by David Foss
January 21, 1994
Two of Zeno’s Paradoxes, Restated
1. The Achilles
Achilles is in pursuit of a tortoise, which moves continuously, although at a slower rate, away from him. Zeno claims that if space is continuous (i.e. infinitely divisible) then Achilles will have to traverse an infinite number of intermediate differentials (the positive distance between Achilles and the tortoise at any given time) before reaching the slow moving animal. Likewise, if time is continuous, Achilles will have to consume an infinite number of durations (the time it takes Achilles to traverse the distance between himself and the tortoise’s initial position, taken successively across the full pursuit) before arriving at the moment of interception. Even worse, it would seem that the tortoise, although stipulatively moving at a constant rate, will never reach a distance equal to the product of the initial distance between it and Achilles and its speed divided by the difference between their two speeds (that is, the geographic point of interception). Of course, the paradox depends upon the presumed unintelligibility (or apparent absurdity) of transcending (or even traversing) an infinitely articulable field (temporal, spacial, or otherwise).
2. The Arrow
An arrow is launched into the air, and Zeno asks us to consider an instantaneous analysis of its motion. Already such an analysis should strike us as quite absurd. Instantaneous motion? Within the instant, the arrow can not of itself have motion. There can not be, within the instant, two distinct instant-stages which each “position” the arrow further upon its path, without violating the stipulative integrity of the instant. And the arrow can not occupy two positions, without thereby failing to remain the self-same arrow. Nor can the arrow “move” during the instant, wherein it might be said to occupy a space larger than itself. Finally, the arrow can not travel “between” instants, without such travel being a sort of quantum velocity. This final point should aptly suggest the core of this paradox. Zeno here relies upon the absurdity of an instantaneous measure of vectored velocity, coupled with the counterintuitive notion of quantum “leaps” as the primitive character of motion. An atomistic reading of either time or space forces a conception of motion (and duration) which (at least on the microscopic level) contradicts our intuitive grasp of the phenomena of momentum, inertia, and material causation generally.
January 28, 1994
Augustine, Leibniz, and Newton
A Puzzle from Augustine:
Where or whence lies duration? If duration is a phenomenon of the future, then it is not yet, and cannot be. If duration is a phenomenon of the past, then it is no longer, and cannot be. If duration is a phenomenon of the present, then it has no span or “length,” otherwise it would have a part which is yet to be as it begins, and a part which is no longer as it ends, such that it would be (present) and yet to be (but not) and is no longer simultaneously. Thus “duration” in the present would have no duration. Therefore, duration does not exist in the future, nor does it exist in the past, nor, finally, does it exist in the present. And yet, it is by duration that we apprehend the temporal location of events with respect to one another. Furthermore, duration must be real to the extent that a thing is in time. For, without a before or an after, a thing cannot be said to change, or exist temporally. This shows that duration must exist if there is change, and duration cannot exist in time. And this may appear strange, if we fail to notice that such an investigation of duration treats a character of temporality temporally (i.e. we are treating duration as an object of time, rather than a character of objects or relations in time).
Leibniz’s ‘Absolute’ Relativism:
Leibniz begins his articulation of the ideality of time, and the relationality of space, by supposing the reality of coincidence and simultaneity. Neither of these are surprising conditions given the logical constraints upon the spacio-temporal orderings internal to the Leibnizian world. Objects in space, the differentiation of which locates the proportional frame of their intervening distance, bear an absolute measure of that proportionality, in spite of the arbitrary nature of any given articulation of the magnitude of that distance. Of two particles, a measure of their distance from one another may be calculated indirectly by reference to a constellation of surrounding particles, locating each individually with respect to the constellation, and then deriving their proximate distance geometrically, or directly by a measure of their distance according to some conventional scale (corresponding to a ratio obtained with reference to a standard measure, like meters or inches, or a local measure of some given particle separation or particle width). In any case, the distance calculated will be mathematically identical (without regard to the particular ratio used). In this sense, place (and derivatively, space) is relationally absolute. Unlike Relativistic distance, the ratios of the distances between objects in a system do not depend upon the differential velocities of constituent subsystems.
