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On Dummett’s Defence of McTaggart and Token-Reflexive Expressions in Space 26.XI.2004 22:28
In his defence of McTaggart’s proof of the unreality of time, Dummett frequently invokes the distinction between the natures of space and time. Specifically, he argues that token-reflexivity (the use of referential words such as I, here and now) are essential to time, but not so to space. In doing so, however, he commits several grave errors. One is the denial of relativity, which physics has proven, and which allows us to perceive multiple times in the same instant. Secondly, he projects onto all possible perception the strictures of human perception, thus limiting the realms of possibility and reality without justification. Dummett’s distinctions between space and time are erroneous, and McTaggart’s criticisms apply equally to both; but these criticisms fail to prove the non-existence of either.

The fundamental component of McTaggart’s proof of the unreality of time is the assertion that A-series events, one of the form (E is past, present or future) contain an inherent contradiction: that each event is all at once, as it exhibits each quality at some point across the whole of time. McTaggart makes no attempt to apply this contradiction to space, and Dummett maintains that this is because token-reflexive expressions are not necessary to enumerate facts about space, but they are about time. This, however, does not seem to hold true.

To show the plausibility, and necessity, of treating time as a bidirectional vector, like the special dimensions, locations in space-time will be referred to as ordered quadruplets of the form (x,y,z,t), with the fourth term being the temporal axis, or t-axis, along which single points are referred to as t-values. This provides a convenient short-hand for relating how changes and relations along the three spatial axes are no different from changes and relations in the t-axis. Furthermore, this notation allows us to track changes along any axis in relation to changes along the other axes.

The first argument Dummett makes to absolve space of McTaggart’s objections is that statements about space do not require token-reflexive words such as here, near or far. ‘I can describe an arrangement of objects in space,’ he says, ‘although I do not myself occupy a position in that space.’ (Dummett, 1960) He cites the example of his visual field, whose arrangement he can describe despite not being in that space.

There is no reason, however, that this argument should not apply equally to time. For example, I do not exist in my visual field just as I do not exist during the life of John Fitzgerald Kennedy. However, I can describe the arrangement of events in his life despite their t-values never overlapping my t-values. I can say that ‘John F. Kennedy died on November 22nd, 1963’ without existing in that time frame or using a token-reflexive expression. My ability to describe events along the t-axis is not reliant on my existence at those points on the axis.

Dummett would now argue, however, that I do not perceive events during the life of Kennedy as I perceive spaces in which I do not exist. I perceive the former only second-hand, through other people’s accounts: I cannot perceive it directly. However, when modern physics, specifically the finite speed of light c, is taken into account, it becomes clear that we are not only capable of perceiving events even though we do not exist at their -values, we are incapable of perceiving events that share the ­t-value of the event of perception. More simply, none of the events we perceive at a given moment share the t-value of our token-reflexive now.

If I look at a pulsating quasar millions of light years away, I perceive, directly, an event whose >t-value is far removed from the range of ­t-values at which I exist. This is because the speed of light limits the rate at which data about one event can be accessed by a perceiver at different spatial co-ordinates from the perceived event. This distance cannot be 0, since no two particles can have the same co-ordinates in space-time, and therefore observer and observed must also have different co-ordinates. Thus, the t-value of the event of perception is always different from the t-value of the perceived event. Token-reflexive expressions are as necessary for time as for space, because the co-ordinates of events we perceive are proportionate along each spatiotemporal axis to the speed of light.

As human observers, we can no more exist outside of space than we can outside of time, and our perceptions of each will always be from the perspective of the co-ordinates of the perception event. This, along with the realisation that token-reflexive expressions have the same use in time as well as space, subjects space to McTaggart’s objections. For McTaggart, for time to exist, there must be an A-series of events, described by token-reflexive expressions such as is past, is present and is future. The B-series consists of relational expressions the form E­­1 is earlier than, later than or simultaneous with E2. The contradiction lies in the fact that when, taking into account all co-ordinates along the t-axis, all events share the contradictory properties of past, present and future.

Equivalents for these can easily be found for the spatial axes. Since time is a single axis, let us also examine a single spatial axis, say the x-axis. Objectively, events can have an x-value greater than, less than or equal to other events’ x-values. If we posit an observer, whose events of perception Ep must by definition have an x-value, we can say that the events he perceives are either ahead of him, behind him, or at his position: respectively having greater, lesser or equal x-values as Ep. These former terms constitute the A-series of the x-axis, and the latter constitute the B-series.

There exist points on the x-axis for which an event is ahead of, behind and at the position of the spatial observer’s token-reflexive here, just as there are events past, and future to the temporal observer’s now. Why, then, do Dummett and McTaggart not dismiss the x-axis by the same mechanism as that by which they dismiss the t-axis? The answer to this is the central error to Dummett’s reasoning: he projects the limited nature onto all possible perception and, by extension, onto all possibility. The reason for this error is the inherent phenomenological guarantee of time for human observers. Put simply, this means that the rate and sequence at which we perceive events will always be proportionate. For example, if we perceive an object at time t1 and then again at t2 the difference between the t-values of the perceived events and the difference between the t-values of the perceiving events will be proportionate. They are commonly held to be equal – if I see a ball, then 5 seconds later I look at the ball again, I would say with confidence that the t-value of the ball was five seconds later – but, because of relativity, they are not equal, but proportionate. At speeds so much lower than the speed of light, this difference is phenomenologically negligible.

