on time and the dichotomy in leibniz

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Studia LeibnitianaZeitschrift für Geschichte der Philosophie und der Wissenschaften

On Time and the Dichotomy in Leibniz*

By

SAMUEL LEVEY (HANOVER, NH)

In April of 1676, Leibniz studies Spinoza’s Letter 12 ‘On the Nature of the Infi nite’ and copies out extracts, including the following passage:

“In order for an hour to pass, it will be necessary for half of it to pass fi rst, and then half the remainder, then half of what remains of this remainder. And if one subtracts half from the re-mainder in this way indefi nitely, one will never be able to reach the end of the hour”1.

In response Leibniz notes:

“From this it only follows that no one can complete the enumeration of parts into which an hour can be divided in continuous proportion. And this is the same thing as saying that no book can be found in which one could write all the numbers of a double geometrical progression. It does not follow from this, however, that an hour cannot pass, but that an hour can only pass in an hour”2.

At most, then, no one can mark out each of the half-intervals of the hour and complete the infi nite series, any more than one could completely write out all the elements of the series 1/2, 1/4, 1/8, 1/16, &c. ad infi nitum; still, the hour itself can pass. This appears to recall Aristotle’s idea that although an actual infi nity cannot be completed in any time, the hour can pass because it contains only a potential infi nity: the infi nite divisibility of time notwithstanding, its passage does not involve the actual division of it into infi nitely many parts. If traversing an actual infi nity were required in order to pass to the end of an hour – as if to complete the enumeration by counting through the divisions – then it would have to be “admitted to be impossible” (Phys. 263b3), according to Aristotle; but an infi nity of merely potential divisions can be traversed (cf. 263b4-6).

* Thanks to audiences in the Philosophy Department and the Mathematics Department at Dartmouth College, the Boston Colloquium for History and Philosophy of Science, at Boston University, and the American Philosophical Association, in Boston, for discussion of ancestor versions of this paper, and to Christine Thomas and Jeffrey McDonough for comments on a recent draft.

1 A VI, 3, 279. Original in Spinoza: Opera, ed. by C. Gebhardt, 4 vols., Heidelberg 1925, vol. IV, p. 58/7-11. Translated in R. Arthur (ed.): G. W. Leibniz: The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672-1686, New Haven 2001, p. 109. Translations of Leibniz’s texts in this paper, nearly all from A VI, 3, generally follow those of Arthur, though I have sometimes modifi ed his translation without comment. Other sources will be noted as they occur.

2 A VI, 3, 279.

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34 Samuel Levey

The echo of Aristotle in Leibniz’s reply is out of place, however, because of a key point of difference between the two philosophers: Leibniz does not reject but embraces the actual infi nite. The hour is not only potentially infi nitely divisible but actually infi nitely divided. The enumeration of its parts is complete even if no one counts them. For time is divided by motion, and motion in any interval however small always contains an actual infi nity of changes, dividing time into distinct parts (cf. A VI, 3, 565-566). Leibniz will need something else to say in reply to the puzzle if he is to explain how the hour can pass.

It turns out that Leibniz does have something else to say, and it comes in material he writes just seven months later, in preparation for his meeting with Spinoza, a dialogue he titles Pacidius Philalethi. The question of the passage of the hour is not raised eo nomine in the dialogue, but Leibniz nonetheless clearly engages with the underlying puzzle – it is, of course, a version Zeno’s “dichoto-my” paradox – in a provocative line of analysis that sheds light on his views of time and motion and suggests something of interest concerning the ancient topic of passage to the limit as well. Passage to the limit is an important concept for solving the puzzle of how the hour can pass on Leibniz’s account, and it marks one way in which the analysis of the nature of time cleaves toward the analysis of motion rather than toward that of space. Time, like, motion, passes, whereas space lacks this aspect of progression. (Or so it appears, at any rate; as in all things, philosophers have sometimes withheld assent on this point.) Perplexity about how a given distance, say, a mile, can be traversed is not traditionally interpreted as implying a challenge to the very existence of the distance itself; even the Eleatic target of the dichotomy appears to be the reality of motion and change rather than that of the interval in which they occur. By contrast, if the hour cannot pass, it clearly cannot exist at all, and neither can any other interval of time, as the same argument will apply to the half-hour, the quarter-hour, and so on. Leibniz’s view of passage to the limit, then, while rooted in his analysis of motion, will speak directly to his conception of time, and so a considerable part of the present paper is devoted to articulating Leibniz’s thought on that subtle topic, a vein of philosophy that as yet is less well known to his readers than many of his ideas about time.

Leibniz’s analysis of the dichotomy paradox and passage to the limit in the Pacidius, it turns out, yields a metaphysical account of time that stands in tension with his celebrated “reductivist” claims about the nature of time to the effect that time is only an “order of successions”, i. e. a construction from concrete events and changes in nature rather than a free-standing feature of reality in its own right. In this respect, for Leibniz, time is like space, for space too is only an ordering of things, an “ordering of co-existences”, whose nature is given a reductive analysis in terms of relations among concrete particulars3. Time, then,

3 For this and the last quoted phrase, see Leibniz’s third letter to Clarke, GP VII, 363. Translation is from The Leibniz and Clarke Correspondence, ed. and trans. by R. Ariew, Indianapolis 2000. Hereafter this work is abbreviated ‘LC’; references are to letter and section.


35On Time and the Dichotomy in Leibniz

is at once akin both to motion and to space in Leibniz’s philosophy. It is worth asking whether Leibniz’s analysis of the dichotomy can be reconciled with his reductivist views of time: whether his account of passage to the limit is compatible with his account of time’s ontological character. I shall argue that the two can be reconciled but only at the price of accepting a certain outré metaphysics of time and temporality – what is lately called “eternalism” about time – that was not in any obvious way a component of Leibniz’s thought. And even this imposes a further cost. In the Pacidius, Leibniz relies on his analysis of the dichotomy to argue that motion contains minimal elements, partless beginnings and ends of motion; this is one of the dialogue’s featured results. Leibniz’s argument for it is rendered circular, however, on the eternalist metaphysics of time required to reconcile his analysis of the dichotomy and his reductivism. So it appears that Leibniz’s arguments on these points cannot be integrated into a stable total position. Patience will be required in what follows, for we have many waypoints fi rst to reach on our journey to that end.

1. The Argument of Pacidius Philalethi for Minima

In the Pacidius motion is provisionally defi ned to be “change of place”4, and the inquiry into the nature of motion considers the structure of the change that motion is supposed to be. Leibniz opens the most profound part of his discussion by invoking the argument of the sorites paradox. He uses the sorites to argue that the change from being poor to not poor, or from being rich to not rich, is a transition that can in principle be effected by the gain or loss of a single penny. The speakers here are Pacidius, who plays the role of Socrates for the dialogue, and Charinus, the principal interlocutor:

– Pacidius: If the wealth of two people differs by only one penny, could one of them be regarded as rich without the same judgment being made about the other?– Charinus: No, I do not believe so.– Pa.: Therefore a difference of one penny does not make one rich or poor.– Ch.: I suppose not.– Pa.: Nor would the gain or loss of one penny make a rich person not rich, or a pauper not poor.– Ch.: Not at all.– Pa.: Therefore no one can ever become rich from being poor, nor become poor from being rich, however many pennies are given to or taken from him.– Ch.: Why is that, may I ask?– Pa.: Suppose a penny is given to a pauper. Does he cease to be poor?– Ch.: No.– Pa.: If another penny is given to him, does he cease to be poor then?– Ch.: No more than before.– Pa.: Therefore he does not cease to be poor if a third penny is given to him?– Ch.: No.

4 A VI, 3, 534.


36 Samuel Levey

– Pa.: The same applies to any other one: for either he never ceases to be poor, or he does so by the gain of one penny. Suppose he ceases to be poor when he gets a thousandth penny, having already got nine hundred and ninety-nine; it is still one penny that removed his poverty.– Ch.: I can see the force of the argument, and I’m surprised I was deluded like this.– Pa.: Do you admit, then, that either nobody ever becomes rich or poor, or one can become so by the gain or loss of one penny?– Ch.: I am forced to admit this5.

Charinus’s “forced admission” is strictly only to a disjunction: either no one ever becomes rich or poor, or else one can become so by the gain or loss of one penny. But the example of becoming rich or poor is just going proxy here for the more general case of change itself, and Leibniz is not going to deny that change actually occurs (“Who would deny it?”; cf. A VI, 3, 541), and likewise in the present case he is not going to deny that anyone ever actually becomes rich or poor. Thus it is the second horn of the dilemma that he embraces: namely, that one can become rich or poor by the gain or loss of a single penny. And indeed, when someone ceases to be poor by getting a thousandth penny, it is in fact just one penny, the very last one, that fi nally removes her poverty.

