2. To fix the idea of duration we might imagine some rigid object—a lamp, say—together with the ticking of a clock. Both are necessary; for if either is missing the image fails. The image is no doubt rather crude, but will perhaps serve to make it clear that duration—what we sometimes call 'the passage of time'—is a combination of unchange and change. Duration and Invariance under Transformation are one and the same.
3. We saw, in Part I, that a thing can be represented by the four symbols, o, o, o, and x, which pair off to define the operation of interchanging o o and o x. This, we found, can be done in three ways, , , and , or by interchange of columns, of rows, and of both together. We do not need, at present, to distinguish them, and we can take interchange of columns, , as representative of the whole. When o o is transformed into o x and vice versa, the thing or operation (o, o, o, x) is invariant—all that has happened is that the symbols have rearranged themselves: has become . This is one unit of duration—one moment. Clearly enough we can repeat the operation, so: . It is still the same operation, namely interchange of columns. (The operation of transforming o o into o x automatically transforms o x into o o—when the old 'o first' becomes the new 'x second', the old 'x second' becomes the new 'o first', as with our journey of §I/6 from A to B—, and each time we are ready to start afresh.) This gives us a second moment; and by continued repetition we can get as many moments as we please, with the thing as a whole remaining unchanged.
4. We know, however, that the structure is hierarchical; and 'a time must come' when the thing as a whole changes—just as becomes , so must become . How many times must the transformation be repeated before the transformation is itself transformed? For how many moments does a thing endure? Let us suppose that it endures for a certain finite number of moments, say a hundred. Then, after a hundred moments the thing changes, and after another hundred moments it changes again, and after yet another hundred moments it changes yet again, and so on. It will be seen that we do not, in fact, have a combination of unchange and change, but two different rates of change, one slow and one fast, just like two interlocking cog-wheels of which one revolves once as the other revolves a hundred times. And we see that this fails to give the idea of duration; for if we make the large cog-wheel really unchanging by holding it fast, the small cog-wheel also is obliged to stop. Similarly, we do not say 'a minute endures for sixty seconds' but 'a minute is sixty seconds'—it would never occur to us to time a minute with a stop-watch. To get duration, the difference between the unchanging and the changing must be absolute: the unchanging must be unchanging however much the changing changes.[j] If a thing endures, it endures for ever. A thing is eternal.
5. A thing changes, then, after an infinity of moments. And since the structure is hierarchical, each moment must itself endure for an infinity of moments of lesser order before it can give place to the next moment. And, naturally, the same applies to each of these lesser moments. It might perhaps seem that with such a congestion of eternities no change can ever take place at any level. But we must be careful not to introduce preconceived notions of time: just as the structure is not in space but of space (amongst other things)—see §I/2—, so the structure is not in time but of time. Thus we are not at all obliged to regard each moment as lasting the same length of absolute time as its predecessor; for we have not encountered 'absolute time'. Naturally, if we regard a given thing as eternal, then each of the infinite moments for which it endures will be of the same duration—one unit. But if this eternal thing is to change (or transform), then clearly the infinite series of moments must accelerate. If each successive moment is a definite fraction (less than unity) of its predecessor, then the whole infinite series will come to an end sooner or later.
