Arabic philosophy

Potential infinite Actual infinite

By addition, by division Recurrent events a parte ante

This sort of actual infinity can be allowed, because even though an

infinity of years or events may have passed, there is no infinite set of

things all present together. In a sense, such an actual infinite is still

at least partially potential. But a problem arises once we admit the

personal immortality of the soul: the souls of all individual humans

that have lived in the past still exist now, and on the assumption

that the world is eternal, they form an infinite set of substances.

Christian and Muslim Aristotelians must now get to grips with a

real actual infinity.

This is one reason, though surely not the whole reason, for

Avicenna’s introduction of a subtle distinction between two types

of infinite sets. Sets of the first type include in themselves their

own rule of construction, their “order” (tart¯ıb). They are infinitely

extendable i.e., potentially, but not actually, infinite (e.g., someone

counting up through the integers, and never of course reaching

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Natural philosophy 299

infinity). Of the second type are sets with no internal rule of construction.

These can be actually infinite.41 The set of the past souls

at any time t is an infinite set of this second type. Potential infinity,

as it were, becomes tied to an intellectual operation (counting,

constructing a geometrical figure, etc.), whereas actual infinity may

exist even in the absence of any mind that could think discursively

through an infinite order.

Avicenna revisits Aristotle’s discussion of infinity not only by

upholding the existence of a “strong” actual infinity, but also by

showing that a certain type of potential infinity is much closer to

actual infinity than orthodox Peripatetics were willing to claim.

The decisive step consists in describing sublunar dynamic motion in

terms of a potential infinity that has much in common with actual

infinity. It is the idea of a dynamic moment that allows Avicenna

to do this. For every sublunar natural motion, there is an infinity of

dynamic states “in actuality.” These states are not purely potential,

since, unlike the points of a line, they have a principle of distinction

(each has amayl-2). But their infinity is not entirely actual, since they

are not all present at the same time. Although Avicenna nowhere

presents a table such as the following, it may represent adequately

the distinctions he introduces in the Aristotelian classification:

“strong” potential infinite “weak” actual infinite

By addition, by division Recurrent events a parte ante

“weak” potential infinite “strong” actual infinite

Sublunar dynamic motion Souls of past men

It is in his Glosses (Ta‘lı¯qa¯ t) that Avicenna sets out in detail the distinction

between sublunar and supralunar motion. In order to do so,

he must explain how the mayl-2, which is characteristic of sublunar

motion, is something real:

The cause of the alteration (al-istih.

a¯ la) that supervenes on natural bodies

endowed with force consists in the places and the positions, insofar as rectilinear

motion is produced by nature and the mobile is not in its natural

state. And the cause of the renewal [reading tajaddud for tah.

addud] and

repetition of its movements, as well as the cause of the alteration (which

tends to the destruction of one force and to the renewal of one another) of

its nature, is the existence of “wheres” and actually determined positions

(wuj ¯ udu uy ¯ unin wa awd. ¯ a‘in mutah.

addidatin bi-al-fi‘l), from the beginning

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300 marwan rashed

of the movement until it comes to a rest. For nature does not cease, at every

instant (fı¯ kulli a¯nin), to be in a new state, different from the previous one.

And what makes these to be states are the changing mayls (wa ha¯dhihi

ah.

wa¯ lun li-al-muyu¯ li al-mutabaddila). This case is similar to the alteration

of this or that quality, e.g., an extraneous temperature of the water, which

does not cease, at every instant, to be altered – increased or diminished –

until the water returns to its natural state. The renewed cause of this process

is the existence of “wheres” and positions actually determined.42

Four points must be emphasized regarding this passage.

(a) The repeated use Avicenna makes of the terms “renewal”

(tajaddud, tajaddada) and “state” (h.

a¯ l) allows us to understand

in all its complexity his position relative to contemporary

kala¯m, and to the school of Abu¯ Ha¯ shim in particular.