This hard line of the logic of vague terms has been very unpopular through the years as a response to the sorites. One can only suspect that it’s a bone in the throat even for its most unfl inching advocates. Leibniz himself shows a few signs of being uncomfortable with the hard line in early drafts of the dialogue, writing in one: “This was a sophism of the Stoics, which they called the Sori-tes…”. And then in another: “This type of argument, which the ancients called the Sorites, is not useless if you use it correctly”6.

2. The Second Sorites

Still, the correct use of the sorites is not far off. To reach it we are to “trans-pose the argument from discrete to continuous quantity”, and immediately Leibniz executes this change to offer his argument, the second sorites, for the existence of minima in reality. The distinction he has in mind between discrete and con-tinuous quantity is this: a discrete quantity is one that has natural boundaries occurring in it – Leibniz sometimes says that discrete quantity is terminatam or “bounded” – and in this context the idea appears to be that discrete quantities come with pre-assigned minimal parts or indivisible units of measure. Poverty is fi nally measured by the penny, baldness by the hair, the heap by the grain. A continuous quantity, on the other hand, does not have such natural boundaries occurring in it – Leibniz will describe it as interminatam or “unbounded” – and here the idea is that continuous quantities do not come pre-assigned with such minima or indivisible units of measure. Rather, the units of a continuous quantity

5 A VI, 3, 539.6 Cf. Arthur (ed.): G. W. Leibniz: The Labyrinth of the Continuum (see note 1), p. 405, note

25. For a broader discussion of Leibniz on the sorites paradox, see S. Levey: “Leibniz and the Sorites”, in: The Leibniz Review 12 (2002), pp. 25-49.



can be taken as fi nely as one wishes. Thus distance can be measured in miles, feet, inches, trillionths of an inch, and so on, to any degree of precision.

Here is Leibniz as he articulates the second sorites in the Pacidius:

– Pa.: Let us transpose the argument from discrete to continuous quantity: for example, if a [movable] point a approaches a point H, then at a certain time it will turn from not being near to being near, as at B, for instance. Shall we not conclude by the same argument as a little while ago that either it never gets near to H, or it does so by the addition of one inch, such as FB? – Ch.: Yes.– Pa.: But couldn’t we have substituted for the inch a hundredth or thousandth of an inch, or any other part, however small?– Ch.: We could, without affecting the force of the argument.– Pa.: Therefore we could have substituted a part smaller than any named by us?– Ch.: Of course.

H•a

B

[not near] [near]

ECF [D]

•

– Pa.: If it is the hundredth part CB of the inch FB that makes the near into the not-near7, then it is not the whole inch which does this.– Ch.: No, it is not: for the fi rst ninety-nine parts FC have not yet made the point near.– Pa.: Then it is clear that the gaining of an inch makes the not-near into the near only because it contains the last hundredth of an inch.– Ch.: And by the same token, the last hundredth CB does not make it near except by virtue of its last part B.– Pa.: But isn’t the last part a minimum?– Ch.: Yes it is, for if it were not a minimum, then something could be removed [rescindi] from it, leaving intact [salvo] whatever produces the nearness. For suppose that the last part of CB is not the minimum B but a straight line DB: this line would not make the near into the not-near of its own accord [per se], but by virtue of some other still smaller part of itself, EB. – Pa.: Therefore we have that either there is no way for something to become near properly and of its own accord [proprie ac per se], or something turns from being near to being not-near by the addition or subtraction of a minimum, so that there are minima in reality8.

Notice the parallel here with the fi rst sorites. The second sorites concludes with what is only strictly a disjunction: either there is no way for something to become near “properly and of its own accord” (more on that qualifi er later), or else it does so by the addition of a minimum. And again, Leibniz’s position is in fact more fully staked out: he embraces the second disjunct. For the example of becoming near from not near is going proxy here for the more general case of change itself, and in particular for the case of motion.

7 Leibniz has ‘near’ and ‘not-near’ reversed here and again in subsequent lines; the translation corrects the slip.

8 A VI, 3, 540.


38 Samuel Levey

The point of transposing the sorites argument from discrete to continuous quantity is to help expose something about the nature of motion by considering a case of continuous change such as a “change of place” would be. The whole dialogue develops Leibniz’s fi rst philosophy of motion – it is subtitled “A First Philosophy of Motion” – and on that fi rst philosophy the reality of motion is not going to be denied. Likewise, Leibniz here is not going to deny that it is possible for something to become near. The movable point a on its journey to the end of the line, does change from being not near to near at B, and it is it possible for the movable point to pass all the way to B. In order for that to occur, there must be a “last part” of the point’s motion that allows the point actually to reach B and become near. Such a continuous change is fi nally effected by the addition of a single minimum.

3. Passage to the Limit

Leibniz’s second sorites in the Pacidius actually incorporates two main lines of argument, only one of which addresses the issue of passage to the limit. The two parts of Leibniz’s argument for minima that I mean to distinguish fall on opposite sides of the diagram in the last passage quoted. The earlier part is simply the sorites “transposed to continuous quantity” or what I have been call-ing the second sorites; the later part is the argument that takes up passage to the limit, and though it too invokes a classical paradox, it is not the argument of the sorites. Some explanation is in order.

The second sorites of the Pacidius addresses change in a continuous quantity, and thus the increments can be taken to any degree of precision. This means that the exact time and place at which the change occurs can likewise be determined to any degree of precision. Here’s a vivid sort of example. James Cargile in his 1969 paper on vagueness9 imagines a rapid-fi re series of photographs being taken of a tadpole turning into a frog. The sorites argument will force us to admit that for some pair of consecutive pictures, pn and pn+1, the creature depicted in pn is a tadpole but the one depicted in pn+1 is not, no matter how close in time those pictures were taken. “And this argument”, Cargile notes, “is obviously independent of the speed of the camera, which could be taken right up to the theoretic limit of camera speed”10.

If that is a familiar thought about the force of the second sorites, Leibniz’s emphasis on the nature of the “theoretic limit” itself is most uncommon. And of course the limit he is concerned with is the interval of change itself, not merely the limit of a mechanical variable such as camera speed that might be involved in recording that change. Leibniz appears willing to say that the change from not-near to near – or in Cargile’s case, from tadpole to non-tadpole – fi nally oc-

9 J. Cargile: “The Sorites Paradox”, in: British Journal for the Philosophy of Science 20 (1969), pp. 193-202. Reprinted in R. Keefe and P. Smith (eds.): Vagueness: A Reader, Cambridge, MA 1996, pp. 88-98. References are to the reprint.

10 Cargile: “The Sorites Paradox” (see note 9), p. 89.



curs in an instant. Moreover, in that fi nal instant the movable point a becomes near by virtue of the addition of a last part of its motion, a minimum of motion.

The sorites itself doesn’t demand any particular understanding of the nature of passage to the limit. As Leibniz uses it on both occasions in the Pacidius, one supposes at the outset that a certain change occurs, and then the sorites forces one to admit that no unit of measure is too small in principle to effect the change. In the case of the fi rst sorites, or the sorites in its application to discrete quantity, it is shown that poverty can in principle be removed by the addition of any number of pennies, down to a single penny. For the second sorites, or the sorites in its application to continuous quantity, it is shown that the not-near can in principle be made into the near by the addition of any increment of distance however small. Yet how the transaction of change or actual passage to the limit is fi nally to be understood is a further matter and a diffi cult one.

It is no surprise, then, to fi nd that most of the text of Leibniz’s second sorites argument in the Pacidius is in fact addressed to the issue of passage to the limit. Only the fi rst six lines (A VI, 3, 539 line 19 to 540 line 5) do the work of applying the sorites to continuous quantity. The following eight (540 lines 5-16) unlimber a distinct line of argument that concerns passage to the limit. (The two are then sewn together in the conclusion at 540 lines 16-18.) The latter line of argument is no longer the sorites, and it is not simply a matter of the logic of the terms involved that forces us to admit that there are minima in motion. Rather, Leibniz is offering a substantial metaphysical argument about the nature of passage to the limit. We shall be asking what it is and how well it stands up to scrutiny.

4. A Fallacy in the Pacidius Account of Passage to the Limit?

It can be tempting to think that Leibniz’s argument involves a recognizable sort of fallacy. When the movable point a reaches B and becomes near, Leibniz appears to say, there must be some last part of its motion by virtue of which it does so, for at any time prior to a’s reaching B, some of the motion still remains to be executed. But this starts to sound like a mistaken shift of quantifi ers:

(1) At every time prior to the end of the motion, there is a part of the mo-tion that still remains to be executed by virtue of which the movable point a actually reaches the near.

(2) There is a part of the motion that still remains to be executed, at every time prior to the end of the motion, by virtue of which the movable point a actually reaches the near.