Likewise, for time, duration depends upon a logic of fixed ratios between successive instants. Time is nothing other than the temporal, or the relative simultaneity of temporal existants. As such, while there is no sense in speaking of an absolute standard [of] time, the temporal ratios between given instants will remain constant across disparate methods of calculation and differing ‘unit’ measures of time. Leibniz has rejected the notion of any zero hour (when time “began”), and of any universal clock (whose minutes are “true” measures of passing time, apart from the relative motions of world-constituents). But he is not entertaining a conception of time under which the rate of change of a phenomenon with respect to proximate phenomenon is sensitive to the relative velocity of the frame of reference (as it is under some relativistic conditions in modern physics). In this sense too, time is relationally absolute.
A vessel holding water is hung from a cord, tightly wound, such that when permitted to move about freely, the vessel will spin rapidly about its vertical axis. At rest, and immediately following the release of the vessel, the water remains level. As the water accelerates to join the rotation of the vessel (brought about by the friction of the vessel walls as communicated through the liquid), it presses outward against the vessel walls (in accord with the centrifugal force) forcing the water at the perimeter to “climb”, until an equilibrium is reached between the centrifugal and gravitational force distributed through the fluid. Equilibrium will be reached when the water rotates at the rate of the vessel (there being no further acceleration imparted to the liquid other than that of the circular motion itself). At this point the water will form a concave figure, falling as the water nears the axis of rotation.
If space and time is relational, our analysis of the forces involved should not vary depending upon the frame of reference. However, if we examine the motion of the water relative to the vessel, if we take our frame to be that of the rotating vessel, then it seems that at the moment of greatest rotational velocity (when the vessel suddenly begins spinning) the water is level, and shows no centrifugal perturbation; and at the moment of least rotational velocity (when the water is rotating at the same rate as the vessel), and is indeed relativistically at rest, shows the curious behavior of clinging to the vessel walls (and falling toward the center). This suggests that the force(s) acting upon a substance or across a field (which is a derivative of acceleration, which in turn is a derivative of motion) cannot be accurately measured with respect to an arbitrary relational frame. Of course, in this case, we might suppose that the problem is particularly acute due to the dynamic motion of the frame itself (where the frame rotates about an axis, and therefore obscures derivative rates of change). By choosing a rotating frame, the frame itself is no longer inertial, but introduces acceleration in any non-coaxial motion.
It might seem, therefore, that Newton has not provided an adequate defense against the use of arbitrary inertial frames of reference, within which relativistic space and time successfully characterize phenomenon thought to require analysis with respect to absolute motion and location. It would not be enough to simply restrict out relativistic frames to non-rotational relata, leaving aside the difficult notion of an inertial frame. In “straight-line” non-inertial framing a similar difficulty arises (although not quite as paradoxical as that of centrifugal force). Consider two cars driving in the same direction, car A at 50km/h and car B at 100km/h. Just as car B, approaching car A from the rear, begins to pass car A, the driver in car B slows to 50km/h. With respect to car B, car A approached at 50km/h and rapidly decelerated (halting) as it drew near. And yet, car A did not expend any energy to achieve such a slowing, or “acceleration.” If we idealize these two cars (eg. as two space-craft), we can imagine car A intertially drifting through the entire operation. And yet, if the frame is fixed to car B, the characterization of the motion of car A reveals a substantial acceleration, and thereby a significant force acting upon it. A quick look at car A’s engine and fuel history, however, reveals no energy consumption coincidental with the action of the force: the acceleration was better than perfectly efficient! Again, the dilemma depends upon a non-inertial frame of reference, with respect to which relative motions are calculated.
February 11, 1994
Two Series, a Passage, and Indexicals
McTaggart’s Puzzle • A Tale of Two Series:
“It was the best of times; It was the worst times.” John McTaggart has isolated and named for us two, presumably exhaustive, characterizations of time-as-real. The A-series, following a roughly phenomenological model, gives a privileged ontological position to the present (as it is, both tenselessly and tensed), and locates time entirely from the “moving” perspective of the Now. The B-series, by contrast following a temporal-logic model, distributes its ontological commitment across time entire[ly], and locates every possible present in terms of its before and after, such that all time may be represented as a transitive asymmetric relation across time/world instants. Ultimately, McTaggart’s claim that time is not real rests upon a rejection of the logical viability of the A-series. The B-series is rejected only because it seems to not be time at all (though it may be some aspect or interpretation of it). But this is a strange way to lay out the problem. He needs something like the B-series in order that the rejection of the A-series be possible: for this rejection is based upon notions of “change” and “truth” which are essentially indexed according to time-of-occurrence or time-of-valuation.
