Gregory of Rimini on the Intension and Remission of Corporeal Forms

GREGORY OF RIMINI ON THE INTENSION AND REMISSION OF CORPOREAL FORMS

Can L. LOEWE

Abstract

The goal of this paper is to provide an account of Gregory of Rimini’s (1300- 1358) theory of the intension and remission of corporeal forms. Under the influence of new Oxford ideas of the fourteenth century Gregory adopts a highly quantitative approach to intensive change. The paper discusses Gregory’s defense of the addition theory, especially in light of Walter Burley’s counter- arguments. It also considers Gregory’s account of the continuity of intensive change, as well as his views on the possibility of the co-presence of contrary qualities in the same subject.

Introduction The goal of this paper is to provide an account of Gregory of Rimini’s (1300-1358)1 theory of the intension and remission of corporeal forms,2 as laid out in book 1, d. 17, qq. 2-4 of his Sentences commentary

1 On the author, see R.L. FRIEDMAN – C. SCHABEL, “Gregory of Rimini,” in: H. LAGER- LUND (ed.), Encyclopedia of Medieval Philosophy. Philosophy Between 500 and 1500, Vol. 1, Dordrecht 2011, pp. 439-444. All references are to the critical edition: GREGORY OF RIMINI, Lectura Super Primum et Secundum Sententiarum, 6. Vols. (= Spätmittelalter und Reformation. Texte und Untersuchungen, vols. 6-11), ed. A.D. TRAPP – V. MARCOLINO – W. ECKERMANN – M. SANTOS-NOYA – W. SIMON – W. SCHULZE – W. URBAN – V. WEND- LAND, Berlin 1979-84. I refer to Gregory’s Sentences commentary as Sent. All translations from the Latin are mine. (NB: I do not necessarily respect the punctuation and orthography of any text I use.) 2 By ‘corporeal forms’ Gregory means those forms that inhere in extended surfaces. He contrasts those with spiritual (or psychological) forms such as charity or courage; see Sent., 1, 17, 2, vol. 2, p. 250. He thinks, however, that his account of the intension and remission of corporeal qualities can be extended without any significant alterations to spiritual qualities (see pp. 277-278 below and n. 17 for references).

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(1346).3 This theory was devised to explain how changes of degree within a given quality occur, e.g., how the shade of a color changes, or how an object becomes hotter. Although this topic initially seems to be a narrowly natural philosophical one, in the Middle Ages it had a theological dimension, too. Specifically, the scholastics were concerned with the issue of how to account for the increase of the theological virtue of charity.4 The discussions of this subject matter were extensive, and it is no great exaggeration when Anneliese Maier calls the debate surrounding intension and remission of forms one “of the highest rank” in scholastic thought.5 Gregory of Rimini’s discussion occupies an important place in this debate. His discussion is one of the longest of the fourteenth century,6 and it is characterized by a level of detail and sophistication that was, in many ways, unprecedented. In Gregory’s times and centuries to follow this did not go unnoticed. Gregory’s views on the topic of intension and remission became so influential that the three questions, which Gregory dedicated to this topic, were made available in a separate treatise entitled Tractatus de intensione et remissione formarum corporalium.7 Despite its great sophistication and influence, Gregory’s account of the intension and remission of corporeal forms has, unlike other

