A Small Discovery: Avicenna's Theory of Minima Naturalia

There has been a long-held misconception among historians of philosophy and science that apart from brief comments in Aristotle and Averroes, the theory of minima naturalia had to await Latin Schoolmen for its full articulation. Recently scholars have shown that far from sporadic comments on minima naturalia, Averroes in fact had a fully developed and well-integrated theory of them. in this study, i complement these scholars’ important work by considering Avicenna’s place in the history and development of the doctrine of the minima naturalia. There is no study to date that mentions Avicenna in connection with this doctrine despite the fact that he dedicated an entire chapter to it in his Physics, yet Avicenna’s account is at least as developed as and even better integrated than Averroes’s presentation. The present study situates Avicenna’s position within the more general history of atomism, and introduces Avicenna’s “new argument” for natural minima. The argument is important not only for its novelty but also because it shows how Avicenna integrated Aristotle’s account of minima naturalia into a theory of mixture as well. keywords Avicenna/ibn Sīnā, Minima naturalia, Atomism, continuity/ continuum, Mixture the theory of minima naturalia is loosely one of atoms;* however, unlike the atoms of Democritus, the minimal parts of Epicurus, or the indivisible substances of the Mutakallimūn, all of which are physically and conceptually indivisible, minima naturalia, while being physically indivisible, can be divided at least conceptually ad infinitum.1 Until recently, historians of science believed that with the exception of some passing remarks in Aristotle and some brief exegesis by Greek commentators in the ancient world and then Averroes in the islamic world, the details of a theory of minima naturalia had to await the Latin Schoolmen of the thirteenth century. * Jon McGinnis is Professor of Philosophy at the University of Missouri–St. Louis. * i would like to thank all the participants at the Avicenna conference in Park city, UT (June, 2010) for their questions and comments on an earlier version of this paper. i am additionally extremely grateful to Taneli Kukkonen, Peter Adamson, and an anonymous JHP reviewer for their incisive and insightful comments. As always, any mistakes are wholly my own. 1 While there is a debate whether Democritean atoms should be considered both physically and conceptually indivisible, there is no question about whether the Epicurean and kalām sorts are both. For a discussion of the debate concerning Democritus, see Sorabji, Time, Creation, and the Continuum, 354–57. 2 journal of the history of philosophy 53:1 January 2015 Ruth Glasner in two groundbreaking works has shown that, far from having just sketchy and sporadic comments about minima naturalia, Averroes had a fully developed and well-integrated theory of them.2 Unlike earlier historians of science, whose understanding of Averroes’s physical theory has been limited to the Latin translation of his great commentary on Aristotle’s Physics, Glasner consulted all three of Averroes’s Physics commentaries—short, middle, and long—which requires an appreciation of not only Latin, but also Arabic and Hebrew. As a result of Glasner’s careful studies, and even more recently that of cristina cerami, we now have a more complete picture of both Averroes’s understanding of the minima naturalia and his place in the history of atomism. in this study, i want to complement this important work by considering Avicenna’s place in the history and development of the doctrine of the minima naturalia. There is no study to date that mentions Avicenna in connection with this doctrine despite the fact that he dedicated an entire chapter to the subject in his Physics.3 This omission is no doubt due in large part to the fact that the section where Avicenna discusses this issue was not translated into Latin. Yet Avicenna’s account is at least as developed and perhaps even better integrated than Averroes’s presentation. (i say ‘better integrated’ because Avicenna was not limited by the commentary genre that Averroes favored, and so had more freedom to bring things together when and where he saw fit.) in order to situate and to appreciate Avicenna’s contribution, then, i begin with a very brief historiography, followed by a short history of minima naturalia, starting with the locus classicus in Aristotle’s Physics and going through Aristotle’s Greek commentators, with a particular emphasis on John Philoponus. The majority of this study is a presentation of the theory and philosophy behind Avicenna’s understanding of the minima naturalia that considers what he took from his predecessors as well as what is novel in his theory. i hope to show that despite the neglect that Avicenna has received concerning his place in the history of atomism, he was in fact a pioneer in this subject and actually laid the groundwork from which Averroes, and indirectly subsequent Latin Schoolmen, developed their accounts of a natural minimum. 