality, as it comes, gets named after and conceptually strung upon this or that element of it which we have already established. The immutability of such an abstract system is its great practical merit; the same identical terms and relations in it can always be recovered and referred to—change itself is just such an unalterable concept. But all these abstract concepts are but as flowers gathered, they are only moments dipped out from the stream of time, snap-shots taken, as by a kinetoscopic camera, at a life that in its original coming is continuous. Useful as they are as samples of the garden, or to re-enter the stream with, or to insert in our revolving lantern, they have no value but these practical values. You cannot explain by them what makes any single phenomenon be or go—you merely dot out the path of appearances which it traverses. For you cannot make continuous being out of discontinuities, and your concepts are discontinuous. The stages into which you analyze a change are states, the change itself goes on between them. It lies along their intervals, inhabits what your definition fails to gather up, and thus eludes conceptual explanation altogether.
'When the mathematician,' Bergson writes, 'calculates the state of a system at the end of a time t, nothing need prevent him from supposing that betweenwhiles the universe vanishes, in order suddenly to appear again at the due moment in the new configuration. It is only the t-th moment that counts—that which flows throughout the intervals, namely real time, plays no part in his calculation…. In short, the world on which the mathematician operates is a world which dies and is born anew at every instant, like the world which Descartes thought of when he spoke of a continued creation.' To know adequately what really happens we ought, Bergson insists, to see into the intervals, but the mathematician sees only their extremities. He fixes only a few results, he dots a curve and then interpolates, he substitutes a tracing for a reality.
This being so undeniably the case, the history of the way in which philosophy has dealt with it is curious. The ruling tradition in philosophy has always been the platonic and aristotelian belief that fixity is a nobler and worthier thing than change. Reality must be one and unalterable. Concepts, being themselves fixities, agree best with this fixed nature of truth, so that for any knowledge of ours to be quite true it must be knowledge by universal concepts rather than by particular experiences, for these notoriously are mutable and corruptible. This is the tradition known as rationalism in philosophy, and what I have called intellectualism is only the extreme application of it. In spite of sceptics and empiricists, in spite of Protagoras, Hume, and James Mill, rationalism has never been seriously questioned, for its sharpest critics have always had a tender place in their hearts for it, and have obeyed some of its mandates. They have not been consistent; they have played fast and loose with the enemy; and Bergson alone has been radical.
To show what I mean by this, let me contrast his procedure with that of some of the transcendentalist philosophers whom I have lately mentioned. Coming after Kant, these pique themselves on being 'critical,' on building in fact upon Kant's 'critique' of pure reason. What that critique professed to establish was this, that concepts do not apprehend reality, but only such appearances as our senses feed out to them. They give immutable intellectual forms to these appearances, it is true, but the reality an sich from which in ultimate resort the sense-appearances have to come remains forever unintelligible to our intellect. Take motion, for example. Sensibly, motion comes in drops, waves, or pulses; either some actual amount of it, or none, being apprehended. This amount is the datum or gabe which reality feeds out to our intellectual faculty; but our intellect makes of it a task or aufgabe—this pun is one of the most memorable of Kant's formulas—and insists that in every pulse of it an infinite number of successive minor pulses shall be ascertainable. These minor pulses we can indeed go on to ascertain or to compute indefinitely if we have patience; but it would contradict the definition of an infinite number to suppose the endless series of them to have actually counted themselves out piecemeal. Zeno made this manifest; so the infinity which our intellect requires of the sense-datum is thus a future and potential rather than a past and actual infinity of structure. The datum after it has made itself must be decompos_able_ ad infinitum by our conception, but of the steps by which that structure actually got composed we know nothing. Our intellect casts, in short, no ray of light on the processes by which experiences get made.
Kant's monistic successors have in general found the data of immediate experience even more self-contradictory, when intellectually treated, than Kant did. Not only the character of infinity involved in the relation of various empirical data to their 'conditions,' but the very notion that empirical things should be related to one another at all, has seemed to them, when the intellectualistic fit was upon them, full of paradox and contradiction. We saw in a former lecture numerous instances of this from Hegel, Bradley, Royce, and others. We saw also where the solution of such an intolerable state of things was sought for by these authors. Whereas Kant had placed it outside of and before our experience, in the dinge an sich which are the causes of the latter, his monistic successors all look for it either after experience, as its absolute completion, or else consider it to be even now implicit within experience as its ideal signification. Kant and his successors look, in short, in diametrically opposite directions. Do not be misled by Kant's admission of theism into his system. His God is the ordinary dualistic God of Christianity, to whom his philosophy simply opens the door; he has nothing whatsoever in common with the 'absolute spirit' set up by his successors. So far as this absolute spirit is logically derived from Kant, it is not from his God, but from entirely different elements of his philosophy. First from his notion that an unconditioned totality of the conditions of any experience must be assignable; and then from his other notion that the presence of some witness, or ego of apperception, is the most universal of all the conditions in question. The post-kantians make of the witness-condition what is called a concrete universal, an individualized all-witness or world-self, which shall imply in its rational constitution each and all of the other conditions put together, and therefore necessitate each and all of the conditioned experiences.
Abridgments like this of other men's opinions are very unsatisfactory, they always work injustice; but in this case those of you who are familiar with the literature will see immediately what I have in mind; and to the others, if there be any here, it will suffice to say that what I am trying so pedantically to point out is only the fact that monistic idealists after Kant have invariably sought relief from the supposed contradictions of our world of sense by looking forward toward an ens rationis conceived as its integration or logical completion, while he looked backward toward non-rational dinge an sich conceived as its cause. Pluralistic empiricists, on the other hand, have remained in the world of sense, either naïvely and because they overlooked the intellectualistic contradictions, or because, not able to ignore them, they thought they could refute them by a superior use of the same intellectualistic logic. Thus it is that John Mill pretends to refute the Achilles-tortoise fallacy.