With the latter’s disciples, Avicenna holds that there

is a renewal of the movement at every instant, and that

the moving thing, at every instant, is in a different state.

This state is characterized by a position (Avicenna’s wad.

‘

corresponds to the h.

ayyiz of the mutakallimu¯ n) and produced

by an impulsion (Avicenna’s mayl, the i‘tima¯d of the

mutakallimu¯ n). These similarities underline the fundamental

difference between the two systems: their interpretation

of continuity. Whereas between any two Avicennian positions,

there exists always a third one (and so on ad infinitum),

Abu¯ Ha¯ shimand his disciples theoreticallymaintain a

series of discrete positions, even if they take great care not to

determine these atomic thresholds quantitatively. Avicenna

is quite skeptical about the discontinuity and finitism of the

kala¯mtheory, but does not seemto reject its notion of tawallud

(“engendering”), to which his tajaddud appears roughly

equivalent. One may thus interpret Avicenna’s doctrine as a

continuist reformulation of the dynamical principles of Abu¯

H¯ashim.

(b) This implies that Avicenna distances himself from Aristotle’s

conception of potential continuity, since every point

of the trajectory has a principle of distinction dictated by its

mayl. At a terminological level, this tension is conveyed by

the word “alteration” (istih.

a¯ la), which Avicenna employs in

order to describe the variations of the movement’s intensity.

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Natural philosophy 301

We know that in Arabic Peripateticism, this word is the

translation of the Greek alloioˆ sis, change in the category of

quality (poiotˆes). This apparent misuse reflects the fact that

Avicenna does not find at his disposal, in the Aristotelian

terminology, a word perfectly suited to the reality he wishes

to describe. The term “alteration” is no longer confined to

the transition from a (qualitative) beginning to a (qualitative)

end, but can also refer to the instantaneous variability of the

movement itself.

(c) Avicenna consciously remains just shy of accepting an actual

infinity. He says twice that the successive positions of the

mobile are actually (bi-al-fi‘l), not just potentially, determined.

Since he obviously accepts that the points, and therefore

the positions, on any stretch AB are not finite in number,

he must conclude that all the elements of a non-finite set are

actually determined. Interestingly, however, Avicenna does

not say that they are actually infinite. For all the trajectory’s

states are not realized together (ma‘an).43

(d) This passage from the Ta‘lı¯qa¯ t permits us finally to understand

Avicenna’s general theory of motion, as it appears

in particular in the Physics of the Shifa¯ ’ (bk. II, ch. 1).44

Avicenna stresses there that we can mean two things when

we speak of “motion”: motion as a trajectory, which pertains

to our imaginative faculty and is conceived of only as

linking a starting point to an end; and motion as an intermediary

state, which must be attributed to each moment of

the trajectory. Motion in the second sense characterizes an

infinitesimal moment, and nothing else. The present text of

the Ta‘lı¯qa¯ t is the only passage where Avicenna draws such a

strong connection between themayl-2 and this second sense

of “movement.” Themayl represents the principle of distinction

of each position of the trajectory. Each substance spatially

or qualitatively removed from its natural state (e.g., a

stone thrown up away from its natural resting place) returns

to it, passing through all intermediary states. Each of these

intermediary states, because it is not the end point of the process,

produces a newmayl,which adds itself to the impulsion

produced by the others. Every moment is thus characterized

by its own kinetic intensity.45

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302 marwan rashed

Avicenna can thus explain the fundamental difference between

sublunar and supralunar motion. Unlike the trajectory of the four

sublunar elements, the trajectory of a heavenly body has no real

principle of distinction for its positions; the principle of distinction

exists only in the imaginative faculty of the celestial substance. In

other words, Avicenna accepts in this case an interpretation of continuity

akin to that of Aristotle, where the potentiality (dunamis) is

hardly to be distinguished from a purely imaginative existence (cf.

Physics, VIII.8): “the reason for the alteration of the celestial body

is not its positions but its imagination and its renewed volition, one

imaginative act after the other.”46

Avicenna thus seems to stand at the crossroads of two traditions.