3 1346 is the date of release. Gregory lectured on the Sentences in Paris in 1343-44. See V. MARCOLINO, “Der Augustinertheologe an der Universität Paris,” in: H.A. OBERMAN (ed.), Gregor von Rimini. Werk und Wirkung bis zur Reformation, Berlin 1981, pp. 127- 194, esp. 170-171; P. BERMON, “La Lectura sur les deux premiers livres des Sentences de Grégoire de Rimini O.E.S.A (1300-1358),” in: G.R. EVANS (ed.), Mediaeval Commentar- ies on the Sentences of Peter Lombard. Current Research, vol. 1, Leiden 2002, pp. 267-285, esp. 268. 4 See A. MAIER, Zwei Grundprobleme der scholastischen Naturphilosophie (= Studien zur Naturphilosophie der Spätscholastik, vol. 2), 3rd ed., Rome 1968, pp. 10-11. 5 See ibid., p. 5. 6 The three questions that Gregory dedicates to this topic (Sent., 1, d. 17, qq. 2-4, vol. 2, pp. 250-417) comprise 167 pages in the critical edition. 7 According to Heiko Oberman, two of the 27 manuscripts of the first book of the Sentences (specifically, manuscripts N, Oxford Bodleian Library, Can. Misc. 177, and X, Venice, Biblioteca Nazionale Marciana MS. VI, 160 [2816]) contain this separate treatise. These manuscripts were for some time in the possession of the Italian philosophers Johannes de Marchanova († 1467) and Nicolettus Vernias Theatinus († 1499). See H.A. OBERMAN, Introduction to Lectura, vol. 2 (d. 7-17), ed. A.D. TRAPP et al., Berlin 1982, p. 5. For more information on the manuscripts of Gregory’s commentary as well as the so-called additiones, i.e., those passages that Gregory did not include in the definitive version of his commentary (ordinatio), see V. MARCOLINO, Introduction to Lectura, vol. 1 (d. 1-6), ed. A.D. TRAPP et al., Berlin 1981, p. 95; BERMON, “La Lectura,” pp. 269-271.

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aspects of his natural philosophy, not received much attention by scholars.8 At least, there has not been, to my knowledge, any study that considers Gregory’s discussion in its entirety.9 In this paper, I will try to fill this gap. What makes Gregory’s discussion particularly interesting is that it is one of the first discussions on the Continent to be extensively influenced by new Oxford ideas, which, as is well known, Gregory was one of the first to introduce.10 Under this influence Gregory adopts a highly quantitative approach to qualities, which, although presented in Aristotelian terminology, is in many ways closer to the account of qualities typically associated with early modern science. To avoid misunderstandings I should say that I do not mean by Gregory’s ‘quantitative approach’ that he uses mathematical formulas to describe physical states of affairs (although some mathematics is present in his account). Unlike the early-fourteenth-century Oxford Calculators, Gregory is not interested in formulating kinematic laws

8 Scholars have studied in depth several of Gregory’s contributions to natural philosophy such as his groundbreaking ideas on the continuum and the infinite. Gregory claimed that actual infinites were not only possible, but in fact existed, and that every continuum was an actual infinite. Furthermore, he claimed that one infinity could be greater than another according to a relation that comes very close to the contemporary set-theoretical relation between set and subset. On Gregory’s theory of the comparison of infinities, see J.E. MURDOCH, “Mathesis in Philosophiam Scholasticam Introducta. The Rise and Develop- ment of the Application of Mathematics in Fourteenth Century Philosophy and Theology,” in: Actes du Quatrième Congrès International de Philosophie Médiévale, Montréal 1969, pp. 215-246, esp. 223-224; On infinity and the continuum see J.M.M.H. THIJSSEN, “Het Continuum-Debat bij Gregorius van Rimini (1300–1358),” in: Algemeen Nederlands Tijd- schrift voor Wijsbegeerte 77 (1985), pp. 109-119; K. SMITH, “Ockham’s Influence on Gregory of Rimini’s Natural Philosophy,” in: V. SYROS – A. KOURES – H. KALOKAIRINOU (eds.), Διαλέξεις. Ακαδημαϊκό έτος 1996-7, Lefkosia 1999, pp.107-142; R. CROSS, “Infinity, Continuity, and Composition: The Contribution of Gregory of Rimini,” in: Medieval Philosophy and Theology 7 (1998), pp. 89-110. 9 Some parts, however, have been discussed. Edith Sylla discusses Gregory’s views on minima and maxima as laid out in q. 2. See E.D. SYLLA, “Disputationes Collativae: Walter Burley’s Tractatus Primus and Gregory of Rimini’s Lectura super primum et secundum Sententiarum,” in: Documenti e studi sulla tradizione filosofica medievale 22 (2011), pp. 383- 464, esp. 404-410. Onorato Grassi and Francesco Fiorentino, in separate articles, explored Gregory’s arguments in q. 4 against Giles of Rome’s theory of intension and remission. See O. GRASSI, “Gregorio da Rimini e l’agostinismo tardo-medievale,” in: Gregorio da Rimini filosofo (Atti del Convegno - Rimini, 25 novembre 2000), Rimini 2003, pp. 66-96, esp. 85-89; F. FIORENTINO, “Gregorio da Rimini a confronto con Egidio Romano e gli egidiani,” in: Analecta Augustiniana 68 (2005), pp. 6-58, esp. 24-35. 10 See BERMON, “La Lectura,” p. 280.

