Abstract
In this paper we will address the question whether a space interval is a set of infinite points . It is a very old problem, but despite its age it is still a live issue, and one we have to confront. We will analyze some topics regarding this question using the most influential objections against it, i.e. The Large and the Small paradox (in particular its Small Horn). We will consider classical contemporary reformulations of the argument (Grünbaum in Philosophy of Science 19:280–306, 1952; Grünbaum in Modern science and Zeno’s paradoxes. Allen and Unwin, London, 1968) and the possible ‘solutions’ to it. Finally, we will propose a new formulation of the paradox and analyze its consequences. In particular, we will bring further arguments supporting the standard thesis that it is possible that a segment of space is composed of a non-denumerable set of indivisible 0-length points.
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Notes
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Other names for the paradox are: Argument from Complete Divisibility; Paradox of Measure; Metrical Paradox.
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See Calosi and Fano (2015) for an analysis of Eleatic Monism.
- 6.
Aristotle, On Generation and Corruption, 316a19. See Aristotle (1984).
- 7.
Note that the paradox features the notion of infinite divisibility rather than divisibility simpliciter. As was to be expected, there has once more been a lot of controversy, at least after Aristotelian physics: see Fano (2012, § II.4). There seem to be at least two fundamental notions of divisibility: physical and conceptual. An object is physically divisible if it is possible to physically separate some of its parts, whereas it is conceptually divisible if it is possible to individuate some parts of it even in the case in which it is physically impossible to separate them (quarks, for instance). In our opinion, Zeno’s argument concerns the latter notion.
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We could apply the paradox to continuous intervals of real numbers, to segments, and to linear stretches of physical space, indifferently. A particle, for example, occupies, in a given instant of time, a certain volume within which we can find a small linear portion of physical space that can be represented both geometrically by means of a segment, and algebraically by a continuous interval of real numbers. So, the paradox applies to the four entities indistinctly. However, these four notions should be distinguished. We shall formulate the paradox taking by into account only the linear stretches of physical space.
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Zimmermann (1996) proposed the term Indivisibilists for the supporters of this thesis and divided them into two groups: extreme and moderate.
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The Large Horn excludes the possibility that the points have finite size. For a very interesting discussion on the Large Horn and its solution see: Calosi and Fano (2015).
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Note that Skyrms’ premise III contains the term “equal” implicitly: “the parts all have positive magnitude, or zero magnitude”. We don’t consider the equality of positive magnitude a correct requirement from Zeno’s point of view.
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See again Skyrms (1983).
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See Salzmann et al. (2007).
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Note that this is only a possible definition of measure, used because it works very well.
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Note that one cannot say the same for the Large Horn, because it is easy to take from the set of non-zero intervals a denumerable infinite number of them, whose union is already of infinite length.
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Note that a set is open if and only if it can be expressed as a countable union of open intervals. Therefore, the outer measure of set A can be rewritten as
$$m^{*} (A) = inf\left\{ {m(O){:}\,A \subseteq O{\text{ and }}O{\text{ is open}}} \right\}$$This leads to the previous definition of the inner measure of A. See Halmos (1950), Le (2010).
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The standard solution to this particular paradox of Zeno uses implicitly the mathematical fact that additivity for Lebesgue measure holds only for countable sets. It has however been pointed out, most notably by Massey (1969), Skyrms (1983) and White (1992), that an ultra-additive measure can be defined, which will allow for example to sum an uncountable number of lengths so that the paradox will again be created. Indeed the measure of an uncountable union of intervals couldn’t be greater than the uncountable sum of the measure of the intervals. And it is difficult to imagine that an uncountable sum of 0’s would yield something other than 0. However, these ultra-additive notions of measure are mathematically uncommon objects, which constitute non-comfortable mathematical frameworks.
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Respectively: c-series and m zeno.
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The attention “to determine the length of an interval by summing the lengths of its degenerate subintervals was the desire to avoid the obvious contradiction that Zeno pointed out long ago” (Sherry 1988, 63). Sherry’s article correctly stresses this view revisiting the question: what is a refutation of Zeno’s metrical paradox? One cannot say that Lebesgue measure is motivated only by this point, since it is very natural and intuitive, but certainly a refusing of Zeno’s metric paradox is implicit in it.
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Remember that the term “not-degenerate” means that the intervals are not points.
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- 27.
See Zimmerman (1996, p. 4).
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Acknowledgements
We would like to thanks our colleagues from the Department of Mathematics at the Polytechnic University of Milan, from the Department of Philosophy at the University of Bologna, from the Department of Philosophy at the University of Cagliari, and from the Department in Pure and Applied Sciences at the University of Urbino for their helpful comments during the presentations of earlier versions of this paper. In particular we would like to thank Claudio Calosi for his helpful suggestions.
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Fano, V., Graziani, P. (2017). Is a Space Interval a Set of Infinite Points? A Very Old Question. In: Catena, M., Masi, F. (eds) The Changing Faces of Space. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-66911-3_12
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