Abstract
I offer a novel solution to Zeno’s paradox of The Arrow by introducing nilpotent infinitesimal lengths of time. Nilpotents are nonzero numbers that yield zero when multiplied by themselves a certain number of times. Zeno’s Arrow goes like this: during the present, a flying arrow is moving in virtue of its being in flight. However, if the present is a single point in time, then the arrow is frozen in place during that time. Therefore, the arrow is both moving and at rest. In “Zeno’s Arrow, Divisible Infinitesimals, and Chrysippus,” White suggests using an infinitesimal value as the length of the present. Contra Zeno, this allows the arrow to be moving in the present, rather than frozen in place. In this paper, I follow the basic outline of White’s solution but argue that his solution suffers from arbitrariness and a related theoretical artificiality in relation to the system of infinitesimals he invokes, viz. in relation to the hyperreal infinitesimals of nonstandard analysis. After arguing that any solution to the paradox must satisfy certain theoretical requirements, I examine White’s solution alongside two nilpotent solutions. One of these solutions is inspired by F.W. Lawvere’s Smooth Infinitesimal Analysis and the other is inspired by Paolo Giordano’s ring of Fermat Reals. I argue that Giordano’s nilpotents supply the best answer to Zeno’s paradox.
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Notes
See Russell’s chapter “The Problem of Infinity,” in Russell (1914).
See “Inconsistencies in Motion” (Priest 1985, p. 340).
Below I will use the term ‘analysis’ numerous times: Nonstandard Analysis, Smooth Infinitesimal Analysis and there is everyday Real Analysis too. Analysis is a branch of mathematics that has at its historical core Newton and Leibniz’s calculus. Notions like limits, derivatives, integrals and similar are at the center of this discipline.
As a simple book-keeping matter, I will avoid the locutions of ‘at a time’ or ‘in a time’ unless directly quoting the argument’s premises. This is a distinction that has appeared in the literature that relies mostly on how those prepositions are used in certain technical contexts. I will attempt to avoid this worry entirely by using the general expression, ‘during a time’ as a way to be deliberately ambivalent towards this debate.
I should be explicit: the contemporary metaphysical notion of exact occupation is not being used here. Intuitively, \(M\) exactly occupies region \(R\) when \(R\) hugs the very contours of \(M\): (i) every subregion is filled and (ii) no part of \(M\) is outside of \(R\). If one were to interpret occupation in this way, then everything would always be at rest.
There is good reason to think that this is how Aristotle saw this premise. See Physics VI.9, 239b30-33.
See Zimmerman (2007).
White suggests that this argument (from Lear’s reading of Aristotle’s presentation of Zeno) equivocates on ‘occupying a space just its own size.’ For White, premise (1) makes sense only if it is interpreted with the temporal qualification added as in \((1')\); whereas, in premise (3), ‘occupying a space just its own size’ must be read unqualified. I am not going to pursue the issue of whether Zeno has equivocated. I am simply going to assume that this argument has the form of the barbara syllogism as it appears. I will deny that it is sound by arguing that one can plausibly deny premise (3).
Since I think Zeno is wrong, not everything he believes can be accommodated. Nevertheless, I side with Jonathan Lear that Zeno should be given a running start—pun intended!
The official “epsilon–delta” definition of a limit is that a function \(f\) has limit \(L\) as \(x\) approaches \(c\) iff\(_{df}\) for any \(\epsilon >0\), there is some \(\delta >0\) such that for any \(x\), if \(|x - c|<\delta \), then \(|f(x)- L| <\epsilon \).
This idea is found in n. 19 of Lear (1981).
To view points as markers is not theoretically stable in combination with the limit-based answer. On the limit-based picture, points are a natural choice for representing the present. Instead, White introduces infinitesimal intervals to play the part of the present.
