WILLIAM CRATHORN S MEREOTOPOLOGICAL ATOMISM. Aurélien Robert

Transcription

1 WILLIAM CRATHORN S MEREOTOPOLOGICAL ATOMISM Aurélien Robert Little is known about Crathorn s life and career, except that he was a Dominican friar who lectured on Peter Lombard s Sentences in Oxford around his only surviving work. 1 He s often considered by recent scholars as a secondary witness to the most important debates in fourteenth-century philosophy, but rarely as an inventive thinker. Nonetheless, isolated arguments have been carefully examined, notably his solution to scepticism 2 and his criticism of William of Ockham s theory of mental language. 3 But, concerning the question of the existence of indivisibles in a continuum, he s generally mentioned as a simple doxographer, even if John E. Murdoch recognized him as a defender of an original theory of speed in such a context. 4 Contrary to this common reading, we would like to show how singular and interesting is Crathorn s atomist position, which can neither be reduced to that of his Oxonian predecessors Henry of Harclay and Walter Chatton, nor identi ed with his Parisian contemporaries Gerard of Odo, Nicholas Bonet and Nicholas of Autrecourt. Indeed, Crathorn doesn t limit himself to mathematical analysis of the divisibility of a continuum, but puts forth the foundations of a genuine atomist physics. In this theory, an indivisible is not conceived as a purely mathematical point anymore, but rather as the ultimate component of reality, from which a conception of motion rivalling the one defended by Aristotle in his Physics can be derived. At the heart of Crathorn s physics, the indivisible thus acquires a new ontological status: it is a thing (res), actually existing (existens in actu) and not only 1 The Questions on the book of the Sentences have been edited by Hoffmann, Quaestionen zum ersten Sentenzenbuch. For biographical elements, cf. Schepers, Holkot contra dicta Crathorn: I. Quellenkritik und biographische Auswertung der Bakkalaureatsschriften zweier Oxforder Dominikaner des XIV. Jahrhunderts; Courtenay, Adam Wodeham: An Introduction to his Life and Writings. For a general survey, see our William Crathorn. 2 Cf. Pasnau, Theories of cognition in the Later Middle Ages. 3 Cf. Panaccio, Le langage mental en discussion: ; Perler, Crathorn on Mental Language. 4 For example, Murdoch, Atomism and Motion in the Fourteenth Century.

2 128 aurélien robert potentially, which possesses a kind of extension (quantitas dimensiva) and a certain nature or perfection (quantitas perfectiva). Moreover, much of Crathorn s effort was directed towards demonstrating that the number of indivisibles in nature is nite, if not countable. Therefore, as he will conclude, the indivisibles are real parts of a continuum. Our aim in this paper is to show how Crathorn brought out an important turn in his natural philosophy, from the indivisible considered as a mathematical point to the atom conceived as a physical entity. Needless to say that he isn t the only one who endeavoured to conceptualize the nature of indivisibles in a more physicalist fashion, but in some respect he s probably one of the most systematic atomistic thinker of the rst half of the fourteenth century even if, as we shall see, some theoretical hesitations still persist in his view. To understand Crathorn s originality, it should be remembered that his ontological analysis of the indivisible is based on two distinctive features: it depends on a special conception of the part-whole relation; and it is supported by a systematic use of the notions of place (locus) and position (situs). For this reason, we may call Crathorn s theory a mereotopological atomism. 5 The notion of place is so central that it will serve as a tool for reconstructing the physical nature of atoms, that is to say their quantity (dimension and perfection); for, as we shall see, indivisibles are primarily de ned by the place they occupy in the world. Consequently, this mereotopological strategy will also allow him to avoid falling into some of the traditional blind alleys of indivisibilism, e.g. the explanation of the composition of a quantity from extensionless points and the problem of contact between indivisibles. Before going through Crathorn s arguments, we must mention how important and recurrent is the topic of indivisibilism in the Questions on the Sentences. 5 We borrow the term mereotopology from contemporary science and philosophy, using it in a slightly different way. For a contemporary de nition, see Smith, Mereotopology: A Theory of Parts and Boundaries, p. 287: Mereotopology... is built up out of mereology together with a topological component, thereby allowing the formulation of ontological laws pertaining to the boundaries and interiors of wholes, to relation of contact and connectedness, to the concepts of surface, point, neighbourhood and so on.

3 william crathorn s mereotopological atomism The Importance of the Continuum Question in Crathorn s Writings The most signi cant occurrence of our topic is to be found in q. 3 of the Questions on the Sentences after a long discussion about the nature of natural knowledge and language (q. 1 and q. 2). As a consequence, one shouldn t be surprised by the gnoseological pretext of this rst long digression: does the wayfarer understand cum continuo et tempore. 6 The few commentators who have studied the continuum question in Crathorn have usually restricted their reading to this sole passage, but many others are relevant in the Quaestiones. As early as q. 1, Crathorn adopts atomist explanations of natural phenomena as the diffusion of light when explaining the multiplication of species through a medium. 7 In q. 4, when questioning the possibility of knowing God s in nity, our topic reappears in the explanation of the in nite. 8 But the most important passages on atoms and continua can be traced in Crathorn s discussion of Aristotle s categories, especially in his long developments in q. 14 on quantity, and in q. 15, entirely dedicated to the quantity of indivisibles. 9 Finally, after developing his own interpretation of Aristotelian categories, Crathorn turns to the nature of time and motion in q. 16, 10 where he de nes more precisely the central elements of what could be the foundations of an atomist physics. Therefore, for a complete reconstruction of Crathorn s theory, one must at least take into account all these passages in which indivisibilism is at stake. To be absolutely exhaustive, one should also look at the series of forty-two quodlibetal questions found in a partially unedited manuscript conserved in Vienna (Cod. Vindob. Pal. 5460). 11 Most parts of the text have been inserted in the Questions on the Sentences, but the manuscript also contains some interesting developments devoted to indivisibles, as 6 In Sent. q. 3, pp : Utrum viator intelligat cum continuo et tempore. 7 In Sent. q. 1, p In Sent. q. 4, pp In Sent. q. 15, pp : Utrum aliqua res omnino indivisibilis possit esse longa, lata et profunda. 10 In Sent. q. 16, pp : Utrum tempus sit aliquid positivum reale vel aliqua res producta a deo vel ab aliquo. 11 For a brief description of the manuscript, cf. Richter, Handschriftliches zu Crathorn.

