5. Zeno's Influence on Philosophy
In this final section we should consider briefly the impact that Zeno has had on various philosophers; a search of the literature will reveal that these debates continue.
The Pythagoreans: For the first half of the Twentieth century, the majority reading—following Tannery (1885)—of Zeno held that his arguments were directed against a technical doctrine of the Pythagoreans. According to this reading they held that all things were composed of elements that had the properties of a unit number, a geometric point and a physical atom: this kind of position would fit with their doctrine that reality is fundamentally mathematical. However, in the middle of the century a series of commentators (Vlastos, 1967, summarizes the argument and contains references) forcefully argued that Zeno's target was instead a common sense understanding of plurality and motion—one grounded in familiar geometrical notions—and indeed that the doctrine was not a major part of Pythagorean thought. We have implicitly assumed that these arguments are correct in our readings of the paradoxes. That said, Tannery's interpretation still has its defenders (see e.g., Matson 2001).
The Atomists: Aristotle (On Generation and Corruption 316b34) claims that our third argument—the one concerning complete divisibility—was what convinced the atomists that there must be smallest, indivisible parts of matter. See Abraham (1972) for a further discussion of Zeno's connection to the atomists.
Temporal Becoming: In the early part of the Twentieth century several influential philosophers attempted to put Zeno's arguments to work in the service of a metaphysics of ‛temporal becoming’, the (supposed) process by which the present comes into being. Such thinkers as Bergson (1911), James (1911, Ch 10–11) and Whitehead (1929) argued that Zeno's paradoxes show that space and time are not structured as a mathematical continuum: they argued that the way to preserve the reality of motion was to deny that space and time are composed of points and instants. However, we have clearly seen that the tools of standard modern mathematics are up to the job of resolving the paradoxes, so no such conclusion seems warranted: if the present indeed ‛becomes’, there is no reason to think that the process is not captured by the continuum.
Applying the Mathematical Continuum to Physical Space and Time: Following a lead given by Russell (1929, 182–198), a number of philosophers—most notably Grünbaum (1967)—took up the task of showing how modern mathematics could solve all of Zeno's paradoxes; their work has thoroughly influenced our discussion of the arguments. What they realized was that a purely mathematical solution was not sufficient: the paradoxes not only question abstract mathematics, but also the nature of physical reality. So what they sought was an argument not only that Zeno posed no threat to the mathematics of infinity but also that that mathematics correctly describes objects, time and space. The idea that a mathematical law—say Newton's law of universal gravity—may or may not correctly describe things is familiar, but some aspects of the mathematics of infinity—the nature of the continuum, definition of infinite sums and so on—seem so basic that it may be hard to see at first that they too apply contingently. But surely they do: nothing guarantees a priori that space has the structure of the continuum, or even that parts of space add up according to Cauchy's definition. (Salmon offers a nice example to help make the point: since alcohol dissolves in water, if you mix the two you end up with less than the sum of their volumes, showing that even ordinary addition is not applicable to every kind of system.) Our belief that the mathematical theory of infinity describes space and time is justified to the extent that the laws of physics assume that it does, and to the extent that those laws are themselves confirmed by experience. While it is true that almost all physical theories assume that space and time do indeed have the structure of the continuum, it is also the case that quantum theories of gravity likely imply that they do not. While no one really knows where this research will ultimately lead, it is quite possible that space and time will turn out, at the most fundamental level, to be quite unlike the mathematical continuum that we have assumed here.
One should also note that Grünbaum took the job of showing that modern mathematics describes space and time to involve something rather different from arguing that it is confirmed by experience. The dominant view at the time (though not at present) was that scientific terms had meaning insofar as they referred directly to objects of experience—such as ‛1 m ruler’—or, if they referred to ‛theoretical’ rather than ‛observable’ entities—such as ‛a point of space’ or ‛1/2 of 1/2 of … 1/2 a racetrack’—then they obtained meaning by their logical relations—via definitions and theoretical laws—to such observation terms. Thus Grünbaum undertook an impressive program to give meaning to all terms involved in the modern theory of infinity, interpreted as an account of space and time.
Supertasks: A further strand of thought concerns what Black (1950–51) dubbed ‛infinity machines’. Black and his followers wished to show that although Zeno's paradoxes offered no problem to mathematics, they showed that after all mathematics was not applicable to space, time and motion. Most starkly, our resolution to the Dichotomy and Achilles assumed that the complete run could be broken down into an infinite series of half runs, which could be summed. But is it really possible to complete any infinite series of actions: to complete was is known as a ‛supertask’? If not, and assuming that Atalanta and Achilles can complete their tasks, their complete runs cannot be correctly described as an infinite series of half-runs, although modern mathematics would so describe them. What infinity machines are supposed to establish is that an infinite series of tasks cannot be completed—so any completable task cannot be broken down into an infinity of smaller tasks, whatever mathematics suggests.
Non-standard analysis: Finally, we have seen how to tackle the paradoxes using the resources of mathematics as developed in the Nineteenth century. For a long time it was considered one the great virtues of this system that it finally showed how to do without infinitesimal quantities, smaller than any finite number but larger than zero. (Newton's calculus for instance effectively made use of such numbers, treating them sometimes as zero and sometimes as finite; the problem with such an approach is that how to treat the numbers is a matter of intuition not rigor.) However, in the Twentieth century Robinson showed how to introduce infinitesimal numbers into mathematics: this is the system of ‛non-standard analysis’ (the familiar system of real numbers, given a rigorous foundation by Dedekind, is by contrast just ‛analysis’). And it has been shown by McLaughlin (1992, 1994) that Zeno's paradoxes can also be resolved in non-standard analysis; they are no more argument against non-standard analysis than the standard mathematics we have assumed here. It should be emphasized however that—contrary to McLaughlin's suggestions—there is no need for non-standard analysis to solve the paradoxes: either system is equally successful. (The construction of non-standard analysis does however raise a further question about the applicability of analysis to physical space and time: it seems plausible that all physical theories can be formulated in either terms, and so as far as our experience extends both seem equally confirmed. But they cannot both be true of space and time: either space has infinitesimal parts or it doesn't.)