The typical route to the existence of the hyperreals is via the Compactness theorem of first-order logic, and this in turn follows immediately by Gödel’s completeness theorem for first-order logic. Nevertheless, the compactness theorem can also be proved using mathematical devices called ultraproducts (Robinson 1974, pp. 13–19). One may directly apply the ultraproduct technique to the set of real numbers to the same effect (Goldblatt 1998, p. 13). For more, see the original (Robinson 1974); alternatively, see Goldblatt’s lucid (1998). A different route developed in Nelson (1977) and more accessibly in the undergraduate text by Robert (1988). In effect, Nelson develops what one might call hyper-set-theory. It is worth noting that his relative consistency proof also uses ultraproducts.
Aristotle suggests in the Physics (232b25) that space and time must have the same structure. The essence of his claim is that variation in the measure of motion can only be captured properly if time and space match in structure. I am simply assuming this Aristotelian thesis of the fit between space and time.
By interpreting (1) this way, White can avoid any difficulties associated with motion during a single point of time. This interpretation is a nod to Aristotle’s view that motion or rest during an instant is not coherent. On Aristotle’s view, motion (rest) requires a span of time over which to move (rest).
A natural choice for this function would be, \(z(t)=-9.8t^2+73 t+1.8\) (where time is in seconds and length is meters). The length of the interval is therefore \(-19.6t_0\varepsilon +73\varepsilon +\varepsilon ^2\), an infinitesimal value.
Along these lines, some ancient atomists developed the view according to which matter, space and time are all composed of extended “atoms” or simples—things with no proper parts—of matter, space and time respectively. The present would count as one of these finitely extended time simples. On certain interpretations, these atomists developed these views as an effort to answer various of Zeno’s paradoxes. For rich discussion, see Sorabji’s Time, Creation and the Continuum: Theories in Antiquity and the Early Middle Ages (1983), especially Part V, or “Atoms and Time Atoms” (1982).
I must be especially careful in this discussion because the hyperreals were traditionally constructed from a metalinguistic point of view that contrasts with the object-language of first-order analysis. On this original conception, the first-order sentences of real analysis are considered “internal” and the metalanguage is something much more powerful like ZFC. In the immediate setting, when speaking of the object language, I mean the whole theory of hyperreals; when speaking of the metalanguage, I mean all of mathematical English used by a practicing mathematician-cum-philosopher, including ZFC or more.
For the moment, I am ignoring the thorny issue of Einstein’s relativity. I am here considering the concepts of motion to which ordinary Newtonian mechanics apply. These are the kinds of mechanics at which Zeno aims his Arrow. To engage the harder questions governing motion at the microscopic or astronomical levels of observation is to jump a ways downstream dialectically.
This discussion is found in Owen (1976).
An anonymous referee raised the concern that infinitesimal solutions do not obviously engage a certain form of the paradox. That form of the paradox goes something like this: (I) The distance the arrow covers during its flight is the sum of the distances covered during the disjoint parts of its flight time; (II) For any part of time during the arrow’s flight, there is a (possibly zero) distance covered during that time; (III) The point-instants during the flight time form a class \(T\) of disjoint parts of time; (IV) For each part \(t\) of \(T\), the arrow covers zero distance during \(t\); (V) Therefore, the arrow covers zero distance during its flight time. In note 12 above and the associated text in Sect. 3, I gesture at what would be needed to respond to this version of Zeno’s Arrow, viz. I imply that premise (III) is false. Any infinitesimal solution has the resources to treat points as mere artifacts of the chosen formalism. This move sidesteps the paradox but only by creating another puzzle. Among the solutions I consider, the fundamental parts of time (time-simples) are lamentably unmeasurable. For example, although the hyperreal monad (defined in Sect. 3.1) does not have measure zero, it is only because it has no measure at all. This is because a monad has neither infimum nor supremum. The paradox can be revised to deliver measurable sums of unmeasurable simples. Comparable difficulties face the nilpotent solutions. There are a few other options available if one denies premise (III). One could arbitrarily select some (infinitesimal?) size and declare by fiat that time-simples just be that size. This is roughly what White has done. This delivers a nonzero sum of measurable, infinitesimal parts. The other option is a so-called gunky solution where all parts have a proper part; i.e. there is no end to the “depth” of parts. This avoids the puzzles of measure for non-arbitary time-simples, whether zero (points) or unmeasurable (monads, etc.). In summary, the models I examine in this paper do not provide wholly satisfying solutions to this version of the paradox. However, there are solutions available for both the friend and foe of infinitesimals.
