SOURCE: Rudavsky, T. M. “Creation, Time, and Infinity in Gersonides.” Journal of the History of Philosophy 26, no. 1 (January 1988): 25-44.

[In the following essay, Rudavsky explains Gersonides's approach to problems involving infinite divisibility and the continuum.]

1. INTRODUCTION

In this paper I should like to examine Gersonides' theory of time and the infinite as developed against the backdrop of his views on creation. Two questions are of paramount importance: the creation of the universe, and the notion of the continuum. Before proceeding to an examination of these two issues, let me first say something about their importance in Gersonides' work.

Gersonides was a Jewish philosopher writing in fourteenth-century France (Avignon, 1288-1344). He spent several years in the papal court in Avignon, and may at that time have come into contact with the views of Ockham and other fourteenth-century scholastics.¹ His major work Milhamot Hashem is a sustained examination of the major philosophical issues of the day: theory of knowledge, divine omniscience and free will, providence and the creation of the universe. In this work Gersonides tries to reconcile traditional Jewish beliefs with what he feels are the strongest points in Aristotle; although a synthesis of these systems is his ultimate goal, the strictures of philosophy often win out at the expense of theology.

The problems of creation and the continuum are both good examples of this attempted synthesis. Aristotle posits an eternal universe in which time is potentially, if not actually infinite.² That is, Aristotle argues that since there can be no “before” to time, time was not created; neither was the universe. Gersonides, however, is committed to the belief that God created the universe. At the same time he wants to accept certain aspects of Aristotle's theory of time and the universe. Hence Gersonides must reconcile for himself a number of strands in Aristotelian thought: most important, he must explain the existence of the universe in time. Since Gersonides will want to argue that both time and motion are finite (and created), he must eliminate Aristotle's notion of infinitely extended time altogether.

The second point concerns the status of the continuum. According to Aristotle, reality is a continuous plenum in which time and matter are infinitely divisible. Hence he must refute Zeno's paradoxes which attempt to demonstrate the feasibility of infinite divisibility. Into a universe in which all potentialities are ultimately actualized, Aristotle introduced the notion of potential infinity in order to explain how time and motion are not finite, but rather potential infinites. Aristotle wants to claim both that extended magnitudes and numbers are finite, as well as that numbers can be infinitely divided. Hence he maintains that numbers are potentially infinite in the following senses: first, that no member of the series in question is infinite, and second, that the members of the series do not all simultaneously coexist.

Gersonides, for reasons having to do with his metaphysics of prime matter, would like to accept Aristotle's notion of the continuum. But this creates problems with divine omniscience. For if time and space are infinitely divisible, and if (as Gersonides will want to argue) God has knowledge of particulars (which are temporally and materially locatable), then it would follow that God must have knowledge of things which are infinitely divisible. This possibility is untenable for reasons we shall examine shortly. Hence Gersonides must introduce a method whereby he can embrace the infinite divisibility of the continuum while rejecting the existence of any type of infinite, actual or potential.

Gersonides, in fact, is one of very few Jewish philosophers to have addressed the problem of infinite divisibility and of time.³ In this paper, I wish to examine the cogency of Gersonides' arguments as developed primarily in his Milhamot.⁴ I will then apply them to Zeno's paradoxes of motion and see whether, as Efros has claimed, Gersonides has offered a viable solution to these paradoxes.⁵ I will first examine Gersonides' theory of time in light of Aristotle's formulation. Then I will turn to the notion of the continuum as reflected in the divisibility of time and space. Finally, I will apply both these discussions to the paradoxes of Zeno. I shall suggest that Gersonides discusses the issues entailed by these paradoxes in an ingenious way, one which adumbrates the spirit of modern discussions of limit theory and denumerably infinite sets.

2. GERSONIDES' STATEMENT OF ARISTOTLE'S THEORY OF TIME

Gersonides' discussion of time is contained primarily in Milhamot VI.1, within the context of his theory of creation. He first lists a number of views of his predecessors (among whom he mentions Aristotle) who propounded the eternity of the universe.⁶ According to Gersonides, Aristotle offered several arguments in support of what I shall refer to as the eternity thesis: the first has to do with the nature of creation, whereas the second is based on the nature of the ‘instant’. Gersonides' statement of Aristotle's first argument can be summarized as follows:

1.1	If time came to be, it would have come to be in time.
1.2	This would imply a time before the original time.
1.3	Time is inseparable from motion.
1.4	Motion is connected to the moved object (that is, the outer sphere).
1.5	This moved object moves in a single, continuously circular motion.
1.6	Since time is inseparable from motion, it too is continuous.
1.7	Hence time is eternal.7

This argument is based on a number of passages in Aristotle wherein he argues that since time is defined in terms of motion, there can be no time without motion. For example, in his work De Caelo Aristotle argued that time is an integral part of the cosmos. He had already postulated that there can be no body or matter outside of the heavens, since all that exists is contained within the heavens. Since, however, time is defined as the number of movement, and there can be no movement without body, it follows that there can be no time outside of the heavens.⁸ A similar argument is propounded in Physics 4.12.⁹

Aristotle's second argument, as stated by Gersonides, is based on his definition of the instant as the middle point between ‘before’ (hakodem) and ‘after’ (hamit'aḥer) and goes as follows: If time came to be, there would have to be an actual instant (‘ata) at which it came to be. But this would entail there being a potential instant before the present instant was actualized. But every part of time has only potential existence, and so no such instant could exist. Hence time could not come to be.¹⁰ The main thrust of this argument, as presented by Gersonides, is that in order to account for the coming into existence of any present instant, there must exist a prior actual instant; but in the case of the first instant, there could be no prior instant, actual or potential.

