Russell on Zeno's Paradoxes PDF

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Russell on Zeno’s Paradoxes Josef A. Kay October 25, 2019 Motion has from an early time been criticised severely, and it has never been defended with much success. - F.H. Bradley, Appearance and Reality 1 Introduction In the 20th century literature on Zeno’s paradoxes, Bertrand Russell is widely rec...

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Russell on Zeno’s Paradoxes Josef A. Kay October 25, 2019

Motion has from an early time been criticised severely, and it has never been defended with much success. - F.H. Bradley, Appearance and Reality

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Introduction

In the 20th century literature on Zeno’s paradoxes, Bertrand Russell is widely recognized for promoting the idea that modern mathematics had provided the resources necessary for responding to Zeno. Many commentators following in Russell’s footsteps have paid homage by presenting one of Russell’s many statements expressing the significance of Zeno’s arguments. For example, Adolf Gr¨ unbaum, in his “Modern Science and Refutation of the Paradoxes of Zeno” (1955), and Wesley Salmon, in introducing his edited collection of essays on Zeno’s paradoxes (1970), both quote a now-famous remark from Russell’s Our Knowledge of the External World : “Zeno’s arguments, in some form, have afforded grounds for almost all the theories of space and time and infinity which have been constructed from his day to our own” (1914, p. 183).1 In claiming that the new foundations of analysis developed by mathematicians such as Weierstrauss, Dedekind, and Cantor provided resources for solving these ancient puzzles, Russell’s discussions of Zeno’s paradoxes in his Principles of Mathematics (1903), the aforementioned Our Knowledge (1914), and several auxiliary publications set the stage for all future work on the topic. But despite acknowledgments of his influence, and occasional critiques of his solutions, there has been no in-depth study of Russell on 1

The opening paragraph of Gr¨ unbaum (1955), which features this quote, reappears in the introduction of Gr¨ unbaum’s book on Zeno’s paradoxes (1967). Meanwhile, the chapter of Russell (1914) from which the quote is taken (Ch. VI, “The Problem of Infinity Considered Historically”) is reprinted in Salmon’s collection.

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Zeno’s paradoxes. Carrying out such a project is of interest, I believe, because in no other era do the paradoxes have the same significance that they have for Russell. Zeno’s paradoxes afford grounds for understanding Russell’s views on space and time during his early idealism, after his conversion to platonism, and again in his later empiricism.2 They also provide an opportunity to evaluate Russell’s exposition of the philosophical significance of the new foundations of mathematics. The discussion that follows is based around the following three questions. First, what significance do Zeno’s paradoxes have for Russell’s ongoing philosophical project? Second, how do Russell’s interpretations of Zeno’s paradoxes, and his solutions to them, change over time?3 Third, are Russell’s solutions successful in responding to Zeno, and do they meet Russell’s own conditions of adequacy? I will argue for the following answers. First, the overarching theme of Russell’s discussion of Zeno’s paradoxes is a Cantorian defense of the mathematical continuum and the infinite as a foundation for our theories of space and time. Before he understood Cantor’s work, Russell held the view that the mathematical continuum, in having an infinite number of terms, is logically incoherent; that is, it gives rise to contradictions. Zeno’s paradoxes are the earliest arguments that appealed to the paradoxical nature of infinity, and Russell endorsed them, alongside other arguments against the reality of space and time, in support of his early idealism. Upon understanding Cantor, Russell uses Zeno to illustrate the historical significance of the new mathematical foundations, taking them to undercut the motivation for idealist positions of all stripes, including Parmenides, Kant, Bradley, Bergson, and his own previous views. Second, though Russell’s views regarding space and time change between his discussions of Zeno, his changes in interpretation are predominantly based on an increased historical sensitivity. The discussion in Principles is enigmatic, ahistorical, and nonstandard. In particular, Russell associates two of Zeno’s paradoxes of motion (the Dichotomy and the Achilles) with paradoxes that he would regard as familiar from his former idealist period. Russell moved to more accurate interpretations upon receiving criticism from others, and the discussion in Our Knowledge is more accessible and historically sensitive, though Russell is still concerned with finding the strongest possible interpretations of Zeno’s arguments. Third, though Russell claims that Cantor’s theory of infinity solves Zeno’s paradoxes, in his most mature discussion of two of the paradoxes (again, the Dichotomy and the Achilles), Russell does not actually go beyond Aristotle’s traditional solutions. In particular, he does not appeal to an actual infinity, as we would expect, given Russell’s focus on Cantor. Combined with the fact that Russell has nonstandard interpretations of the 2 3

These labels for Russell’s views on space and time are borrowed from (Hager 1987, p. 10). I do not intend this in the Zenonian sense.