To be specific, a field is a set \(F\) with binary operations \((+, \cdot )\) satisfying the following:
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1.
Closure For any \(x, y \in F\), both \(x+y \in F\) and \(x\cdot y \in F\).
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2.
Associativity For any \(x,y,z\), \((x+y)+z=x+(y+z)\) and \((x\cdot y)\cdot z=x\cdot (y\cdot z)\).
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3.
Commutativity For any \(x, y\), \(x+y=y+x\) and \(x\cdot y=y\cdot x\).
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4.
Distributivity For any \(x,y, z\), \(x \cdot (y+z)=(x \cdot y)+(x \cdot z)\).
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5.
Identities For any \(x\), there are unique \(e\) and \(o\) such that \(x+o=x\) (o is like 0) and \(x \cdot e=x\) (e is like 1).
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6.
Additive Inverses For any \(x\), there is some unique \(y\) such that \(x+y=o\). (Often this \(y\) is written \(-x\).)
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7.
Multiplicative Inverses For any \(x \ne o\), there is some unique \(y\) such that \(x \cdot y=e\). (Often this \(y\) is written \(\frac{1}{x}\).)
The relevant fact about classical fields follows mainly from principle 7.
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1.
There are two philosophical schools that take intuitionistic logic very seriously. The first one is from the mathematician L.E.J. Brouwer. On this view, mathematical concepts and statements are a matter of mental construction. Generally, in order to assert any mathematical disjunction \((\varphi \vee \psi )\), one must have produced either a proof of \(\varphi \) or a proof of \(\psi \). One may not simply assert for any statement \(\varphi \), \(( \varphi \vee \lnot \varphi )\) since at any time there are many yet unsettled mathematical statements. The other school is often referred to as semantic anti-realism, an updated form of verificationism, developed along similar lines by each of Michael Dummett, Dag Prawitz and Neil Tennant. On this view, all truth—not just mathematical truth—is bound up with verification. As a result, these thinkers’ eschewal of excluded middle involves understanding logical constants such as disjunction in a manner similar to the one described above. One is entitled to assert some disjunction only if she is already entitled to assert at least one of the disjuncts. A nice discussion of these views can be found in Tennant (1997). Neither of these historical motivations for intuitionistic logic is behind sia, but rather a completely different strand of thought that emerged by developing mathematical logic from within Category theory.
I draw extensively from Bell’s lucid monograph (2008).
The reason the kip is written this way is that \(0 \in \varDelta \) and so one cannot simply divide by all of the \(\varepsilon \in \varDelta \) as in the usual expression of the derivative as a quotient.
This pecularity can be stated so as to directly undermine the law of excluded middle. In particular, in sia, it is provably false that for any nilsquare \(\varepsilon \), either \(\varepsilon =0\) or \(\varepsilon \ne 0\).
Recall from Sect. 3.2, that I am not concerned to settle the structure of space but rather deny Zeno’s contention that the concept of motion is incoherent.
Strictly speaking, to make this move, I am assuming that micro-neighborhoods are topologically connected. Therefore, if \(\varDelta _{t_0}\) is topologically connected, then both \(t \in \varDelta _{t_0}\) and \(z(t)=z_0\) imply \(t=t_0\).
The most persuasive argument for intuitionism comes from the semantic anti-realist school described in footnote 23 above. As indicated in that note, one may only introduce a given logical connective into conversation under certain circumstances; similarly, one may only eliminate a given logical connectives from conversation under certain circumstances. This perspective on the logical connectives is rooted in inferentialist semantics, where all concepts (not just logical ones) are governed by conversational norms codified in the form of introduction and elimination rules. Intuitionism enters when one requires that the introduction rules are in certain sort of harmony with the elimination rules. All common binary connectives satisfy this requirement, but classical negation elimination is not in harmony with negation introduction. A consequence of this is that lem is unprovable (even though its double-negated counterpart is). That being said, I find neither inferentialist intuitions nor these principles of harmony particularly compelling. At the very least, they are less compelling than maintaining a commitment to lem. For more on these matters, see Dummett (1978), Prawitz (1974) and Tennant (1997).