4 130 aurélien robert another version of the question on the continuum 12 and some discussions on the in nite. 13 In this short paper, we ll limit ourselves to the edited Questions on the Sentences, without ignoring the aforementioned relevant passages. To begin with, we must examine the mereological principles on which the atomist edi ce is built. 2. The Mereological Composition of the Continuum The gnoseological excuse for discussing the divisibility of continuous quantities helps Crathorn to reveal a rst apparent paradox in Aristotle s analysis. If a continuum were composed of an in nite number of parts, shouldn t we accept that the wayfarer s intellect should understand the in nite when thinking of a continuous body? If not, could we still say, as Aristotle himself asserts in the De memoria et reminiscentia, 14 that the intellect knows cum continuo et tempore? 15 Though it may seem sophistical, the formulation of the problem chosen in this context is not totally innocent and already indicates his nitist presuppositions (that a continuum is composed of a nite number of parts) and his reductive mereology (that a continuum is nothing but its parts). It is also noteworthy that when these two claims are combined, parts cannot be just potential ones, but rather are actual components of the whole as such. This is the reason why Crathorn presupposes that the wayfarer s intellect could know (de jure) all the actual parts of a continuous being. 12 In Sent. q. 8: Utrum continuum componatur ex indivisibilibus, id est ex punctis. This question has been partially edited in Wood, Adam de Wodeham, Tractatus de indivisibilibus, pp In particular q. 9 on the nature of instants (ff. 39va 40rb: Utrum instans secundum substantiam maneat idem toto tempore ) and two others implying the notion of in nite (q. 22 ff. 55vb 57va: Utrum ex in nitate motus extensiva possit concludi in nitas virtutis intensiva in primo motore ; q. 23 ff. 57va 59ra: Utrum causa prima possit producere extra se aliquem effectum actu in nito ). 14 Aristotle, De memoria et reminiscentia, 1, 450a In Sent. q. 3, p. 206: Utrum viator intelligat cum continuo et tempore. Quod non videtur. Nullum continuum est intelligibile a viatore. Igitur viator non intelligit cum continuo et tempore. Consequentia patet. Probatio antecedentis, quia si aliquod continuum posset intelligi a viatore, sit illud a; aut est nitum aut in nitum. Si in nitum, non potest intelligi. Si nitum, cum continuum non sit aliud quam partes, sequitir quod viator intelligens a intelligeret omnes partes eius et per consequens intelligeret in nita, quia omne continuum divisibile est in in nitum, igitur continet in se in nitas partes. Ad oppositum est Philosophus in primo libro De memoria et reminiscentia, ubi dicit quod intelligimus cum continuo et tempore.

5 william crathorn s mereotopological atomism 131 This leads Crathorn to a distinctive mereology, according to which a whole is nothing more than the sum of its parts. The main argument supporting this assertion is the following: if, from an ontological point of view, a whole were different from its parts, one should consider it as a distinct entity; but two absolute things can exist separately without any contradiction, at least thanks to God s absolute power. It would follow from these premises that God could conserve the whole as existing while destroying its parts, which seems self-contradictory. 16 It looks rather as if the whole had no ontological existence, for a whole must be identi ed with the mere continuous addition of its parts. 17 This is therefore the rst principle of Crathorn s mereology: P1: a whole is nothing but the sum of its parts Philosophers have traditionally opposed to this principle the aporia of diachronic identity. Indeed, shouldn t we conclude from P1 that a whole is not the same after losing or gaining one of its parts? Socrates s essence would have changed with one of his hairs being torn off. Anticipating this kind of objection, Crathorn envisaged the existence of essential parts. 18 Taking essential parts into account, one should be able to distinguish the identity of a whole W in a given time t, and the identity of a whole W through time, i.e. through a series of instants 16 In Sent. q. 3, p. 206:... totum est suae partes. Hoc probo sic: si totum non est suae partes et totum est aliquid, igitur totum ponit in numerum cum partibus. Consequentiam probo, quia sit a multitudo partium, sit b totum, tunc sic: b non est a nec aliqua pars ipsius a. Igitur si b est vera res, b ponit in numerum cum a et cum qualibet parte illius a. Igitur per potentiam dei b totum posset esse non existente multitudine aliqua nec aliqua pars illius, quod falsum est. As usual in the fourteenth century, God s absolute power is considered as a logical principle in such a context, because God s omnipotence is only limited by the principle of non-contradiction. 17 In Sent. q. 3, p. 217: Propter praedicta videtur mihi quod totum non est alia res ab omnibus suis partibus sibi invicem continuatis. 18 Crathorn often uses the notion of partes essentiales and though he strongly criticized the Aristotelian categories notably the very idea of substance he nonetheless seems to maintain the general distinction between essence and accident. Cf. In Sent. q. 13, pp For a use of the notion of essential part in the continuum debate, see In Sent. q. 3, p. 207 et 223. Essential parts are sometimes called naturae coextensae. Cf. In Sent. q. 13, pp : Istud nomen substantia derivatur ab isto verbo substo substas ; unde illud proprie vocatur substantia quod stat sub alio vel aliis; sed nihil est in isto ligno de quo proprie possit dici quod stet sub aliquo alio, quod est in ligno. Licet enim in ista re sunt multae naturae coextensae, tamen una illarum non est magis sub alia quam econverso. Igitur nulla illarum potest proprie dici substantia. (italics mine). This precisely means that some parts are essential, but also that all parts have an essence as we shall see.