The basis for this argument can be found both in the Physics and Metaphysics, in which Aristotle develops the notion of the instant as a basic feature of time. The instant is defined as the middle point between the beginning and end of time. Since it is a boundary or limit, it has no size and hence cannot be considered to exist. This characterization of time in terms of the limit leads Aristotle to ask whether time is real. Since instants do not in and of themselves exist, it might be argued that time itself does not exist. That is, the past and future do not now exist, and the present instant is not a part of time since, as we have already noted, it is sizeless. In Physics 8.1 Aristotle claims that because the extremity, or limit, of time resides in the instant, time must exist on both sides of the instant.¹¹ And in Metaphysics 12.6 Aristotle claims that there can be no ‘before’ or ‘after’ if time does not exist, for both terms imply the existence of relative time.¹²

3. THE FINITUDE OF TIME

In order to uphold the finitude of time, Gersonides must refute all of these arguments. In order to do so, however, he first makes a number of observations pertaining to the general characteristics of time. First, he argues that time falls in the category of continuous quantity. We speak, for example, of the parts of time as being equal or unequal; time is measured by convention (be-hanaḥah) as opposed to by nature; and its limit is the ‘instant’ which itself is indivisible.¹³ Incidentally, echoing Aristotle, Gersonides points out that time cannot be comprised of ‘instants’ because the instant measures time, but is not a part of time. Unlike time which is divisible, the instant is indivisible.¹⁴

Further, Gersonides claims that time can be construed both as separate from its substratum and as residing in it. That time resides in its substratum is demonstrated from the fact that it has distinguishable parts: that is, present time is distinguished from both past and future time. Were these parts not distinguishable, argues Gersonides, then any part of time would equal the whole of time. Hence, time must reside in that which it measures. At the same time, it is separable from any substratum; for if it were in its substratum, there would be as many times as there are substrata. But we know that there is only one time and not a multiplicity of times. Hence time must not reside in its substratum.¹⁵

According to Gersonides time is partly potential and partly actual. Aristotle had argued that the past, in being a potency, was infinite. Gersonides however claims that potency refers only to the future and not to the past.¹⁶ If the past were potential, then, Gersonides argues, contrary possibilities would inhere in the past as well as in the future; this however is absurd, since we know that the past has already occurred.¹⁷ Hence, Gersonides concludes that only future time carries within itself potency. It might be noted in passing that these comments are consistent with his statements elsewhere regarding the nature of future contingents.¹⁸

Having laid out these general characteristics of time, Gersonides now demonstrates that time must have been generated. We have seen that time is contained in the category of quantity. Gersonides will argue that just as quantity is finite, so too is time. A number of arguments are based on the nature of body, the spheres, and the regularity of the eclipse.¹⁹ These arguments reflect the influence of Islamic Kalam thought and center on the Aristotelian principle that no infinite can be greater than another.²⁰ The basis of these arguments has been analyzed in Davidson's article “John Philoponus as a Source of Medieval Islamic and Jewish Proofs for Creation” and will not be discussed here.²¹

Gersonides does, however, base a number of arguments on his concept of time. One argument for the finitude of time utilizes the nature of the ‘when’ (matai) and can be restated as follows:

2.1	No ‘when’ is infinite, since every part of time is finite and the ‘when’ measures the finite.
2.2	Since no ‘when’ is infinite, no part of past time is an infinite distance to the present now.
2.3	But if time were infinite, given any past ‘when’, the relation of the time before it to the time between it and the now would be the relation of infinite to finite.
2.4	But time is homogeneous; that is, all its parts have the same ontological composition.
2.5	So 2.3 is impossible, since it implies that time is both finite and infinite.
2.6	Hence time must be finite.22

It should be noted that in this argument Gersonides uses the notion of a part of time to correspond to the Aristotelian sense of a ‘span’. Medieval Aristotelians generally reflected Aristotle's distinction in the Categories between two senses of the term ‘part’: the first is an interval of time (i.e., instant), whereas the second is a stretch of time (i.e., span). Gersonides uses the second of these two meanings in his argument.²³

A second argument of Gersonides is based on the notion of the future. We have already seen that potency resides in the future alone. Gersonides now argues that if time were infinite, it would have to occur with respect to the future because only the future is potential. But the future cannot become infinite in quantity. And if the future cannot become infinite, the past can surely not become infinite. And so Gersonides concludes that time must be finite.²⁴

4. GERSONIDES' REFUTATION OF ARISTOTLE

Having shown to his satisfaction that time is finite, Gersonides must now refute the original arguments proffered by Aristotle in support of the infinity of time. Aristotle's first argument was that since all generation must take place in time, there can be no beginning to time. Hence phrases like “beginning of time” have no reference. Gersonides, however, refutes this argument by distinguishing two types of generation. The first is based on Aristotle's notion of a change from contrary to contrary and takes place in time. The second, however, is what Gersonides terms absolute generation and is atemporal; that is, it is instantaneous and does not take place in time. It is in this second sense that Gersonides argues that time was generated. Before this absolute generation, there was no time. Phrases such as “beginning of time,” Gersonides argues, must be understood in an equivocal sense.²⁵

Aristotle's second objection centered around the notion of the instant as the limit between the past and future. In answer to this argument, Gersonides makes a number of points. To Aristotle's objection that we cannot imagine an ‘instant’ before which there is no time, Gersonides claims that there are many truths which we cannot imagine (just as many imaginable things are not true). Reminiscent of the Kalam controversy over the doctrine of admissibility, Gersonides' point is that the non-imaginability of a claim is not a sufficient condition for rejecting its truth.²⁶

Gersonides' major objection, however, centers on Aristotle's formulation of the notion of the instant. In contradistinction to Aristotle, Gersonides distinguishes two roles of the instant:

3.1	an initial instant which does not yet constitute time
3.2	subsequent instants which demarcate ‘before’ from ‘after’.