1 . h i s t o r i o g r a p h y Historians of science have not represented the contributions of natural philosophers working on the notion of minima naturalia in Arabic adequately.4 Pierre Duhem in his grand Le système du monde has only one sentence in which he mentions Averroes’s supposed lack of interest in the topic,5 and the usually thorough Anneliese Maier wholly neglects the contribution of Muslim thinkers 2 Glasner, “ibn Rushd’s theory of minima naturalia,” and Averroes’ Physics. 3 Avicenna, Physics, iii.12; the numeric references to Avicenna’s works are book number (in Roman numerals), followed by a period and then chapter number (in Arabic). in those cases where this number is followed by a comma and then another Arabic numeral, the second number refers to the paragraph number in the editions of Avicenna’s works from The islamic Translation Series. 4 For a brief study of the historiography of minima naturalia, see Murdoch, “The Medieval and Renaissance Tradition of Minima Naturalia,” 91–131, esp., 91–96. 5 Ce texte ne semble guère avoir retenu l’attention d’Averroès, car celui-ci se borne à écrire, en son commentaire: “il est manifeste de soi que le volume de la chair est limité en grandeur comme en petitesse” (Duhem, Le système du monde, Vii.42). 3 avicenna’s theo ry o f m i n i m a n at u r a l i a on this topic.6 Andrew G. van Melsen dedicated approximately two pages of his study on atomism, From Atomos to Atom, to Averroes’s account available in Latin.7 norma Emerton likewise reserved about as much space in her work The Scientific Reinterpretation of Form to Averroes’s theory of minima naturalia, but now with an eye to how that theory was linked with another important medieval debate—namely, the theory of mixtion or primary mixture.8 Ruth Glasner is the first to begin giving philosophers working in the medieval islamic world their due on this subject. in both an article and chapter of her book, Averroes’ Physics, she chronicles Averroes’s place in the history of the minima naturalia. ostensibly, Averroes’s account of the minima naturalia was, Glasner tells us, an attempt to reconcile seemingly contradictory claims in Aristotle’s Physics. The first such claim is Aristotle’s criticism of Democritean atomism, which committed Aristotle to the belief that all natural magnitudes are continuous and so potentially divisible infinitely. The second is Aristotle’s critique of Anaxagoras, in which he maintains that there is a limit beyond which natural substances cannot be further divided.9 Even more recently cristina cerami, while challenging some of Glasner’s claims, has further extended our knowledge of Averroes’s theory of the minima naturalia.10 in none of the works of this set of historians, starting with Duhem and going through Glasner and cerami, is there ever mention of Avicenna’s place in this history. Before i can tell his story, however, i need to begin by relating the origins of the idea of minima naturalia in, first, Aristotle, and then some of his later Greek commentators, most notably John Philoponus. 2 . a h i s t o r y o f m i n i m a n a t u r a l i a i n t h e g r e e k w o r l d Aristotle introduces the idea of natural minima (elachista) at Physics 1.4, which became the locus classicus for the doctrine of minima naturalia. in that passage, he critiques Anaxagoras’s principle that “everything is in everything.”11 Arguably, Aristotle’s most important proof concerning minima naturalia runs thus: if the part can be indefinitely big or small—i call a ‘part’ the ingredient into which the whole is divided—then necessarily the thing itself can be too. So if it is impossible that an animal or plant be indefinitely big or small, then clearly the part is not such as [to be indefinitely big or small] either, for the whole also will be such. now, flesh, bone, and the like are parts of an animal, and fruits [the parts] of a plant. Hence, it is clear that flesh, bone, and the rest cannot be indefinitely big in the direction of greater or smaller. (Physics 41.4, 187b13–21)12 Aristotle’s thesis is that the parts that constitute natural kinds—parts such as flesh, blood, bone, fruits, and the like—have a definite limit with respect to their size, 6 Maier, Die Vorläufer Galileis im 14. Jahrhundert, 179–96. 7 Van Melsen, From Atomos to Atom, 58–60. 8 Emerton, The Scientific Reinterpretation of Form, 87–88. 9 Glasner, “ibn Rushd’s Theory of mimina naturalia,” 10–14. 