6 132 aurélien robert t,..., tn. In a given time t, the identity of W is nothing but the totality of its parts, as an instantaneous ontological photograph, but the identity of W through time is the subset of its essential parts only. But this does not evacuate P1, Crathorn contends, because even at the level of essential parts, one cannot conceive the whole formed by the essence as distinct from its parts. 19 Then, even if one wanted to distinguish the essence of a thing from its accidental parts, it should be recognized that P1 works in the same way for the whole set of essential parts. What is at stake here is the generalization of the mereological principle P1, which is applicable to every kind of continua, and not only to natural substances and artefacts. As we shall see, it can also be applied to time and space and in general to all continuous quantities. Among the several arguments offered by Crathorn, 20 one must be carefully examined, because it contains the keystone of the theory, namely, that the composition of spaces is parallel to the composition of things existing in space. 3. The Mereological Composition of Space To convince his readers, our Oxford master invites them to think about P1 within the category of place. How could we conceive of the place of a whole and its parts if not in a mereological fashion? When a body is divided into its parts, the total place of the whole body is also divided into proportional parts of space. For example, half of a body occupies half of the place of the whole body, and so on if we could divide it again several times. As a corollary of that point, one has to consider that the division of places is strictly parallel to the division of the things existing in those places. Further, it might be noted that this new element strengthens the argument for P1. This new argument runs as follows: if a whole W were different from its parts, then the place occupied by W must be distinct from the place occupied by its parts for something always exists in 19 In Sent. q. 3, p. 207: Deus potest conservare omnes partes istius ligni non destruendo totum lignum. Sed forte aliquis posset dicere quod quia istae partes sunt de essentia istius ligni, ideo non potest destruere istius ligni, ideo non potest destruere partes istius ligni, nisi destruat totum lignum. Contra: eadem ratione non potest conservare partes istius ligni, nisi conservaret totum lignum; sed partes ligni sunt de essentia ligni. Crathorn declines this argument in many different versions. Cf. p There are twenty-eight different arguments.

7 william crathorn s mereotopological atomism 133 a place; but this sounds nonsensical, otherwise my house could be in London and its parts in Paris. 21 Likewise, if we were to divide W in actu, each part would occupy a distinct place and once gathered they would occupy again the whole space of W. This is an important theoretical move from the composition of a continuum to the composition of space. 22 Crathorn thus holds a second mereological principle: P2: the space occupied by a whole thing is composed of parts of space corresponding to the thing s parts This is an important shift that will serve as the basis of Crathorn s attacks against the divisibilists. 23 Indeed, if it is conceivable that a thing is in nitely divisible, it is far more dif cult to understand an in nite division of space. Without determining the number of parts in a continuum for the time being, Crathorn assumes that P1 and P2 imply a third principle concerning the proportional parts in division. 21 Ibid., p. 212: Si continuum non est suae partes sed res distincta, implicaret contradictionem unum corpus esse in loco, nisi plura et distincta corpora essent in eodem loco. Et hoc probo sic: in eodem loco est corpus et suae duae medietates, sicut hoc lignum et suae medietates, et hoc loquendo de toto loco totius ligni. Sed totum lignum est aliud a suis medietatibus. Igitur in eadem loco sunt distincta corpora occupantia totaliter locum illum, scilicet hoc corpus et suae medietates, quia certum est quod totum lignum occupat suum totum locum et duae medietates occupant eundem locum totum. Igitur impossibile est quod aliquod totum occuparet totaliter totum locum, nisi aliqua alia occuparent totaliter eundem locum. Dicitur quod duae medietates non occupant totum locum, sed partes loci, scilicet duas medietates totius, quia sicut totum locatum est alia res a suis medietatibus, sic totus locus non est suae medietates. Et ideo licet duae medietates totius occupent suas medietates loci, non potest concedi ex hoc quod occupant totum locum, sed illud quod occupat totum locum est totum componitur ex duabus medietatibus locati, et non sunt duae medietates... Igitur totus locus non est aliud quam suae duae medietates, et per consequens omne corpus locatum est suae duae medietates. Igitur eadem ratione omne totum est suae partes. 22 We use indifferently space and place, because as we shall see later, Crathorn de nes the place of a thing as the space it occupies. Cf. In I Sent. q. 14, p We will turn to this point in the forthcoming sections. 23 Here, it may be suggested that the main source of this theory is the Liber sex principiorum, which is often mentioned by Crathorn, and where a similar approach to space can be found. Cf. Liber sex principiorum [Minio-Paluello] pp : Ubi autem aliud quidem simplex aliud vero compositum; simplex quidem est quod a simplici loco procedit, compositum autem quod ex coniuncto.