According to Gersonides, these two notions of the instant serve different functions. 3.1 delimits a particular portion of time, namely continuous quantity, and is characterized in terms of duration. 3.2, on the other hand, reflects the Aristotelian function of the instant as characterizing division. Gersonides claims that if there were no difference between these two functions of the instant, we could not distinguish between any two sets of fractions of time, for example three hours and three days, because our measure of the two sets would be identical. Since each period of time would be divided by the same kind of instant, there would be no way of distinguishing three days from three hours.²⁷ …

If we construe the instant in the sense of 3.2, then the intervals on AB would be identical to those on CD and EF, since all we have to go by are the number of instants and their corresponding intervals. Further, the instants on AB would be interchangeable with those on CD and EF. But we know that a year is not equivalent to a day or an hour. Gersonides points out that, technically speaking, i₁ on AB is not identical to that on CD, but rather takes on the characteristics of its temporal span. Understanding the instant in sense 3.1, then, allows us to see that the duration of these sets of intervals differs in each case.

But how does this distinction resolve the original problem? Gersonides' point is that Aristotle's original objections to the finitude of time obtain only if the instant is understood in sense 3.2. When the instant is taken in sense 3.1, that is, as an initial instant of a temporal span, we see that there can be a ‘first instant’ without contradiction. Hence Gersonides' point is that the instant taken in the sense of duration need not be preceded by a past time.²⁸

A final argument concerns the potentiality of the instant. Gersonides attributes to Aristotle the idea that the instant can only have potential existence since it has only a hypothetical beginning and end. Because the instant divides time hypothetically, it turns out that past and future time are relative given their position with respect to other times. Gersonides however claims that the instant is only potential in the sense that it does not exist in the present; when I allude to the present instant it has already receded into the past. The instant is not potential in the sense of a point, viz., that it is simultaneously both beginning and end of the measure in question. Hence we can posit the potential instant as the beginning of time.²⁹

5. DIVISIBILITY AND THE CONTINUUM: ZENO'S PARADOXES

Having argued that time is finite, Gersonides must now deal with those arguments which pertain to the infinite divisibility of space and time. These arguments are part of a continuing dialogue between what F. Miller has termed the nihilistic and atomistic strands of the continuum debate.³⁰ Before embarking upon Gersonides' discussion, let me say a few words about this debate.

The main question in the debate concerns the characterization of the smallest units of space and time: are they composed of divisible or indivisible units? Another way of stating this question is by asking whether a magnitude is divisible everywhere, i.e., perpetually divisible into smaller units, or divisible only down to some atomic magnitude, beyond which subdivision is no longer possible. In answer to this query, there were two schools of Greek thought: the nihilists believed in the infinite divisibility of space and time, whereas the atomists believed that time and matter were made up of smallest basic units. Further, the nihilists denied the existence of a vacuum and argued that matter is continuous, whereas the atomists supported the notion of a vacuum, which supplies the space in which their indivisible atoms move.³¹

It is into this controversy that Zeno introduced his celebrated paradoxes of time and motion. Zeno's purpose (although unstated in the paradoxes themselves) is to demonstrate the untenability of both atomism and nihilism. His main contention is that if time is construed as infinitely divisible, then change or motion are impossible. But if atomism is adopted, then notions of place and change become untenable. In short, Zeno constructs these paradoxes in order to undermine any pluralist ontology. Zeno's first paradox of motion can be summarized briefly as follows: Imagine Achilles attempting to traverse the distance AB. In order to reach B, he must first traverse half that distance, or AC. But in order to reach C, he must traverse half that distance, or AD. Assuming an infinite number of subdivisions between A and B, Achilles will never be able to reach his goal B.³²

These paradoxes of motion are in turn dependent upon Zeno's Paradox of Plurality, which can be stated as follows:

4.1	If extended things exist, they must be composed of parts.
4.2	There is therefore a plurality of parts.
4.3	These parts in turn have parts.
4.4	Since the process of division is indefinitely repeatable, there must be an infinite number of parts.
4.5	These ultimate parts must have no magnitude, else they can be further subdivided.
4.6	But an extended object cannot be composed of parts which have no magnitude, since no matter how many zero-magnitude parts are combined, the result is zero-magnitude.
4.7	Therefore these parts must have magnitude.
4.8	But the addition of an infinite number of magnitudes (greater than zero) yields an infinite magnitude.
4.9	Extended objects are therefore “so small as to have no magnitude and so large as to be infinite.”33

When Aristotle turns his attention to solving these paradoxes, his main concern is to refute the atomistic horn of the dilemma. In the De Generatione et Corruptione Aristotle poses the dilemma as follows: “are the basic units which grow, things which are indivisible or divisible?”³⁴ The problem as he sees it arises with respect to the second possibility, for if things are infinitely divisible, there is nothing which will ultimately survive the division. Assume, says Aristotle, that body is everywhere divisible. Each of these parts is then everywhere divisible. But then body would be comprised either of indivisible points, or of nothing, neither of which Aristotle allows. As Furley has pointed out, the resulting segments of an infinite division could neither have any magnitude nor be without magnitude; Aristotle's reason for saying they could not have any magnitude is simply that it would contradict the hypothesis that a division everywhere had been completed.³⁵ Hence it would appear that body must be comprised of indivisibles; that is, of ultimately irreducible atoms.