These two limitations – that of perceiving events in a sequence of increasing t-values, and that of perceiving events with t-values compared to our now proportionate to their x,y,z-values compared to our here – place a severe restriction on human perception. A consequence of these two is the restriction, that we cannot simultaneously perceive two objects with identical spatial co-ordinates but different temporal co-ordinates. However, there is no reason that this restriction should be placed on all possible perception in the universe. Nevertheless, this is what Dummett erroneously does.

He asks us, at one point, to imagine an observer that can perceive events with multiple t-values simultaneously, without the restriction of perceiving only simultaneous events, in the traditional sense. This observer could then, for example, perceive two events at the same spatial location with different t-values. Dummett then contends that the observer could ask, ‘which of these events is happening now?’ (Dummett, 1960) However, this very question links space and time inextricably: for the observer, the now has the same value as the here has for a human observer: it is that point at which the event of perception takes place, but not necessarily the point at which the perceived events take place. Just as we can observe events that are ahead of, behind or at our position on the x-axis, so this non-human observe can do with the t-axis.

It is here that Dummett lapses into error; while he affirms that the observer might see simultaneously multiple events at disproportionate times, i.e. spatially equidistant events from the observer could have different t-values, he neglects the possibility of the observer seeing two non-simultaneous events at one location. Thus, he states that, ‘what [the observer] observes can only be a model of the sequence of events in our three-dimensional space, not that sequence of events itself.’ (Dummett, 1960) He says this because, as with human perception, each x,y,z-value observed can have only one t-value mapped to it, which would indeed reduced to a three-dimensional model. However, there is absolutely no justification for such a stricture to be made.

Following the same line of reasoning, he dismisses the ability to model events over multiple t-values, because he states that there is an underlying assumption of temporal movement that is cannot be represented within the model itself. This, however, is also erroneous for a being who can perceive the world as four distinct vectors. Human perception only allows three; even though the events we perceive have different t-values they are mathematically interrelated, just as a two-dimensional creature might perceive a conic section as a parabola. However, when we view what we term as ‘sequence’ as being a range along an axis, we can model things along the t-axis, and even give them shape, as long as we do not take into account one of the other four vectors.

Let us consider the three-dimensional shape of a rectangular prism. This prism has the property that its height, its upper bound along the z-axis, is proportional to its t-value. This is to say that if a unit change along the -axis cannot exist without an equal change along the z-axis. Crucially for this thought-experiment, the x,y-bounds of the object stay constant. Since they are constant, for an accurate description of the object, we can ignore one of the other dimensions in our model, keeping in mind its constancy. Eliminating the y-value, we see that the model does indeed have a shape: it is that of a wedge, covering the range of ordered pairs (x,z,t) such that z = t. This is a model of an object with multiple t-values that can be understood by human perception; extending it to the 4-dimensional reality of the universe by re-inserting the constant y-value is a simple matter, albeit an inconceivable one in terms of mental modelling.

Another way in which human perceptive limits affect discussion on the subject of time is exhibited in this model: the perception that all ‘motion’ must be relative to the t-axis. In giving a synopsis of the above model, most people would say that the object’s z-value changes at a rate of one z-unit per one t-unit, that it is a change of height over time. However, for an observer not limited by human perception, it would be just as correct to say that its time changes over height, perhaps at a rate of three seconds per metre. Human prejudice indicates that time precedes motion, but this is not the case any more than the sky being above the ground precedes the grounds being below the sky. They are interrelated facts, inextricably linked and one cannot be true without the other.

Dummett’s article, even though it was written after the advent of relativity, continues as though time is an absolute, continuing at a steady rate of change. Relativity, however, has shown this to be untrue: all motion, be it along spatial or temporal axes, is only measured in relation to the rest of space-time. Just as a period in time cannot be described as being before or after the other moments in time, so locations in space can only be described in terms of their relation to other locations in space. Finally, motions between these points is describable only relative to the other points in space-time, because motion in time changes just as motion in space does at different speeds. Redshift, the light-wave equivalent of the Doppler Effect, is an example of this principle: events, occurring at the source at a constant rate, are perceived at different rates depending on the observer’s motion relative to the source of the events. Time is relative just as space.

Dummett asserts that any reality must be describable in its entirety, and he cites this tenet as a final argument in favour of McTaggart’s criticisms of space, but not of time. The reality of a rock, he says, can be described independent of any observer, without recourse to token-reflexive expressions. But can it? The rock, taken in isolation, can only be described in terms of the relation of points in the rock’s space to other points in that space, in the same way that events can be described in time only in relation to other events. To be located in space as a whole, it must again be relative to objects within that space, which in turn can only be located relative to other objects, and so on regressively. Location and description in space, just as in time, is totally regressive, and the token-reflexive expression remains the easiest anchor point for such descriptions. Nothing can be fully described in space-time except relative to everything else.

Descriptions of spatiotemporal objects are inherently relativistic, and a complete description of reality such as the one Dummett desires is impossible without resort to token-reflexive expressions. However, these token-reflexive events exist also relative to all other events and, conceivably, a full description of reality can be achieved in terms of such relation. By McTaggart’s definition, it seems neither space nor time is real; yet if reality consists of describability, such a definition does not preclude reality itself.

Space and time cannot be regarded as distinct kinds with distinct properties. They are both manifestations of the same dimensionality, and it is our human perceptions which limit our perception to only three of those at a time, and restrict our t-axis perception immeasurably. However, Dummett makes the error of applying this restriction to all possible perception, and thus arrives at an erroneous distinction between space and time, concluding the former can be described in absolute terms. However, once the notion of absolute time is abandoned, as physics proves it must be, the arguments upon which Dummett relies are greatly weakened, and time and space are unified. Space is no more independent of token-reflexive expressions than time, and no more or less real.
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