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such as the middle degree theorem.11 Nor does he provide anything like a geometrical representation of intensities as can be found in Nicole Oresme.12 What I mean by Gregory’s ‘quantitative approach’ is rather that Gregory treats intensities ontologically as if they were quantities, and that, in this ontological respect, his account is close to that of early modern science (although early modern science drops the as if and, basically, treats intensities as quantities tout court). It is one goal of this paper to shed light on the ontology behind the as if. Section 1 of this paper will provide a brief overview of the structure of Gregory’s discussion of intension and remission in light of the Aristotelian and the medieval background. Section 2 will then discuss Gregory’s account and defense of the then prominent addition theory of intension and remission and his refutation of rival theories (corresponding to q. 4, and part of q. 3 of his d. 17). I shall here deal in detail with Gregory’s discussion of Walter Burley’s arguments against the addition theory. Section 3, finally, will deal with Gregory’s account of the continuity of intension and remission (with special attention to his use of some of Richard Kilvington’s ideas13), as well as with his account of the relation of contraries (corresponding to q. 2 and the other part of q. 3 of d. 17).

1. Gregory’s discussion: its historical background and structure First, some historical background information. In book I of his Sen- tences, d. 17, Peter Lombard raises the question as to whether charity

11 The theorem is that “the distance traversed by a uniformly accelerated motion is the same as that traversed by a uniform motion in the same time interval at the mean degree between the two extreme velocities of the first motion”. See E.D. SYLLA, “The Oxford Calculators’ Middle Degree Theorem in Context,” in: Early Science and Medicine 15 (2010), pp. 338-370, at 338. 12 On Oresme’s theory of intensities see M. CLAGETT (ed.), Introduction to Nicole Oresme and the Medieval Geometry of Qualities and Motions. A Treatise on the Uniformity and Difformity of Intensities Known as Tractatus de configurationibus qualitatum et motuum, Madison 1968, pp. 3-49, esp. 14-37. 13 I will not discuss Gregory’s use of Kilvington’s ideas with regard to the comparison of motions because this topic has already been analyzed in detail. See S. KNUUTTILA – A.I. LEHTINEN, “Plato in Infinitum Remisse Incipit Esse Albus: New Texts on the Late Medieval Discussion on the Concept of Infinity in Sophisma Literature,” in: E. SAARINEN (ed.), Essays in Honour of Jaakko Hintikka: On the Occasion of His Fiftieth Birthday, Dordrecht 1979, pp. 309-329, esp. 311-312.

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can be increased.14 Having identified charity with the Holy Spirit, Peter answers the question in the negative. The later scholastics were more or less unanimous in their rejection of Lombard’s identification of charity and the Holy Spirit. They took charity to be a created habit. So, for them, an increase of charity was not something impossible.15 How could this increase be explained, though? To this end, the scholastics drew on certain ideas of Aristotle’s as well as the commentary tradition (Boethius and especially Simplicius since 1268). Much of what Aristotle had to say about this topic concerned corporeal qualities. The scholastics thus often investigated the nature of changes in degree in corporeal qualities first, in order to then elucidate how spiritual qualities such as charity change in degree. Gregory is a case in point. He writes: “[…] so as to have a clearer picture about the increase of charity and of other spiritual qualities, we first need to look at the increase or intension of corporeal qualities.”16 Indeed, Gregory develops his general account of intension and remission in his discussion of gradual corporeal changes (d. 17, qq. 2-4) such as changes in color or temperature, and then goes on to apply this account to spiritual forms such as charity (d. 17, q. 5).17 So, even though this paper focuses on Gregory’s discussion of