An inference from (1) to (2) would not automatically be truth-preserving, and so the argument for a last part of motion would be in error if that is how Leib-niz is reasoning. Further, if Leibniz is committing that sort of fallacy in his second sorites, it would not be without precedent in his writings of this period. One precedent in particular comes to mind when, a few lines before launching


40 Samuel Levey

his two sorites arguments in the Pacidius, he writes of other sorts of change in continuous quantity:

“While water is hot even if it is getting hotter and hotter, there must be some moment when it changes from being not-hot to hot, or vice versa, just as there must be a moment when a line changes from being straight to oblique”11.

The reference to a line’s changing from being straight to oblique at a mo-ment seems to recall an argument Leibniz offers earlier in that same year for the thesis that an infi nite line is immobile, and that earlier argument does seem to commit a quantifi er-shift fallacy. Here it is:

A

B

D

C

F

E

“Let there be a line AB, infi nite in the direction B, which is to be translated by a motion onto AC. Between B and C let there be a line DE parallel to AC. When it reaches AC, it will be completely below DE; but when assigned at any point whatever, for example at AF, there will be an infi nite part of it above DE. Therefore if AB is assumed to be perfectly unbounded, i. e. if there is no last point, it is necessary that the whole of that unbounded line should at the last descend below DE at the same time, and the whole of it would complete the intervening space at the same time, i. e. be in several places at the same time. Ατοπον. [Absurd.] Hence it seems to be proved that an unbounded body is immobile”12.

As the oblique line AB is rotated in its approach to the straight line AC, there is always an infi nite subsection of AB – an infi nite ray starting at the point of intersection of AB and DE – that remains above the line DE, and hence at a fi xed fi nite distance above AC. The oblique line fi nally coincides with the straight line only at the terminal moment of its motion. At every time prior to the terminal moment, an infi nite subsection of AB lies above DE. Leibniz’s worry here is that there will, therefore, be an infi nite subsection of the line AB that lies above DE at every moment prior to the terminal one, and that this remaining subsection will have to leap across the fi nite distance between AC and DE in that last moment in order for the oblique line AB actually to become straight or coincide with AC.

11 A VI, 3, 539.12 “An Infi nite Line is Immobile”, 3 January 1676. A VI, 3, 471. The second occurrence of

‘AB’ in the translation corrects Leibniz’s obviously mistaken ‘AC’.



As Michael White has observed, the conclusion doesn’t follow13. The point of intersection of AB and DE moves progressively off to the right without bound, and though at every time prior to the terminal moment there is an infi nite subsec-tion of AB left above DE, it will be a different subsection of AB – a different ray with a different initial point on DE – that is left above the line at each moment. And as the oblique line AB approaches the straight line AC, each and every ray of AB manages actually to descend below DE at a fi nite time before the end of the motion. In this way, each and every ray of AB is eventually eliminated from having any claim to being left above the line DE at the terminal moment. There is no residual “limit ray” in the motion.

With the fallacy of that argument in mind, we might wonder whether Leibniz isn’t doing the same thing again in the second sorites argument of the Pacidius: inferring from the fact that every moment before the end of the motion leaves some part of the motion unfi nished, to the claim that there is a certain part of the motion left unfi nished at every moment prior to the end, the alleged “last part” of the motion. That verdict would be too quick, however, for the case of the point a changing from being not near to near differs crucially from the case of the motion of the infi nite line.

In the case of the infi nite oblique line becoming straight, Leibniz wrongly supposes that there is a fi nite distance that won’t be crossed by a given subsection of the moving line until the limit of the motion is actually reached. That could only hold true, however, if the series of points of intersection of the oblique line AB with the line DE tended toward a fi xed limit point as the angle between the oblique line and the straight line tended towards zero. And of course that condition doesn’t hold. In fact, the series of points of intersection of AB with DE proceeds off to the right without limit. The series of “rays” or subsections of the infi nite line AB that are formed starting at DE, and hence a fi nite distance above the straight line AC, likewise does not tend towards a “limit ray”. Leib-niz’s mistake here involves supposing that the series of points of intersection, or the series of rays of the line AB bounded by those points of intersection, must eventually converge to a limit; but they do not.

By contrast now, in the case of the moving point becoming near, the motion of the point does tend towards a fi xed limit point: it becomes near at B, the end of the interval traversed by the motion. And, moreover, Leibniz doesn’t repeat the earlier misstep of supposing that there will be some fi nite distance that won’t be crossed by the moving point until the limit of the motion is reached. The spe-cifi c conditions leading to error that were assumed in the argument against the mobility of an infi nite line are not in force in the second sorites of the Pacidius.

All Leibniz argues for in the second sorites is that the motion of the point only actually reaches the near by virtue of its having a last part that actually reaches the limit. The series of initial parts of motion that come closer and closer to the near simply won’t reach the near without this last part, even if they are all

13 Michael J. White: “A Puzzle from Leibniz’s Zettel”, in: History of Philosophy Quarterly 12 (1995), pp. 405-409.


42 Samuel Levey

completely played out. Indeed, the motion of the point “doesn’t make the not-near into the near except by virtue of the last part”. The argument of the second sorites may be in error, but if so, its error is not that of assuming a series to have a limit when it has none; the error, if there is one, still remains to be diagnosed.

To get a better handle on Leibniz’s second sorites, we should not be turning to the fallacious argument of “An Infi nite Line is Immobile”, but rather to an even earlier one Leibniz offered in an effort to prove that there are infi nitesimals in the continuum. In “On Minimum and Maximum”, a document dated to early 1673 or so, Leibniz gives the following argument, again with a diagram:

bcdea

“Let there be a line ab, to be traversed by some motion. Since some beginning [initium] of motion is intelligible in that line, so also will be a beginning of the line traversed by the beginning of the motion. Let this beginning of the line be ac. But it is evident that dc can be cut off [resecari] leaving intact the beginning [salvo initio]. And if ad is believed to be the beginning, from it in turn ed can be cut off [resecabitur] leaving intact the beginning, and so on ad infi nitum. For even if my hand is unable and my soul unwilling to pursue the division to infi nity, it can nevertheless be understood all at once that everything that can be separated [abscindi] leaving intact the be-ginning does not reach the beginning. And since the separation can be done to infi nity – for the continuum, as others have demonstrated, is divisible to infi nity – it follows that the beginning of the line, i. e. that which is traversed in the beginning of the motion, is infi nitely small”14.

This is much the same basic argument that appears in the second sorites in the Pacidius, though inverted to show that there must be a beginning of motion, a fi rst part, rather than an end or last part of motion. The conclusion Leibniz draws here, however, is somewhat different: the beginning of motion is claimed to be infi nitely small, where in the Pacidius the end of motion is only said to be a minimum, something without parts. But we should not be too impressed by that difference.

In the interim between 1673 and late 1676, when the Pacidius is fi nished, Leibniz had come to reject the idea that there could actually be infi nitesimal quantities of any kind15. So it is no surprise that the second sorites argument of the Pacidius claims the existence of a minimal “last part” without saying that it would be infi nitely small. This is not just a change of doctrine: the second sorites improves on this earlier argument by concluding that the terminal part must be a minimum. For consider that even if infi nitesimal parts were possible, it would not automatically follow that such things were indivisible. An infi nitesimal piece of

14 “On Minimum and Maximum, on Bodies and Minds”, 1673. A VI, 3, 98-99.15 See R. Arthur: “Actual Infi nitesimals in Leibniz’s Early Thought”, in: Mark Kulstad, Mogens

Laerke, and David Snyder (eds.): The Philosophy of the Young Leibniz (= Studia Leibnitiana, Sonderheft 35), Stuttgart 2009, pp. 11-28, and S. Levey: “Archimedes, Infi nitesimals and the Law of Continuity: On Leibniz’s Fictionalism”, in: U. Goldenbaum and D. Jesseph (eds.): Infi nitesimals Differences: Controversies between Leibniz and his Contemporaries, Berlin – New York 2008, pp. 107-134.



motion might well be an infi nitesimal line that could be cut into smaller parts, in which case “something could be removed from it while leaving intact whatever produced the nearness”, and thus the infi nitesimal piece “would not make the not-near into the near properly and of its own accord but by virtue of another still smaller part of itself”. The logic of the second sorites argument demands, so to speak, a minimum of motion, independently of its scale. And in fact this is what the logic of the earlier inverted version of the argument actually demands too; Leibniz was drawing the wrong conclusion from his own argument in 1673. By 1676 he has refi ned his understanding.

I said earlier that Leibniz’s additional line of argument in the second sorites of the Pacidius was not the sorites itself, but rather another classical argument. It is the same one that Leibniz advances in 1673 for infi nitely small beginning-parts of motion, and it is the mother of all arguments concerning passage to the limit: Zeno’s dichotomy paradox.