8 134 aurélien robert 4. The Proportions in Division According to Crathorn, it follows from P1 and P2 that there should be the same number of parts in the two halves of a thing, and that there shouldn t be as many parts in a quarter than in the rest of that very thing for example. 24 Generally speaking, the number of proportional parts be it nite or in nite is conserved through division. It also works out when comparing two different continuous magnitudes: if a piece of wood is twice as big as another, the former has double the number of parts of the latter. 25 The third principle is therefore the following: P3: if a whole W is naturally composed of n parts, if we divide W in x parts, then each of the x parts will be composed of n/x parts Considered separately, these three principles do not support a particular version of atomism, for the number n of natural parts could still be nite or in nite. Furthermore, Crathorn has not yet given an answer to Aristotle s anti-atomist critiques. At least, we may assume that if there were an in nite number of indivisibles in a continuum, one should accept according to P3 that there are unequal in nities. 26 Implicitly, Crathorn seems to reject this claim, because he uses P3 in the series of arguments tending to prove that there is a nite number of indivisibles. 27 Indeed, he seems to consider that parts should be countable in some way (at least by God, it might be said). But this is not enough 24 For example, In Sent. q. 3, p. 225: Secunda conclusio est quod non sunt tot partes in medietate continui quot in toto continuo. Primo quia totum continuum est duplum respectu suae medietatis, sed totum continuum est multitudo omnium partium continui et medietas totus est multitudo partium medietatis, igitur multitudo partium totius continui est dupla respectu multitudinis partium suae medietatis... and p. 226: Quarta conclusio est quod multitudo omnium partium medietatis continui est aequalis multitudini omnium partium alterius medietatis. 25 Ibid. p. 226: Quinta conclusio quod generaliter qualis est proportio continui ad continuum, talis est proportio multitudinis partium unius continui ad alterius. Unde si unum continuum sit duplum ad aliud, multitudo partium illius continui est dupla ad multitudinem partium alterius et sic de aliis proportionibus. 26 Henry of Harclay, for example, accepts such a possibility. Cf. Murdoch, Henry of Harclay and the In nite. For the medieval debates on in nity, see Côté, L in nité divine dans la théologie médiévale ( ) and Biard & Celeyrette, De la théologie aux mathématiques. L in ni au XIV e siècle. 27 In Sent. q. 3. pp

9 william crathorn s mereotopological atomism 135 to prove the nite constitution of the continuum. Hence, in support of his nitist view, Crathorn adds another stone to the edi ce: Everything which is a part of a continuum is either something actual (actualiter aliquid) or not. One cannot say rationally that it is not something actual, because one cannot understand that what is an actual part of a continuum and belongs to its essence is not something actual or a certain thing. Therefore one must say that a part of a continuum is something actual. 28 Indeed, how could a thing be composed of non-things? There are no potential parts in a continuum, but only actual ones, according to P1, because parts constitute the whole as its essence. This fourth principle could be summed up as follows: P4: the parts of a whole W are all actual parts The terms composition and part should thus be taken in a strong sense. The parts in question are the components of reality and can, in principle, exist independently: But the same things that are called different [pieces of ] wood when they are discontinuous, these very same are called one wood and one whole when continuously joined together, in such a way that the expressions the whole wood and one wood signify nothing more than the essence of the parts, except from the continuation of these things with each other. 29 If we add the compositionality of spaces (P2) to that claim, each part of a continuum is an actually existent thing occupying a single place. Indivisibles are parts of bodies ( partes indivisibiles) and not only mathematical points. 30 Therefore, as we shall see in much more detail further down, parts must be conceived in a very strong sense as actual 28 Ibid. p. 227: Omne id quod est pars continui, vel est actualiter aliquid vel non. Non potest dici rationabiliter quod non sit actualiter aliquid, quia istud non est intelligibile quod id quod est actu pars continui et de essentia continui, non sit actualiter aliquid nec res aliqua. Igitur oportet dicere quod pars continui sit actualiter aliquid. 29 Ibid. p. 217: Sed illae eaedem res numero, quae dicuntur plura ligna, quando sunt discontinuata, illae eaedem dicuntur unum lignum et unum totum quando sibi invicem continuantur, ita quod isti termini totum lignum vel unum lignum nihil aliud signi cant ultra essentiam partium nisi continuationem illarum rerum ad invicem. 30 Ibid., p. 237: Septimo conclusio est quod indivisibile est pars corporis continui... Nullum corpus continuum nitum est divisibile in in nitum; igitur cuiuslibet corporis continui niti est aliqua pars indivisibilis.

10 136 aurélien robert things, with a certain quantity equal to the part of space in which they are located. On this point, Crathorn parts from his colleague Walter Chatton, who strongly rejects the existence of indivisibles in actu. Chatton s opinion is particularly well expressed in the Reportatio of his Commentary on the Sentences, where he criticizes a point of view similar to Crathorn s. 31 According to Chatton, a continuum can t be composed of actual indivisibles, for if it is the case, then the whole wouldn t be a totum per se unum but a mere aggregate of indivisible parts. 32 In Chatton s view, indivisible parts exist are only potentially in the whole, but they would be actual if really divided. On the contrary, the existence of actual indivisible things in the continuum is one of the main arguments of Crathorn s nitist theory, for one cannot understand an in nity of actual parts in a nite continuum. 5. The Number of Indivisibles From the elements previously posited, Crathorn infers his critique of Aristotle s view about the in nite divisibility of a continuum, as well as Henry of Harclay s atomist version of it. 33 Crathorn clearly asserts that no nite continuum can have an in nite number of proportional parts. 34 Considering P1 and P4, it follows that if a nite continuum were composed of an in nite number of parts, it would be a self-contradictory claim, and the continuum should be actually in nite, for its 31 Walter Chatton, Reportatio super Sententias [Etzkorn e.a.] II, d. 2, q. 3, pp : Modo volo ego declinare ad hoc quod continuum componatur ex indivisibilibus in potentia, non ex indivisibilibus in actu. 32 Ibid. p. 126: Dico quod continuum componi ex talibus in actu includat contradictionem, quia eo ipso quod continuum et contiguum differunt oportet quod partes continui uniantur et faciant per se unum, quod si non, non facerent continuum sed contigua esse tantum. Chatton ascribed the position he is challenging to Democritus (cf. ibid. p. 125). 33 Harclay accepts the existence of indivisibles, but they are in nite in a continuum. Crathorn frequently refers to Harclay by name. For example, In I Sent. q. 3, p In Sent. q. 3, p. 226: Nulla multitudo nita est in nita, sed si aliqua multitudo partium alicuius continui esset in nita, aliqua multitudo in nita esset nita vel aliqua multitudine nita esset in nita; igitur nullius continui multitudo partium est in nita...