With the mathematicians, he recognizes that every one of the infinite

points on a spatial interval AB, without perhaps being perfectly

real, is however notionally and qualitatively distinct from every

other point. But with the mutakallimu¯ n, he sees in a dynamic of

impetus the efficient principle of such a distinction. Thus, starting

from a classificatory project of the different types of impetus,

Avicenna arrives at a complex – because partially “ontological” –

doctrine of instantaneous motion. This combination of the kinematics

of the geometers and the dynamics of mutakallimu¯ n deeply

influenced Avicenna’s successors in the East and theWest. It is probably

Avicenna’s main achievement in natural philosophy that after

him, for every lucid reader, the discussion of motion must focus on

what happens at an infinitesimal level.

post-avicennian kal ￣am: an overview

We have already seen that the great mutakallimu¯ n of the tenth

century did not hesitate to appeal to the authority of Euclid in

defense of their atomism. But because of the finitist principles of

their ontology, they limited themselves to assimilating their indivisibles

to the points of the geometers. After Avicenna, and probably

under the influence of his doctrine of continuity and the infinite,

the mutakallimu¯ n seem ever more eager to extend their appeal

to geometry from the model of the point to that of the line. This

shift is made possible only by concentrating, even more than previously,

on the question of motion, and above all by putting tacitly

aside the finitist considerations that were characteristic of the

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Natural philosophy 303

school of Abu¯ al-Hudhayl.47 The modern historian is struck by the

impression that thinkers after Avicenna, apart from rather verbal

polemics on some refined – and sometimes extremely interesting –

points, share a more or less common doctrine of motion as a realized

set of punctual moments. But whereas orthodox Avicennians

insist that the moments of the trajectory belong to a continuum,

the mutakallimu¯ n stress that each kinetic point is totally and perfectly

realized. What makes the discussion somewhat scholastic is

that the latter more than ever avoid emphasizing the finite character

of this set of points, while the former, as we have seen, refrain

from admitting clearly that what we have here is nothing other

than a pure actual infinity. They seem rather to consider sublunar

motion as a false potential infinity, or, so to speak, a virtual actual

infinity.

By far the most interesting discussion on these topics appears

in the sixth book of the Mat.

a¯ lib al-‘a¯ liyya of Fakhr al-Dı¯n al-Ra¯zı¯

(d. 1210). He dedicates lengthy chapters to the opposition between

continuism and atomism, and carefully and honestly presents the

“geometrical proofs” that each doctrine uses as support. Two aspects

of al-R¯az¯ı’s approach are striking. First, he is dealingwith the foundations

of geometry, since the discussion of atomism leads him to discuss

such questions as the generation of geometrical objects through

motion and the fifth Postulate (in both cases, al-R¯az¯ı levels criticisms

at the mathematicians). Second, atomism is no longer simply

opposed to geometry,48 but is taken to be confirmed by at least some

geometrical postulates.

It is impossible to summarize here the numerous arguments and

counterarguments presented by al-R¯az¯ı. Very broadly, we can distinguish

two main intuitions in the argumentation of the atomists.

First, they rely on the generation of simple geometrical figures by

motion, in particular the generation of the line by the motion of a

point. A line perpendicular to a surface, moved in a direction parallel

to this surface, will trace a line on the surface. This shows that at

every instant, the line is in contact with the surface in one distinct

indivisible. Second, they appeal to tangent lines. A line can be in

contact with a circle at one single point only if indivisible parts do

exist. It is worth noting that these reflections are permitted by the

re-evaluation of the epistemic status of the imagination, which as

mentioned above is typical of classical kala¯m.