14 PETER LOMBARD, Sententiae in IV Libris Distinctae, 3rd ed., vol. 1, Grottaferrata 1971, I, d. 17, c. 5, 64, p. 146. NB: The division into distinctions is not Peter’s own. It was likely introduced by Alexander of Hales. See I. BRADY, “The Distinctions of Lombard’s Book of Sentences and Alexander of Hales,” in: Franciscan Studies 25 (1965), pp. 90-116. 15 On this question, see J. CELEYRETTE – J.-L. SOLÈRE, “Jacques de Lausanne, censeur et plagiaire de Durand de Saint-Pourçain: Édition de la q. 2, dist. 17 du l. I de son Com- mentaire des Sentences,” in: K. EMERY, Jr. – R.L. FRIEDMAN – A. SPEER (eds.), Medieval Philosophy and Theology in the Long Middle Ages. A Tribute to Stephen F. Brown, Leiden 2011, pp. 855-889; G. ZUIJDWEGT, “Utrum caritas sit aliquid creatum in anima. Aquinas on the Lombard’s Identification of Charity with the Holy Spirit,” in: Recherches de Théolo- gie et Philosophie médiévales 79 (2012), pp. 39-74. 16 Sent., 1, 17, 2, p. 250: “[…] ut plenius videatur de augmento caritatis et aliarum spiritualium qualitatum, videndum prius est de augmento seu intensione qualitatum corporalium.” Gregory did indeed take the task of getting a clear picture of the intension and remission of corporeal forms very seriously. As Edith Sylla noted, Gregory’s discussion of the intension and remission of corporeal forms considerably exceeds in length his treatment of the problem of the increase of charity. See SYLLA, “Disputationes,” p. 403. 17 The conclusiones that Gregory puts forth concerning the intension and remission of charity (indeed, of all spiritual habits) in Sent., 1, 17, 5 neatly correspond to those he puts forth concerning the intension and remission of corporeal forms. He claims that corporeal forms can be intensified continuously (Sent., 1, 17, 2, p. 252), and he says the

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corporeal forms, the general account that emerges is one that is not restricted to physical changes. What then did Aristotle have to offer that was of relevance to the discussion of the intension and remission of forms? Aristotle’s longest remarks about the problem of a quality’s gradual change can be found in his Categories.18 Aristotle there explains that qualities (unlike quan- tities19) admit of more or less. Someone can be more or less pale, more or less just, or more or less healthy. Aristotle is unsure, however, as to what the bearer of the more and the less is. Is it the quality or the substance qualified by the quality? Aristotle claims that it makes sense to say that the quality of pallor itself is more or less present in a thing. As he notes, however, “some people claim” that problems arise when we apply this sort of account to dispositions (διαθέσεις20) such as justice or health. It seems, on the view of these people, that health and justice stay the same, and that they differ in intensity only due to subjects who are more or less just or healthy. Aristotle neither fully accepts this view, nor does he reject it. The scarce remarks about gradual change found in the Physics do not provide a definitive answer either.21 Commentators in the Middle Ages were split on this issue. Some (primarily Franciscans) maintained that a quality itself has a range of intensive variations,22 or a ‘latitude’ as the scholastics called it.23

same about charity (Sent., 1, 17, 5, p. 431). He tells us that corporeal forms do not change in degree by the admixture of contraries (Sent., 1, 17, 3, p. 331), and he makes the same point about charity (Sent., 1, 17, 5, p. 421). Finally, we learn that the intension and remission of corporeal forms occurs by the addition of a new form (Sent., 1, 17, 4, p. 380) without the corruption of the preceding one (Sent., 1, 17, 4, p. 395), and Gregory also thinks this is true for charity (Sent., 1, 17, 5, pp. 423-424). 18 Categories, 8, 10b27-11a14. I refer to the following translation of Aristotle’s works: ARISTOTLE, The Complete Works. 2 vols., revised Oxford Translation, ed. J. BARNES, Princeton 1984. 19 The point that quantities do not admit of more or less is that a given quantity, e.g., a stick that is a meter long, cannot be more or less of a meter long. Either it is a meter long or it is not a meter long. 20 For the Greek text I rely on ARISTOTLE, Categories, in: Categoriae et Liber De interpretatione, ed. L. MINIO-PALUELLO, Oxford 1949. 21 ARISTOTLE, Phys., V, 2, 226b1-2; VII, 2, 244b6-12; VII, 3, 246b15-17. 22 I intend “range of intensive variations” to be a translation of Maier’s term “inten- siver Spielraum.” See her An der Grenze von Scholastik und Naturwissenschaft (= Studien zur Naturphilosophie der Spätscholastik, vol. 3), 2nd ed., Rome 1952, p. 26. 23 As Edith Sylla has shown, the term ‘latitude’, understood as a range of gradual variation, likely originated in the medical works of Galen and Avicenna. These two authors speak of a latitude of health. See E.D. SYLLA, “Medieval Concepts of the Latitude