10 cerami, “Mélange.” 11 For a discussion of Anaxagoras’s principles, see Drozdek, “Anaxagoras and the Everything in Everything Principle.” 12 All translations, whether Greek or Arabic, are my own. 4 journal of the history of philosophy 53:1 January 2015 both in greatness and smallness, namely, a minimum and maximum. The argument for this thesis—limiting the present discussion merely to the idea of minima—is straightforward enough. Aristotle takes it as impossible that cats, for example, should be indefinitely small. Thus, any premise that leads to the possibility of indefinitely small cats must be false. if Anaxagoras is correct, and everything is in everything such that it is possible, at least in principle, to extract out from a given substance indefinitely small cat flesh, cat blood, and all the sundry cat organs and bits, then these indefinitely small cat parts should, again in principle, be able to constitute an indefinitely small cat. Since the assumption is that indefinitely small cats are impossible, the premise that gave rise to the absurdity, namely, that there are indefinitely small traces of everything in everything, Aristotle concludes, must be false. Later in the same passage, the existence of minima naturalia plays an essential role in two other arguments against Anaxagoras, but in those passages Aristotle takes the existence of minima naturalia as demonstrated, presumably on the strength of the argument that i have just presented.13 According to the medieval Arabic bibliographers, al-nadīm and al-Qif. ti, commentaries on book i of the Physics by the later Greek commentators, Alexander of Aphrodisias, Themistius, and John Philoponus were available in Arabic translation. Thus, in addition to Aristotle’s own Physics, these are likely the Greek works to have influenced the understanding concerning minima naturalia of subsequent natural philosophers working within the medieval islamic milieu. of these, Alexander’s commentary is no longer extant in either Greek or Arabic, with Simplicius preserving only fragments of it in his own Physics commentary. Additionally, there are a few scattered fragments of Alexander’s commentary extant in Arabic but none from book 1.14 As for Themistius’s Paraphrase, it remains fairly close to Aristotle’s text. in contrast, Philoponus’s commentary exists both in Greek and Arabic and is somewhat extensive.15 in addition to the Physics 1.4 passage, which is Aristotle’s clearest statement concerning minima naturalia, scholars also see the doctrine hiding behind Aristotle’s comments in On Generation and Corruption 1.10 and in On Sense and Sensibilia 6. On Generation and Corruption 1.10 includes Aristotle’s theory of primary mixture—what historians of science sometimes refer to as mixtion. As will become apparent, this passage seemed to exert some influence on Avicenna’s theory of minima naturalia. As for the fate of On Generation and Corruption in the Arabic world, unlike the Physics, the Arabic translation of that text is no longer extant. in addition to the work itself, al-nadīm and al-Qif. ti mention the commentaries of, again, Alexander of Aphrodisias, Themistius, and John Philoponus. of these, Themistius’s commentary is not extant. Alexander’s commentary was thought to be lost, but an Arabic translation of On Generation and Corruption 2.2–5 was preserved in the alchemical work, Kit̄ab al-Ta. srīf (Book of Transformation), of Jābir ibn . Hayyān (721–815), but again it would seem nothing concerning our passage remains.16 13 See Aristotle, Physics 187b27–32 and 187b35–188a2. 14 See Giannakis, “Fragments from Alexander’s Lost commentary on Aristotle’s Physics.” 15 Philoponus (=Ya. hyá), In Phys. 16 See Gannagé, “Le commentaire d’Alexandre d’Aphrodise In de generatione et corruptione,” and On Aristotle’s on coming-to-be and Perishing 2.2–5. 5 avicenna’s theo ry o f m i n i m a n at u r a l i a Finally, while the Arabic translation of Philoponus’s commentary of On Generation and Corruption is no longer extant, the complete text still exists in Greek.17 As for On Sense and Sensibilia, al-nadīm claims that the work was unknown even at the end of the tenth century. Still, there appears to be evidence that Avicenna had access to some form of it, although whether this text influenced his theory of minima naturalia is perhaps impossible to say.18 in light of the foregoing, Philoponus’s discussions of the relevant passages are clearly the most complete and latest of the Greek commentaries, and thus incorporate many of the earlier advances on the topic of the minima naturalia. Moreover, Philoponus’s commentary (certainly his Physics commentary) had the greatest apparent influence on Avicenna’s understanding of Aristotelian physics. Hence, i focus primarily upon Philoponus’s