6. Now we see that three levels of the hierarchy are involved: on top, at the most general level of the three, we have a thing enduring eternally unchanged; below this, we have a thing changing at regular intervals of one unit of duration, one moment; and below this again, in each of these regular intervals, in each of these moments, we have an infinite series of moments of lesser order accelerating and coming to an end. We have only to take into account an eternal thing of still higher order of generality to see that our former eternal thing will now be changing at regular intervals, that the thing formerly changing at regular intervals will be accelerating its changes (and the series of changes repeatedly coming to an end at regular intervals), and that the formerly accelerating series will be a doubly accelerating series of series. There is no difficulty in extending the scheme infinitely in both directions of the hierarchy; and when we have done so we see that there is no place for anything absolutely enduring for ever, and that there is no place for anything absolutely without duration.[k]
7. We can represent a thing by O. This, however, is eternal. To see the structure of change we must go to the 4-symbol representation , where o and x are things of the next lower order of generality. From §3 it will be seen that O is the invariant operation of interchange of columns: becomes , and then becomes , and so on, to infinity. But now that we have found that moments (or things) come to an end, some modification in this account is needed. In , o is 'this' and x is 'that' (i.e. 'not-this'), as we saw in Part I. When the moment marked by one interchange of columns comes to an end, 'this' vanishes entirely, and we are left just with 'that', which, clearly, is the new 'this'. The o's disappear, in other words. Thus when has become we shall not, contrary to what we have just said, have the same operation simply in the opposite sense, i.e. , since all that remains is . In the repetition of the operation, then, x will occupy the same position as o in the original, and O (i.e. 'interchange of columns') will now be represented by . The second interchange of columns will thus be , the third interchange will be , and the fourth , and so on. It will be evident that, while O is invariant (eternally), the symbols at the next lower level of generality will be alternating between o and x. (For convenience we may start off the whole system with the symbol o at each level, though in different sizes, to represent 'this'; and we may then allow these to change to x as the system is set in motion. But we can only do this below a given level, since if only we go up far enough we shall always find that the system has already started. We cannot, therefore, start the system at any absolute first point—we can only 'come in in the middle'. It will be seen, also, that the system is not reversible: future is future and past is past. But this will become clearer as we proceed.)
8. Disregarding other things, consciousness of a thing while it endures is constant: and this may be counted as unity. We can regard consciousness of a thing as the thing's intensity or weight—quite simply, the degree to which it is. In §I/12 (f) we noted that any interchange is equivalent to the other two done together. Thus, to pass from 1 to 4 it is necessary to go by way of both 2/3 and 3/2, so: . The intensity or weight must therefore be distributed among the four symbols in the following way: , or . This will mean that the intensity of o is two-thirds of the whole, and of x, one-third. (A moment's reflexion will verify that 'this' is necessarily more intense than 'that'. Visual reflexion will do here; but it must be remembered that visual experience, which is easy to refer to, is structurally very complex—see §I/4—, and visual evidence normally requires further break-down before revealing aspects of fundamental structure. It is usually less misleading to think in terms of sound or of extension than of vision, and it is advisable in any case to check the evidence of one sense with that of another.) When vanishes we shall be left with x, whose intensity is only one-third of the whole. But just as stands to x in the proportion of intensity of 2:1, so of a lesser order stands to o of the same lesser order in the same proportion, and so on indefinitely. Thus we obtain a hierarchy of intensity , , , , ,... to infinity, the sum of which is unity. The total intensity at any time must be unity, as we noted above; and when the first term of this hierarchy, , which is the total intensity, vanishes, it is necessary to increase the intensity of the rest to compensate for this loss; and to do this we must make x, when it becomes , be (or exist) correspondingly faster. This is achieved, clearly enough, by doubling the rate of existence (i.e. halving the relative length) of each successive moment. (When the first term of + + + + ... vanishes, it is only necessary to double the remainder, + + ++ ..., to restore the status quo.)