The vagueness of the terms in the second sorites isn’t essential to the issue of passage to the limit, and it will simplify matters to shift from Leibniz’s example of the movable point a becoming near a fi xed point H to Zeno’s more familiar example of a runner’s motion across a mile. In fact, Leibniz’s ‘a’ might just as well be an abbreviation for ‘Achilles’, and we might suppose that a becomes near to H at precisely the one-mile mark, all without affecting the force of the argument. Or, to conform to the puzzle about time from the top of our inquiry, we might equally imagine Achilles counting out the half-intervals of the hour, reaching its end at H. The two are interchangeable here, for it is precisely by motion that time and space are actually divided, in Leibniz’s view.

The second half of Leibniz’s second sorites is the dichotomy, and it is the dichotomy, infused with some metaphysical principles about the nature of motion, that brings Leibniz to the limit. After a brief review of the dichotomy itself, we shall turn to the metaphysical principles that are at work in Leibniz’s argument and thereby bring his early understanding of the nature of time, motion and pas-sage to the limit into sharper focus.

5. Leibniz and the Dichotomy

In its traditional form the dichotomy presents a challenge to the possibility of motion by pointing out that the approach of a runner through the series of half-distances of the mile cannot succeed in actually bringing him to the end of the mile. To traverse the mile, half the distance or 1/2 mile must fi rst be tra-versed, then half of what remains or 1/4 mile, and again half of what remains or 1/8 mile, and so on ad infi nitum. An infi nite series of such distances must be successively traversed if the runner is to reach the end of the mile. Though the runner may pass through any number of the intervals prior to the end of the mile, it is impossible for him actually to reach to the end of the mile. And since the same argument applies, mutatis mutandis, to any distance whatsoever, it is clearly impossible for motion ever to occur.


44 Samuel Levey

In the ancient tradition, the perplexity is understood to arise from the fact that the series of half-distances is infi nite and thus inexhaustible. In order to pass through all of them it appears that the runner would have to complete an actually infi nite series of half-distances one by one – as if to count completely through the infi nite series of numbers and exhaust them all. As we noted earlier, such a feat is impossible according to Aristotle, whose solution to the dichotomy is to hold that the mile does not contain an infi nity of half-distances actually, but only potentially (cf. 263a28 and b3-9), for it is not actually divided into an infi nity of parts but only potentially divisible into parts ad infi nitum. The runner can thus reach the end of the mile without having to exhaust an actual infi nity of half-distances, and the hour can pass without having to contain an actual infi nity of parts.

Leibniz’s view about the infi nite in nature is quite at odds with Aristotle’s. The mile and the hour are not merely potentially infi nite, they are actually infi nitely divided. But it is not thereby impossible for time to pass or motion to occur. The runner traverses an actual infi nity of half-distances in his journey across the mile. Also, the motion of the runner is not merely potentially divisible into parts ad infi nitum, it is actually divided into an infi nity of parts – and since those parts develop one after another, it seems that the motion of the runner is not merely infi nite potentially and “by division” but it is infi nite actually and “by addition”. Likewise for time: the half-intervals of the hour form an actually infi nite suc-cession. Far from being admitted to be impossible, this is entirely the ordinary course of things in Leibniz’s account; as he puts it in 1693 to Simon Foucher, “I am so much in favor of the actual infi nite that instead of admitting that nature abhors it, as is often vulgarly said, I hold that it affects it everywhere”16. When Achilles traverses the mile – i. e. when the movable point a traverses the interval to B, and there fi nally become near to H – he will pass through an actual infi nity of actually distinguished subintervals.

For Leibniz the diffi culty, and interest, of the dichotomy lies elsewhere. When he introduces it, both in 1673, in “On Minimum and Maximum”, and in 1676, in the Pacidius, Leibniz conducts his discussion under the hypothesis that the end of the interval is reached. Suppose Achilles runs right to the end of the mile. What must be true about his motion in order for this to have occurred? In particular, what about his motion makes it the case that he actually reaches the end of the mile and does not simply approach it? Leibniz’s answer is that mo-tion must contain an indivisible last part by virtue of which Achilles passes to the limit, actually reaching the end of the mile.

6. Some Principles of Passage to the Limit

Leibniz’s thoughts about passage to the limit are given slightly different expression in the two pieces in which he unlimbers the dichotomy, and the contrast is instructive.

16 GP I, 416.



In the 1673 document his focus is on the parts of motion and their role in effecting the passage to the limit. This remark in particular is illuminating:

“It can be understood all at once that everything that can be separated [abscindi] leaving intact the beginning [salvo initio] does not reach the beginning”17.

And that naturally suggests the following principle:

(*) Everything that can be separated salvo initio does not reach the begin-ning.

‘Separated’ here translates ‘abscindi’ and should be understood to mean “sepa-rated off” rather than, say, “separated into parts”. (‘Abscindi’ is roughly equivalent to ‘cut off’ or ‘cut loose’.) The sense in which something “can be separated” from the beginning of the motion is not entirely the ordinary one. For those parts that can be separated leaving intact the beginning cannot, say, be prized apart from it or moved to another location. Nor are even those parts of the motion that are far down the line from the beginning “separable” from earlier parts of the motion in the sense of being independent of them: presumably the later parts of the motion would not even exist if the earlier parts had not existed. It suffi ces for a part of motion x to be separable salvo initio that it be disjoint from the beginning, i. e. that there be at least some part of the motion, y, that contains the beginning but which does not “overlap” x, or has no part in common with x.

In Leibniz’s 1673 example of the motion ab (from a to b), the second half of the motion, cb, can be separated salvo initio and thus does not reach the begin-ning. Presumably the same general point applies to the far end of the motion as well. Everything that can be separated leaving intact the end or the limit, salvo termino, does not reach the limit. For instance, the fi rst half of the motion, ac, can be separated salvo termino and so does not reach the end. The generalized version of principle (*) can be codifi ed as:

(0) Everything that can be separated salvo termino does not reach the limit.

Principle (0) at fi rst looks straightforward. None of the fi rst ninety-nine parts of Achilles’ motion reaches the end of the mile, since each can be separated leaving the end intact in the hundredth part. But in fact the universal quantifi er in (0) needs to be given a fairly strong reading in order to secure Leibniz’s result – what Russell and Leibniz would call a “collective” reading. Here is why.

The cutting edge of the 1673 dichotomy is carried by the understanding of what falls under the description ‘everything that can be separated salvo termino’. Roughly speaking, Leibniz holds that all of those things, each of which can be separated salvo termino, can all at once be separated salvo termino. Thus Leib-niz’s principle tells us that all the separable parts, even taken collectively, do not reach the limit; we can see that this is what is intended when he writes: “it can be understood all at once that everything that can be separated leaving intact the beginning does not reach the beginning [potest tamen semel in universum intel-

17 A VI, 3, 99.


46 Samuel Levey

legi, id omne ad initium non pertinere, quod salvo initio abscindi potest]”. This strong principle will allow us to conclude, for instance, that the entire infi nite series of half-distances, 1/2 + 1/4 + 1/8 + 1/6 + &c. ad infi nitum, fails to reach the end of the mile. And this is indeed what Leibniz’s 1673 dichotomy claims.

Further premises are needed to complete the argument for a minimum in motion, and they are not too hard to reconstruct given the way Leibniz handles the diagram in his discussion. For instance, any proper initial part of the motion will count as being separable salvo termino, and so on. But for now I am go-ing to leave the details aside and assume we can see how the 1673 dichotomy is supposed to work. As Achilles traverses the series of half-distances, he is by his motion essentially discharging successive proper initial parts of the remain-ing “halves” of the mile. Each half-distance he crosses can be separated salvo termino, precisely because the end of the mile continues to lie ahead. And if Achilles’ motion is indeed infi nitely divisible in the traditional sense, then there are infi nitely many such proper initial parts, each of which can be separated salvo termino. Still, there is a remainder of motion never captured in the series of half-distances. Only an indivisible motion that belongs to none of those half-distances can fi nally close the gap to the end of the mile.

Leibniz’s discussion in the 1676 dichotomy gives us more explicit material to work with, so we’ll focus a bit more closely on it. Two lines of the discussion express an important idea concerning the way motion can reach the end of an interval:

(a) If it is the hundredth part of the inch that makes the near into the not-near, then it is not the whole inch which does this18.

(b) The gaining of an inch makes the not-near into the near only because it contains the last hundredth of an inch19.

These principles concern explanatory or ontological priority in passage to the limit, and priority is falling here to the parts of motion by which the limit is actu-ally attained. The whole of the motion, which also contains the fi rst ninety-nine parts “that have not yet made the point near”, does not make the not-near into the near; it is not what carries Achilles to the end of the mile; it does not reach the limit. Or rather, it does so only secondarily, by way of containing the part or parts that are primarily responsible for reaching the limit.