However, Aristotle conceives of reality as a continuous plenum in which space, time and matter all have the quality of continuity. Hence he will want to maintain that no continuum can be made up of indivisibles. He defines a continuum as “those limiting extremes of the two things in virtue of which they touch each other become one and the same thing and (as the very name indicates) are ‘held together’, which can only be if the two limits do not remain two but become one and the same.”³⁶ In Physics 6.1 Aristotle offers an argument to the effect that “any continuum is divisible into parts that are divisible without limit.”³⁷

How then does Aristotle avoid both horns of Zeno's dilemma? In Physics 6.2 Aristotle distinguishes two senses of infinity: infinity with respect to divisibility and infinity with respect to extent. He argues that time is infinite in the first sense, and then claims that this distinction allows us to maintain that Achilles has an infinite amount of time in which to traverse an infinite distance.³⁸

However, in Physics 3.6 and 8.8 Aristotle acknowledges the inadequacy of this response. He argues that those who had posited indivisible magnitudes in order to respond to Zeno had confused two senses of indivisibility. These two senses can be distinguished as follows: infinite₁ refers to a complete divided state, whereas infinite₂ refers to the process itself of dividing. The first Aristotle terms actual infinity, and the latter he terms potential infinity. The atomist argument in favor of indivisibles relied upon the notion of actual infinity. According to Aristotle, however, a solution to Zeno's paradox depends upon the second sense of infinity. That is, according to Aristotle, the infinite with respect to division exists potentially but not actually; infinite divisibility is a continuing process and so the potency in question is actualized whenever the process is in effect.³⁹

Aristotle's point, intimated already in De Generatione et Corruptione, and elaborated in Physics 3.6, is that no continuum can be divided everywhere simultaneously. Once we realize that the continuum can be only potentially and not actually divided, we can accept the infinite divisibility of the continuum.⁴⁰ This is an important point, the implications of which will emerge more fully in our discussion of Gersonides.

6. GERSONIDES ON THE CONTINUUM

When Gersonides turns to the problem of infinite divisibility, he does so primarily against the backdrop of Aristotelian arguments. His discussion of the continuum is contained primarily in Books 3 and 6 of Milhamot, in the context of his discussion of divine omniscience and creation respectively. The issue of infinite divisibility is raised in a curious context, namely that of God's knowledge of particulars. According to Gersonides “some recent philosophers”⁴¹ used the claim that continuous quantity (hacamut hamitdabek) is divisible into what is indivisible in order to demonstrate that God does not know particulars. The general argument of these philosophers was that if God knows particulars, then if he has knowledge of what is divisible, then he knows all its parts. But in accordance with what is entailed by divisibility, these parts would be infinitely divisible. Hence God would know those parts which are infinitely divisible. But by definition God's knowledge is only of bounded objects. Hence He cannot know divisibles. But since there are no indivisible magnitudes, these philosophers concluded that God does not know particulars.⁴²

From this argument it would seem to follow that either God does not know particulars, or that continuous quantity is divisible into indivisible parts. Gersonides however wants to claim both that God does know particulars and that there are no indivisibles. He therefore rejects the argument of these philosophers by arguing that God knows the division of body according to its nature: “He knows that everything which is divisible can be divided (further) in that it is an instance of quantity (ba'al camah), (but) He does not know the limit of the divisibility which by nature is limitless.”⁴³ In other words, Gersonides claims that although God knows the nature of quantity qua quantity (hacamut bemah shehu camah), namely that it is infinitely divisible, nevertheless He does not know of each particular instance of quantity what its infinite divisibility comprises.

This point is further emphasized in Gersonides' rejection of attempts to explain God's knowledge by claiming that God knows the possible division of magnitude into parts, but not its actual division. Again, Gersonides argues that God knows that every part into which a quantity is divided is capable of division, but He does not know the limit of what is limitless by nature: “we may say (that God knows) all parts only because by this statement we render (in thought) universal what is not (in actuality) universal, because what is infinite is not universal.”⁴⁴ It is here that Gersonides is already sowing the seeds for rejecting any notion of potential infinity.

Gersonides now turns to the problem of the divisibility of the continuum. He first states what he takes to be Aristotle's formulation of the atomist argument showing that body must be divisible into indivisibles. Gersonides' summary of this atomist position can be reformulated as follows:

5.1	Assume that a body is divisible completely.
5.2	It is logically possible that this body be divided (that is, that the division be completed).
5.3	If this complete division is assumed, then body must be divided into either divisible or indivisible parts.
5.4	If these parts are divisible, body would not be divided into every possible part (since the results of the division would be further divisible).
5.5	But this is impossible in light of 5.2
5.6	Hence it follows that body must be divided into what is indivisible.
5.7	Hence indivisible parts (atoms) must exist.45

We have already seen that Aristotle himself resolved this dilemma by distinguishing two senses of divisibility: actual and potential divisibility. Averroes, however, understood Aristotle's solution somewhat differently. In his commentary on De Generatione et Corruptione, Averroes argues that since the actual division of a body is impossible, 5.1 is an illicit assumption. Averroes emphasizes the impossibility of simultaneous division and argues that just as a person cannot acquire knowledge of all the sciences simultaneously but must do so successively, so too a body cannot be divided at all of its points simultaneously. Simultaneous division, according to Averroes, could occur only if the points in question were contiguous.⁴⁶ Gersonides summarizes Averroes' argument as follows:

6.1	A body can be divided at all its points simultaneously only if all the points meet each other.
6.2	But according to Physics 6, one point cannot be immediately next to another.
6.3	So when we divide body at point A, it is not possible for division to occur at A1 (when A1 is contiguous to A).
6.4	Before the division occurred at A, division was possible at both A and A1.
6.5	But when the division occurred at A, division at A1 was precluded.47

The main thrust of Averroes' argument is summarized by Gersonides in Milhamot as follows: “Thus, when we take a point, it is possible to divide the magnitude (at that point) at any place we want, but when we divide the magnitude (at that point) in that place, it is not possible for us to divide it at a second point at any place we want, since it is impossible for us to divide it at a point (immediately) next to that (first) point.”⁴⁸ Gersonides, however, disagrees with Averroes on several counts. First he points out that Averroes must assume the very thing he sought to deny: for if, as Averroes thought, division is not possible at any point A₁ contiguous to A, and if body is infinitely divisible, then there is an infinite number of points at which division is not possible. Hence, on Averroes' argument, it follows that body is comprised of indivisibles—the very condition that he sought to deny.