9. If we go to the 16-member representation it will be clearer what is happening. This representation, , combines two adjacent levels of generality: it is a combination of and . But this combination, we see, can be made in two ways: and . Alternatively, however, we can regard the combination of and not as that of two adjacent levels of generality, but as that of the present and the future on the same level of generality; and, clearly, this too can be made in these two ways. If, furthermore, we regard the first of these two ways in which the combination of and can be made as the combination of two adjacent, equally present, levels of generality, we must regard the second way as the combination of the present and the future, both of the same level of generality; and, of course, vice versa. This means that, from the point of view of , can be regarded either as present but of lower order or as of the same order but future. (And, of course, from the point of view of , can be regarded either as present but of higher order or as of the same order but past.) In other words, the general/particular hierarchy can equally well be regarded—or rather, must at the same time be regarded—as the past, present, and future, at any one level of generality. (A simple illustration can be given. Consider this figure:
It presents itself either as a large square enclosing a number of progressively smaller squares all within one plane at the same distance from the observer, or as a number of squares of equal size but in separate planes at progressively greater distances from the observer, giving the appearance of a corridor. A slight change of attention is all that is needed to switch from one aspect to the other. In fundamental structure, however, both aspects are equally in evidence.) This allows us to dispose of the tiresome paradox (noted, but not resolved, by Augustine) that, (i) since the past is over and done with and the future has not yet arrived, we cannot possibly know anything about them in the present; and (ii) there is, nevertheless, present perception and knowledge of the past and of the future (memory is familiar to everyone,[l] and retrocognition and precognition are well-known occurrences; though it is clear that awareness of movement or of change of substance provides more immediate evidence[m])—the very words past and future would not exist if experience of what they stand for were inherently impossible.[n]
10. Past and future (as well as present) exist in the present; but they exist as past and as future (though what exactly the pastness of the past—'this is over and done with'—and the futurity of the future—'this has not yet arrived'—consist of will only become apparent at a later stage when we discuss the nature of intention). And since each 'present' is a self-sufficient totality, complete with the entire past and the entire future, it is meaningless to ask whether the past and the future that exist at present are the same as the real past or future, that is to say as the present that was existing in the past and the present that will be existing in the future: 'the present that existed in the past' is simply another way of saying 'the past that exists in the present'.[o] From this it will be understood that whenever we discuss past, present, and future, we are discussing the present hierarchy, and whenever we discuss the present hierarchy we are discussing past, present, and future. The two aspects are rigorously interchangeable:
11. In §3 we took the interchange of columns as representative of all three possible interchanges: (i) of columns, (ii) of rows, and (iii) of both together. We must now discriminate between them. Neglecting the zero operation of no interchange, we may regard a thing as a superposition of these three interchanges (§I/13). We saw in §8 that ('this') has twice the intensity or weight of ('that'), and this is obviously true of each of the three possible interchanges. But this imposes no restriction whatsoever on the intensities of the three interchanges relative one to another: what these relative intensities shall be is a matter of complete indifference to fundamental structure. Let us, therefore, choose convenient numbers; let us suppose that the weight of interchange of columns, , is one-half of the total, of interchange of rows, , one-third, and of interchange of both, , one-sixth, the total being unity. Then, in interchange of columns, 'this' will have the value , and 'that' the value ; in interchange of rows, 'this' will have the value , and 'that' the value ; and in interchange of both, 'this' will have the value , and 'that' the value . It will be observed that the three 'this' are indistinguishable, whereas the three 'that' and are not; and that consequently we simply have one single 'this', of value or , and three separate 'that', of respective values , , and , totalling . No matter what the relative weights of the three interchanges may be, the weight of 'this' is always twice the combined weights of the three 'that'. This means, in effect, that however much the relative weights of the three 'that' may vary among themselves, the weight of 'this' remains constant.
12. The question now arises, which of these three possible interchanges is the one that will take place when the time comes for 'this' to vanish and 'that' to become 'this'. We said, in §7, that a thing, O, is the invariant operation of interchange of columns to infinity. This, however, is equally true of interchange of rows and of both columns and rows. In other words, O is simply the invariant operation of interchange, no matter whether of columns, of rows, or of both. Any or all of these interchanges are O. It will be seen, then, that the invariance of O is unaffected by the distribution of weight among the three possible interchanges that can take place. A simplified illustration may make this clearer. Suppose my room contains a chair, a table, a bed, and a wardrobe. If there is no other article of furniture in the room, the chair is determined as the chair by its not being the table, the bed, or the wardrobe. In other words, the piece of furniture in my room that is not-the-table, not-the-bed, and not-the-wardrobe, is the chair. But so long as all these determinations are to some extent present it matters not at all where the emphasis is placed. The question of degree, that is to say, does not arise. If, when I am about to sit down and start writing, I pay attention to the chair, it will present itself strongly to me as being not-the-table, but perhaps only faintly as not-the-wardrobe, and hardly at all as not-the-bed; but if I pay attention to it when I am feeling sleepy, it will be most strongly present as not-the-bed, and much less as not- the-table and not-the-wardrobe. In either case the chair keeps its identity unaltered as 'the piece of furniture that is neither table, bed, nor wardrobe'.