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Others (especially those following Aquinas) argued that it is the subject bearing the form, e.g., a hot or white thing, and not the form itself, that admits of more or less of this form. A second issue raised by Aristotle and prominent in the medieval debate of intension and remission of forms was the problem of the continuity of qualitative change. In his Physics Aristotle claims that all motions (i.e., all non-substantial changes in the categories of quality, quantity, and place) are continuous. This means, more specifically, that motions are infinitely divisible,24 and this in two ways: (i) according to the time they take, and (ii) according to the parts of the magnitude that is in motion (e.g., if your finger is moving, then the infinitely many parts that your finger could be divided into are moving too).25 Aristotle denies that a continuum can consist of infinitely many point-like entities such as instants. His reason is this. On Aristotle’s view, x is continuous with y only if x and y have a common boundary (ἔσχατον), i.e., if x and y touch.26 This boundary has to be distinct from the item it bounds, Aristotle claims.27 Hence, all items standing in a relation of continuity must admit of a division into boundary and what is bounded. Now, instants do not admit of this division because they are, on Aristotle’s view, extensionless and therefore indivisible beings.28 Thus, instants cannot constitute a continuum. Aristotle also claims that one instant cannot “immediately succeed” another instant either.29 By this he means that it is impossible for one

of Forms: The Oxford Calculators,” Archives d’Histoire Doctrinale et Littéraire du Moyen Âge 40 (1973), pp. 223-283, esp. 226-227. On the latitude of forms in Antiquity and the reception of Aristotle’s remarks in Categories 8 by Porphyry and Simplicius, see R. SORABJI, “Latitude of Forms in Ancient Philosophy,” in: C. LEIJENHORST – C. LÜTHY – J.M.M.H. THIJSSEN, The Dynamics of Aristotelian Natural Philosophy from Antiquity to the Seventeenth Century, Leiden 2002, pp. 57-63, esp. 61. 24 Phys., VI, 2, 232b24-5. 25 Phys., VI, 3, 234b22-25. 26 Phys., VI, 1, 231a22. For the Greek text I rely on ARISTOTLE, Physics, Revised Greek Text with Introduction and Commentary by W.D. ROSS, Oxford 1936. 27 Phys., VI, 1, 231a28. 28 Phys., VI, 1, 231a25-30. 29 Phys., VI, 1, 231b6-10. The term ‘immediately succeed’ is not Aristotle’s (who simply speaks of ‘succeed’), but Norman Kretzmann’s. What Aristotle means by ‘succession’, however, is better conveyed by ‘immediate succession’ (i.e., having nothing of the same kind in between). See N. KRETZMANN, “Incipit/Desinit,” in: P.K. MACHAMER – R.G. TURNBULL (eds.), Motion and Time, Space and Matter. Interrelations in the History of Philosophy and Science, Columbus 1976, pp.101-136, esp. 102.

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instant tn to come after another instant tm without there being some 30 further instant tl (m

30 Phys., V, 3, 226b34-227a5. 31 Phys., VI, 6, 237b20-2.

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How do intension and remission enter the picture here? Aristotle seems to suggest that qualitative changes (alterations), of which intensions and remissions form a sub-class, differ from other changes with respect to continuity. First, Aristotle explains that not all alterations are continuous. In his Physics as well as in De Sensu et Sensato, he adduces water’s freezing as an example of an instantaneous alteration.32 Furthermore, Aristotle also points out that those alterations that are divisible and, therefore, continuous, are divisible only in a special sense. They are not divisible per se, but only per accidens.33 From this he concludes, rather cryptically, “that only in qualitative change can there be an indivisible in itself”.34 In his Long Commentary on the Physics, Averroes took Aristotle here to actually deny the continuity of alteration.35 This is not an implausible reading. As we have just seen, a continuum cannot, for Aristotle, consist of indivisible entities. But if there are indivisibles in qualitative change, then, how could qualitative change be continuous? This has an important consequence for the understanding of qualitative change. If alterations are not continuous, then it is not ruled out that alterations have first instants. After all, Aristotle’s argument against first instants holds only for continuous changes. Averroes’s commentary was very influential in the Middle Ages. The scholastics thus had to determine whether qualitative changes are instantaneous or if some could also be continuous. A third issue raised by Aristotle, which was of interest to the debate on intension and remission, concerned the contrary terms involved in gradual change. Aristotle maintains that all motion occurs between contraries, i.e., between, as Norman Kretzmann puts it, “mutually