Second, Gersonides claims that divisibility at each point is possible. “[W]ith respect to the nature of quantity, divisibility at each of its points is possible. I mean to say that division at a point is possible with respect to division occurring at the point immediately next to it.”⁴⁹ That is, according to Gersonides, division at A₁ is possible even if division has already occurred at A. Even were he to grant Averroes his assumption, Gersonides claims, the dilemma would not be resolved.

Gersonides therefore rejects Averroes' solution and offers his own solution to Aristotle's original dilemma, arguing that “from our positing that when continuous quantity is divided it is divided into parts which are (themselves) divisible, as was explained in various places, it necessarily follows that it is impossible that it can be divided into parts which are not capable of (further) division.”⁵⁰ In other words, Gersonides is distinguishing here two senses of infinite divisibility:

7.1	the claim that a continuum is divisible into parts which are themselves further divisible
7.2	the claim that a continuum is actually divisible into indivisible parts.

The main thrust of Gersonides' discussion is to distinguish between what we might call infinite and endless division: a continuum is not actually infinitely divisible, but rather endlessly, or potentially, divisible. The first refers to a complete divided state, whereas the second refers to the process itself of dividing.⁵¹

7. THE CONTINUUM AND THEORY OF NUMBER

Gersonides' distinction between endless and infinite division is brought out even more forcefully in his discussion of number. This discussion occurs in two contexts. In Milhamot 6 and 3 Gersonides has characterized matter in terms of continuity (hitdabkut), i.e., that by virtue of which matter can be infinitely divided and its parts still retain their continuity.⁵² In Milhamot 6 Gersonides poses a possible objection to his characterization as follows: suppose one were to argue that inasmuch as number is augmentable, so too is quantity augmentable, or infinite. Gersonides' response is that the endlessness we find in number is not an endlessness of quantity, but rather of the process of division and augmentation. That is, Gersonides distinguishes between quantity itself and the act of increase/diminution, which is based on quantity. Although the act of augmenting is never exhausted, quantity itself remains finite.⁵³ This is my understanding of Gersonides' statement that “quantity is infinitely finite.”⁵⁴

In Milhamot 3 the question is posed in terms of whether God can have knowledge of a non-augmentable number. Gersonides summarizes the argument of those who argue that there might exist a non-augmentable number which is known to God, and claims that such a position is absurd.⁵⁵ The issue which emerges from this is how number and divisibility of the continuum are related.

Averroes had argued that in a body there is an infinite number of points which exists potentially, and division is possible at each point.⁵⁶ Gersonides dismisses Averroes' position as absurd on the grounds that if a body did contain finite dimensions, and these dimensions were potentially infinite, then they could not be in contact. Why not? According to Physics 6.1, between any two points there would be another point. Since there is an infinite number of points between any two points, the measure between any pair of dimensions would be infinite in number. But Gersonides points out that this would result in a finite body being infinite, for the reason that if all the distances within the finite body were totalled, the result would be an infinite number.⁵⁷ …

Imagine the finite body, AB which is divided at C and D. The distance from AB = f, which is a finite number. The distance CD = i, which is infinite, however, since there is an infinite number of points between C and D. So the distance of CD, which is a part of AB and hence smaller than it, would nevertheless be infinitely large and greater than the finite distance of AB.

In order to avoid positing the existence of the infinitely large, Gersonides concludes therefore that “continuous quantity does not contain dimensions which are infinite in number, neither potentially nor actually.”⁵⁸ So how does he account for the fact that the continuum can be infinitely divided, or number infinitely augmented? He claims that infinity with respect to divisibility is found not in the number of divisions undertaken but in the very act of dividing. The number of divisions in any continuum is always actually finite, although this number may be agumented indefinitely. For example, Gersonides claims, take a continuum which is divided into two parts, each of which is divided into two more parts, and each in turn divided so that there are now eight parts. Gersonides states that “the number of parts is continually augmented, while (that number) always (remains) a finite number.”⁵⁹

The main thrust of Gersonides' discussion is that a finite quantity cannot contain infinite parts. We have seen that according to Gersonides quantity by its very nature is finite: “the nature of quantity necessitates that it be finite.”⁶⁰ So too is number. If, Gersonides argues, quantity had an infinite number of parts, it would follow that a finite quantity would be infinite, for each of its potential parts would be comprised of quantity.⁶¹ In order to avoid this conclusion, Gersonides therefore characterizes infinite divisibility of the continuum in terms of the possibility of subdividing its parts. Thus, just as no number can be infinite, so too no continuum can contain either parts or dimensions which are infinite in number.

7. CONCLUSION

We are now in a position to summarize our findings. Working within a framework which posits both an eternal universe as well as the infinity of time, Artistotle was able to account for infinite divisibility of time and space by distinguishing potential infinity from actual infinity. Time and space were both perceived by Aristotle as potentially infinite in the sense that not all their parts were divisible simultaneously. In this way Aristotle felt that he avoided the atomistic consequences of Zeno's paradoxes.

Gersonides as well wished to avoid an atomistic ontology, but for him this ontology took the form of Islamic atomism, as mediated through the works of Maimonides. However, unlike Aristotle who had recourse to the potential infinite, Gersonides denied the reality of infinity in any guise—potential or actual. In order to eliminate the ontological status of infinity, he therefore argues that Aristotle's attempt to create a separate category for time and motion fails. Gersonides attempts to destroy the analogy between time and numbers by claiming a discontinuity between the two types of infinity. As we have seen, Gersonides argues that in the case of numbers, it is the act of division/augmentation which is without limit, not the magnitude itself. But in the case of time or motion, the infinity would have to inhere in their very essence.