The A-series is found incapable of handling “truth” across time, while maintaining time-as-change. And if the “real” is going to correspond at all with the “true” this should be treated as something of a problem. But McTaggart is anything but clear in most of his analysis. “Change” it seems requires an inversion of a truth value. Something is true at one time, and false at another. (Already we should notice that this is a definition of change native to the B-series, not the A-series.) Quickly, however, McTaggart time-indexes these truths, such that being-true-at-time-t1 and being-false-at-time-t2 localize and restrict the truth conditions of temporally variable propositions, and proclaims the death of “change”! How could there be change, he asks? These truth valuations “never” change.
There is a great deal going on in McTaggart’s paper. Unfortunately, much of it has the appearance at least of logical sloppiness, confusion between categories of time, temporality, and tense, and a dogmatic equivocation between “time” (generally understood) and time-as-an-A-series. None of these may ultimately be real flaws in McTaggart’s analysis. But they do make for some rough running. In the end, however, his analysis depends upon time (in the form of time-indexing truth valuations) to dismiss time (in the form of change with respect to propositional truth value). In other words, McTaggart needs time, in order to “freeze” facts in the way he wants to. And this, I believe, is a flaw neither generated by the ambiguity of his style of presentation, nor reversible by the discovery of further arguments lurking somewhere in the subtext. (He believes that his mysterious C-series can do the trick, without itself being called “time”.)
An important final point is the fundamental role the theory of truth plays here. The interplay between the “real” and the “true” is hardly as obvious as many a logician would have it. “Truth” need not be viewed as a sort of “real-world correspondence”, or as a propositional model of the real. Do we loose the metaphysical project if we embrace a pragmatic or conversational truth semantics (or any truth-theory which does not in itself reduce the World to a rigid or flowing cluster of platonic “facts”)? McTaggart’s analysis deeply relies upon the super-reality of facts, with respect to which the natural realm is a sort of shadow world. What sort of “reality” would we have if we replaced this ‘platonic’ truth definition with a semantics of truth-talk? (I suspect it would be one far more familiar and “natural”, than the “strange” world of truths-as-things.)
Williams’ Architectonic Manifold:
Donald Williams casts time a bit differently: As but one axis of the ontological geometry of the world, which he calls the “manifold”. It is a sort of four dimensional map, where we may point to any space-time location, and say of it true or false things, but in which we may not literally “move” freely from one position to another. The strangeness of the “time-dimension” is clearly overstated in many cases. Still, is does seem strange that Williams should think that such a manifold, rigid in its totality even if locally “sensitive” to apparent causal relations (i.e., adjacent chunks of space-time “obey” the rules of this special 4-dimensional geometry which are mathematically identical with our “laws” of motion, change, etc.), is so obviously neutral with respect to questions of free will, destiny, or the phenomenological unity of the self. Setting aside the complex issue of reifying a four-dimensional trans-sensory geometry (an admittedly powerful tool in terms of temporal truth definitions), it seems that theories of free will which rely upon existential indeterminacy, or demand absolute spontaneity in an agent in order that the agent be subject to moral evaluation, will be ruled out; not because they are somehow internally inconsistent, but because the ontology of the manifold literally seals our fate. There will be “manners of speaking” which get around this difficulty, and in the end, I’m not sure it’s one which threatens the basic theory. On this point, in any case, Williams ought to have been a trifle bit more honest about the real differences the manifold would make in terms of fact-like (the inter-subjective as opposed to the objective) realms of discourse.
More problematically, however, the manifold almost entirely mystifies the phenomenon of temporal embeddedness. As a road map, it is beautiful. But I am not a map. And a map is not, by any stretch of the imagination, “the real me.” The manifold might be relegated to the status of a convenient heuristic device, capable of making sense of my experience as a timely creature. But it seems strange to suppose that it is, furthermore, the reality of which I am (speaking of me now, of course) only a part. If it is the reality, and I am just a part, then what is this consciousness that only “sees” its successive time-locations, limited in a way it is not with respect to its space-locations. The phenomenon of the I, the most immediate and intimate of all experience (or as some would have it), becomes a terribly strange thing in this world of the manifold.