32 Phys., VIII, 3, 253b26; De sensu et sensato, 6, 447a1-2. 33 Phys., VI, 5, 236b6. 34 Phys., VI, 5, 236b17-18. On instantaneous alterations and their accidental continuity in Aristotle, see R. SORABJI, Time, Creation, and the Continuum. Theories in Antiquity and the Early Middle Ages, London 1983, p. 411; ID., “Latitude of Forms in Ancient Philosophy,” p. 58. 35 AVERROES, Phys., VI, comm. 49, in: Aristotelis Opera Omnia cum Averrois Com- mentariis, vol. 4, Venice 1562 (repr. Frankfurt 1962), fol. 278vK: “[…] transmutatio in quanto est transmutatio in continuo per se, et declaratum est quod continuum semper diuiditur in diuisibile, econuerso transmutationi in qualitate […] non est in continuo.” See also AVERROES, Phys., VIII, comm. 23, fol. 359vI-360F, to which Gregory explicitly refers in Sent., 1, 17, 2, p. 256.

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exclusive, but non-exhaustive” states such as cold and hot, or white and black.36 There are two ways in which a change between contraries may occur, as Aristotle sees it. A thing may move from one contrary to another simpliciter (ἁπλῶς), e.g., when it changes from being black to being white. It may, however, also move from one contrary to another “in a way” (πῃ).37 This is what happens in intension or remission. For instance, when a white thing becomes whiter, Aristotle thinks, the thing does not move away from blackness as such – since it is not black –, but nonetheless “in a way”, because the whiteness it gains is farther away from blackness.38 The same holds mutatis mutan- dis for remission. When a thing becomes less white then it does not move to blackness as such. It does not become black. Yet, it gains a color that is closer to blackness. So, the moving away from a contrary in intension Aristotle understands to be the contrary’s becoming more and more absent in the moving thing. Similarly, the moving toward the contrary in remission he understands to be the contrary’s becoming more present in the moving thing.39 Aristotle thus seems to hold that, in some sense, contraries are both present in processes of intension and remission. Indeed, in the Topics he suggests that intension is just a process of a quality’s being less mixed with its contrary, while remission is the process of its becoming more mixed with its contrary.40 Simplicius would develop a theory on intension and remission on exactly this basis, now called the ‘admixture theory’.41 For the debate of the intension and remission in the Middle Ages this account of contraries had a profound impact. It became crucial to discuss whether intension and remission occurred by admixture, and if and how contraries could be co-present in the changing subject.

36 Phys., V, 1, 225b1-5. See N. KRETZMANN, “Continuity, Contrariety, Contradic- tion, and Change,” in: ID. (ed.), Infinity and Continuity in Ancient and Medieval Thought, Ithaca, NY 1982, pp. 270-296, at 271. 37 Phys., V, 2, 226b3. 38 Phys., V, 2, 226b3-6. 39 Phys., V, 2, 226b8. 40 Topics, III, 5, 119a 27-28. 41 SIMPLICIUS, In Aristotelis Physicorum Libros Quattuor Posteriores Commentaria (= CAG, vol. 10), ed. H. DIELS, Berlin 1895, pp. 863, 10 - 864, 14; see also In Aristotelis Catego- rias commentarium (= CAG, vol. 8.), ed. C. KALBFLEISCH, Berlin 1907, p. 219, 17-18 and p. 284, 20-24. Only this latter text was known in the Latin West through Moerbeke’s translation.