The question, then, is how to account for the infinite divisibility of the continuum. Gersonides is unwilling to deny infinite divisibility altogether, as evidenced in his characterization of matter. And he is aware of its importance with respect to the addition/division of number. Hence his solution is to deny the reality of the infinite while at the same time affirming the possibility of the process of infinite division/addition. That is, according to Gersonides, what we mean by the infinite is a process which by nature is never-ending; the members of the process, however, are finite. Thus, for example, the natural number series is infinite in the sense that its members can be ever-augmented; but each member of the series is finite.

How can this solution be applied to Zeno's paradoxes? In Milhamot Gersonides does not directly address the issue.⁶² However, I would like to suggest that, given what Gersonides has said about infinite divisibility of the continuum, his solution in this work is surprisingly modern. In fact, I would venture to suggest that Gersonides has captured the spirit of modern mathematical notions of infinity. Two points stand out. First, with respect to the infinite divisibility of a continuum, we have seen that Gersonides disagrees with Aristotle's distinction between the actual and potential infinite in order to resolve Zeno's paradox. Gersonides' solution is in fact closer to modern limit theory. According to this theory, mathematicians speak of an infinite converging series (i.e., the sum of subdivisions between 0 and 1) having a finite sum which approaches 1. It is important to note that the finite sum functions as a limit and is not part of the converging series itself. Gersonides is at pains to point out that infinite numbers cannot have a finite sum. However, his characterization of infinity as a process rather than as a state leads ultimately to similar consequences. For what both systems amount to is that the most we can say is that an infinite series converges to a finite sum. Limit theory by itself does not resolve Zeno's paradoxes; and this, I think, is the point of Gersonides' discussion. As Max Black has pointed out in his discussion of infinity machines, the operation of summing the infinite series will tell us where and when Achilles will catch the tortoise if he can catch the tortoise, but that is a big if.⁶³

The second point had to do with the augmentability of number. We have seen that in answer to the question “Can God have knowledge of a nonaugmentable number?” Gersonides has argued that given the nature of God's knowledge, such a state is impossible. At most God can know that the number series is indefinitely augmentable, but there is no ‘last’ number in the series to know. In this discussion Gersonides foreshadows more recent discussions of Cantor and others.⁶⁴ In light of his contention that God only knows that a continuum is infinitely divisible, but not its individual parts, Gersonides might be construed as arguing that the most God can know is a denumerably infinite set: that is, he can count the members of the set, that is, he knows ‘how many’ members of the set there are, but he does not know the ‘infinitieth’ member of that set, since there is no such number to know. Of course, Gersonides differs from Cantor in that he would deny that such a set comprises an actual infinity, for as we have seen he denies the completability of an infinite series. But in terms of the implications of such a set for God's knowledge, Gersonides would accept the description of such a set. In short, Gersonides' discussion of time and the continuum is both a sophisticated and noteworthy attempt to resolve problems which to this day plague philosophers.⁶⁵

Notes

1. The questions raised by Levi ben Gerson (Gersonides, 1288-1344) are contained in his major work Milḥamot Hashem (MH; Wars of the Lord). In this paper, reference will be made primarily to the this Hebrew edition, which was reprinted in Leipzig in 1866. References will be to treatise, chapter and page number. Unless otherwise specified, all translations from the Hebrew are my own. In addition, the following recent English translations of portions of Gersonides' works should be noted: S. Feldman, trans. and ed., The Wars of the Lord (Book I) (Philadelphia: Jewish Publication Society of America, 1984); C. Manekin, “The Logic of Gersonides: An Analysis of Selected Doctrines, with a Partial Edition and Translation of The Book of the Corrected Syllogism” (Columbia University Dissertation, #DA8427427); N. Samuelson, trans., Gersonides: The Wars of the Lord; Treatise Three: On God's Knowledge (Ontario: Pontifical Institute, 1977); J. Staub, The Creation of the World According to Gersonides (Brown University, Scholars Press, 1982). For an extensive bibliography of scholarly works on Gersonides, see M. Kellner, “R. Levi ben Gerson: A Bibliographical Essay” in Studies in Bibliography and Booklore 12 (1979): 13-23. References to specific articles will be made in the present essay when relevant; however, the following works should be noted in particular for their treatment of Gersonides' theories of time and creation: I. Efros, The Problem of Space in Jewish Medieval Philosophy (Ithaca, N.Y.: Cornell University Press, 1917); S. Feldman, “Gersonides' Proofs for the Creation of the Universe”, Proceedings of the American Academy for Jewish Research 35 (1967): 113-37; S. Feldman, “Platonic Themes in Gersonides' Cosmology” in Salo Baron Jubilee Volume I (Jerusalem, 1974), 383-405; C. Touati, La Pensée Philosophique et Théologique de Gersonide (Paris: Les Éditions de Minuit, 1973).

2. Aristotle's discussion of the eternity of the universe is contained in several places, most notably De Caelo 1, Physics 8.1, and Metaphysics 12.6. For a discussion of these and other relevant passages, see R. Sorabji, Time, Creation and the Continuum (Ithaca, NY: Cornell University Press, 1983), 276ff.

3. For a brief examination of other Jewish philosophical discussions of the continuity of time and space, see I. Efros, Problem of Space 46ff. In recent years, attention has been paid to Scholastic discussions of the continuum; see, for example the collection of articles in N. Kretzmann, ed., Infinity and Continuity in Ancient and Medieval Thought (Ithaca, NY: Cornell University Press, 1982); and the articles by J. E. Murdoch, E. D. Sylla and J. A. Weisheipl in N. Kretzmann, et al, ed., The Cambridge History of Later Medieval Philosophy (Cambridge: Cambridge University Press, 1982). Although not concerned with Jewish discussions, these articles explore issues common to both Jewish and Christian medieval thinkers.

