Zeno's Paradoxes (Salmon edition)

The collection is structured around a single question: do the resources of modern mathematics—specifically the theory of infinite series and the calculus—actually resolve Zeno's paradoxes, or do they merely describe the same situation in a different notation without addressing the underlying metaphysical worry? Bertrand Russell's essay argues that the problem of infinity was solved by nineteenth-century mathematics, making Zeno's arguments obsolete. But Max Black, James Thomson, and Paul Benacerraf shift the debate toward 'super-tasks': whether an infinite series of discrete actions can be completed in finite time. Salmon's introduction and appendix on sets and infinity frame these contributions as a sustained inquiry into the adequacy of mathematical solutions to ancient philosophical puzzles, rather than a mere historical survey .

Aristotle's response in the Physics, which denies that a continuous magnitude is actually composed of infinitely many parts, serves as the historical baseline against which the modern essays are measured. Several contributors argue that Aristotle's solution, while internally coherent, relies on distinctions that modern mathematics has rendered unnecessary or unsustainable. G. E. L. Owen's essay 'Zeno and the Mathematicians' is particularly important here: it argues that Aristotle misidentified the target of Zeno's arguments, treating them as attacks on motion when they are better understood as attacks on plurality and the reality of parts. The anthology thus presents the modern debate as a corrective to Aristotle's framing, even as it acknowledges that Aristotle preserved the arguments we possess .

Grünbaum contributes three related essays and is the collection's most sustained defender of the view that modern science and mathematics decisively refute Zeno. His distinctive move is to argue that the paradoxes rest on an outdated, pre-Cantorian conception of infinity, and that once we accept the rigorous theory of infinite sets, the contradictions Zeno derived simply evaporate. He also addresses the 'metrical paradox of extension,' the puzzle of how a line segment with zero-length points can have positive length, arguing that this too is resolved by measure theory. Grünbaum's essays represent the strongest case for the thesis that Zeno's paradoxes are historical curiosities, a position that other contributors to the volume challenge .

The Achilles paradox assumes that a complete run can be decomposed into an infinite series of half-runs, and that if the series can be summed mathematically, the motion is thereby explained. James Thomson's essay on super-tasks challenges this assumption by asking whether an infinite series of discrete actions—switching a lamp on and off infinitely many times, for instance—can be completed at all. If such a series cannot be completed, then the mathematical summation of the series does not describe a possible physical process. Paul Benacerraf's reply argues that Thomson's puzzle conflates the mathematical description of an infinite series with the physical performance of infinitely many acts, and that no contradiction follows from the former. This exchange, reprinted in Salmon's volume, remains the standard entry point to the super-task literature .

The Stanford Encyclopedia of Philosophy explicitly recommends Salmon's collection—specifically the 2001 Hackett reprint—as 'the place to begin any further investigation' of Zeno's paradoxes, praising its selection of essential articles and its comprehensive bibliography of twentieth-century English-language scholarship . The Hackett edition updates the bibliography while retaining the 1970 essays, which remain the canonical formulations of the analytic debate. More recent work, such as the SEP's own entries on infinitesimals and non-standard analysis, has extended the discussion, but the core positions articulated by Russell, Black, Thomson, Benacerraf, Owen, and Grünbaum are still best approached through this volume. It is not a comprehensive historical treatment of Zeno in antiquity, but it is unsurpassed as a guide to the modern philosophical reception.

The most common misunderstanding is that it is a primary source collection containing Zeno's own fragments or a translation of his lost book. It is neither. It is an anthology of twentieth-century secondary essays, and the attribution to 'Zeno of Elea' in library catalogues can mislead readers into expecting ancient texts. The volume presumes some familiarity with the paradoxes and with elementary calculus; it does not introduce Zeno historically or survey the ancient sources. Readers seeking Zeno's fragments should turn to Diels/Kranz or to H. D. P. Lee's edition. Salmon's book is a record of how professional philosophers and mathematicians in the analytic tradition argued about Zeno, not a guide to what Zeno himself wrote or meant in his own context.