11 william crathorn s mereotopological atomism 137 parts are always actual according to P4. 35 Of course, no Aristotelian philosopher would accept the existence of such an actual in nite. 36 In attempting to explain this argument, Crathorn recalls that following P1 and P2, there are as many parts of extension as [parts] of the extended thing, 37 which means that if a continuum were composed of an in nite number of parts, it would not only be composed of an in nity of places, but it would also consist in an in nite extension. 38 As a corollary to this rst philosophical consequence, it should be noticed that to accept the in nite division of places would imply the idea of an in nite number of places inside the continuum according to Crathorn, i.e. the existence of an in nite place by composition and addition. 39 Moreover, one must also admit the possible existence of an in nite body, for if there were an in nite number of proportional parts in a nite continuum, this nite body according to the dimension of place would stretch to the in nite. 40 Admittedly, such an in nite body can t exist according to the Aristotelian cosmology. 41 In such a context, it is clear that P2 and the notion of locus are the most important requirements for Crathorn s position. The aforementioned strategy consists in showing that from a mereotopological point of view, the only conceivable in nity of parts is actual 35 Ibid.: Si igitur in quolibet continuo sint actualiter in nitae partes, quod oportet dicere si in quolibet continuo sint in nitae partes, sequitur quod quodlibet continuum est actualiter in nitae partes. Igitur eadem partes continui sunt actualiter nitae, quia sunt continuum nitum, et sunt actualiter in nitae, quia in quolibet continuo sunt in nitae partes... igitur eadem partes sunt nitae et non nitae. 36 Aristotle s critique of the possibility of an actual in nity is well-known and it is not necessary to review it in this context. Cf. Physics, book III and VI in particular. For the medieval background, see Murdoch, In nity and Continuity. 37 In Sent. q. 3, p. 227: Tot sunt partes extensionis quot rei extensae. 38 Ibid. p. 227:... sicut tota res extensa correspondet toti extensioni, sic medietas rei extensae medietati extensionis, et secunda pars proportionalis rei extensae correspondet secundae parti proportionali extensionis, et sic de aliis partibus proportionalibus. Igitur si continuum est in nitae partes, est in nitae partes extensae. Igitur si continuum nitum est in nita secundum multitudinem, ista est in nita secundum extensionem. 39 Ibid. p. 228: Aliter potest dici quod partes continui sunt in nitae secundum multitudinem sed nitae secundum locum, quia sunt in loco nito. Sed contra: locus non est aliud quam partes loci; sed partes loci sunt in nitae secundum se si partes rei locatae sint in nitae, quia medietas totius locati est in medietate totius loci et medietas medietatis locati est in mediate medietatis loci; igitur quot sunt partes proportionales illius locati tot sunt partes proportionales illius loci. Sed multitudo partium proportionalium illius locati est in nita; igitur multitudo partium proportionalium illius loci est in nita. 40 Ibid. p. 229: Si in aliquo continuo nito essent partes proportionales in nitae, illud corpus nitum secundum omnem dimensionem loci extenderetur in in nitum. 41 See for example, Aristotle, De caelo, I, 5 et 7.

12 138 aurélien robert and not just potential, because if we start thinking from the point of view of the part-whole relation we cannot consider parts as mere potential units. Here, the most original feature of the argument rests on the use Crathorn makes of the notion of place. Indeed, beyond his adherence to the four principles P1 to P4, it follows that to each indivisible part corresponds a single place and a single position (situs punctualis), and therefore a kind of quantity, as we will see in the next chapters. If not, the arguments of the in nite extension and the in nite body would fail. It is illuminating to compare Crathorn s solution to similar arguments given by some of his contemporaries. As an example, Gerard of Odo in Paris, who wrote his Commentary on the Sentences and other small tracts on the continuum a few years earlier than Crathorn, also uses the argument of actual in nities for his own nitist theory. 42 Its form is indeed quite the same: if we suppose that a continuum is composed of an in nite number of parts, this continuum should be itself actually in nite. 43 Odo s followers and critics will also report this argument as a major one. For example, it is to be found in John the Canon s Questions on the eight books of the Physics 44 and in Gaetano of Thiene s Collection on the eight books of the Physics. 45 Even in nitists such as Nicholas of Autrecourt and Nicholas Bonetus used this argument. 46 Whether or not Crathorn is the rst to have used this argument, he nonetheless has a particular place in this story, for it isn t likely that he was in uenced by Gerard of Odo or John the Canon, who both were 42 See Sander de Boer s contribution in this volume for the context of the argument. 43 Gerard of Odo, De continuo, Ms. Madrid, Bibl. nac. 4229, ff. 179rb va (quoted by S. de Boer in this volume): Omne totum compositum ex magnitudinibus multitudine in nitis, sicut componitur cubitus ex duobus semicubitis, est magnitudo actu in nita. Sed non est dare magnitudinem actu in nitam. Ergo nullum continuum est divisibile in in nitum. 44 John the Canon, Quaestiones super octo libros Physicorum Aristotelis [Venice, 1520], f. 59vb: De totum compositum ex partibus vel magnitudinibus multitudine in nitis, sicut componitur cubitus ex duobus semicubitus, est magnitudo in nita; sed nullum continuum est magnitudo actu in nita. 45 Gaetano of Thiene, Recollectae super octo libros Physicorum [Venise, 1496], f. 38vb:... si continuum componitur ex semper divisibilibus ipsum in in nitum excedit aliam magnitudinem et est actu in nitum For example, Nicholas of Autrecourt, Exigit ordo [O Donnell], p. 212, ll : Dicerent autem contra hoc forsan: si punctum additum puncto faciunt majus et extensionem quamdam et tres faciunt majus quam duo et sic semper et ibi sint in nita, sequitur quod ibi sit in nita extensio. On this point, cf. Grellard, Les présupposés méthodologiques de l atomisme: la théorie du continu de Nicolas d Autrécourt et Nicolas Bonet.