4. It should be noted that this investigation is concerned primarily with Gersonides' discussions in Milḥamot, and not with other works of his, such as his Supercommentaries on Averroes' Commentaries on the Physics. These latter works will form the basis for a separate study.

5. Efros, Problem of Space 57. “His [Gersonides'] most original contribution to the problem of infinite divisibility is his solution of Zeno's puzzle … we are now to see how Gersonides finally solved it—a solution well worth serious consideration on the part of present day thinkers.”

6. MH 6.2.2, 294. In addition to the view of Aristotle, Gersonides lists the following views as well: that the world comes into existence and passes away an infinite number of times; the view attributed to Plato that the world was created one time out of some thing (nitḥadesh midavar) and the view attributed to the Kalam and to Maimonides, that the world was created out of absolute nothing (nitḥadesh milo' davar be-muḥlat). It is interesting to note that Maimonides' view is characterized by Gersonides as a version of Plato's theory in that, according to Gersonides, neither one advocates creation ex nihilo. For a recent discussion of Maimonides' theory of creation, see H. Davidson, “Maimonides' Secret Position on Creation”, in I. Twersky, ed., Studies in Medieval Jewish History and Literature (Cambridge, Mass: Harvard University Press, 1979), 16-40.

7. This reconstruction of Gersonides' understanding of Aristotle's argument is based on the text in MH 6.1.3, 298.

8. See De Caelo 1.9 279a 8 ff: “It is obvious then that there is neither place nor void nor time outside the heaven, since it has been demonstrated that there neither is nor can be body there.”

9. In Physics 4.12 Aristotle demonstrates that inasmuch as time is the measure of motion, those things which are subject to generation and corruption are necessarily in time.

10. MH 6.1.3, 295.

11. See Physics 8.1 251b ff: “Since the instant (τò νυν) is both a beginning and an end, there must always be time on both sides of it” (251b 25).

12. See Metaphysics 12.6. 1071b 6ff: “For there could not be a before and an after if time did not exist.”

13. The various characteristics of time are elaborated in MH 6.1.10, 329ff.

14. For a discussion of this point, see J. Staub, Creation, 30.

15. MH 6.1.10, 329-30.

16. MH 6.1.10, 330-1.

17. MH 6.1.10, 331.

18. Gersonides' notion of the contingency of the future is elaborated in MH 3, in the context of the issue of divine omniscience. For further discussion of this issue, see T. M. Rudavsky, “Divine Omniscience and Future Contingents in Gersonides” Journal of the History of Philosophy 21 (1983): 513-36; N. Samuelson, “Gersonides' Account of God's Knowledge of Particulars”, Journal of the History of Philosophy 10 (1972): 399-416.

19. These arguments are contained in MH 6.1.11, 340-41. For further elucidation of these arguments, see S. Feldman, “Gersonides' Proofs,” 130-31.

20. See, for example, MH 6.1.11, p. 342: “It follows that the number of one infinite is neither larger nor smaller than the number of another infinite; for in order for the one to be larger than the other, it would necessarily have to be finite.” The term Kalam refers to a particular system of thought which arise in Islam prior to the philosopher al-Kindi (d. 873); its exponents (mutakalimun) were contrasted with straightforward philosophers. For a detailed discussion of standard Kalam doctrines, see M. Fakhry A History of Islamic Philosophy (New York: Columbia University Press, 1983); H. A. Wolfson, The Philosophy of the Kalam (Cambridge: Harvard University Press, 1976).

21. H. Davidson, “John Philoponus as a Source of Medieval Islamic and Jewish Proofs for Creation,” Journal of the American Oriental Society 89 (1969): 357-91.

22. MH 6.1.11, 343.

23. This distinction is made in Aristotle Categories 4. A similar distinction is made in Plato's Parmenides where Plato distinguishes similarly between the now (τò νυν) and the instant (τò ξαίφναs). I owe this latter reference to Prof. E. A. Browning.

24. MH 6.1.11, 345

25. MH 6.1.21, 385-86. See S. Feldman “Gersonides' Proofs,” 134-35 for a discussion of these pages.

26. MH 6.1.21, 386. For a description of the Kalam notion of admissibility, see M. Maimonides, The Guide of the Perplexed, trans. by Sh. Pines, (Chicago: Univ of Chicago Press, 1963), III.15. See the following works for a critical analysis of Maimonides' exposition: Z. Blumberg “Ha-Rambam al Musag al-Tajwiz beshittatam shel ha-mutakalimun,” Tarbiz 39 (1970): 268-76; A. Ivry, “Maimonides on possibility” in J. Reinhartz et al, eds., Mystics, Philosophers, and Politicians (Durham: University of North Carolina Press, 1982), 77ff; and H. A. Wolfson, The Philosophy of the Kalam (Cambridge, Mass.: Harvard University Press, 1976), 43ff.

27. MH 6.1.21, 387. Wolfson points out that this distinction can be traced back to Aristotle's Physics 6.11. 219a22-30. Cf. H. A. Wolfson, Crescas' Critique of Aristotle, (Cambridge: Harvard University Press, 1929), 653.

28. MH 6.1.21, 387-88. This diagram is based on that of C. Touati's discussion; cf. Pensée de Gersonide, 236.

29. MH 6.1.21, 389-90

30. F. Miller, “Aristotle Against the Arabs” in N. Kretzmann, ed., Infinity and Continuity, 90.

31. For further elaboration of this point, see Sh. Sambursky, The Physical World of Late Antiquity (London: Routledge and Kegan Paul, 1962), 1ff.

32. The major source for Zeno's paradoxes is Aristotle, most notably in Physics 6.9. For an introductory examination to considerable recent literature dealing with these paradoxes, see J. Barnes, The Presocratic Philosophers (London: Routledge and Kegan Paul, 1982); W. Salmon, Zeno's Paradoxes, (New York: Bobbs-Merrill Co., 1970); and R. Sorabji, Time, 321ff.