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Dichotomy and the Achilles in POM, this leads to the conclusion that Russell never provided a solution to Zeno’s paradoxes of motion that lived up to his claims. Thus, my main argument is as follows: when Russell uses modern mathematics to resolve “Zeno’s paradoxes,” his targets are not Zeno’s actual arguments; and when Russell accurately interprets Zeno’s arguments, he does not appeal to modern mathematics (with the exception of the Arrow argument). Thus Russell’s claim that Cantor’s theory of infinite sets is necessary to address Zeno’s paradoxes is not witnessed in Russell’s own work. As mentioned before, Russell discusses Zeno’s paradoxes comprehensively on two occasions: first, in his Principles of Mathematics (1903), and second, in Our Knowledge of the External World (1914). Before each of these treatments, Russell had written smaller articles that feature some discussion of Zeno. The treatment in Principles was written just before “Recent Work on the Principles of Mathematics” (1901), which contains a discussion of the Achilles and the Arrow arguments.4 The discussion in Our Knowledge is predated by a discussion of Zeno’s Arrow argument in an 1912 article, “The Philosophy of Bergson.”5 In order to present the relevant background on Zeno while remaining focused on Russell’s discussions, I have chosen not to follow their chronological order. Instead, I will begin with Russell’s definitive presentation in Our Knowledge, as it includes the most historical and philosophical background. After evaluating this presentation, I turn to Russell’s earlier treatment of Zeno’s Dichotomy and Achilles arguments in his (1901) and (1903). This order of doing things will ensure that we are familiar with the more standard interpretations of the paradoxes before encountering Russell’s peculiar reconstructions.

Zeno in Russell’s Empiricist Period (1912-1914)6

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Russell engaged in historical scholarship on Zeno between September and November of 1913, during the preparation of his Lowell lectures, which were later published as Our Knowledge of the External World.7 In this work, Russell addresses how our sensory experience of the world is related to the mathematical descriptions of space, time, and matter that appear in physics. Zeno is used to illustrate another major theme of the book, 4

This article was later reprinted with some minor commentary with the new title “Mathematicians and the Metaphysicians” in the collection Mysticism and Logic (1918). The original 1901 article is also included in Vol. 3 of Russell’s Collected Works (1994). 5 This article was later published as a book in 1914, and featured a defense of Bergson by Carr, along with Russell’s response. Most of the original article reappeared in Russell’s History of Western Philosophy (1946) as the section dedicated to Bergson . 6 Russell adopted empiricist views only sometime in 1913 and not earlier, but I am including Russell’s discussion of Zeno in “The Philosophy of Bergson” in this period. 7 During this period, Russell maintained a correspondence with Ottoline Morrell, and sent her drafts of the lectures. There are several letters from Russell to Morrell which mention Zeno.