13 william crathorn s mereotopological atomism 139 in Paris at that time, or by Nicholas of Autrecourt who wrote his Exigit ordo at the very same time. It seems more likely that Crathorn and his contemporaries were inspired by their Oxonian predecessors Henry of Harclay and Walter Chatton. The Chancellor of the University of Oxford, Henry of Harclay, af rmed that a continuum is composed of an in nite number of indivisibles, but he accepted only one sort of actual in nite: the straight line. 47 But according to him it is not rational to extend this for a surface or a body. As a consequence, Harclay considered indivisibles as potential parts and would probably have refused Crathorn s mereological principles. Nevertheless the attack on actual in nitism already existed in embryonic form in Harclay s position, and Crathorn probably found his starting point in it, adding to this intuition his mereological principles. More probable still is Walter Chatton s in uence on Crathorn, 48 for he will use a similar argument based on the actual in nite, although he didn t attach as much importance to the notions of locus and situs. 49 One of the reasons for this difference of opinion is that indivisibles are still considered as potential parts of the whole continuum in Chatton s view. 50 They cannot exist separately, therefore they cannot have a single place on their own. From this point of view, Crathorn s position is quite original in the atomist family. While using a common matrix of arguments, he is the only one who emphasizes both the mereological composition of 47 On this point, cf. Murdoch, Henry of Harclay and the In nite (in particular p. 233). 48 It might be noticed that Crathorn doesn t agree with Chatton on several points. For references to Chatton, see In Sent. q. 3. pp. 261 and Walter Chatton, Quaestio de continuo, in. Murdoch & Synan, Two questions on the continum: Walter Chatton(?), O.F.M. and Adam Wodeham, O.F.M., p. 258:... responsio ad primum: dico quod sunt nita. Ad philosophum dico quod continuum esse divisibile in in nitum potest <dupliciter> intelligi: vel quod in continuo sint actu in nite partes, quarum quelibet est extra aliam et nulla est alia, que possunt dividi ad invicem, et hoc est falsum et contra rationem See the texts quoted above at the end of the previous section. See also Walter Chatton, Quaestio de continuo, in. Murdoch et Synan, Two questions on the continuum: Walter Chatton(?), O.F.M. and Adam Wodeham, O.F.M. p. 246: Istis suppositis, teneo 3 conclusiones. Prima: quod non componitur continuum ex indivisibilibus in actu, quia inter terminos est contradiccio continuum et indivisibile in actu quia, si sit continuum, igitur partes eius sic se habent quod nulla est in actu per se existens separata ab alia; et si sit indivisibile in actu, est per se existens separatum ab alio eiusdem rei. Oppositum dicit Democritus ponens continuum eri ex athomis, tantum per congregationem quandam continuatis...

14 140 aurélien robert things and spaces as well as the actuality of parts. 51 But his singularity comes from his use of the notion of locus. Indeed, the arguments of the actual in nity of parts and in nite extension require P2 and the coincidence between indivisible parts and single places for its ef cacy. Moreover, Crathorn contends that whole and part are only imposed to signify the local conjunction of indivisible places. 52 In this respect, he is probably one of the more consequent atomists, because if an indivisible were not considered as an indivisible part, with a kind of extension resulting from the minimal place it can ll, all the previous argumentation would fail. Gerard of Odo, for example, whose theory is very close to Crathorn s, is not consistent when he seems to consider indivisibles as unextended. 53 Here again, Crathorn is indebted to Harclay s analysis of contact between indivisibles when he tries to de ne contiguity and continuity of atoms thanks to the central notion of place. 6. Contiguity and Continuity Aristotle s well-known paradox formulated in the sixth book of the Physics arises when asking the atomist how two atoms can generate an increase in size. 54 It raises the problem of the contact between indivisibles, for two things can touch together parts to parts, whole to whole, or parts to whole, Aristotle said. Having no parts, atoms can 51 Except from Crathorn, there were no other philosophers at Oxford, as far as I know, who held this view about the actuality of indivisible parts of a continuum. In Chatton s Reportatio, the reportator ascribes this position to some contemporaries. Cf. Walter Chatton, Reportatio super Sententias [Etzkorn e.a.] II, d. 2, q. 3, p. 136: Sed secundum usum modernorum, potest componi ex actu indivisibilibus, quia vocat actu tale quod est tale extra animam et extra causam, quantumcumque non sit separatum ab alio; componitur ergo ex actu indivisibilibus, id est ex non habentibus partes. (italics mine) It s impossible to know to whom Chatton was refering in this text (written around ). If we believe Gregory of Rimini, the nitist position, with actual or potential indivisibles, was common among his contemporaries. Cf. Gregory of Rimini, Lectura super secundum Sententiarum [Trapp], t. II, d. 2, q. 2, p. 278: Nec potest dici, sicut communis dicunt tenentes huiusmodi compositionem ex indivisibilibus, quod componatur ex nitis tantum indivisibilibus. We don t know whether Gregory had Chatton, Odo or Crathorn in mind, or some other unknown philosopher. At the end of the fourteenth century, Wyclif held a position very similar to Crathorn s own thesis. For Wyclif, see the references in the next sections. 52 In Sent. q. 3, p. 224:... hoc nomen totum et hoc nomen pars,... imponuntur a locali coniunctione rerum, quae coniunctio non est aliud quam loca vel partes loci. 53 As it is revealed in the texts quoted by Sander W. de Boer in this volume. 54 Aristotle, Physics VI, 1, 231a29 b6.