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These then are, in broad outline, three crucial issues that the scholastics inherited from Aristotle’s discussion of gradual changes: (i) the problem of the bearer of more or less (the quality, or the qualified substance), (ii) the problem of whether intension and remission are continuous or instantaneous changes, and (iii) the problem of the admixture and co-presence of contraries. These are perhaps not the only issues, but they correspond precisely to Gregory’s three quaestiones on the intension and remission of corporeal forms. Thus, in d. 17, q. 2, Gregory discusses whether intension and remission are continuous, in q. 3 whether contraries can be co-present in the same subject as in the admixture theory, and finally in q. 4 whether the quality itself or the bearer admits of more or less, and whether this occurs by a process of addition of a new quality. Although the structure of Gregory’s discussion is firmly rooted in Aristotle’s remarks on the topic of gradual change, it is nonetheless expressive of a common trend in fourteenth-century discussions of the problem of intension and remission. The structure of Gregory’s discussion is very similar to that of many other mid-fourteenth- century discussions such as those of John Buridan, Albert of Saxony, and Marsilius of Inghen.42 Is there a reason for this?

42 For Buridan the headings are: “utrum qualitates contrariae […] possint se compati simul in eodem subiecto secundum aliquos gradus ipsarum; utrum qualitas secundum quam est alteratio per se proprie dicta continua et temporalis acquiritur tota simul vel pars post partem; utrum in alteratione pars qualitatis quae prius acquiritur manet cum parte quae posterius acquiritur” (see Acutissimi philosophi reverendi Magistri Johannis Buridani Subtilissime questiones super octo Physicorum libros Aristotelis, diligenter recognite et revise a Magistro Johanne Dullaert de Gandavo, in edibus Dionisii Roce, Paris 1509 [repr. Frankfurt a.M 1964], Liber III, qq. 3-5, fols. 42vb-48ra). For Albert the headings are: “utrum qualitates contrariae secundum aliquos gradus earum possint se simul compati in eodem subiecto; utrum in intensione qualitas quae acquiritur acquiratur tota simul vel secundum partem post partem; utrum in intensione alicuius qualitas quae primo acquiritur maneat cum qualitate quae posterius acquiritur” (see ALBERT OF SAXONY, Quaestiones in Aristotelis Physicam, in: Expositio et Quaestiones in Aristotelis ‘Physicam’ ad Albertum de Saxonia attributae, vol. 3, ed. B. PATAR, Louvain 1999, Liber 5, qq. 9-11, pp. 825-864). For Marsilius the headings are: “utrum in alteratione qualitas acquiratur subito; utrum in intensione pars primo acquisita maneat cum parte secundo acquisita; utrum formae contrariae possint esse simul” (see JOHANNES MARCILIUS INGHEN, Super octo libros Physico- rum, per Johannem Marion, Lyon 1518 [repr. Frankfurt a.M. 1964], Liber III, qq. 3-5, fols. 37vb-40ra. For an account of the structure of these commentaries see S. CAROTI, “Some Remarks on Buridan’s Discussion on Intension and Remission,” in: Vivarium 42 (2004), pp. 58-85, esp. 59.

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In part, this is perhaps explained by the influence of Walter Burley’s Tractatus Secundus, written in the early 1320s.43 In the first chapter of this tract, Burley offered a battery of arguments against the addition theory, which Gregory aims to defend in q. 4.44 In the second chapter of the Tractatus Secundus Burley seeks to refute the doctrine, then espoused, e.g., by Walter Chatton (and later, after the Tractatus Secundus had already appeared, by Adam Wodeham), that contraries can be co-present in the same subject at the same time, which explains Gregory’s extensive treatment of this question in q. 3.45