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one that carries over from Russell’s earlier discussion of Zeno: the idea that Cantorian foundations of the infinite definitely resolved philosophical problems of space and time. Though Ch. VI of Our Knowledge is dedicated to the topic, the clearest expression of Russell’s attitude towards Zeno can be found in a 1913 article, “The Philosophical Importance of Mathematical Logic,” where Russell writes, The paradoxes of Zeno the Eleatic and the difficulties in the analysis of space, of time, and of motion, are all completely explained by means of the modern theory of continuity. This is because a non-contradictory theory has been found, according to which the continuum is composed of an infinity of distinct elements; and this formerly appeared impossible (p. 483-484). Our Knowledge is best known for (1) exhibiting Russell’s method of substituting logical constructions in place of inferred entities, which superseded the approach advanced in The Problems of Philosophy, and (2) Russell’s proto-positivist approach to philosophy, which is expressed in the following statement: “Every philosophical problem, when it is subjected to the necessary analysis and purification, is found either not to be really philosophical at all, or else to be, in the sense in which we are using the word, logical” (Russell 1914, p. 42). Both play an implicit, background role in Russell’s treatment of Zeno. Before addressing Zeno, Russell sketches a construction of points and instants out of classes and series of sense-data, which forms a foundation for the application of mathematical theories to the empirical world.8 This undercuts the motivation for one possible response to Zeno, one that Russell attributes to Henri Bergson: the idea that space and time are not composed of points and instants (p. 184).9 Russell does not take a stand on whether space and time have the same structure as the mathematical continuum, for he views this as an empirical problem (and thus ‘not really philosophical at all’).10 But for Russell, our use of mathematical physics to model space and time is justified on the basis of the mathematical consistency of these theories, along with the method of constructing the required elements from what is given in experience. On this basis, Russell holds that Zeno’s paradoxes, as ‘logical’ problems, are no longer tenable in the wake of the new foundations of infinity.11 As Russell reminds us, biographical information about Zeno, and records of his ar8

See e.g. the passage from 124-129, where Russell concludes that “If such constructions are possible then mathematical physics is applicable to the real world, in spite of the fact that its particles, points, and instants are not to be found among actually existing entities.” 9 In other occasions, Russell attributes to Bergson the position that motion cannot be analyzed into a series of states or “immobilities (p. 157, 162). Gr¨ unbaum (1967) takes Bergson’s view seriously and examines the relationship between our psychological experience of “becoming” and the mathematical structure of the continuum. 10 In fact, Russell is not even willing to assume that actual space and time are compact. See p. 138-139. 11 As a logicist, Russell would include higher mathematics within the domain of logic.

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guments, are passed down to us only indirectly. Zeno is present in Plato’s dialogue Parmenides, where he is portrayed as a student of the Eleadic school of Parmenides. His four famous paradoxes of motion are addressed by Aristotle in the Physics. Part of Aristotle’s solution was to deny the existence of an actual, completed infinity, thus setting an agenda for mathematics that would last millenia. Aristotle’s commentator, Simplicius, provides more arguments from Zeno, including several so-called paradoxes of plurality. In Our Knowledge, Russell includes the original passages from Aristotle in footnotes, but relies on paraphrases in the main text. A passage from Simplicius appears as well, providing one of Zeno’s paradoxes of plurality. It appears that, unlike his earlier treatment in Principles, Russell is concerned with historical accuracy.12 For most of the arguments, he considers multiple interpretations of the original text, seeing how they affect the soundness of the argument. Like most commentators, Russell is most concerned with determining the target of Zeno’s arguments, as their soundness depends on which background assumptions are being targeted. Zeno’s arguments were presented to support the view of his teacher Parmenides, who held that reality is comprised of a single, unchanging object, and that all appearances of plurality and change are illusory. The strategy of Zeno’s paradoxes is to grant some assumption proposed by his opponents (e.g. that there are many things, or that things change), and then show that these assumptions lead to contradiction. Parmenides can then claim that the prima facie absurdity of his doctrine is no argument against it, for the views of his opponents are equally beset by absurdity. Not surprisingly, Russell finds similarities between the views of Parmenides and those of the British Idealist F. H. Bradley, who influenced Russell in his early philosophical career. Russell describes Bradley’s philosophical project as intent on proving that “what is real is one single, indivisible, timeless whole, called the Absolute,” and “this is established by abstract logical reasoning professing to find self-contradictions in the categories condemned as mere appearance” (1914, p. 16). While Russell had once envisioned a similar project, he had “revolted” against such monistic idealism and defended a pluralistic realism for many years since.

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The Targets of Zeno’s Paradoxes of Motion

Russell addresses Zeno (both in Our Knowledge and earlier) after decades of lively debate in the philosophical community over the target of Zeno’s arguments. Much of this debate took place in French journals; Cajori (1915) provides a summary of this period, which involved mathematicians alongside philosophers. Interestingly, it is Tannery (1885), not Russell, who first brings up Cantor’s theory of the infinite in the context of Zeno’s argu12

In a letter to Morrell, dated October 25, 1913, Russell writes, “I wish I had a staff of trained underlings—I want some one to collect all references to Zeno in antiquity.”