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304 marwan rashed

On the other side, the continuists appeal again and again to the

incommensurability of the side and the diagonal of the square. If both

the side and diagonal contain a finite number of indivisibles – let us

say p and q respectively – then the ratio p/q ought to be a rational

number. But of course, this is not the case. The only escape for the

adversary would be to postulate that there is vacuum between the

indivisible parts – which is, mathematically speaking, no escape at

all. The rhetoric of these polemics aside, we have already alluded to

the fact that the positive argument of the atomists tacitly renounces

the traditional finitism of atomism. The “indivisible parts” of the

latemutakallimu¯ n becomemore andmore akin to “positions” in an

Avicennian sense. We ought however to realize that in taking this

physical turn, the mutakallimu¯ n are simply bringing out a latent

aspect of classical (pre-Avicennian) kala¯m, to which Avicenna too

had been sensible.

This suffices in any case to show that from the eleventh century

C.E. onward, all parties recognize the validity, in sublunar physics,

of a theory of infinitesimal positions characterized by dynamic

moments. It is probable that these decisive transformations of the

Aristotelian doctrine of continuity, and the positing of a new relationship

between imagination and reality that made these transformations

possible, deeply influenced Latin preclassical physics49 and

European scholars of the sixteenth and seventeenth centuries.

notes

1 The exceptions are extremely rare. One thinks of al-J ¯ah. iz.

, partially

transmitted because of his literary skill, and of al-Kind¯ı’s philosophical

treatises, preserved in one Istanbul manuscript.

2 On what follows, see also Rashed [199].

3 The mutakallimu¯ n intensely debated the nature of the relation of

the minimal body to its indivisibles. See, e.g., Al-Ash‘arı¯, Maqa¯ la¯ t alislamiyy

¯ın, ed.H. Ritter, 3rd edn. (Wiesbaden: 1980), 302.16–306.13; Ibn

Mattawayh, Al-Tadhkira f¯ı ah.

k¯ amal-jaw¯ ahir wa al-a‘r ¯ ad.

, ed. S.N. Lut.f

and F. B. ‘Un (Cairo: 1975), 47–8, 193.7ff.

4 It is true that al-Naz.z.

a¯m objects to Abu¯ al-Hudhayl that parts without

extension cannot produce an extended body (see Ibn Mattawayh,

Tadhkira, 189.4–5). This is not, however, meant to prove that there are

no indivisible parts at all, but only that Abu¯ al-Hudhayl has not carried

his atomism of motion as far as he could have done. Otherwise, he

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Natural philosophy 305

would have realized that atomism is properly a theory about motion, or

kinetics, and not about “chemical” composition. Briefly put, al-Naz.z.

¯am

recognizes the existence of unextended indivisibles, but denies that they

play any role in the constitution of bodies. They are rather to be understood

as positions.

5 See D. Furley, Two Studies in the Greek Atomists (Princeton: 1967),

117ff.

6 See S. Dasgupta, A History of Indian Philosophy, 4 vols. (Cambridge:

1922), vol. I, 326–30; S. Radhakrishnan and C. H. Moore, A Sourcebook

in Indian Philosophy (Princeton: 1957), 412. Cf. S. Pines, Beitra¨ ge zur

islamischen Atomenlehre (Berlin: 1936), 112–23; for English translation

see Pines [198].

7 See M. Aouad and M. Rashed, “L’ex’eg`ese de la Rh´etorique d’Aristote:

recherches sur quelques commentateurs grecs, arabes et byzantins, 1`ere

partie,” Medioevo 23 (1997), 43–189, at 89–91.

8 See Dhanani [193], 106.

9 See al-Sharı¯f al-Murtada¯ , Ama¯ lı¯ al-Murtada¯ : Ghurar al-fawa¯ ’id wa

durar al-qala¯ ’id, ed. M. A. Ibra¯hı¯m, 2 vols. (Cairo: 1998), vol. I, 182.1ff.

10 There are however some very interesting transformations, in particular

concerning atomic motion and the continuity of time.