Thomas Nagel (1997) makes a similar point, though not about intuitionistic logic. He argues that postmodern “arguments” from sociology designed to undermine meaning and reason presuppose stable notions of meaning and reason. I think his arguments are fatal and mine are far from fatal. However, in this same spirit, I think lem is further upstream conceptually than the principles summoned to undermine it.
First, a set \(A\) is open just in case for every member \(x \in A\), there is some \(r>0\) such that the corresponding neighborhood of \(x \big (N_r(x)=\{y: |x-y|<r\}\big )\) is included in \(A\). Furthermore, I use a generic letter \(R\) here since the property could be stated either for either \(\fancyscript{R}\), \(\mathbb {R}\) or in this case, Giordano’s ring \({^{\bullet }\mathbb {R}}\). Of course, it is false for \(\mathbb {R}\), since \(D=\{0\}\) in \(\mathbb {R}\) and so \(m\) is not unique.
Giordano’s ring has some cancellation laws that mimic invertibility: For nonzero \(x \in {^{\bullet }\mathbb {R}}\) and \(a,b \in \mathbb {R}\)
$$\begin{aligned} xa=xb \implies a=b \end{aligned}$$In other words, provided that \(a,b\) are ordinary real numbers without any infinitesimal “fringe,” the \(x\)’s can be cancelled. Also, as long as \(x \in {^{\bullet }\mathbb {R}}\) is not infinitesimal, then \(x\) has a multiplicative inverse. Another natural cancellation law follows immediately from this. For any \(x, a, b\in {^{\bullet }\mathbb {R}}\), if \(x\) is not an infinitesimal (but might be in \(^{\bullet }\mathbb {R}\setminus \mathbb {R}\)), then again
$$\begin{aligned} xa=xb \implies a=b. \end{aligned}$$Let the reader note that because Giordano’s ring is multiplicatively commutative, these cancellation laws could have been presented with \(ax=bx\) as the first clause. The reason why these cancellations laws are slightly weaker than the typical ones is that the nilsquares provide direct refutation of the ordinary unqualified principle. For any two distinct nilsquares, \(d_1, d_2\), their product is zero. Hence, there are cases where given some further nilsquare \(d_3\), \(d_3 d_1 = d_3 d_2\) but \(d_1 \ne d_2\) by stipulation.
Projective geometry employs points at infinity. For the more pedestrian purposes here of an arrow’s flight, if one can eschew points at infinity without significant loss, all the better.
It might be tempting to argue that this is also true of the nilsquares since they are defined using \(2\) and \(0\) in the open sentence ‘\(x^2=0\)’. Define \(\varPhi (x) := \ulcorner x^2=0 \urcorner \). So, one may define the nilsquares to be \(\{ x\in {^{\bullet }\mathbb {R}}: \varPhi (x)\}\). Note, however, that \(\varPhi (x)\) can be rewritten as \( (\exists ! y)[(\forall z)(y+z=z+y=z)~ \& ~ x \cdot x =y]\). Such algebraic control is unavailable in the hyperreals, specifically because of the fact that the real numbers and hyperreals satisfy all the same first-order statements in the language of real analysis. This latter property is often referred to as the Transfer Principle.
Reeder argues (in forthcoming) that Giordano’s nilpotents are actually smaller than the invertible hyperreal infiniteimsals. In other words, even ignoring the intrinsicality feature, Giordano’s ring as the edge over the hyperreals.
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Acknowledgments
Many thanks to Ben Caplan, Stewart Shapiro and Neil Tennant for their comments on earlier drafts and versions. Thanks also to Matthew Davidson, Philip Ehrlich, Michael Miller, Tony Roy, David Sanson and Declan Smithies for helpful discussion. Lastly, I appreciate the challenges posed by the anonymous referees.
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Reeder, P. Zeno’s arrow and the infinitesimal calculus. Synthese 192, 1315–1335 (2015). https://doi.org/10.1007/s11229-014-0620-1
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DOI: https://doi.org/10.1007/s11229-014-0620-1