This priority of part to whole is signifi cant not only for the argument in favor of the existence of a minimal last part of motion, but also for exposing one aspect of Leibniz’s underlying view of the nature of motion. Insofar as the parts are prior to the whole in bringing about the passage to the limit, motion is, metaphysically, a discrete quantity rather than a continuous one (cf. A VI, 3, 502).

The picture is fi lled out further in Leibniz’s little reductio argument, in closing, for the claim that the “last part” of motion must be a minimum. If the

18 A VI, 3, 540, lines 6-7.19 A VI, 3, 540, lines 9-10.



motion that fi nally makes the not-near into the near were not a minimum (i. e. without parts),

“[…] then something could be removed [rescindi] from it leaving intact whatever produced the nearness. For suppose the hundredth part of CB is not a minimum but a straight line DB: this line would not make the near into the not-near of its own accord [per se], but by virtue of some other still smaller part of itself, EB”20.

Any piece of motion from which “something could be removed” leaving intact the part that “produces the nearness” does not itself make the not-near into the near – it does not reach the limit – per se, of its own accord. Rather, it does so only in virtue of the still smaller part of itself that produces the near-ness or reaches the limit. The qualifi er ‘per se’ is expanded in the argument’s concluding lines:

“Therefore we have that either there is no way for something to become near properly and of its own accord [proprie ac per se], or something turns from being near to not-near by the addition or subtraction of a minimum, so that there are minima in reality”21.

The force of ‘proprie’ is diminished somewhat by its translation in this context as ‘properly’. It typically means something like “personally” or “in one’s own name”. The qualifi er ‘proprie ac per se’ puts a very fi ne point on the way in which the part of the motion that produces the nearness does so. It does so personally, not just by means of some intermediary. No motion from which something can be removed leaving intact whatever actually reaches the limit can do this, for such a motion merely dispatches “some other still smaller part of itself” to see to it that the limit is reached. Or rather, since it is the “still smaller part” that has priority in bringing it about that the limit is actually reached, the larger whole can claim credit for the limit’s being reached only secondarily or – now to introduce the traditional contrast – per accidens.

Though Leibniz does not quite furnish a direct thesis about passage to the limit prior to his conclusion in the 1676 dichotomy, a few principles seem to fall naturally out of his discussion. In particular, in the last few remarks quoted from Leibniz’s text there appears to be the following link conceived between separable parts of motion and passage to the limit:

(1) No motion from which something can be separated salvo termino reaches the limit properly and of its own accord.

I take it that Leibniz does not mean to conclude only that minima are necessary for motion to reach the limit “properly and of its own accord”, though that is all he explicitly mentions in the concluding lines of his argument. The rest of the dialogue evidently proceeds on the understanding that minima are neces-sary for motion to reach its limit tout court. What is needed here to fi ll the gap is a principle to the effect that no motion can reach the limit unless some part of it does so properly and of its own accord. Such a principle would appear to

20 A VI, 3, 539.21 A VI, 3, 540.


48 Samuel Levey

be in keeping with Leibniz’s expressed views about the priority of the parts to the whole in bringing about passage to the limit. The whole does not reach the limit except by virtue of containing a part that does; and any part of motion from which something can be separated salvo termino can reach the limit only secondarily. In order for this secondary means of reaching the limit even to be possible, it might now be suggested, there must be something contained in the motion that reaches the limit primarily. Presumably Leibniz’s position will be that this is achieved only by something that reaches the limit proprie ac per se. This idea can be articulated as a priority thesis about the composition of motion:

(2) No motion reaches the limit except by virtue of containing something that reaches the limit properly and of its own accord.

From (1) and (2) there results a principle strong enough to ensure that in order for passage to the limit to occur motion will require the existence of minima:

(3) No motion reaches the limit except by virtue of containing something from which nothing can be separated salvo termino.

Only a minimum situated at the limit of the motion can satisfy the condition ‘is something from which nothing can be separated salvo termino.’ And thus the argument of Leibniz’s 1676 dichotomy, or the second half of the second sorites in the Pacidius, is complete. There must be minimal “last parts” contained in motion.

7. A Mathematical Fallacy in the Dichotomy?

Leibniz’s two dichotomy arguments for the existence of a minimal end of motion differ subtly in the metaphysical principles upon which they rely, but for now let us treat them as connected on a single important point. Both dichotomies hold that after all the initial parts of motion have been exhausted, there still has to be an indivisible remainder of the motion in order for motion to reach the limit. Could there be motion without the minimal remainder? Could a motion reach the end even if it contains no minimal “last part” by virtue of which it passes to the limit?

During the twentieth century a commonplace view of Zeno’s dichotomy paradox was that its solution had been delivered by contemporary mathematics, and in particular by the theory of infi nite convergent series. The series of half-distances in the mile exhaustively accounts for the mile itself, since the sum of the infi nite series 1/2 + 1/4 + 1/8 + 1/16 + &c. ad infi nitum is equal to 1. The mile itself is therefore equal to the sum total of the fi rst 1/2 mile and the next 1/4 mile and the next 1/8 mile, &c., and by traversing through each of the half-distances in succession, the runner covers a distance of one mile.

This proposed resolution of the dichotomy immediately suggests a reply to Leibniz’s argument for the existence of minima in motion. Just as the infi nite convergent series contains no last element and yet exhausts the mile, Achilles’



motion across the mile need have no last part – minimal or otherwise – to reach the end of the mile. His motion might be completely understood as nothing more than an infi nite series of parts none of which reaches the end per se, and each of which can be separated from the end salvo termino. Indeed, his motion to the end of the mile could be understood as a convergent series of partially overlapping proper initial parts of the motion: the motion to the 1/2 mile, the motion to the 3/4 mile, to the 7/8 mile, to the 15/16, &c. ad infi nitum. Taken together, those partial motions – each separable salvo termino – complete Achilles’ motion to the end of the mile, with no minimal remainder necessary to achieve passage to the limit.

There would be some irony in this reply to the dichotomy as well, for Leibniz above all should have been able to see it in advance, having discovered the theory of infi nite convergent series in the fi rst place22. Just months before writing the Pacidius, Leibniz remarks in the document “Infi nite Numbers”:

“Whenever it is said that a certain infi nite series of numbers has a sum, I am of the opinion that all that is being said is that any fi nite series with the same rule has a sum, and that the er-ror always diminishes as the series increases, so that it becomes as small as we would like”23.

This perfectly anticipates Cauchy’s defi nition of the sum of an infi nite con-vergent series as the limit of its partial sums. Cauchy’s criterion of convergence makes fully explicit what appears without a precise analysis in Leibniz’s remarks. The series a1, a2, a3, …, an, … converges to the limiting value L if and only if, for any positive rational number ε, there is a neighborhood of terms surround-ing L, constituting an interval of length ε, that contains more elements of the sequence than lie outside it. Only fi nitely many elements of the series an could lie outside any such neighborhood of L. The limit of the series an, as n → ∞, is L if and only if | L – an | is less than ε for all n suffi ciently large: n ≥ N. In Leibniz’s words, “The error always diminishes as the series increases, so that it would become as small as we would like”.

Later in “Infi nite Numbers” we fi nd Leibniz stating, with full clarity of mind about the signifi cance of his remark, that “in fact there is no last number in the [infi nite numerical] series, since it is unbounded” (A VI, 3, 503). Expecting the rejoinder that an infi nite numerical series will eventually run into a fi nal infi nite or infi nitesimal term, he further notes: “Thus if you say that in an unbounded series there exists no last fi nite number that can be written in, although there can exist an infi nite one: I reply, not even this can exist, if there is no last number”. And fi nally,

“[I]t is clear that even if fi nite numbers are increased ad infi nitum, they never…reach infi nity. This consideration is extremely subtle”24.

22 For discussion and references see S. Levey: “Leibniz on Mathematics and the Actually Infi nite Division of Matter”, in: The Philosophical Review 107 (1998), pp. 49-96.

23 A VI, 3, 503.24 A VI, 3, 504.


50 Samuel Levey

These were no passing insights. Two and a half decades later, Leibniz puts the same points to Bernoulli in their dispute about the nature of infi nite numeri-cal series, with Zeno’s dichotomy series openly in discussion:

“Let us suppose that in a line its 1/2, 1/4, 1/8, 1/16, 1/32, etc., are actually assigned, and that all the terms of this series actually exist. You infer from this that there also exists an infi nitieth term. I, on the other hand, think that nothing follows from this other than that there actually exists any assignable fi nite fraction, however small you please”25.

