Abstract
Van Bendegem has recently offered an argument to the effect that, if time is discrete, then there should exist a correspondence between the motions of massive bodies and a discrete geometry. On this basis, he concludes that, even if space is continuous, it should nonetheless appear discrete. This paper examines the two possible ways of making sense of that correspondence, and shows that in neither case van Bendegem’s conclusion logically follows.
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Notes
A different view is offered by Aronov (1971), according to whom space and time are discrete in the sense that different physical laws are predominant within spatio-temporal regions of different scales. Newton-Smith (1980: 113) discusses the idea that time could be discrete in the sense of being a lattice of finitely many durationless instants. More recently, Dummett (2000) has outlined, though not endorsed, a fuzzy account of discrete time aimed at avoiding the possibility of abrupt instantaneous changes. However, we shall thereafter restrict our attention to the sole atomistic conception.
I shall be using ‘time’ and ‘space’ is a somewhat wide sense, so as to include the temporal and spatial dimensions of space-time, respectively.
For the purposes of the discussion to follow, it is immaterial if times and places are set-theoretical members of intervals or rather mereological parts of them; accordingly, let us prefer the more neutral constituents.
Things get even more complicated if we consider that, few lines after having formulated his argument, van Bendegem explicitly says that discrete time implies discrete space (van Bendegem 2011: 153). This claim is evidently at odds with his purported conclusion that discrete time forces continuous space not to become discrete, but only to appear so. However, this might just be taken as a much unfortunate way of phrasing the same conclusion, so we shall not consider it any further.
These are only to include (in each possible world) the actual positions (at that world) of the physical objects. This is because there is no apparent reason why any given object could not have been at whatever other place at the time considered. But since space is continuous by hypothesis, it follows that the positions that an object could have had at each chronon are more than denumerably many, and thus cannot be put into a one-one correspondence with the hodons of any discrete geometry.
Notice that, having assumed that space is continuous, and since we are dealing with pure conceptual possibilities, there is nothing which could prevent us from imagining x as a massive point. Given the hypothesis that nothing could be at more than one place at a time, the position of x at each time will then be unanalysable because dimensionless. Notice, too, that this would be the case independently of the microscopic structure of time.
Admittedly, discrete spaces and dense spaces are both endowed with a discrete topology. Thus, one could preserve van Bendegem’s thesis from this objection by contending that all geometrical properties are reducible to topological properties, and that in consequence geometry itself is in fact nothing more than topology. This claim, however, would be highly debatable, as recently illustrated by Maudlin (2010).
This objection was suggested to me by an anonymous referee.
Russell explicitly demands that a request for continuity should be in-built into this analysis of motion. However, he also recognizes that ‘this is an entirely new assumption, having no kind of necessity, but serving merely the purpose of giving a subject akin to rational dynamics’ (Russell 1938: 473). Extending the at-at account of motion to the present case only requires dropping this admittedly non-necessary assumption.
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I would like to thank two anonymous referees for their detailed, fruitful and encouraging feedback.
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Mazzola, C. Can discrete time make continuous space look discrete?. Euro Jnl Phil Sci 4, 19–30 (2014). https://doi.org/10.1007/s13194-013-0072-3
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DOI: https://doi.org/10.1007/s13194-013-0072-3