1. Zeno of Elea
This Presocratic philosopher (5th cent. BCE), inventor of dialectic (i.e. the art of refutation) according to Aristotle (fr. 65 Rose, DK 29 A 10), must not be confounded with Zeno of Citium (4th -3rd cent. BCE), the founder of Stoicism. Given the account in Plato’s Parmenides (127B ff.), Zeno of Elea was about two decades older than Socrates and, hence, contemporary with such leading figures in post-Parmenidean cosmology as Anaxagoras, Empedocles and (perhaps) Leucippus. The pioneers of the Sophistic movement, Gorgias and Protagoras, may have been a decade younger. Being a disciple of Parmenides, Zeno published a series of arguments which, according to Plato (Parmenides 127E ff.), were designed to indirectly support the Parmenidean claim that Being is one by deriving contradictory conclusions from the assumption that there are many things. Other Zenonian arguments, such as the paradoxes of motion reported by Aristotle, may also have been directed against plurality.
Verbatim quotations have only survived (via Simplicius, 6th cent. CE) from two arguments on plurality, demonstrating that, “if there are many things,” they must be
(N) both “limited” and “unlimited” in number (DK 29 B 3), and
(M) ”both small and large; so small as not to have magnitude, so large as to be unlimited” (DK 29 B 2 and B 1),
respectively. Only secondary reports are available in all other cases, including the paradoxes of motion which, following Aristotle, may be referred to and summarized as follows.
(D) Dichotomy (or Stadium). There is no motion since “before reaching the goal” the runner must arrive at the half-way point,” and so forth ad infinitum (Phys. VI 9, 239b11-13, see also Top. VIII 8, 160b8-9 and Phys. VI 2, 233a21-23). Two variants of this argument may be distinguished (see Phys. VIII 8, 263a4-11).
(DG) In Dichotomyg, infinite division takes place towards the goal of the race-course. The runner first traverses half the race-course, then another quarter, and so forth, thus (if the race is from 0 to 1) successively arriving at
0, 1/2, 3/4, 7/8, … .
(DS) In Dichotomys, by contrast, infinite division takes place towards the starting point of the race-course. Before having traversed the whole course, the runner must have traversed its first half, and before that its first quarter, and so forth, thus successively arriving at
… 1/8, 1/4, 1/2, 1.
Aristotle first comments that there is no question of passing through an infinity in finite time since the infinities involved are the same concerning space and time (Phys. VI 2, 233a24-31). Later, he adds that it makes a difference whether the divisions in question are taken as potential or actual: difficulties arise only in the latter case which, however, requires that the movement in question be interrupted (Phys. VIII 8, 263a15-b9). In both cases, an infinity of actions are exhibited which the runner must perform. Dichotomyg makes it hard to see how the overall task can be completed. Dichotomys, by contrast, makes it hard to see how it can be taken up at all since there is nothing to be done first. In a sense, therefore, Dichotomys is particularly puzzling.
(AC) Achilleus. “In a race, the slowest is never caught up by the quickest” since “the pursuer must first reach the point where the pursued started, so that the slower must always hold a lead” (Phys. VI 9, 239b14-18). It is generally agreed that this is a mere restatement of Dichotomyg.
(AR) Arrow. Assume that (i) “everything either is at rest or moves whenever it occupies a position equal to itself” and (ii) “the moving thing is always in the now.” Then the flying arrow is (iii) “motionless” and, therefore, (iv) “stands still” (Phys. VI 9, 239b5-7, b30). It may be fairly agreed that instantaneous motion is a contradiction in terms and, hence, (v) “nothing moves in the now” (Phys. VI 3, 234a24). (ii) and (v) entail that (iii) the arrow is always “motionless” (and evidently occupies a space equal to itself). Taken together with (i), (iii) entails that (iv) the arrow “stands still”. Aristotles comments that (vi) “time is not composed of nows” (Phys. VI 9, 239b8, b30-31). His point is that, on the one hand, “always” in (ii) and hence in the whole argument (insofar as it is valid) only refers to “nows,” i.e. indivisible positions in time. But since, on the other hand, “time is not composed of nows,” nothing follows concerning the extended lapses of time required by motion and rest. In particular, instantaneous rest is as much as instantaneous motion a contradiction in terms. For instance, neither motion nor rest take place in the very moment when something has finished its movement, and will thereupon be at rest (Phys. VI 3, 234a31-b9). Since at that moment the thing in question undeniably occupies a space equal to itself, (i) is false and, hence, Zeno’s argument is fallacious.