account of natural minima. The most notable advancement in Philoponus’s commentary is that he shifts the discussion from natural substances and their parts to the form (eidos) of those substances and their parts. This shift does not appear to be new to Philoponus. Themistius occasionally mentions form in his exposition of our passage.19 in fact, the introduction of form may go back as far as Alexander of Aphrodisias, since Simplicius suggests that Alexander and Themistius interpreted Aristotle similarly.20 The introduction of form is probably part of the commentators’ larger project of developing an integrated Aristotelian physics. So, for example, in his On Generation and Corruption 1.10 commentary, Philoponus situates Aristotle’s discussion of mixture within the broader context of refuting Anaxogoras’s principle, “Everything is in everything,” and Philoponus’s theory of mixture developed there presupposes the account of form presented in arguing for minima naturalia, a point to which i return below.21 Turning to Philoponus’s version of the argument for minima naturalia, he begins by claiming, “Every form naturally subsists in some definite quantity, and it is neither naturally augmented to just any degree of largeness nor naturally diminished to just any degree of smallness, but rather there is a certain boundary to greater and smaller beyond which the form does not exist.”22 Philoponus next attempts, in perhaps a not altogether successful way, to defend his newly modified major premise, namely, that the form cannot exist beyond a certain natural maximal and minimal quantity. This defense itself goes beyond Aristotle, for Aristotle merely asserted that natural substances could not be of 17 Philoponus, In Gen. et Corr. 18 See Peters, Aristoteles Arabus, s.v. “Liber Sensus et Sensati,” 45–47. 19 Themistius, In Aristotelis Physica Paraphrasis, 14, 25–26 and 15, 13–16. 20 Simplicius, In Aristotelis Physicorum Libros Quattuor Priores Commentaria, 169, 5–25. Unfortunately, while Simplicius quotes Alexander extensively, there is no mention of Alexander’s using form in his interpretation. Still, such an absence need not be an indicator that Alexander did not re-frame the argument in terms of form, since Simplicius’s emphasis in quoting Alexander is on the apparent fact that Alexander’s version of Aristotle’s Physics differed from that of Simplicius’s own. 21 For the significance of Anaxagoras’s principle, see Philoponus, In De Gen. et Corr., 192, 10–16; and the same for the attenuation or reduction of form; and Philoponus, In De Gen. et Corr., 198, 18–19, for form’s need of a minimum quantity. Discussions of Philoponus’s theory of mixture can be found in De Haas, “Mixture in Philoponus,” and Wood and Weisberg, “interpreting Aristotle on Mixture.” 22 Philoponus, In Phys., 96, 27–30. 6 journal of the history of philosophy 53:1 January 2015 just any size whatsoever. in the case of a maximum, Philoponus claims that the form fades, or becomes attenuated (exitēla), the more it must spread throughout a given quantity. indeed no human [for example] would come to be a hundred feet or equal to the size of the cosmos; for we are not assuming some human in thought but in reality. clearly, then, [a real human] cannot be augmented to just any size and beyond all. instead, there is a certain limited size beyond which [a real human] cannot be augmented, for the form extended over a large subject becomes attenuated. (Philoponus, In Phys., 97, 4–9; emphasis added) Presumably, then, there are decided maxima beyond which forms simply perish, as the drop of wine perishes as it spreads throughout 10,000 gallons of water.23 Philoponus next posits that just as there is a maximum, so there must be a minimum quantity beyond which form cannot exist. it should be noted, however, that he merely asserts this last point and does not provide an argument for it; for the attenuation argument that he used to justify a natural maximum clearly is inapplicable to minima, and, if anything, there would be a concentration of the form. Philoponus next turns to a mathematically motivated objection to the idea of minima naturalia. This objection presupposes Aristotle’s discussion of whether magnitudes are continuous or discrete from Physics 6, and so we should briefly linger over this issue. The driving question of Physics 6 is whether magnitudes such as distance, motion, and time are continuous, and so are potentially divisible ad infinitum, or whether they are discrete such that a process of division ultimately terminates in certain indivisible parts or atoms. Despite what we have seen in Aristotle’s Physics 1.4 concerning his belief in minima naturalia, Aristotle unequivocally denies that magnitudes are composed of atoms, and instead holds that