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ments.13 As previously mentioned, Zeno’s overall strategy begins by granting an assumption of his opponent: that there are many things. In the context of his arguments on motion, this assumption allows for the division of space and time into regions and durations. But is this division allowed to continue on infinitely, or is there a limit? Zeno’s contemporaries held conflicting views on such matters. The four arguments against motion are intended to demonstrate absurd consequences of every such view about the structure of space and time. The standard interpretation of Zeno’s strategy is that the Dichotomy and the Achilles target the idea that space and time are continuous, while the Arrow and the Stadium target the idea that space and time have an atomic structure. In the terminology that Russell adopts, these ideas correspond, respectively, to the doctrine of infinite divisibility and to the doctrine of indivisibles. Before diving into Russell’s reconstruction of Zeno’s master strategy, we can express it as follows. Suppose that space and time can be divided. Then they can be divided until we reach points and instants. The number of points and instants must be either infinite or finite. If they are infinite, then the Dichotomy and Achilles both aim to show that this results in absurdities. If they are finite, then the Arrow and Stadium are used for similar purposes. Thus, according to Zeno, space and time are not composed of points and instants—and so cannot be divided at all.14 In the section just previous to the discussion of Zeno, while describing the Pythagorean discovery of incommensurables, Russell lays out these doctrines of indivisibles and infinite divisibility, though without much content. The doctrine of indivisibles is the idea that “space is composed of indivisible points, while time is composed of indivisible instants.” Do they have length and duration? On the standard interpretation, the Arrow targets the view that they do not; the Stadium targets the view that they do. Russell implicitly recognizes this in his discussion of these arguments. Russell never defines infinite divisibility while discussing Zeno, but he describes the notion in the previous chapter, while illustrating the notion of continuity.15 He makes it clear that infinite divisibility is admissible from a contemporary standpoint, and that it does not produce infinitesimals. By infinite divisibility, Russell has in mind the denseness property that holds of the series of rational numbers: between any two rationals, another can be found. (But, we might object, this isn’t actual infinite divisibility, which would produce the continuum of real 13

While Tannery likely saw Cantor’s theory as relevant to the debate on Zeno’s paradoxes, it is unclear whether Tannery endorses Cantor’s theory as providing a solution to Zeno, and so Russell probably deserves credit for making the connection more concrete—though not, as we shall see, for carrying out any such solution. 14 This reconstruction of Zeno’s master argument was provided by Owen (1957) and has become widely accepted after appearing in Salmon (1970). 15 p. 141. He also requires infinite divisibility in his constructions of instants, on p. 127.

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Argument Dichotomy Achilles Arrow Stadium

Target (Doctrines held by Ancient Greeks) Indivisibles (valid); infinite divisibility (invalid) Indivisibles (valid); infinite divisibility (invalid) Indivisibles, durationless Indivisibles, finite duration

Table 1: The target of each of Zeno’s arguments, according to Russell. numbers. It is merely potential infinite divisibility.) After reviewing the literature, Russell lands near the standard interpretation: he recognizes that Zeno’s main target is plurality, and he recognizes that the Arrow and Stadium are directed at the doctrine of indivisibles. However, he hesitates on assigning a fixed target to the Dichomoty and the Achilles, even though he reports No¨el (1893) as maintaining that “the first two arguments refute infinite divisibility, while the next two refute indivisibles” (p. 174). Instead, Russell allows that these arguments could be either directed at the doctrine of indivisibles (alongside the Arrow and Stadium), or directed at the doctrine of infinite divisibility (see Table 1). This seems to be motivated by Russell’s belief that Zeno’s arguments come out valid on the assumption of indivisibles, and thus by a principle of charity.16 According to Russell, the doctrine of indivisibles itself is not the culprit. The problem is that the doctrine was jointly held with the claim finitude: “the number of points in any finite area or of instants in any finite period must be finite.” This latter claim was undermined by the existence of incommensurables (p. 164). As a result, it became requisite to instead hold infinitude: “every finite length must contain an infinite number of points” (p. 169). Zeno’s arguments provide additional reasons to deny finitude; on that assumption, Zeno’s arguments come out valid, and thus their conclusions must be accepted. While the Dichotomy and Achilles may h...