(MR) Moving rows. This argument is particularly difficult to reconstruct from Aristotle’s discussion (Phys. VI 9, 239b33-240a18). It may be dismissed here since it plays no role in Whitehead.
2. Zeno’s Influence
Reactions to Zeno are already traceable in contemporary cosmology (Anaxagoras and especially the Atomists) and in the Sophists. The major part of Plato’s Parmenides is a dialectical “exercise” formed of a series of Zeno-like arguments. Aristotle’s analysis in Physics VI of motion and the continuum is evidently designed to avoid the difficulties exhibited by Zeno’s paradoxes. Diodorus Cronus, on the other hand, is reported to have developed Zeno’s arguments and explicitly endorsed the formula “never moves, but has moved” which in Aristotle indicates the absurdity to which the assumption is reduced that time and magnitude are composed of indivisible parts (Diodorus Cronus in Sextus Empiricus, Adv. math. 10,85 ff.; Aristotle, Phys. VI 1, 232a10-1).
Subsequent philosophy was usually aware of Zeno’s arguments. In particular, the “new science” of Galileo and his followers required a reconsideration of the infinities involved in continuity. “The whole labyrinth about the composition of the continuum,” wrote Leibniz (Loemker, 159), “must be unravelled.” Kant’s antinomies, in his Kritik der reinen Vernunft reflect Zeno’s Dichotomy. Given the contradictions exhibited by “the old dialecticians,” Hegel was happy to conclude that “motion is contradiction in actu” (“der daseiende Widerspruch,” Logik II, 76).
Modern scholarship, on the one hand, has been deeply influenced by Tannery’s claim that Zeno’s arguments were not directed against common sense but, rather, against a Pythagorean doctrine describing space and time as composed of indivisible units. Only after the 1950s was this interpretation seen to be ill-founded (see Vlastos 1967, 366 ff.). On the other hand, Aristotle’s eliminative strategems against Zeno were successfully resumed. Thus, Russell and, more recently, Grünbaum and others argued that modern mathematics, based on set theory, provides consistent accounts of continuity and motion, including the infinities involved. But it should be also noted that modern mathematics gives rise to such novel paradoxes as Cantor’s proof that the concept of cardinal number does not apply to the universe (i.e., in mathematics, the class of all classes). Surprisingly, the similarity between this result and Zeno’s paradox of number is rarely observed.
In particular, Russell pointed out that Zeno’s argument that “there is no such thing as a state of change” (CP 3, 370) does not prevent a body from being “in one place at one time and in another at another” and, hence, to “move” in the only relevant sense of that term (CP 3, 371 f.). Bergson objected that this “cinematographical” description is inevitable in retrospect but fails to account for the unity of the movement which spans a duration of time and is only grasped by “installing oneself in the change” (L’évolution créative, 307 ff.). For Bergson, Zeno’s arguments boil down to rendering absurd the notion of movement being “made of immobilities” (ibid.). Similarly but in a far less sophisticated way, James employed Zeno to confirming his view that, just as perceptual experience “grows by buds or drops,” so do time, change, etc. (Some Problems …, p. 80-95; quotation on p. 80). James mentioned Zeno’s Arrow in passing (p. 81) and amply discussed the Achilleus (p. 81-82, 87, and 91-93: unsuccessfully attacking Russell).
3. Zeno in Whitehead
The relevant passages are: a section in Whitehead’s Harvard Lectures for 1924-25 (March 31—April 11; Ford 1984, 275-286), SMW 124-127, and PR 68 ff. In what follows, I will first examine these passages (in reverse order) in their relation to the traditions and topics described above (Section 3.1). In the second place, then, I will describe Whitehead’s use of Zeno’s arguments, starting in the Harvard Lectures (Section 3.2) and successively including SMW (Section 3.3), and PR (Section 3.4).
3.1. Relevant Passages
In Process and Reality, Whitehead rightly dismisses James’s argument concerning the Achilleus as not “allow[ing] sufficiently for those elements in Zeno’s paradoxes which are the product of inadequate mathematical knowledge” (PR 68.14-16) and, in particular, of Zeno’s “ignorance of the theory of infinite convergent numerical series” (PR 69.4 f.). His own “consideration of Zeno’s arguments” (PR 68.6) solely relies on Dichotomys (cf. PR 68.18-69.2 and 69.17-26), which he, however, mistakes for the Arrow (cf. Chappell, p. 72). James mentions the Arrow only in passing (p. 157/81) and amply discusses the Achilleus (p. 157 ff., 171, 179 ff. / 81 f. 87, 91 ff.; unsuccessfully attacking Russell). One gets the impression that Whitehead’s citation of the Arrow (instead of Dichotomys) is a slip that serves the purpose of finding something in James’ “argument from Zeno” to “agree with” (if only “in substance,” cf. PR 68.13 f.).