15 william crathorn s mereotopological atomism 141 only touch whole to whole, and this makes no increase in size but a mere superposition; now, superposition non facit maius. 55 When replying to this famous objection, Henry of Harclay claims that Aristotle missed the point, because the Stagirite used to consider indivisibles in one and the same position (situs), but according to him, indivisibles can touch according to different positions (secundum distinctos situs) and this way they can cause an increase in size. 56 Henry of Haclay doesn t develop this point further, but Crathorn will explain this point very clearly. 57 To my mind, the interesting idea that emerges from these topological elements is the idea of a reference landmark, thanks to which atoms can be located by their respective positions. This is not totally extraneous to Aristotle s cosmology, since similar tools can be found in the De caelo for example. 58 Moreover, Aristotle sometimes de nes a point as a substance with a position, as in the Posterior analytics. 59 But the appeal to locus and situs of indivisibles is problematic for philosophers 55 On this argument and the consecutive medieval debates, see Murdoch, Superposition, Congruence and Continuity in the Middle Ages. 56 Henry de Harclay, Quaestio de in nito et continuo, Mss Tortosa Catedral 88, f. 89r and Florence, Biblioteca Nazionale, Fondo principale, II. II. 281, f. 98r v (quoted in Murdoch, Henry of Harclay and the In nite, p. 244): Sed, licet hec responsio suf- ceret ad hominem, non tamen est realis responsio. Et ideo pono aliam et dico quod indivisibile tangit indivisibile secundum totum, sed potest hoc esse dupliciter: vel totum tangit totum in eodem situ, et tunc est superpositio sicut dicit Commentator, et non faciunt in nita indivisibilia plus quam unum.... Et ideo dico quod non propter indivisibilitatem quod unum indivisibile sic additum indivisibili non facit maius extensive, sed quia additur ei secundum eundem situm et non secundum distinctum situm. Si tamen indivisibile applicetur immediate ad indivisibile secundum distinctum situm, potest magis facere secundum situm. 57 Gerard of Odo, for example, will follow Harclay s text to the letter. Cf. Gerard of Odo, De continuo, Ms. Oxford, Bodleian Can. Misc. 177, f. 230v (quoted in Murdoch, Superposition, Congruence and Continuity..., p. 435): Dico quod totum indivisibile tangit totum aliud indivisibile, non tamen secundum omnem differentiam situs, sed secundum unam tantum, scilicet secundum ante vel secundum retro et sic de aliis. Unde si unum indivisibile tangeret aliud indivisibile secundum omnem differentiam loci, scilicet secundum ante et retro et secundum alias differentias omnes, tunc bene sequitur quod indivisibilia non essent loco discreta nec constituerent aliquod maius. Sed si unum tangit reliquum secundum unam differentiam loci, ideo sunt loco discreta et constituunt aliquod maius. 58 Cf. Aristotle, De caelo, II, Aristotle, Posterior analytics, I, 27, 87a36. As Rega Wood points out in her chapter in this volume, this surprising assertion by Aristotle has been ignored by a majority of medieval philosophers, although Robert Grosseteste paid some attention to it in his commentary. Cf. Robert Grosseteste, Commentarius in Posteriorum analyticorum libros, 1, 18 [Rossi], p. 258.

16 142 aurélien robert such as Henry of Harclay or Gerard of Odo, who do not conceive of indivisibles as occupying places. In fact, how could indivisibles occupy a place if they are neither extended nor actual things? Again, Aristotle himself raised this point explicitly. When criticizing the atomists in De generatione et corruptione (I, 2) Aristotle repeated the argument from the Physics saying that it is not possible for a quantity to come from non-quantitative things. 60 He immediately added: Furthermore, where will the points be? 61 When he turned back to the problem of the continuum in chapter 6 of book I, Aristotle insisted on the notion of contact, thanks to which continuity can be conceived. But, as he contended, the notion of contact cannot be understood without the notion of position, which in turn relies on the notion of place. Nevertheless, contact in its proper sense belongs only to things which have a position, and position belongs to those things which have also a place. 62 Touch always occurs between two situated things because things in contact need to be in a certain spatial relation. These touching things, Aristotle said, also need to have a certain discrete magnitude, for if not they cannot touch. 63 So, when reading the De generatione et corruptione, the medieval philosophers could nd the key to the problem of contiguity and continuity: to ascribe a place and a certain magnitude to the indivisibles. But other sources could be used, as the Liber sex principiorum which is sometimes cited by Crathorn and where one can nd a similar notion of indivisible places for points and body s minima. 64 Due to the need to consider indivisibles as occupying a single place, Crathorn is able to reconstruct contact between things and their continuity from the notions of locus and situs. As he told us in a passage of his Questions on the Sentences: 60 Cf. Aristotle, De generatione et corruptione, I, 2, 316b5. 61 Ibid. 62 Aristotle, De generatione et corruptione, I, 6, 322b30 323a5. 63 Ibid. 64 Liber sex principiorum [Minio-Paluello] pp : Locus autem simplex est origo et constitutio eius quod continuorum est, locus vero (ut dictum est quidem) compositus habet particulas quidem ad eundem terminum copulatas ad quem et corporis particule coniunguntur, corporis vero partes ad punctum. Loci ergo partes iuxta punctum necesse eri; erit itaque locus simplex in quo punctum adiacere constabit, loci vero particule soliditatis particulas claudunt; etenim loca quidem simplicia minimi corporis occupativa sunt.