With this grasp of the infi nite series as “unbounded” in the sense that it has no last term, it seems that Leibniz should have had little diffi culty in seeing that the dichotomy paradox might be resolved simply by an application of the theory of infi nite convergent series without further postulating any last term or last part of motion. When he instead insists in the Pacidius that there must be a minimal last part of motion in order for passage to the limit to be achieved, it would then appear that he has lost sight of his earlier theoretical advance in mathematics, or at least failed to grasp its signifi cance for understanding the nature of motion. The dichotomy then awaits its proper resolution for another two hundred years, when in the nineteenth-century revolution in mathematics Cauchy rediscovers the theory of infi nite convergent series.

The ironic history of the dichotomy paradox is a false history, however, for in fact the theory of infi nite convergent series has never provided a satisfactory response to the dichotomy. And it is precisely in Leibniz’s handling of the di-chotomy that this becomes most evident. Nowhere in his dichotomy arguments does Leibniz deny, for example, that the sum of the series 1/2 + 1/4 + 1/8 + 1/6 + &c. ad infi nitum is equal to 126. Neither does Zeno. The dichotomy does not address whether the measure of the mile is exhausted by the series of its half-distances. The question, as Leibniz sees it, is whether the series of half-distances reaches the end of the mile. It is open to Leibniz to allow the fi rst while denying the second. For the question about the metrical properties of the mile is distinct from the question about its topological properties, and it is perfectly consistent to hold that the series of half-distances “add up” to the same length as the length of the mile but that the space covered by those half-distances fails to include the end of the mile. Indeed, this seems to be exactly what contemporary mathematics tells us. The sum of the series 1/2 + 1/4 + 1/8 + 1/16 + &c. ad infi nitum is 1, because 1 is the limit of the partial sums 1/2, 3/4, 7/8, 15/16, &c. But the space covered by the union of the intervals [0, 1/2], [1/2, 3/4], [3/4, 7/8], [7/8, 15/16], &c., is not

25 GM III, 535-536.26 In fact, it is a part of Leibniz’s philosophy of mathematics that such an equation as ‘1 = 1/2

+ 1/4 + 1/8 + 1/6 + &c. ad infi nitum’ is not rigorously true, since in his view the infi nitary expression on the right-hand side is not admissible, strictly speaking. Such devices are only useful abbreviations. Again, for discussion see Levey: “Leibniz on Mathematics” (see note 22). But this element of his philosophy of mathematics is independent of his dichotomy arguments and is not relied on in them. Even if he were to accept the equation as rigorously true, it still would not follow that he must hold the motion across the series of half-distances actually to reach the limit.



the “closed” interval [0, 1] – that is, the interval from 0 to 1 including the termini at 0 and 1 – but rather the “open” (or “half-open”) interval [0, 1), which does not include the terminus at 127. The open interval and the closed interval are metrically indiscernible: their lengths are assigned the same measure, in this case one mile. But topologically they differ in the crucial respect that only the closed interval actually contains the limit towards which the half-intervals approach. The open interval does not. Motion that completely covers a closed interval actually passes to the limit of that interval; motion that covers only an open interval does not.

If Achilles’ motion contains only the series of half-distances of the mile, he will never reach the end of the mile by means of it. If somehow he should pass to the limit and complete the interval [0, 1], his journey will be extended only by a single point beyond [0, 1), and this will not increase the measure of his motion. The limit point itself is metrically superfl uous. Yet it is not thereby inessential to the possibility of motion. Covering the open interval is by no means “close enough” to reaching the end of the mile. If Achilles is admitted not to be able to pass to the limit of the closed interval [0, 1], it will follow that he must likewise be admitted not to be able cover the open interval [0, 1), and thus not to be able to complete the mile in any sense, metrical or topological. For in order to cover the open interval [0, 1) he must be able to cover the prior closed interval [0, 1/2] along the way, but now, mutatis mutandis, the dichotomy will show that he cannot cover that closed interval either. The result is perfectly general. There can be no motion across any distance that does not include at least some motion across a closed interval. A motion that, per impossibile, included only a series of half-distances across an interval must fail to reach the limit – the theory of infi nite convergent series notwithstanding. And failure to reach the limit implies failure to achieve any motion at all.

Leibniz does not articulate the distinction between metrical and topological features of the interval of motion as cleanly as we just have. Still, it is clear that his dichotomy arguments in fact address the topology of motion: they are conducted in terms of intervals of motion, their parts and their limits. This seems generally to be true of the dichotomy paradox. It is striking that in the actual history of the dichotomy, the concern about the possibility of motion across the infi nite series of half-distances seems never really to have been whether the sum of the lengths of the intervals in the series is equal to the length of the mile. Or rather, only in the twentieth century was this raised as a concern, and virtually always under the name of identifying a simple mathematical fallacy in the paradox; calculus textbooks especially have seemed enamored of attacking a straw dichotomy that denies that an infi nite series of fi nite magnitudes could have a fi nite sum as a segue to the theory of infi nite convergent series, which could then be introduced as the “solution”.

27 It is not essential that the intervals chosen here be closed intervals; the same result follows even if we take open intervals, throwing the successive endpoints either to the right, so that the series is [0, 1/2), [1/2, 3/4), [3/4, 7/8), &c., or to the left, so that the series is [0, 1/2], (1/2, 3/4], (3/4, 7/8], &c. Both series will only cover the space [0, 1), which is open at 1, and thus fail to cover the closed interval [0, 1].


52 Samuel Levey

Instead, the dichotomy’s point of interest has been whether motion could reach the end of the mile by passing through the infi nite series of half-distances. Zeno’s view, presumably, was that the infi nite series cannot ever be completed, and thus that the end of the mile cannot be reached. Aristotle’s more refi ned view was that an actually infi nite series cannot be completed, though motion might still be possible on the hypothesis that the mile is by nature only potentially infi nite and does not contain an actual infi nity of half-distances to be traversed. Leibniz in his own writings pushes the dialectic further by admitting the actual infi nite into the nature of motion, while yet recognizing that even traversing through each and every half-distance in the infi nite series still is not suffi cient to guarantee that the limit will be reached. Passage to the limit requires something still more of motion.

In his classic 1964 study, José A. Benardete arrives at a view quite close to Leibniz’s:

“Zeno’s misgivings are now seen to be vindicated in an unexpected way. Achilles undertakes to travel from the earth to the moon in one minute. During the fi rst half minute he travels half the distance, during the next 1/4 minute he travels the next quarter of the distance, &c. ad infi nitum. Assuming that he actually exhausts the entire infi nite series of rational intervals, must he then be found on the moon at the end of the minute? Not necessarily. It is quite possible that at the end of the minute Achilles will be found on the earth or Mars or perhaps nowhere at all. He will indeed have satisfi ed Cauchy’s criterion by penetrating into every rational neighborhood of the moon, but he may well have failed actually to touch the surface of the moon at any time. Merely because one succeeds in actually approaching arbitrarily close to the moon, does not in and of itself logically entail that he will ever reach it. If we are to guarantee the success of Achilles in actually touching the moon, we must presuppose some kind of postulate of continuity”28.

A postulate of continuity would guarantee contact with the moon, presum-ably, since by any usual mathematical defi nition a continuous motion will include all its limits and thus be “complete in itself”. The postulate Benardete has in mind would not just be a mathematical assumption, though, but a metaphysical principle that makes the mathematical description true of Achilles’ motion and explains why it is true. What metaphysical principle could that be?

I am not aware of Leibniz addressing this question. In one way, of course, he proposes an answer: motion includes minimal ends (and beginnings). These minima are exactly the elements of motion that constitute it as continuous in the sense required by the postulate of continuity. Still, this answer has not really provided the explanation that is needed. For even if Leibniz’s arguments – sup-posing them successful – show that there must be minimal elements in motion to negotiate passage to the limit, they do not yet amount to a philosophy of mo-tion that resolves the dichotomy paradox. What they show is only the fact that there must be a minimal end of motion. For, otherwise, Achilles’ motion could not cover the whole mile or any other interval. Leibniz supposes that motion is possible and then stipulates that it must have the topological feature required for it to pass to the limit: the minimal end. But we still do not know why motion has that structure. We should want to understand the nature of motion in a way that illuminates, from within, the reason why motion is so constituted as to carry the

28 J. Benardete: Infi nity: An Essay in Metaphysics, Oxford 1964, pp. 277-278.



elements needed to transact passage to the limit. If, for example, we had some way of understanding motion as being complete in itself so that it was always guaranteed to contain its limits, we might then have insight of the sort we need to see why it satisfi es a postulate of continuity. Perhaps a theory that pressed the ancient distinction between kinesis and energeia, or exploited the idea of entelechia, could try to provide a metaphysical explanation of the topology of motion as a quantity complete in itself. Certainly such concepts are familiar to Leibniz. So far this sort of account is one that we have not found in his work. But with those few hints I must leave this subject to others and turn now to the question of the compatibility Leibniz’s analysis of motion and passage to the limit in the Pacidius and his view of time, the last part of our journey.