all magnitudes must be continuous, and so potentially divisible infinitely. Moreover, unlike his passing remarks concerning minima naturalia, Aristotle spends all of Physics 6 engaged in, at times, highly technical argumentation that magnitudes must be continuous. Given the seeming discrepancies between Physics 1.4 and book 6, the objection that Philoponus considers is a pressing one. it runs thus:24 since all magnitudes are continuous and so potentially divisible ad infinitum, let the purported minimum amount of flesh, for example, be divided. The resultants of the division are either themselves flesh or they are not flesh. if they are flesh, then there are quantities of flesh less than the minimum amount of flesh, which is absurd. “if the divided things are not flesh, then how will they produce the composite flesh again? if the flesh is homoeomerous, clearly the parts of this would be flesh too.”25 This last horn of the dilemma, to which i return again when considering Avicenna’s account of minima naturalia, can be framed thus: if the resultants of the division are not flesh, then a composite of flesh would not be uniformly flesh through and 23 Philoponus does not make this final point here, but the example and the assertion that forms require a certain quantity are given in Philoponus, In De Gen. et Corr., 198, 11–19. 24 Philoponus, In Phys., 98, 13–21. 25 Philoponus, In Phys., 98, 17–19. 7 avicenna’s theo ry o f m i n i m a n at u r a l i a through, but instead would be a collection of distinct non-flesh parts, which the objector and Philoponus find absurd.26 Philoponus’s response to this objection again draws on his introduction of form to explain minima naturalia, for, says Philoponus, flesh can be considered either qua form or qua magnitude. Qua magnitude the flesh is continuous and so potentially divisible infinitely, in which case there is no minimal magnitude.27 This is all that Philoponus says here about division qua magnitude and unfortunately he leaves the notion of division (diairesis) underdetermined.28 Division here could mean physically dividing the magnitude into smaller and smaller portions or merely conceptually dividing the magnitude, as in the mathematical series 1, 1/2, 1/4, 1/8, 1/16, . . . 1/2n . . . ∞. Since the mathematicians (hoi apo tōn mathmētōn) raise the puzzle, one might suspect that the latter form of division is meant, and yet when Philoponus turns to division qua form, it would seem that he intends physical division. i return to the suggestion that there are two distinct kinds of division when discussing Avicenna on division. As for division qua form, the flesh qua form is not infinitely divisible. Any division that results in a quantity less than that required for the subsistence of the form brings about the destruction of the form. To make his point graphically, he has us take as an example a human. on the one hand, we can consider the human qua magnitude as, for instance, 6 feet tall. in this case the individual can be divided into indefinitely smaller magnitudes: 3 feet, 18 inches, 9 inches, and so on. While there are smaller and smaller magnitudes, magnitude is never completely destroyed but always remains. on the other hand, if we consider the human qua (human) form, then, when we divide our ill-fated victim into feet, legs, torso, and head, we do not have smaller instances of the human form. We simply no longer have the form of human at all but instead a dismembered corpse. Species forms, in short, cannot survive division the way magnitudes can. Similarly, says Philoponus, if there is to be flesh at all, that flesh requires the presence of the form of flesh, and the form of flesh is dependent upon some minimal quantity of matter. The inference is perhaps a bit too quick, for while it seems fairly obvious that the form of human is not fully localized in any part of the person, like the head, it is not equally clear that the form of flesh is not fully present in any bit of flesh regardless of how small. The problem is that even if one is convinced that Philoponus’s arguments work at the level of the species form, it is not as obvious how those arguments translate at the level of the forms of mixtures. Whatever the limits of Philoponus’s argument, he has certainly gone beyond Aristotle and helped motivate Aristotle’s original argument. natural substance must have natural minima, since there are minimum quantities required for the subsistence of forms. As for how Philoponus’s version of the argument is an advancement over Aristotle’s original version, consider the following. Aristotle 26 in fact, Aristotle also commits himself to the position that flesh, blood, and the like are uniformly flesh, blood, etc., through and through, when he develops his theory of primary mixture in On Generation and Corruption 1.10. 