17 william crathorn s mereotopological atomism 143 What continuity in fact is will become clear below when asking whether God is ubiquitous (q. 15). There it will be said, indeed, what is place, and once the nature of place is understood, one can easily see what is continuity and discontinuity, and what is contiguity. 65 John E. Murdoch has already noticed that the notion of situs used by Harclay implies a physicalist version of the indivisible, even if Harclay doesn t develop his view in this way. 66 If Harclay didn t take the plunge, Crathorn seems to have thought of indivisibles in this physicalist fashion. Crathorn s goal is to rede ne the problematic notions in Aristotle s critique of atomism. If two things are continuous, according to Aristotle, their limits have to be one, and if they are contiguous, the extremities of the things must be together. 67 The strongest attack against atomists comes from this characterization of continuity and contiguity. Therefore, Crathorn suggests an alternative de nition: [contiguity] has to be de ned in this way: contiguous things are situated and located things, between which there is no intermediary place or position, and this de nition suits for bodies, surfaces, lines and points. 68 It should be objected to this de nition that contiguity never explains continuity. 69 But since Crathorn adds the clause of non-existence of interparticulate spaces between indivisibles, contiguity becomes continuity, because they produce a new thing without any vacuum in it. Therefore, properly speaking, a continuous thing is full of matter, there is no empty space in it, no void. This is true, in slightly different senses, for every kind of entity, including time. 65 In Sent. q. 4, p. 218: Quid vero sit continuitas patebit infra, cum quaeretur utrum deus sit ubique; tunc enim dicetur quid est locus, et intellecto quid est locus, cito potest videri quid est continuatio et quid dicontinuatio et quid contiguitas. 66 Murdoch, Henry of Harclay and the In nite, p. 244: It is clear that in claiming that indivisibles can touch according to distinct positions Harclay was considering these indivisibles not as absolutely extensionless entities they really were, but as if they were physical things. It is not absolutely evident that Harclay was not aware of this physicalist consequence. Nevertheless, it would be incompatible with his in nitism. On the contrary, it is clear enough that Crathorn endorsed consciously these physicalist implications. 67 Aristotle, Physics, VI, 321a In Sent. q. 3, p. 255: Sed debet sic de niri: contigua sunt situata vel locata, inter quae non est locus vel situs medius, et ista de nitio competit corporibus et super ciebus, lineis et punctis. 69 We nd this argument in Chatton s Reportatio for example. Cf. Reportatio super Sententias [Etzkorn e.a.], d. 2, q. 3, p. 126.

18 144 aurélien robert For a thing to be continuous is nothing but its parts being joined together in a local or a temporal way, without any intermediary place or time [between them], and these parts being joined, they hold together or mutually succeed in place or in time without subtraction of place or time. It follows that the continuity of a body, of a line or a surface, is the continuity of the parts of place, because parts are said to be continuously located according to the continuity of the parts of place From the beginning, Crathorn presupposes that an indivisible occupies a single place and that it must consequently have a certain quantity. This is implied by the argument from actual in nity and also from the aforementioned de nition of contiguity and continuity. Aristotle himself thought that position and place should imply weight or lightness in some sense. 71 Indeed, Aristotle s target was mainly the possibility of touching unextended points, but not of touching things (i.e. something with a place, a position and a certain extension). Crathorn s reappraisal of contiguity and continuity seems to work as follows: Aristotle was right in claiming that extensionless points cannot touch, but if points or indivisibles were given a certain place and extension, they could touch according to their contiguous positions. Crathorn s theory is in some sense a reinterpretation of Aristotle, paying attention to the physical possibilities of his cosmology. The keystone of Crathorn s atomist theory is thus to have considered that indivisibles may have a sort of quantity and extension. How are the notions of place, extension and quantity linked in Crathorn s thought? 7. Extended and Qualified Atoms In a long discussion on the category of quantity (q. 14) and on the quantity of indivisibles in particular (q. 15), Crathorn incidentally 70 In Sent. q. 16, pp : Rem esse continuam non est aliud quam partes illius rei sibi invicem coniungi localiter vel temporaliter sine loco vel tempore medio et tales partes sic coniunctas simul teneri vel sibi invicem succedere vel loco vel tempore sine interceptione loci vel tempori. Unde continuitas corporis vel lineae vel super ciei est continuitas partium loci, quia continuitate partium loci dicuntur partes locatae continue Aristotle, De generatione et corruptione, I, 6, 323a10: Now, since position belongs to such things as also have a place, and the primary differentiation of place is above and below and other such pairs of opposites, all things which are in contact with one another would have weight and lightness, either both of these qualities or one of them.