Similarly, only Zeno is cited but something similar with Dichotomys is employed in Science and the Modern World (Chapter 7) to exhibit an inconsistency in Kant’s attribution to space and time of both extensiveness (i.e., to be adequately represented by “producing all its parts one after the other,” KrV A162 f. / B203) and continuity (i.e. that “space consists of spaces only, time of times,” KrV A169 / B211; as quoted SMW 125 f.). It should be noted that Whitehead does not only misrepresent Kant by neglecting the assignment to intuition (“Anschauung”) and to reality (“das Reale…”, KrV A166), respectively, of extensiveness and of continuity. He also claims that Kant’s account of space and time as continuous “is in agreement with Plato as against Aristotle” (SMW 127) which, of course, is quite the opposite of the truth.
Still earlier are Whitehead’s Harvard Lectures of March 31‑April 11, 2005 (Ford 1984, 275-86) where Zeno is credited with “something permanently true” (Ford 1984, 277) but no particular argument of Zeno’s is cited. Whitehead’s starting points are relativity theory (Ford 1984, 275) on the one hand and Bergson (Ford 1984, 276) on the other; quotations from Kant (Ford 1984, 277 and passim) are the same as in Chapter 7 of SMW. “Aristotle’s idea” (in dealing with Zeno’s question: “how is generation possible?” and/or with a “Pythagorean difficulty” concerning “infinity” and “limitations”) is described as follows. “We ought to start with points and moments and avoid all these difficulties. Points and moments with external relations” (Ford 1984, 278).
Both in the Harvard Lectures and in SMW, Whitehead presents himself as being totally unaware of Aristotle’s discussion, in Phys. VI, of Zeno’s arguments. It is hard to believe that he ever had a piece of relevant contemporary scholarship before his eyes. His neglect of scholarship contrasts conspicuously with Russell’s painstaking discussion in Chapter 6 of his Our Knowledge of the External World (of which, again, no traces are to be found in Whitehead). Russell thus presents himself as really caring about the issues raised by Zeno’s arguments. Whitehead, by contrast, seems to merely harness Zeno for his own purpose, i.e. to exhibit “the epochal character of time” (SMW 126).
3.2. Zeno in the Harvard Lectures for 1924-25
Three interrelated difficulties are presented in the first lecture, of March 31 in the relevant part of Whitehead’s Harvard Lectures for 1924-25. First, difficulties in “the idea of alternative time systems” with which relativity theory replaces “the linear idea of becomingness” (Ford 1984, 276). Second, difficulties in Bergson’s claim that “Durée is indivisible” (Ford 1984, 276). Third, such difficulties in “the old idea of the flux of time” as are exemplified by the above-mentioned quotations from Kant and by Zeno’s arguments. Whitehead’s remark concerning the former points to the desideratum of “hav[ing] some theory of the parts and the wholes” (Ford 1984, 277). A modification of the latter is adumbrated by Whitehead’s remark that “Zeno made an unfortunate choice in dealing with motion and space—muddling up time and space together” (Ford 1984, 277). Rather, Zeno should have dealt with time alone (see the corresponding passage in SMW, 127). Accordingly, the point in Zeno’s arguments is represented by Whitehead as follows.
Zeno: How are you going to move forward into the future? How is process possible? If you conceive it unter the guise of a temporal transition into the non-existent, you can’t get going. There is nothing you can point to into which there is a transition, or is there and then created (Ford 1984, 277).
In the lecture of April 2 Whitehead remarks that “Kant’s statement that the parts are antecedent to the whole” (that is, his description as extensive magnitudes of space and time, ) leads into a “vicious regress” (Ford 1984, 277; both the lecture of April 9, p. 283, and the corresponding passage in SMW 126, attribute this objection to Zeno). The rest of the lecture is mainly devoted to Whitehead’s claim that temporal relations are “internal” (Ford 1984, 278) and, accordingly, “that moment” is duly equated with “that particular concrete relatedness of that past to that future” (Ford 1984, 279).
The next lecture, of April 4, presents itself as a series of historical remarks of which the relevant two have been already mentioned (Ford 1984, 279-281).