Mellor’s Tenseless Temporality:
Again, here is the voice of truth-at-a-time, and whether such truth follows from the phenomenological succession of “Nows” or vice-versa. If this sounds like McTaggart, it should. Only, now rather than claiming that a dismissal of the A-series is fatal to “time,” Mellor finds it merely fatal to “tense.” For the sake of space (I’ve run on too long already) I will only mention briefly here that, once again, a dismissal of the reality tense (broadly understood in terms of the phenomenology of temporal experience) is founded upon a non-neutral theory of reality as a set of truth-objects, whose ontological status is intended to explain the naturalistic world. This may not be a thesis easily dismissed or defended, but it is one which should be admitted. Time-sensitive indexicals only further a sense of neo-plantonism with respect to these mathematical and logical devices which seem to sharpen our sense of truth-across-time. Ultimately, however, we must ask whether a given theory of temporal truth helps or hinders our appreciation and understanding of the human experience as such, and I am not entirely convinced that calling tense or the Now “unreal” contributes significantly in this respect.
February 25, 1994
First Assignment: True Time and Real Time
How can something be the case, without it being now? I do not have in mind the sorts of necessary truths with make up mathematics or logic, although these too present something of a challenge to an essentially temporal ontology. Rather, how can it be the case that something occurred, or will occur, and yet not be (now)? The suppressed “now,” parenthetically conceded, in the framing of the dilemma suggests that the paradox is merely due to a confusion with respect to the sense of Being comported by the latter conjunct. We might be urged to restate the proposed puzzle: How can it be the case that something occurred, or will occur, and yet not occur now? This is almost trivially resolvable, having removed almost every aspect of potential conflict. John M. E. McTaggart frames the difficulty similarly, “... every moment, like every event, is both past, present, and future.” (McTaggart, p.48) And, similarly, we might (following an Aristotelian analysis of contradiction) insist that McTaggart restate the seeming paradox as: every moment, like every event, is both past, present, and future, in the same sense, simultaneously. The simultaneity requirement seriously jeopardizes the integrity of his point. The difficulty arises by confusing a non-temporal “is” with a temporal one. In order for there to be a contradiction, the same sense of “is” must be applied across the senses in which an event or moment is past, present, and future. Relieving the paradox in such a manner, however, does not clarify what senses of “is” are involved, and therefore cannot, as yet, avoid the possibility that an ontological commitment to any non-instantaneous segment of time is inconsistent. Indeed, it is this commitment which McTaggart is directly concerned with attacking. So long as the “is” of future, the “is” of the now, and the “is” of past, involve an ontological commitment with respect to some enduring “fact”, then the predications of “future”, “present”, and “past” are subsequently being made of a thing I am ontologically committed to by virtue of these predicates. Because each predicate confirms my commitment to the ontological viability of an occurrence in precisely the same manner (i.e. as temporally proximate to the now of consciousness by some relational distance), there is a sense in which I am predicating all three (logically) indiscriminately. There is a sense in which the predications are “simultaneous” (in the sense of “time-less”, rather than “coincident”).
This paradoxical aspect of time-language rests upon a confusion suggested above, but imperfectly framed as “different senses of is.” There is a substantial confusion of the is of truth, with the is of necessary ontological commitment. That these are not the same, that there is an authentic distinction between the “true” and the “real”, is to some extent suggested by this problem. The A-series and the B-series are in part constructed in terms of either the experiential, or the true respectively.
To the extent that an A-theorist asserts only sheer ontological commitment by the phenomenological facticity of the now, I take her to be correct. To the extent that a B-theorist asserts only a commitment to truth across the B-series, I take him to be correct. However, the A-theorist cannot assert (legitimately) an ontological commitment to any phenomenological artifact which departs from the now, due to the temporal present-ness of the basis for ontological commitment generally within an A-theory, although she may speak truthfully of the content of expectation and memory. Similarly, the B-theorist cannot assert (legitimately) an ontological commitment to any part of the B-series merely on the basis of truth-in-the-model, although some legitimacy may be possible concerning the “truth” of ontological claims. The reason for this division of labor (where the B-series concerns truth, and the A-series concerns ontological commitment), and the restricted nature of legitimate ontological commitment, will become clear as the formal details of each series is more clearly articulated.
Briefly stated; the only sense in which a moment of future time (or past time) “is”, is in terms of the B-series, or in terms of the appropriateness of a truth claim “about” a future (or past) occurrence. The only sense in which a “moment of time” (to speak loosely) now “is”, is in terms of the A-series, where “is” is understood in terms of the ontological conditions of consciousness.
