Abstract
The paper explores a neglected conception in the foundations of spacetime theories, namely the conception of gunk, point-free spaces inaugurated by De Laguna and Whitehead. Despite the epistemological merits of the proposal they argue that this would have rather unwelcome consequences for the description of motion that is provided by most of our physical theories, even simple ones such as classical mechanics. The tension, they claim, is generated by the following facts: (i) classical mechanics crucially adopts the notion of a point-particle in its description of motion; (ii) sets of (constructed) points in these Whiteheadian spaces turn out to be non-connected; (iii) connectedness is a necessary condition for continuity.
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Notes
- 1.
See however the responses to Grünbaum given by Shamsi (1989) and Ringel (2001). Grünbaum (1953, 220) himself acknowledges that in Process and Reality Whitehead does not require all the regions used in the extensive abstraction method to be perceivable. Grünbaum does not seems to fully understand this new approach, which has become clear in the formulation of Gerla and Tortora (1992, 1996). This new rendition is thus immune from Grünbaum’s criticism.
- 2.
- 3.
It should be noted that Gerla and Miranda (2008) provide a formal rendition also of the notion of extensive abstraction, one based upon the notion of inclusion. They further address the formal rendering of the relation of connection that Whitehead takes from De Laguna (1922). We will focus only on the latter, which is both more intuitive and effective.
- 4.
In order to appreciate the importance of the succession Sn for Zeno’s Dichotomy consider again series (4.1): \(1/2, 1/4, 1/8,\ldots\). Its generic term is 1∕2n. As n approaches infinity the series measures how much time is left for O to get to b. We know that ab = 1 m, so \(S_{n} = 1 - 1/2^{n}\) represents the space covered by O. It is easy to prove that, if n goes to infinity, Sn approaches 1 since 1∕2n tends to 0. Thus an infinite sum of intervals is not necessarily infinite and O can get to its final destination.
- 5.
We do not address the logical problem of supertask.
- 6.
It may be possible to characterize rigorously the Aristotelian notion of “infinite divisibility”, where the modality involved in the second word is crucial, in mereological terms, through a statement of atomlessness.
- 7.
See for example Munkres (2000, 98).
- 8.
As far as continuity of space is concerned, it should be taken into account that recent attempts to unify gravity and Quantum Theories have yielded to the hypothesis that space, at the Planck scale, is not continuous after all, but rather discrete. These hypotheses are still under scrutiny. If they turned out to be correct, the problem of motion as a supertask would vanish.
- 9.
We follow the clear exposition given in Pecoraro (2006). We refer to this for many details about the formal rendition of the numerous assumptions proposed by Whitehead.
- 10.
All the formulas are intended to be universally closed for any free variable unless otherwise specified.
- 11.
- 12.
- 13.
See, for example, Landau and Lifshitz (1976, V.I,1).
- 14.
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Acknowledgements
We are very grateful to Adriano Angelucci, Caludio Calosi, Massimo Sangoi and anonymous referees for their comments on the draft of this paper.
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Fano, V., Graziani, P. (2014). Continuity of Motion in Whitehead’s Geometrical Space. In: Calosi, C., Graziani, P. (eds) Mereology and the Sciences. Synthese Library, vol 371. Springer, Cham. https://doi.org/10.1007/978-3-319-05356-1_4
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