In the lecture of April 7, “an atomic theory of time” (Ford 1984, 281) is presented. Whitehead’s starting point is the “distinction between temporality and extensiveness” brought out by the observation that “the idea of extension doesn’t include time-direction” (recall that relativity theory “presents us with the notion of alternative progressions in time,” 275). Extension is only temporalized
via realization of the potential, i.e. the individualization of each event into a peculiar togetherness […]. An event as present is real for itself. It is this becoming real which is temporalization (Ford 1984, 281).
Here, Whitehead continues, “we bump up against the atomic view of things, also the subjective view” (Ford 1984, 282). A “subject” is on the one hand, “a parallelogram” in the extensive structure described by relativity theory. On the other hand,
its reality is the realization of something as entering into its own being. “The pulling together of a duration from its own viewpoint, i.e. as entering into its own essence.” […] The subject is what that grasping together is (Ford 1984, 282f).
The clue to “atomicity,” then, is Whitehead’s claim that “the becoming real is not the production via the parts of the duration—contradicting Kant” (Ford 1984, 282); that is, contradicting Kant’s description of time as extensive, with successive parts “antecedent to the whole” (Ford 1984, 277), which, Whitehead claims, is inconsistent with Kant’s own description of time as continuous (Ford 1984, 283). Accordingly, “the time transition” must not be conceived as a succession in becoming but, rather, as “a transition within what is already there. […] There is no relation between something and nothing” (Ford 1984, 283). Zeno is not mentioned in this lecture. But the result just stated corresponds to the principle, attributed to Zeno by Whitehead, that “process” must not be conceived “under the guise of a temporal transition into the non-existent” (Ford 198, 277).
The impact of Zeno’s arguments is only adumbrated by a remark in the lecture of April 9 which summarizes the previous lectures as follows:
Starting with events, and bringing the future and past into it, didn’t give enough differentiation. Had to introduce ‘reality’ as ‘real togetherness’, bringing in the time-idea. If you take time as merely generating the event, Zeno gets at you. There is no such thing as a moment. What must be real is the togetherness of the content of the event (Ford 1984, 283).
3.3. Zeno in Science and the Modern World
How and why does Zeno “get at you”?—The extant lecture notes are silent on this. In the corresponding passage in SMW, just after quoting Kant on extensive and continuous magnitudes, Whitehead claims that
Zeno would object that a vicious infinite regress is involved in the former description. Every part of time involves some smaller part of itself, and so on. Also this series regresses backwards ultimately to nothing; since the initial moment is without duration and merely marks the relation of contiguity to an earlier time. Thus time is impossible, if the two extracts are both adhered to. I accept the later, and reject the earlier, passage (SMW 126).
I take it that “Zeno gets at you” in the Harvard Lectures just in the same way as he gets at Kant in SMW. Accordingly, I suggest that in the Harvard Lectures, the description of “time as merely generating the event” (April 9, 283) corresponds to the claim, attributed to Kant, that “the becoming real is […] the production via the parts of the duration” (April 7, 282).
It should be noted, however, that Whitehead “is in complete agreement with the second extract” only “if ‘time and space’ is the extensive continuum.” (SMW 126). The qualification is essential since Whitehead is, of course, not at all willing to equate time with extension. Rather, Whitehaed affirms the doctrine of the Harvard Lectures (p. 281) that extension is only temporalized “via realization of the potential” as follows.
Realization is the becoming of time in the field of extension. Extension is the complex of events, qua their potentialities. In realization the potentiality becomes actuality. […] Temporalization is realization. Temporalization is not another continuous process. It is an atomic succession (SMW 126).
In order to understand this passage, Whitehead’s observation in the Harvard Lectures that “the idea of extension doesn’t include time-direction” should be adduced. I take it that “time-direction,” for Whitehead, must take the form of there being “succession,” i.e. of there being earlier and later events. Given Whitehead’s claim that extension is only temporalized and, hence, time-direction is only imposed on the extensive continuum “via realization of the potential” (Ford 1984, 281), two candidates present themselves for the succession in question. (i) Assuming that “the becoming real is […] the production via the parts of the duration” (Ford 1984, 282) a succession of temporal parts might be supposed to correspond to that production. (ii) The transition in question is in the nature of what has become real […] and, hence, is a transition from one duration to another. If (i) is refuted by Zeno’s arguments, “the time transition” is easily seen to give rise to a succession of durations each of which is atomic, that is, hasn’t become real because of the [i.e.: any internal] transition (Ford 1984, 283).