Commitment to the truth of a proposition does not entail a commitment to some sort of propositional ontology, where it is facts and not things (as extra-linguistic entities, whether happenings or natural systems) which exist. By “true,” I mean nothing more complicated than “correctly assertible.” From this, it is possible for the B-series to be true without is possessing extra-linguistic potency. By design, the B-series is a model for the truth valuation of temporally sensitive propositions. The truth of a proposition in the B-series need not entail the non-temporal or necessary existence of that proposition (or its corresponding model). By claiming that the proposition “Socrates teaches Plato at time t [where time t is approximately 2500 years prior to the utterance of this proposition]” is true, I am committed to asserting a battery of entailed commitments if challenged (eg. “Plato is taught by Socrates at time t”, or “Time t is 2500 years prior to my [David Foss’] utterance of the claim...”, etc.), none of which concern an ontological commitment to the proposition in itself. I may be committed to the claim “Socrates exists at time t,” but I am not generally taken to be committed to the rather curious claim “The proposition [or fact] ‘Socrates exists at time t’ exists now.”
This exercise is merely meant to emphasize a substantial distinction between truth-in-a-model and ontological commitment. Truth is not sufficient, although it may be necessary, for such commitment. This is to say, commitment to a truth-claim may be necessary for the legitimate commitment to an ontological-claim. Nevertheless, the proprietary (linguistic) necessity of truth for Being need not be (and often is not) of causal or ontological significance itself. The claim the truth is metaphysically prior to Being (in time) is not entailed by the claim that the commitment to the proposition ‘X is [now]’ (or ‘X exists’) entails (or is entailed by) commitment to the proposition ‘X is [true]’ (or ‘It is true that X exists’).
From such a perspective, the B-series may be true, and therefore a B-theory correct (in some sense of correct), without it being the case that a B-series possesses some kind of extra-mental existence. The B-series may be true, without it being real. Am I able to claim that a B-theory may still be correct? A B-theorist would presumably be committed to the claim that the B-series, by its truth (by its “being the case”), is real. What could it mean to say that the B-series, as a system, is true, without implying the “real-ness” of its representational form?
The B-series models the formal character of time, in the same way in which mathematics (geometry) is said to model the formal character of space. It is an ideal representation of the ordering properties of (linguistic) consciousness. Nothing of the ordering can demonstrate an extra-mental status to the formal content of time consciousness. Nothing within language can demonstrate (in terms of a model-specific verificational analysis) trans-linguistic correspondence or formal ontological correctness. Correctness and truth, as intra-linguistic notions (about propositional commitments, and proprietary compliance with grammatical and justificatory norms), are only mistakenly applied to the trans-linguistic notion of the “real”.
While the “real” is the principle object of temporality for the A-series, an A-theory presents a series of rather different difficulties (from the constraints placed upon truth-talk). If truth-talk is native to the B-series, how are we to understand the phenomenological-talk of the A-series? Central to an A-theory is the prima facia Being of becoming. This is to say, the A-series represents the bare fact of change, motion, flux, the phenomenological occurrence of the now. Commitment to these notions may begin to look a great deal like truth-style commitment. If it is taken as such, a B-series analysis (of the A-series) can quickly model the full content of the A-series, in spite of substantially different senses of “change” between the two models. This would be a mistake. In order that the phenomenological/”real” claims of A-series talk to be legitimate, they must be modeled as commitments to something other than the appropriateness (or correctness) of assertion. Ontological commitment is not a form of truth-commitment, although a truth analysis of an ontological commitment is possible. Rather, ontological commitment, in terms of the A-series, is expressive of an existential stake the speaker has with respect to the content of the linguistic commitment. While the B-series might be said to represent the formal character of time, the A-series represents the material character of time.
The “realness” of the A-series, then, is the existential complicity of consciousness in the Now. Ontological commitment is appropriate in terms of the A-series only because it is a purely material commitment. This is to say, ontological commitment in terms of the A-series concerns the facticity of the now, and not the formal structure of time as represented to linguistic consciousness. The B-series is a formal representation of time-for-linguistic-consciousness. While the B-series is a powerful heuristic tool for the truth-valuation (or justificatory assessment) of claims in and about time, it cannot provide a frame for the evaluation of ontological commitments about time or temporal things. Without grounds for the ontological “simultaneity” of past, present, and future, neither the B-series nor the A-series poses a threat to the intelligibility of time. Correctness being a matter of degree here, an A-theory possesses insight into ontological commitment which is absent to a B-theory; and a B-theory provides a powerful model for the appropriate analysis of truth-in-time, making sense of non-ontological truth commitments in a way in which a strictly A-theory cannot.