13. Let us consider two adjacent levels of generality, O and o, where O endures for one moment while o undergoes an infinity of transformations in an accelerating series. But the symbols O and o simply give the immediate thing (§I/15), and we need to see the structure of the thing. We must therefore write each thing in the form and expand accordingly. We also need to see the structure of the two adjacent levels at the same time. This will give us the figure of §I/16 (h), viz: .
(This figure is out of scale: it should be one-quarter the size.) We see that O is represented by and o by . (Note that D, for example, is simply with interchange of both columns and rows, i.e. , and similarly with B and C.) Let us suppose that, at the lower level, repeated interchange of columns (a-b, c-d) is taking place. This, naturally, will be taking place in all four quarters, A, B, C, and D. Let us also suppose that, to begin with, the relative weights of the three possible interchanges of O are 1(A-B) : 2(A-D) : 3(A-C). We have seen in §7 that whenever an interchange, say, takes place, it is actually not simply an interchange, but a disappearance of leaving just x. This x is then the fresh , which in its turn becomes o, and so on. In other words, each time what we have represented as an interchange takes place, things lose a dimension. This statement can be inverted, and we can say that the present, each time it advances into the future, gains a dimension, with the consequence that immediately future things, when they become present, will necessarily appear with one dimension less. Though, from one point of view, O remains invariant throughout the series of interchanges (it is the series of interchanges, of any or all of the three possible kinds), from another point of view, each time an interchange takes place O vanishes and is replaced by another O differing from the earlier O only in that having been future to it (or of lower order—see §9) it has, relative to it, a second dimension. We must at once qualify this statement. The loss of a dimension takes place at the level, not of O, but of o, which is at a lower level of generality; and properly speaking we should say that O loses an infinitesimal part of its one dimension each time there is the loss of a dimension at the level of o. Similarly, O's successor is only infinitesimally future or of lower order. In other words, O's dimension is of a higher order than that of o. But consideration of O's possible interchanges takes place at the level of o, as we may gather from the necessity, noted above, of writing O in the reflexive form . It must therefore be understood that when we say that each future O has one more dimension than the present O, the dimension in question is a dimension of o, not of O. The original O, then, while present, has one dimension: its successor, so long as it is future, has two dimensions: and when this becomes present it appears as having one dimension, just as its predecessor did when present. But the original O now has no dimension; for it has vanished. (That is to say, o has vanished: O is actually no more than infinitesimally closer to the point of vanishing—which means that it remains absolutely the same, in the ordinary meaning of that word. But we have to remember that changes in a thing's internal distribution of weight—the weight, that is, of its determinations—do not affect it.) Relatively speaking, then, each next future O has one more dimension, at the level of o, than the present O, even though it has but one dimension when it is itself present. If, therefore, the relative weights of the possible interchanges of the original O are in the proportions 3:2:1, the relative weights of the succeeding O, when it becomes present, will be in the proportion 9:4:1, that is, with each number squared. Following that, the next O will have relative weights 81:16:1, and so on. It is obvious, first, that the most heavily weighted of the possible interchanges will tend more and more to dominate the others and, in a manner of speaking, to draw all the weight to itself; and secondly, that it can only draw the entire weight to itself after an infinity of squarings, that is, of interchanges at the level of o. As soon as one of the three possible interchanges has drawn the entire weight to itself and altogether eliminated its rivals, that interchange takes place (at the level of O).[p] In the case we are considering there will be interchange of rows, i.e. of A and C, and of B and D. Notice that this interchange is quite independent of the kind of interchange that is taking place at the next lower level: interchange of rows at the level of O does not in the least require that the interchange at the level of o should also have been of rows.