8. Time and the Ends of Motion: Limits of the Argument

In defending what we might call Leibniz’s ‘dichotomy argument’ for minimal elements in motion, I earlier relied on the distinction between open and closed intervals. Approaching the end of the mile by penetrating into every rational interval containing the endpoint of the mile is not “close enough” to reaching the end, because without an assurance that the end of the mile itself is reached, we cannot safely suppose that Achilles traverses the half-mile either. So the motion must reach the end itself and not merely approach it without bound. But the distinction between open and closed intervals assumes the existence of minimal elements, namely, points in space or moments in time. One can reason-ably object now that this assumption is too strong if it is to be part of the defense of Leibniz’s argument for minimal elements in motion. If space and time are assumed to have minimal elements, surely it is no surprise that motion too will then turn out to require them if the mile is to be traversed. The argument for the ends of motion then depends on a premise that is nearly as strong all by itself. What justifi es the premise?

In the Pacidius the existence of moments in time and points in space is taken for granted. This is not a strange view to adopt. Nor is the assumption strange that space and time are continuous – though, for reasons that will not detain us here, the exact character of space and time as continuous quantities is subject to fairly radical reconfi guration in the dialogue. (The outcome of that reconfi gura-tion does not affect whether space or time can be described as continuous in the sense required for the application of mathematical formulae to the natural world: the language of the calculus, with its assumptions of continuity, can still safely be used to describe the motions of bodies29.) These are assumptions standardly made. To the extent that they are safe, the assumption of minimal elements in space and time and the assumption of the continuity of space and time – so that

29 For the discussion see S. Levey: “The Interval of Motion in Leibniz’s Pacidius Philalethi”, in: Noûs 37 (2003), pp. 371-416, and S. Levey: “Leibniz on Precise Shapes and the Corporeal World”, in: D. Rutherford and J. Cover (eds.): Leibniz: Nature and Freedom, New York – Oxford 2005, pp. 69-94.


54 Samuel Levey

miles and hours and other such intervals must have topological ends and not merely measures – will leave Leibniz’s argument for minimal elements in mo-tion in good stead. Yet these assumptions are not without metaphysical weight and philosophers may be skeptical of making them.

It is quite notable in Leibniz’s case, then, that elsewhere in his philosophy he is at pains to say that space and time, conceived as mathematically continu-ous quantities, are not “real” quantities but “ideal” ones, beings of reason that are not genuine components of the natural world30. The argument that Achilles’ motion across the mile must contain a minimal element in order to reach the end of the mile presupposes, it seems, that the mile itself is there, with all its topological features, waiting to be traversed. It is not just the idea that somehow there must be a boundary separating vaguely delineated intervals – the near and the not-near – that is troublesome, but, given Leibniz’s own views, the very idea that intervals of space and time exist as delineated entities at all.

Of course Leibniz does not deny that there are spatial or temporal intervals. Famously Leibniz defends a reductive account of space and time in terms of relations among individual objects or features of them31. Space is the “order of co-existences”, and time the “order of successions”32. Spatial and temporal relations hold among actual objects, and taken all together these constitute the spaces and times of the actual world. One might call this system of relations ‘real’ space and time. When further taken together with the system of all possible such relations – that is, as conceived among actual and possible objects – and ignoring the distinction between the actual and the possible, the abstract result could be called ‘ideal’ space and time, understood as mathematically continuous quanti-ties of the sort studied in geometry but not perfectly instantiated in nature itself. Real spaces and times, by contrast, are discrete quantities that approximate to continuity without being continuous properly speaking, with boundaries of actual individuals concretely marked out in specifi c ways. As Leibniz writes to Clarke:

“The parts of time and place considered in themselves are ideal things, and therefore they per-fectly resemble each other as abstract units. But it is not so with two concrete ones, or with two real times, or two spaces fi lled up, that is, truly actual”33.

Thus in nature there is a real spatial interval between Achilles’ starting line and, say, the surface of the moon. The abstract description of Achilles as travers-ing a mile can only be shorthand for his traversing some particular distance from one given point to another, each defi ned by the boundary of some actually exist-

30 For the discussion, see G. Hartz and J. Cover: “Space and Time the Leibnizian Metaphysic”, in: Noûs 22 (1988), pp. 493-519.

31 There is an extensive literature on the topic. For a few sources, see R. Arthur: “Leibniz’s Theory of Time”, in: K. Okruhlik and J. Brown (eds.): The Natural Philosophy of Leibniz, Dordrecht 1985; J. Cover: “Relations and Reduction in Leibniz”, in: Pacifi c Philosophical Quarterly 70 (1989), pp. 185-211; and M. Futch: Leibniz’s Metaphysics of Time and Space (= Boston Studies in the Philosophy of Science 258), New York 2008, chap. 2.

32 GP VII, 363/LC 3.4. 33 GP II, 395/LC 5.27. Compare also GP II, 278-279.



ing “real” being. This perhaps gives purchase to the defense of his argument that distinguishes open from closed intervals: if Achilles does not touch the surface of the moon, he does not complete the journey, does not cover the interval; and likewise for the mile, the half-mile, and so on. Thus motion requires reaching the ends of intervals, and, by Leibniz’s analysis, reaching the ends requires minimal parts of motion by which to do so proprie ac per se.

This suggests, though, that the dichotomy argument for minima might not work if the interval to be traversed by motion does not have a pre-assigned end. Suppose Achilles endeavors to run a mile into empty space. With no fi nishing line, must we say his motion is made complete by a minimal element of motion? Or might it cover a mile yet have no end? Presumably Leibniz can resist this by denying the coherence of the case. There is no such thing as a mile of empty space in the absolute sense of a distance without terminal bodies to stand as the relata of spatial relations. Space, conceived as a free-standing entity, is only an ideal being, an entia rationis, and not something to be traversed by motion. If Achilles succeeds in running a mile into empty space this can only mean that at some time his body, or some boundary of it, comes to be a mile from his starting place. The motion by which he carried himself along fi lls this interval completely, and so has defi ned boundaries, ends, of its own. But prior to his run, there is no actual pre-existing bounded spatial interval awaiting him. To describe the yet-to-be-run mile in empty space is only to describe certain possibilities for Achilles’ movements, not an actually existing “real” being.

Perhaps the foregoing refl ections indicate how Leibniz’s idealism about space might be reconciled with his argument for the existence of minimal ends of mo-tion. Intervals of space need not be understood as real beings independently of the bodies that defi ne them or the motions that traverse them, in order to deploy the dichotomy paradox to argue for minima in the composition of motion. But if so, the parallel stratagem used for the passage of the hour may precipitate more radical consequences. In order for an hour to pass, it will be necessary for half of it to pass fi rst, and then half the remainder, then half of what remains of this remainder and so on. Now if one subtracts half from the remainder in this way indefi nitely, one will never be able to reach the end of the hour – unless, following Leibniz, the hour contains a fi nal, minimal moment. If one subtracts that too, in addition to all the half-intervals, then the end of the hour is reached after all. Yet what is this minimal moment, the “end of the hour”? Again we must not lapse into thinking of the hour as an independently existing real being with its own given boundaries. The hour must be reduced to relations among other, actually existing things. Moments would then seem to be the beginnings or ends of temporally extended things – say, actions, motions, or other time-fi lling processes. The passage of time taken purely in the abstract is nothing at all. Actual intervals of time such as a given hour or minute measure particular occurrences in the natural world, changes of things from one state to another.

Unlike Achilles’ journey to the moon, however, it not clear what pre-existing boundaries the passage of an hour may link together. Of course applied to events that have already played out, one can readily identify the passage of an hour


56 Samuel Levey

from the beginning to the end of a fi nished journey. For time passing now, on the other hand, what temporal boundary is it that must be reached in order for the hour to pass? It seems we are pressed to posit some future boundary – a change in things, the beginning or end of some enduring state – that does not yet exist, as if the hour needs to have not merely some last moment it eventually reaches but some telos towards which it strives. Even with Achilles’ run into empty space we could supply the real end-marker of his mile-long motion, some part or boundary of his body at a given time. The passage of the hour into an “empty” future, however, provides no obvious counterpart.

We do not remain entirely without recourse; there is a more extraordinary measure one might still take. If we equate the past and future with the present as existing domains and populate them throughout with objects or events, as if they were simply neighboring regions of a single country, it will be easy enough to supply boundaries for intervals of time: the beginnings and ends of those objects or events themselves could serve as the markers for temporal intervals, whether they fall in the past, present or future. How drastic a step would this be?