3.4. Zeno in Process and Reality
The “epochal theory of time” of SMW is reaffirmed at PR 68.3. In Process and Reality, “Zeno’s method” is employed to prove that, though “there is becoming of continuity”, “there can be no continuity of becoming” (PR 35.32-4) and to bring out “the principle that every act of becoming must have an immediate successor” (PR 69.17 f). The “epochal theory of time” of SMW is thus reaffirmed (PR 68.3).
 Modern interpreters usually follow Zeller in deleting from (i) the clause “or moves” (ê kineitai, b6).
 Two more mentions of “Zeno’s method” are PR 35.32 and 307.22.
 It should be noted that only W.E. Hocking’s notes of Whitehead’s lectures, with scarce verbatim quotations (marked in the sequel by […]) survived.
 A telling exception is Heath’s Euclid in Greek (cited SMW 127n1). This book (by an eminent scholar, of course) contains a Greek text of Elements, Book I, with introduction and notes. It is addressed to a general public (p. vii: “senior boys at school,” etc.). The notes, being designed “to make the schoolboy […] think” (viii), contain much additional information about ancient philosophy and mathematics, but the author has found it appropriate to omit all references. In the “note on Points” (i.e., to Euclid’s definition of “point”) mentioned at (SMW 127n1), Heath tacitly quotes Aristotle’s Metaphysics, I 9, 992a19-24 as follows: “Plato, we are told, objected to recognising points as a separate class of things at all, and regarded them as a ‘geometrical fiction.’ He preferred to conceive a point as being merely ‘the beginning of a line;’ alternatively, he spoke of ‘indivisible lines.’ But, as Aristotle says, even indivisible lines must have extremities: hence an indivisible line […] must contain at least two points […].” (Heath 1920, 115) The only thing Whitehead seems to have learned from this is that Plato, as opposed to Aristotle, denied the existence of points—which, of course, is misleading since Aristotle only insists that points are no more abstract as Plato’s lines and planes are. But Whitehead is even unaware of Heath’s description in the same note (114) of Aristotle’s account of continuity.
 In his earlier work on Zeno, by contrast, Russell relied on one article grasped from the debate among French scholars (i.e. Noel 1893, cf. Russell 1903, 348n).
 Zeno is also represented as asking: “how is generation possible?” (April 2; p. 278). The context, however, is different.
 See also Ford’s summary (1984, 54).
 The relevant passages in Kant are KrV A162 f. / B203 and KrV A169 f. / B211, respectively. See p. 277 in the lecture notes; SMW 125ff has the full quotations.
 The addenda presented in the lectures of April 9 and April 11 (p. 283-86) throw no additional light on Whitehead’s interpretation of Zeno.
 Nothing is added to this by a preliminary mention of Zeno at SMW 125 and by the summary at SMW 127.
 Unfortunately, there is no more space here to go into any details; see my webpage at www.uni-kassel.de/philosophie.
Works Cited and Further Readings
Aristotle. 1924. Metaphysics (Oxford: Clarnedon Press).
Aristotle. 1957. Physics (Harvard University Press)
Bergson, Henri. (1907). Creative Evolution.
Ford, Lewis S. 1984. The Emergence of Whitehead’s Metaphysics 1925-1929 (Albany, State University of New York Press Press)
Kant, Immanuel (1781). Critique of Pure Reason.
Loemker, Leroy (1976). Gottfried Wilhelm Leibniz: Philosophical Papers and Letters (Springer).
Plato (1926). Parmenides (Harvard University Press).
Russell, Bertrand. 1993 (1900-02) The Collected Papers of Russell Bertrabnd Vol. 3: Toward the Principles of Mathematics. Ed.:Moore, Gregory H.. (New York NY: Routledge).
Zeno: The standard edition of the evidence concerning Zeno is DK, Chapter 29 (A: testimonia, B: fragments). More comprehensive editions, with translation and commentary, are Lee  and Caveign . Both KRS and Mansfeld [1983-86] present good selections of, and introductions to, the evidence. None of the many survey articles available goes without any complaint, but Vlastos  and Makin  nevertheless are outstanding.
Universität Kassel, FB 01, Philosophie, 34109 Kassel
How to Cite this Article
Heinemann, Gottfried, “Whitehead’s Interpretation of Zeno”, last modified 2008, The Whitehead Encyclopedia, Brian G. Henning and Joseph Petek (eds.), originally edited by Michel Weber and Will Desmond, URL = <http://encyclopedia.whiteheadresearch.org/entries/thematic/metaphysics/whiteheads-interpretation-of-zeno/>.