[j] This will clearly permit different relative rates of change, or frequencies, at the same level. The ratios between such frequencies would seem to be arbitrary, but it is clear that they can change only discontinuously. In other words, the substance of my world (real and imaginary) at any time is not dictated by fundamental structure, and vanishes abruptly. (See RŪPA [c].) The only change considered by the main body of this Note, in its present incomplete form, is change of orientation or perspective. Duration does not require change of substance, though the converse is not true. (Might it not be that with every change of orientation in the world of one sense there is a corresponding change of substance in the world of each of the others? This is partly observable at least in the case of intentional bodily action; which, indeed, seems to change the substance also of its own world—as when the left hand alters the world of the right. But this supposition is not without its difficulties.) The 'unchange' that is here in question is on no account to be confused with what is described in ATTĀ as an 'extra-temporal changeless "self"'. Experience of the supposed subject or 'self' (a would-be extra-temporal personal nunc stans) is a gratuitous (though beginningless) imposition or parasite upon the structure we are now discussing. See CETANĀ [f] . (Cf. in this connexion the equivocal existentialist positions discussed by M. Wyschogrod in Kierkegaard and Heidegger (The Ontology of Existence), Routledge & Kegan Paul, London 1954.) [Back to text]
[k] It would be a mistake to attempt to take up a position outside the whole system in order to visualize it as passing from the future into the past through a 'present moment' in a kind of universal time. At any given level of generality, the 'present moment' lasts for one whole eternity relative to the next lower level, and there is thus no such thing as a 'present moment' for the system as a whole; nor has the system any outside (even imaginary) from which it may be viewed 'as a whole'. [Back to text]
[l] All memory involves perception of the past, but perception of the past is not in itself memory. The question of memory, however, does not otherwise concern us in these Notes. (The attention we give to whatever happens to be present will, no doubt, permanently increase its weightage relative to all that does not come to be present.) [Back to text]
[m] Neither movement nor change of substance is fundamental: fundamental structure is necessary for them to be possible, and this is true also of their respective times (see §4 (j)). In other words, the time (past, present, future) that is manifest in movement and in change of substance is dependent upon, but does not share the structure of, the time that is discussed in these pages. Thus, in movement, the time is simply that of the hierarchy of trajectories (see PATICCASAMUPPĀDA [c]), and its structure is therefore that of the straight line (see §I/13): the time of movement, in other words, is perfectly homogeneous and infinitely subdivisible. In itself, therefore, this time makes no distinction between past, present, and future, and must necessarily rest upon a sub-structure that does give a meaning to these words. In fundamental time, each unit—each moment—is absolutely indivisible, since adjacent levels are heterogeneous. [Back to text]
[n] McTaggart has argued (op. cit., §§325 et seq.) that the ideas of past, present, and future, which are essential characteristics of change and time, involve a contradiction that can only be resolved in an infinite regress. This regress, he maintained, is vicious, and change and time are therefore 'unreal'. It is clear enough that perception of movement, and therefore of time, does involve an infinite reflexive (or rather, pre-reflexive) regress. We perceive uniform motion; we perceive accelerated motion, and recognize it as such; we can perhaps also recognize doubly accelerated motion; and the idea of still higher orders of acceleration is perfectly acceptable to us, without any definite limit: all this would be out of the question unless time had an indefinitely regressive hierarchical structure. If this regress is vicious, then so much the worse for virtue. But see §I/15 (g), which indicates that it is not in fact vicious. [Back to text]
[o] These remarks do not imply that the present that will be existing in the future is now determined; on the contrary (as we shall see) it is under-determined—which is what makes it future. Similarly, the past is now what is over-determined. [Back to text]
[p] §I/4 (d) would seem to imply that three different frequencies are involved, all converging to infinity together. This will complicate the arithmetic, but can scarcely prevent the eventual emergence of one dominating interchange. (If they are not all to be squared together, the relative weights a : b : c must be made absolute before each squaring: .) [Back to text]