One element of this proposal is clearly at home in Leibniz’s own account. Leibniz holds that every change of state supplies the end of one temporal inter-val and the beginning of another, and he argues explicitly in the Pacidius that change always involves a pair of immediately neighboring or “indistant” moments serving as the boundaries of temporal intervals containing opposite states34. When someone dies, there is no single “middle” moment of dying that belongs commonly to life and to death, but in fact the act of dying is composed of two adjacent moments: the last of life, and the fi rst of death35. So each such change induces, as it were, temporal boundaries, beginnings and ends of intervals of time, into nature. More doubtful, however, is the other element of the proposed account: the equation of past and future times to the present. For to posit the existence of a domain of changes across all times – not merely an ontology that says that there is always change but one that says there are such things as past and future changes whose existence must be recognized in addition to changes occurring in the present – in order to guarantee the ubiquity of temporal minima, becomes a massive metaphysical commitment, and not something easily counted as simply a consequence of the idea that the hour passes.

Might it yet be a commitment Leibniz is willing to embrace? Certainly many philosophers have found attractive an “eternalist” view of time, i. e. one in which any objects and events that should ever exist at any time, past, present or future, have an equal claim to existence at all times, even times which they do not “occupy” or which happen not to be present. And it is a natural analog to the “possibilist” view of modality that countenances possible worlds constituted by “possible individuals” similar to what Leibniz envisions. But Leibniz himself

34 A VI, 3, 541 and 557.35 A VI, 3, 535-536. Compare A VI, 4 A, 568-569 and 869. For the discussion, see Levey:

“The Interval of Motion” (see note 29).



may actually incline away from eternalism about time. There are at least two reasons to suspect this.

First, in some texts Leibniz claims that time and duration themselves do not even exist apart from the present; for instance, in 1716 he writes in his fi fth letter to Clarke, “Whatever exists of time and duration, being successive, per-ish continually”36. The metaphor of “perishing continually” certainly suggests that the present moment is the exclusive domain of reality, and it would seem to rule out the existence of non-present past or future beings along with those non-present times to which they belong. (For good measure, it should also be remembered that Leibniz likewise tends to describe merely possible individuals as not existing; part of his defense of the contingency of the actual world is that “not all possibles exist”37. The now-familiar, subtle distinction between actual-ity and existence is not quite perfected in his texts. For Leibniz actual things are created individuals whereas possible things are only concepts or essences or ideas in the mind of God rather than real individuals of the same kind as actual ones, just existing in some distinct logical space38. His embrace of possible worlds is thus not a paradigmatic case of “possibilism” that puts all actual and possible things on equal footing, and so too it is not quite a perfect analog of eternalism about time.)

Secondly, eternalism is not a natural ally for the idea that time passes; rather, eternalism is often paired with a debunking attitude toward temporal passage according to which it is only a subjective phenomenon: the asymmetry of in-formation individuals experience at any given time about prior and posterior times and the relative differences in that information across subsequent times; a function of consciousness, memory and ignorance39. Again, Leibniz’s metaphor of time “perishing continually” gestures in the opposite direction, stressing the passage of time itself. Other texts also seem without irony to underscore the idea of temporal passage and a real difference between past, present and future, for example when Leibniz writes:

“Whatever is future will be present, whatever is present will be past, whatever is past will always be past, whatever is future up to this time will be future, but not always, whatever is past was future, and whatever is past was present”40.

As noted at the outset, the idea of passage connects time and motion in a way that distinguishes them from space. To the extent that eternalism tends to “spatialize” time by making its parts all equal in terms of their mode of exist-

36 GP VII, 402/LC 5.49.37 Grua I, 263.38 Compare GP II, 45 and 55. 39 For one example among many in the literature, see D. Mellor: Real Time II, London – New

York 1998, p. 122. For a different approach to explaining the appearance of passage while denying its reality, see S. Prosser: “Why Does Time Seem to Pass?”, in: Philosophy and Phenomenological Research 85 (2012), pp. 92-116.

40 A VI, 4 A, 908. Translated in Futch: Leibniz’s Metaphysics of Time and Space (see note 31), p. 89.


58 Samuel Levey

ence, it will seem a less natural fi t with a philosophy that looks to hold out a special privilege for the present and to embrace the reality of temporal passage.

The texts are not defi nitive on the point, however, but only suggestive. In some places, particularly when promoting the idea of possible worlds, Leibniz writes in ways that suggest an eternal standpoint over the whole of history as a single domain, for instance in section 8 of the Theodicy: “I call ‘World’ the whole succession and whole agglomeration of existent things”41. When taking the totality of existent things into view as a single world to explore the concepts of modality, a god-like eternal perspective is a natural one to adopt; how far this ramifi es into a thorough-going metaphysics of time for Leibniz, however, is at best a further question. Leibniz’s degree of sympathy or antipathy toward eternalism is thus debatable42. (Likewise purely for the arguments themselves: eternalism does not automatically contradict the reality of temporal passage or an objective distinction between past, present and future, even if its most natural form rejects them.) We need not overstate the case. It is enough to note that it is not clear whether Leibniz has a considered view of the subject, nor, in particular, one that comfortably accommodates an eternalist account of time. So to introduce eternalism as an interpretative device for reconciling his reductiv-ist view of time with his dichotomy argument for minima imputes to Leibniz’s philosophy a major doctrine that does not obviously belong to it.

We should also take care to note that the dichotomy argument of the Pacidius belongs to quite an early period in Leibniz’s writings, whereas the reductivism familiar in the Leibniz-Clarke correspondence is a distinguished feature of much later one; roughly forty years separates the canonical texts on these subjects, and the tension between them might not face Leibniz’s philosophical thought as it was constituted at any one time. We shall remain agnostic on this matter, and so no settled verdict of inconsistency in Leibniz’s own position is to be in the offi ng here. Still, the conceptual issue is worth considering in its own right.

The question before us is whether Leibniz’s dichotomy argument for minima is compatible with his reductive view of time. If all times exist equally and are populated with changing objects and events, it then seems there are also pre-assigned boundaries available in nature to mark the beginnings and ends of temporal intervals. Without that metaphysical assumption, and considering a time passing now, it is unclear whether anything will be available to stand as the boundary that an interval of time must reach in order for an hour to pass. Without a limit to be reached, Leibniz’s dichotomy argument for minima seems to lack a crucial premise. For in this case the series of half-intervals might be removed without thereby “leaving intact the end”; or at least we would have lost Leibniz’s ground for saying the whole of the hour must fail to pass if it does not contain

41 G. W. Leibniz: Theodicy. Essays on the Goodness of God, the Freedom of Man and the Origin of Evil, transl. by E. M. Huggard and ed. by Austin Farrar, La Salle, IL 21985, p. 129.

42 For discussion of the question of eternalism or its rival, presentism, in Leibniz, see Futch: Leibniz’s Metaphysics of Time and Space (see note 31), chap. 6.



some fi nal limiting moment. A non-reductive view of time could re-establish the argument if it introduces moments as independent elements of reality; then the hour can be said not to be able to pass without actually reaching its limits. But a reductive view of time seems to require something existing outside the present to supply the needed boundaries in the case of an interval of time pass-ing now. If Leibniz’s philosophy of time is to uphold both his reductivism about time and his dichotomy argument for minima, it may be pressed into accepting eternalism in order to do so.

A last concern remains as well. Even if eternalism were to be invoked in order to reconcile the dichotomy argument for minima with reductivism about time, the argument could no longer be used here to defend the existence of minima in motion. For in this reconciled account, the beginnings and ends of motions, or other time-fi lling processes, are now being presupposed in order to make the case that the hour contains minima. Thus an asymmetry arises between the spatial and temporal aspects of motion and the dichotomy argument for minima. (Our early assumption that Achilles’ running the mile and his counting through the half-intervals of an hour are interchangeable interpretations of a’s approaching H in Leibniz’s own example turns out to have been premature.) Motion may be shown to require minima in order to explain how Achilles can actually reach the surface of the moon and not merely approach it. These are spatial minima in motion: the ends of a motion covering some space. A reductive account of space has to presuppose something with boundaries to underpin the proof, and perhaps the surfaces of bodies can serve in this role without spoiling the argument. But the temporal counterpart to that line of thought cannot provide temporal bounda-ries without presupposing exactly what the dichotomy argument is intended to prove, namely, the existence of minimal beginnings or ends of time-fi lling processes such as motion. This does not mean that the dichotomy argument is powerless to show that the hour can pass only if it contains a minimal end, nor that it cannot show that motion can occur only if it contains a fi nal temporal end. But in the presence of a reductive account of time, those two arguments cannot be defended at once; at least one of their conclusions must be presupposed in order to support the other. The grounds for the claim left merely presupposed, whichever it should be, will then remain still to be secured.

Prof. Dr. Samuel Levey, Department of Philosophy, Dartmouth College, 6035 Thornton Hall, Hanover, NH 03755, USA, [email protected]


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on time and the dichotomy in leibniz

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