Hudson's argument is by way of the metaphysical construction of a supposedly material object dubbed `Quick'. Quick lives up to its name by superluminally traversing a spatial distance of just over two-billionths of a light-second (about two feet) in the following way. The distance constitutes the height of a 3-D object `Cone'Foot note 1_1 (one dimension is an hour's temporal duration), which itself is comprised of a nondenumerable cross-section stack of 2-D space-slice discs that taken together make up the so-called `Disc Set'. These discs are ordered in relation to the temporal extension of Cone so that they can be matched with appropriate instantaneous moments in a familiar asymmetric temporal manner. Hudson then posits an interval coincident with that of Cone named `T' which is just over one-billionths of a second in duration. Then Hudson set-theoretically maps instantaneous moments of T -- the `T Set' -- with members of the Disc Set in the aforementioned well-ordered manner so that the result mirrors time traversal throughout T corresponding with space movement across Cone. The result of this mapping generates members of another set -- the `Quick Set' -- the fusion of which produces the aforementioned material object `Quick'. Since Quick is a moving object (composed of spatiotemporal parts), and Quick traverses the length of Cone in just over one-billionth of a second, Quick moves at twice the speed of light [2002, pp. 203-204].
Hudson admits throughout this argument to a number of controversial metaphysical assumptions: that at least one n-D object may have (n-1)-D cross-sectional spatial parts, that any extended object has spatiotemporal parts, that fusions of the members of well-ordered spatiotemporal-parts-sets (such as the Quick Set) result in whole objects. He maintains, however, that these are not incredible propositions, and thus that superluminal motion is not merely possible, but a fact (if the assumptions are true) [2002, pp. 203-204].Foot note 1_2
However controversial these assumptions are, and even if they are true as Hudson allows, I will show that Hudson's argument, when fully explicated, entails consequences that are counterintuitive if not outright false. Further, under a (perhaps most) plausible interpretation of what fusions of spatiotemporal-parts-sets are, I will argue that this form of argument in general cannot entail that Quick and its ilk exist.
To see the full scope of Hudson's reasoning, review the following variation on it. Let Cone and the Disc Set remain as before. Say, however, that T* is an interval one trillion times longer than T -- around 17 minutes duration, but still within Cone's time of existence. (Since Hudson places no constraint on the selection of a time interval within Cone's existence parameters, it would appear that T*, and any such interval less than or equal to Cone's temporal duration, can be used in the following way.) Now map T*'s instantaneous-moment set to the Disc Set as T's was before. The result is (what I must call) the `Tortoise Set' -- though the resultant fusion-object `Tortoise' would lose a race to all able-bodied tortoises, I suspect. So Tortoise is much, much slower than Quick, taking nearly 17 minutes to traverse the same space -- yet Tortoise is constructed in the same kind of way. Clearly then, within the two-billionths-light-second distance and one-hour time limits of Cone's existence, one can construct a very large number of movers (and even an infinite number, given that temporal intervals within Cone's hour of existence are subsets of the nondenumerable set of Cone's instantaneous moments), bounded on one extreme by asymptotically near-instantaneous movers (like Quick, but even quicker and quicker and ...) and on the other by the slowest hour-crawler (like Tortoise, but taking the full hour to move the distance). Therefore, Hudson's revised point should be that material objects that move at every speed possible within any given well-ordered spacetime volume can be proven to exist (again, with his admitted assumptions).
Given the intuitively vast array of Cone-like spacetime volumes where such metaphysical construction might occur, Hudson's argument, when applied to the universe at large, indeed yields quite a prodigious progeny of moving objects (certainly enough to induce the infamous Lewisian stare).
Further, since distinct time intervals of any given Cone-like object can overlap -- T and T* might well overlap by T's duration, for example -- then resultant fusion-objects of time-and-space sets such as `Quick set' and `Tortoise Set' themselves are in such cases coincidently existing and moving things.Foot note 1_3 While it is quite understandable that real objects might overlap in this way -- consider lightspeed cosmic rays that might happen to coincide with my movement with the Earth by flashing through me within a certain interval of terrestrial motion -- such an overlap is typically interpreted as an accident of nature, not a metaphysical requirement of existence. Thus every one of the magnificent offspring that result from fertile argument, fast and slow alike, always exist in the company of a staggering assemblage of fellow travelers, all of which exist merely in virtue of sharing their various times with the same space. While this is certainly not logically impossible, filling spaces with a plenum of such movers is at best an excess of ontological exuberance.