11 Ibn Mattawayh, Tadhkira, 162.8–11.

12 Cf. Euclid, Elements, V, def. 4.

13 Ibn S¯ın¯a [Avicenna], al-Mub¯ ah.

atha¯ t, ed. M. Bı¯da¯ rfar (Qom: 1413 A.H./

1992 C.E.), 363–4, §1136.

14 Ibn Mattawayh, Tadhkira, 162.12–14.

15 Euclid, Elements, I, def. 1.

16 See in particular W. Detel, Aristoteles: “Analytica Posteriora” (Berlin:

1993), 189–232.

17 For a good synthesis of the traditional arguments against the use of

imagination by the mutakallimu¯ n, see the “warning” (tanbı¯h) in Ibn

Maymu¯ n, Dala¯ la al-h. a¯ ’irı¯n, ed. H. Atay (Ankara: 1974), 209.21–211.25.

18 Ibn Mattawayh, Tadhkira, 163.1–4.

19 Ibn Mattawayh, Tadhkira, 163.5–8.

20 A list of the preserved fragments is to be found in D. Gimaret,

“Mat’eriaux pour une bibliographie des Gubb¯a’¯ı,” Journal asiatique 264

(1976), 277–332, at 312. To this can be added al-B¯ır ¯ un¯ı, Tah.

dı¯d niha¯yat

al-Am¯ akin li-tas.h.

ı¯h. masa¯ fa¯ t al-masa¯kin, ed. V. Bulgakov and I. Ah.mad,

2 vols. (Cairo: 1964), 185–6.

21 See the discussion (with further literature) in C. Wildberg, John

Philoponus’ Criticism of Aristotle’s Theory of Aether (Berlin: 1988),

28–37.

22 See van Ess [44], vol. III, 418–45 and below.

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306 marwan rashed

23 SeeG.W. Leibniz, Pacidius Philalethi, Academy Edition, 6th ser., vol. III

(Berlin: 1980), 528–71, at 568.1–3: “Hincmirifice confirmatur quod praeclare

olim a Theologis dictum est conservationem esse perpetuam creationem,

huic enim sententiae affine est quod a te [sc. Leibniz] demonstratur

mutationem omnem quandam esse transcreationem.” We learn

from other documents that Leibniz was deeply interested in the theories

of the mutakallimu¯ n. I present and investigate the relevant material in

my French translation of the Pacidius Philalethi, to appear in 2004.

24 See Avicenna, Risa¯ lat al-adh. awiyya fı¯ al-ma‘a¯d, ed. and Italian trans.

in Avicenna, Epistola sulla vita futura, ed. F. Lucchetta (Padua: 1969),

114–15.

25 Cf. al-J ¯ah. iz.

, Kita¯b al-h. ayawa¯n, ed. A.S.M. Harun, 7 vols. (Cairo: 1938–

45), vol. V, 113.8ff.

26 See van Ess [44], vol. III, 428–45 and vol. VI, 76–8.

27 See Th¯abit ibn Qurra, Answers to the Questions of Ibn Ussayyid, in A.

Sabra, “Th¯abit ibn Qurra on the Infinite and Other Puzzles,” Zeitschrift

fu¨ r Geschichte der arabisch-islamischen Wissenschaften 11 (1997),

1–33.

28 See M. Rashed, “A ‘New’ Text of Alexander on the Soul’s Motion,” in

Aristotle and After, ed. R. Sorabji (London: 1997), 181–95.

29 See R. Morelon, Th¯abit ibn Qurra: OEuvres d’astronomie (Paris: 1987),

LXXVIII–LXXIX.

30 See Rashed [201], 9–14.

31 See M. Naz. ı¯f, “A¯ ra¯ ’ al-fala¯ sifati al-islamiyyı¯n fı¯ al-h. araka wa

mus¯ahamatuhum f¯ı al-tamh. ¯ıd il ¯a ba‘d ma‘ ¯an¯ı ‘ilm al-d¯ın¯am¯ık¯a alh.

adı¯th,” Al-jam‘iyya al-mis. riyya li-ta¯ rı¯kh al-‘ulu¯m 2 (1942–3), 45–64.