27 Philoponus, In Phys., 98, 21–2. 28 For a discussion of division within Philoponus, see De Haas, John Philoponus’ New Definition of Prime Matter, 116–20. 8 journal of the history of philosophy 53:1 January 2015 frequently notes that substance (ousia) is said in three ways: matter, form, and the composite.29 Moreover, of these different ways of speaking of substance, there is a sense that substance-as-form is primary.30 Aristotle’s original argument, however, was solely in terms of natural (i.e. composite) substances. Thus, at the very least, Philoponus’s argument in terms of form is a generalization of Aristotle’s original argument, recasting it now in terms of the more general or basic notion of substance-as-form. Still, there is also a sense in which Philoponus’s presentation might not be merely commentary but a new argument altogether. As noted above, a key element in Philoponus’s version of the argument is that the form fades and becomes attenuated. This conception of a form’s being able to fade is not unique to Philoponus’s Physics commentary but is found in other works by him as well. For example, in commenting On Generation and Corruption 1.10, Philoponus explains the difference between generation (genesis) and mixture (mixis/krama) thus: concerning generation, the matter of the air [for example] is potentially air but actually water, while in a mixture what is mixed subsists potentially, not the matter itself but rather the very form is reduced [kekolasmena]. Because of this it is in potency, since it is neither pure nor such as it was before the mixing. (In De Gen. et Corr., 192, 10–14; my emphasis) it would seem that Philoponus has some idea of the intension and remission of (species!) forms.31 it would go well beyond the scope of the present paper to adjudicate as to whether Philoponus in fact has a doctrine of intension and remission of forms as well as whether the historical Aristotle may have held that theory. nonetheless, it is safe to say that if Philoponus’s theory of form is substantively different from Aristotle on this point (and my suspicion is that the two are different), then the present argument represents a creative moment in the history of atomism.32 3 . t h e a v i c e n n a n b a c k g r o u n d t o t h e m i n i m a n a t u r a l i a The preceding discussion provides roughly the theory of minima naturalia as it appeared at the end of the late Antique Period and as it would have been passed on to the medieval islamic world. Turning now to Avicenna, his theory of minima naturalia comes at the end of book iii of his Physics, which as a whole is dedicated to the topic of discrete and continuous magnitudes. consequently, Avicenna’s account of minima naturalia presupposes two things: an understanding of his rejection of atomism—both as atomism was inherited from the Greek tradition 29 Aristotle, De anima 2.1, 412a6–9; Metaphysics 7(Z).3, 1028b33–1029a3; and Metaphysics 8(H).1, 1042a24–31. 30 See, for instance, Metaphysics, 7(Z).3, 1029a26–33. 31 While the later medieval tradition certainly found the doctrine of insensio et remissio in the works of Aristotle, it was with respect to accidental forms. For the medieval Latin context see Sylla, “Medieval concepts of the Latitude of Forms: The oxford calculators,” and The Oxford Calculators and the Mathematics of Motion, 1320–1350: Physics and Measurement by Latitudes; also see Dumont, “Godfrey of Fontaines and the Succession Theory of Forms.” 32 one study that touches on Philoponus’s theory of forms in a significant way is Macierowski and Hassing, “John Philoponus on Aristotle’s Definition of nature.” 9 avicenna’s theo ry o f m i n i m a n at u r a l i a and more importantly as contemporary Muslim theologians conceived it—and an appreciation of his defense and (somewhat novel) understanding of the continuum. Thus, let me provide some background to his view concerning discrete and continuous magnitudes. Perhaps the most important thing to bear in mind when considering Avicenna’s critique of atomism is the nature of the atoms that he wants to reject. He happily concedes that there might be bodies for which there are no physical means to divide them further. He thus recognizes that certain substances may as a matter of fact be indivisible, and so in a literal sense be atoms. The philosophically dubious atoms, at least by Avicenna’s lights, are those that are not only physically indivisible, but also, and more importantly, conceptually indivisible. These are the minimal parts of Epicurus and the indivisible parts of the islamic speculative theologians (al-juz’ alladhī lā yatajazza’u).33 Despite the fact that these atoms were believed to be conceptually indivisible, they were