But there is an additional sense in which Hudson's argument is, as it were, unsafe at any speed. Clearly the above extension of the argument does not constitute a reductio or anything of the sort. Given assumptions and perspective, it only serves to populate the universe with objects that move at all logically possible speeds (if one additionally assumes that there exists at least one infinite spacetime Cone-like stretch, to allow for infinite sloth in that case). But what objects? Hudson says, as he does of Quick, that these are material objects [2002, p. 204]. But why should anyone accept this claim? After all, even given a fusion of the members of the `Quick' Hudson's and and likewise sets, these are still sets, the members of which are the result of logical mapping of arbitrarily selected dense time-lapses to equivalently dense space-extensions (it is this Cantorean equipotence of all such space and time sets that gets Hudson' `Tortoise's argument off and running, so to speak). As such, I would argue that fusions of the members of these sets constitute at best logically possible objects in spacetime-perdurant objects that must, if they exist, move at such speeds.Foot note 1_4 A recognizable example of such a possible object construction is that of Santa Claus delivering gifts to the world's children in one night. Sophisticated versions of this scenario estimate that this would require a Santa-sleigh-velocity of many thousands of miles per second. What is done here is to assign Santa's work to one night, map that time to the needed distance of travel, and voila! -- a logically possible Santa. Presumably, Hudson would not seriously want to argue that a speedy temporal-parts Santa really exists. On the other hand, nothing in Hudson's form of argument prevents one from concluding that the Jolly Old Elf, conceived as such a moving object, exists (largely due to Hudson's failure to address identity questions about fusion objects -- see note 4). That alone suggests a laxity within Hudson's argument that undermines the force of his conclusion that superluminally moving objects must exist, since they are on a metaphysical par with a speedy but presumably mythical Santa.
Ultimately, the ingenuity of -- and the problem for -- Hudson's argument is that it is Zeno turned inside out. Instead of producing paradoxes of motion over distance and time, Hudson uses (or, as I argue, should have used) the continua of distance and time to produce a surfeit of movers at all possible rates of motion relative to a space traversed. The fact that Hudson restricts his argument to proving that only superluminal movers exist indicates that either he is unwilling to acknowledge, or unaware of, the breathtaking existential commitment of its complete scope should he be right. If, however, as argued above, Hudson's fusion-object movers constitute only possible objects, then whether such objects are existentially instantiated is in fact a further question beyond any fanciful entertaining that they do.Foot note 1_5 Hudson's Zenophilic argument is not sufficient warrant in itself to conclude that any movers, super- or subluminal, exist.
Hudson, Hud. 2002. «Moving faster than light». Analysis 62: 203-205.
V. Alan White
University of Wisconsin Manitowoc
awhite@uwc.edu | awhite@lsol.net
[Foot Note 1_1]
I note that this distance is only an intelligible quantity in the context of a background space/spacetime within which Cone exists. Thus there may be hidden assumptions about such a space/spacetime that further complicate Hudson's argument, but I will try to avoid these issues in my analysis.
[Foot Note 1_2]
Afterward Hudson expands this argument by quantizing the space and time mappings into four dimensions with `ThickQuick' [2002, pp. 204-205], though this expansion in no way dispenses with a reliance upon the essentials of his earlier `Quick' argument (or its accompanying deficiencies).
[Foot Note 1_3]
And apart from an additional assumption that Hudson's time-space mappings must not overlap spacetime intervals distinct from Cone-e.g., as would `Coneahead', which contains Cone but includes an additional anterior spacetime of about .4 inches and another minute's duration (to preserve scale with Cone)-the overpopulation problem here worsens! Such an assumption, however, seems entirely arbitrary, and in fact contrary to intuitions about spacetime continuity.
[Foot Note 1_4]
Though I won't pursue it here, I should register my protest against Hudson's assumption that Quick and its relatives (including `Tortoise') are properly objects at all, since they are fusions of nothing more than times correlated to spaces apart from any properties instantiated within these times and spaces that could satisfy trans-spatiotemporal identity conditions of what counts as the same object. At best I would call the result of fusion of the Quick, Tortoise, etc. sets abstract objects. Still, for simplicity's sake I will yield to Hudson to the extent of calling such `bare' spatiotemporal-parts fusion-objects logically possible ones.
[Foot Note 1_5]
In this way my criticism reflects something of the same criticism Aristotle leveled at Zeno-namely that the latter's arguments conflate questions of potentiality and actuality.