33. The text for this paradox is quoted by Simplicius as follows: “In his [Zeno's] book, in which many arguments are put forward, he shows in each that a man who says that there is a plurality is stating something self-contradictory. One of these arguments is that in which he shows that, if there is a plurality, things are both large and small, so large as to be infinite in magnitude, so small as to have no magnitude at all. And in this argument he shows that what has neither magnitude nor thickness nor mass does not exist at all. For, he argues, if it were added to something else, it would not increase its size; for a null magnitude is incapable, when added, of yielding an increase in magnitude. And thus it follows that what was added was nothing. But if, when it is subtracted from another thing, that thing is no less; and again if, when it is added to another thing, that thing does not increase, it is evident that both what was added and what was subtracted were nothing.” Cf. W. Salmon, Zeno's Paradoxes, 13.

34. Aristotle, De Generatione et Corruptione, 1.2 316a 26.

35. D. Furley, “The Greek Commentators' Treatment of Aristotle's Theory of the Continuum”, in N. Kretzmann, Infinity and Continuity, 34.

36. Aristotle, Physics 5.3.227a14. A similar definition is given in Categories 4b 22.

37. Aristotle, Physics 6.1.231b 19.

38. Aristotle, Physics 6.2.33a 21-233b 32: “For there are two senses in which a distance or a period of time (or indeed any continuum) may be regarded as illimitable, viz., in respect to its divisibility or in respect to its extension. Now it is not possible to come in contact with quantitively illimitable things in a limited time, but it is possible to traverse what is illimitable in its divisibility; for in this respect time itself is also illimitable.”

39. Physics III.6. 206a18ff. “magnitude is not actually infinite. But by division it is infinite … the alternative then remains that the infinite has a potential existence” (206a16). In Physics 8.8 Aristotle maintains that he had formerly misconstrued Zeno's challenge, and now claims that the notion of potential division is crucial for understanding Zeno: “if we are asked whether it is possible to go through an unlimited number of points, whether in a period of time or in a length, we must answer that in one sense it is possible, but in another not. If the points are actual it is impossible, but if they are potential it is possible” (263b2-8). For a discussion of these passages, see D. Furley, “The Greek Commentators' Treatment,” 32-33.

40. For an application of infinity to the eternity of time, see Physics 4.4-7.

41. The term contemporaries (ha-mit'aḥerim) could be used by Gersonides to refer to any number of philosophers. For further discussion of this term, see N. Samuelson, trans. Gersonides: The Wars of the Lord; Treatise Three: On God's Knowledge (Ontario: Pontifical Institute, 1977), 122. Samuelson's translation will be referred to as Wars.

42. MH 3.4, 143. Cf. Wars, 258-59.

43. MH 3.4, 143. Cf. Wars, 260.

44. MH 3.4, 144. Wars, 262-63 (Samuelson's translation).

45. MH 3.4, 144. Cf. Wars, 264.

46. For Averroes' discussion of the divisibility of body, see Averroes, On Aristotle's De Generatione et Corruptione: Middle Commentary and Epitome ed. and trans. by S. Kurland (Camb. Mass: Medieval Academy of America, 1958), 12ff.

47. MH 3.4, 145. Wars, 286 (Samuelson's translation).

48. MH 3.4, 145. Wars, 286.

49. Ibid.

50. MH 3.4, 145 Wars, 270-71. (Samuelson's translation).

51. Wars, 271. See also MH 3.4, 145.

52. See MH 6, 345 for an elaboration of the continuous quality of matter. For a discussion of Gersonides' theory of matter, see C. Touati, Pensée de Gersonide, 243ff; T. M. Rudavsky, “Individuals and the Doctrine of Individuation in Gersonides,” The New Scholasticism 51 (1982): 30-50.

53. MH 6.1.11, 334. See C. Touati, Pensée de Gersonide, 224 for further discussion of this point.

54. MH 6.1.11, 345. For a slightly different interpretation of this statement, see I. Efros, The Problem of Space, 101.

55. MH 3.4, 145. Cf. Wars, 274-75. Aristotle makes a similar comparison between the divisibility and augmentability of number in Physics 2.6. 206b4ff.

56. For a discussion of Averroes' argument, see C. Touati, trans. Les Guerres du Seigneur, Livres III et IV (Paris, 1968), 93.

57. MH 3.4, 146. Wars, 277.

58. Ibid.

59. MH 3.4, 146. Cf. Wars, 278.

60. MH 6.1.11, 333.

61. MH 6.1.11, 334.

62. I am hoping in subsequent research to explore Gersonides' discussions of infinity and the continuum in his Supercommentary on Averroes' Commentary on the Physics (Middle Commentary); preliminary investigations suggest that Gersonides' critique of Averroes will shed light on his own views of Zeno's paradoxes.

63. Max Black's article “Achilles and the Tortoise” is contained in W. Salmon, Zeno's Paradoxes, 67 ff. Black's article has led to wide discussion and controversy among contemporary philosophers. For a representative sampling of the issues, cf W. Salmon, Zeno's Paradoxes, 67-138.

64. For a discussion of Georg Cantor's groundbreaking theories of infinity and the continuum, cf. Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, trans. P. E. B. Jourdain, (New York: Dover Publ.); Stephen Korner, The Philosophy of Mathematics, (London: Hutchinson University Library, 1960). Salmon has a valuable introduction to the main elements of Cantorian views on pages 251ff. For a different reading of Gersonides' views with respect to contemporary accounts, see the discussion by S. Feldman in “Gersonides' Proofs,” 130, and in “Platonic Themes,” 404.

65. I would like to thank Professors I. Boh, E. Browning, S. Feldman, A. Ivry, C. Normore, and two anonymous referees for their helpful comments on earlier versions of this paper.