nonetheless thought to be space occupying (muta. hayīz). For many it might seem problematic to predicate of atoms both that they occupy space—and so, it would seem, are extended—and yet that they are conceptually indivisible. Be that as it may, there were a number of puzzles, involving purported absurdities surrounding continua and their infinite divisibility, that were quickly resolved if one had an ontology of discrete, that is, atomic, magnitudes. As for the actual arguments that Avicenna uses to dismiss atomism and embrace continua, these need not bother us here.34 Suffice it to say that Avicenna believed that any purportedly indivisible magnitude could be divided into conceptually distinct parts. Moreover, he argued that without positing continua, the conclusions of Euclidean geometry could not even be approximated, and yet geometry was the most well established science of the time. Finally, as for the absurdities that seemingly arise from the infinite divisibility of continua, Avicenna maintained that once one properly understands the nature of the continuum, those absurdities are seen for the sophistries that they are. (i return to this final point at the end of this section.) Thus, turning to Avicenna’s theory of the continuum, a complete account would require a discussion of his theory of matter, corporeality—or more exactly, the form of corporeality (. sūrat al-jismīya)—and three-dimensionality, most of which would take me far afield from the issue of minima naturalia.35 instead, allow me to focus primarily on Avicenna’s fullest treatment of the continuum as it appears at Physics iii.2, with the occasional side remark about the other issues to clarify his discussion.36 While Physics iii.2 is Avicenna’s most detailed discussion of 33 For Epicurus, see Furley, Two Studies in the Greek Atomists, and for the mutakallimūn, see Dhanani, The Physical Theory of Kalām, 90–140. For Avicenna’s knowledge of both Greek and islamic atomism, see Avicenna, Physics, iii.3. 34 See Avicenna, Physics, iii.4. For a study of Avicenna’s refutation of atomism, see Lettinck, “ibn Sīnā on Atomism.” 35 For discussions of the relation between matter and corporeality, see Avicenna, Physics, i.2, 4, and Metaphysics, ii.2–3. For studies see Hyman, “Aristotle’s ‘First Matter’ and Avicenna’s and Averroes’ ‘corporeal Form,’” and Stone, “Simplicius and Avicenna on the Essential corporeity of Material Substance.” 36 For a discussion of Aristotle’s various treatments of the continuous, which influenced both positively and negatively Avicenna’s own understanding, see Glasner, “ibn Rushd’s Theory of Minima Naturalia,” 11. 10 journal of the history of philosophy 53:1 January 2015 continuity, taken alone it can be disjointed in places. Fortunately, certain remarks that he makes about the continuum in a letter to the Vizier Abū Saʿ d significantly smooth out the Physics discussion.37 Both texts observe that continuity (itti . sāl) is an equivocal notion, and in the Physics, Avicenna identifies three ways in which one might speak of the continuous.38 Two senses of ‘continuous’ are understood relative to something else (Avicenna mentions only one of these senses in his letter to Abū Saʿ d), while the third concerns the continuous considered in itself. one of the relative senses of being continuous identified in the Physics—the one omitted in the letter to Abū Saʿ d—is said of an object inasmuch as it is moving. This form of continuity occurs, Avicenna tells us, when . . . one side of the continuous thing is moved in a direction away from the other, the other follows it. . . . The two extremities can be two in actuality, and there can be something actually contiguous after adhering during the motion. The extremity of what is continuous and that with which it is continuous can be one, but it is termed continuous in the present sense not inasmuch as its extremity and that of the other are one, but only inasmuch as it follows it during the motion in the aforementioned way. (Avicenna, Physics, iii.2, 9) Here continuity is relative to the motion. Thus, for example, if one considers a train, the engine, caboose, as well as any intervening boxcars do not share a common limit, and yet they move together such that relative to that motion one can say of them that they make up a continuous moving whole. The second relative sense, which Avicenna considers, in both his Physics and the letter to Abū Saʿ d, is the continuous relative to a limit (.taraf ), which he finds in Aristotle’s Physics 5.3 227a11–12, namely, “that whose limit is the same as a limit of something else.” This case occurs when something is continuous relative to a shared limit that is one and the same for two parts. (in his own Physics, Avicenna unfortunately gives the impression that there are two subspecies of this relative notion of continuity: absolute and accidental. in the letter to the Vizier Abū Saʿ d, however, it is evident that the accidental continuity that Avicenna discusses in the Physics is in fact the continuous in itself.)39 An example of this second relative sense of being continuous is two lines forming an angle, since each line has one and the same common point at the angle’s vertex, and so can be said to be continuous relative to the limit according to Aristotle’s Physics 5.3 account of continuity. nonetheless, the two parts are actually distinct even though they share one and the same common limit. Avicenna’s account of the continuous in itself, which he perhaps misleadingly also labels as accidental continuity, is, he tells us, “the definition that is mentioned in [Aristotle’s] Categories [6, 5a1–2], namely, ‘that for whose parts a common limit can be found at which they meet.’”40 in his Physics, Avicenna describes the continuous in itself as that magnitude that in itself has no parts (lā juz’ ),41 but in which one 37 For the text and a French translation, see Avicenna, Letter to Abū Sa‘d. 38 Avicenna, Physics, iii.2, 8–10. 39 Avicenna, Letter to Abū Sa‘d, 42–44. 40 Avicenna, Letter to Abū Sa‘d, 43. 41 The description of the continuous in itself having no parts is explicitly made at Avicenna, Physics, iii.3, 1. 11 avicenna’s theo ry o f m i n i m a n at u r a l i a can posit limits in it in an accidental sense, a point to which i return shortly. The most obvious instance of parts that the continuous in itself must lack is that of physical parts, as in the case of, for example, the articulation of the bones of the arm, and, more generally, actual parts, as in the two sides of the angle mentioned above. Additionally, the continuous should not be thought to have even latent parts present within it waiting to emerge into actuality. This theory of latency (kumūn) is most commonly associated with ibrahim al-na. z. zām (c. 775–845). Avicenna clearly wants to distinguish his theory of continuity from al-na.z. zām’s, who believed that an actual infinity of parts are latent within a continuous object.42 For Avicenna, the continuous only ever has accidental parts, parts that result through a psychological act of positing accidental limits within the continuous object. Avicenna describes these accidental limits thus: [They are] like what happens when our estimative faculty imagines or we posit two parts for a line that is actually one, where we distinguish one [part] from the other by positing. in that way, a limit is distinguished for [the line] that is the same as the limit of the other division. in that case, both are said to be continuous with each other. Each one, however, exists individually only as long as there is the positing, and so, when the positing ceases, there is no longer this and that [part]; rather, there is the unified whole that actually has no division within it. now, if what occurs through positing were to be something [really] existing in the thing itself and not [merely] by positing, then it would be possible for an actually infinite number of parts to exist within the body (as we shall explain), but this is absurd. (Physics, iii.2, 8) Accidental limits thus occur within the continuous in itself when the single unified continuous whole is distinguished into two (or more) conceptual parts through some act of positing (far. d), such as pointing toward a uniform surface and saying, “this side,” while pointing to the right, and “that side,” while pointing to the left. The limit in this case—and Avicenna is adamant about this point—arises only as an accidental result of the positing, and in fact that limit ceases once the positing ceases.43 indeed, to maintain that the part still remains after the pointing stops, says Avicenna, is tantamount to saying that the pointing itself remains when the pointing has stopped. it is simply false, warns Avicenna, to think that the limit in this case really exists in the continuum. Moreover, he warns his reader not to mistake the description (rasm) of the continuous in Aristotle’s De caelo (1.1, 268a5–6)44 which is given in terms of “that which can be divided into things always susceptible to [further] division,” as constituting the essence (māhīya) of the continuous in itself.45 in other words, it is not the essence of the continuous to have a potential infinity of divisions within it. instead, this description is at most a concomitant of the continuous, which must be demonstrated to belong to it necessarily. in other words, one must be careful even when one speaks of potential limits inhering within the continuous in itself, if by ‘potential’ one means, again, something latent within the continuum waiting