43 I shall refer to this work as TS. I refer to the Venice edition: WALTER BURLEY, Tractatus de intensione et remissione formarum per Bonetum Lucatellum, Venice 1496. On Walter Burley’s Tractatus Secundus, see H. SHAPIRO, “Walter Burley and the Intension and Remission of Forms,” in: Speculum 34 (1959), pp. 413-427; MAIER, Zwei Grund- probleme, pp. 315-352; E.D. SYLLA, The Oxford Calculators and the Mathematics of Motion. Physics and Measurement by Latitudes, New York 1991, pp. 95-111. On the relation of the Tractatus Secundus to the Tractatus Primus, see L.M. DE RIJK, “Burley’s So-called Tractatus Primus, with an Edition of the Additional Quaestio “Utrum contradictio sit maxima oppositio,” in: Vivarium 34 (1996), pp. 161-191; SYLLA, “Disputationes,” pp. 383- 424. 44 Gregory is not the first to react to Burley. One of the earliest reactions seems to have come from Girald of Odo in d. 17 of his commentary on book 1 of the Sentences (written in the 1320s). Not everyone in the fourteenth century seemed to have taken Burley’s arguments seriously, though. Shapiro mentions a scribe who wrote: “plus non scribo de hoc tractatu, quia frustra esset cum nunc nihil teneatur de eo, quod ipse ponit” (SHAPIRO, “Walter Burley,” p. 413). Maier mentions John of Ripa, who in his commentary on the Sentences (ca. 1350-55) refused to consider Burley’s arguments (MAIER, Zwei Grundprobleme, p. 349). (For the date of Ripa’s commentary, see P. BAKKER – C. SCHA- BEL, “Sentences Commentaries of the Later Fourteenth Century,” in: G.R. EVANS [ed.], Medieval Commentaries on the Sentences of Peter Lombard. Current Research, Vol. 1, Leiden 2002, pp. 425-464, at 434; “Appendix C: Biographies of Medieval Authors,” in: R. PASNAU – C. VAN DYKE [eds.], The Cambridge History of Medieval Philosophy, Vol. 2, Cambridge 2010, pp. 833-996, at 912.) In the twentieth century scholars also took a rather critical stance as regards Burley’s Tractatus Secundus (one exception being Shapiro). Maier states that Burley’s Tractatus Secundus is “voll von Ungenauigkeiten und Irrtümern, absurden Behauptungen und unlogischen Argumentationen” (MAIER, Zwei Grundprob- leme, p. 315). Edith Sylla calls Burley’s Tractatus an “unbalanced work” and considers the first three chapters (which contain the arguments against the addition theory) to be “nearly useless” (SYLLA, The Oxford Calculators, p. 110). 45 WALTER CHATTON, Reportatio Super Sententias Liber I, dd. 10-48, ed. J.C. WEY – G.J. ETZKORN Toronto 2002, d. 17, q. 2, a. 1, pp. 62-63. Wodeham does not defend this view in his Lectura, it seems, but he defends it in his Ordinatio. See ADAM WODEHAM, Super quattuor libros sententiarum: Abbreviatio Henrici Totting de Oyta, ed. JOHN MAIR, Paris, 1512, Liber I, d. 17, q. 4, a. 1, fols. 58vb-59rb. It should be noted, however, that while Chatton held the co-presence of contraries to be naturally possible, Wodeham argued that this was only possible by God’s absolute power.

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What about the issue of the continuity of intension and remission, which Gregory discusses in q. 2? Interestingly, there was no comparable debate as to whether intension and remission could be a continuous process. Most assumed that it was.46 However, they needed to square this with Aristotle’s remarks about alteration. Moreover, there was a debate about some of the concepts associated with that of continuity such as the above-mentioned concept of the first instant, as well as the concept of an intensive minimum, which I shall briefly elucidate here. The concept of an intensive minimum is based on an Aristotelian idea. On Aristotle’s view, forms cannot be instantiated in a magnitude of any arbitrary extension.47 There is a minimum quantity a magnitude needs to have in order for a form of, say, flesh or fire, to be instantiated in it. These minimum quantities are so-called minima naturalia. This idea of Aristotle’s, which concerns extensive magnitudes, was, in the Middle Ages, transferred to intensive magnitudes.48 Medieval thinkers raised the question as to whether there were mini- mal increments by which a quality of, say, heat could be raised. The problem posed by these intensive minima was that, if they existed, they could not be acquired continuously over the course of a period of time, but only in an instant. For, if they were acquired over a period of time, then parts of this minimum would be acquired in the intervals into which the time period is divisible. There are, however, no parts of minima, since such a part would be less than a minimum, which is contradictory. Hence, it became a task to show whether and how the existence of minima is compatible with the continuity of intension and remission. We now have a sense of the intellectual backdrop against which Gregory developed his account of the intension and remission of

46 See E.D. SYLLA, “Infinite Indivisibles and Continuity in Fourteenth-Century Theo- ries of Alteration,” in: KRETZMANN (ed.), Infinity and Continuity, pp. 231-257, esp. 231. 47 Phys., I, 4, 187b35-188a2. 48 For this theory see A. MAIER, Die Vorläufer Galileis im 14. Jahrhundert (= Studien zur Naturphilosophie der Spätscholastik, vol. 1), 1st ed., Rome 1949, pp. 179-196; J.E. MURDOCH, “The Medieval and Renaissance Tradition of Minima Naturalia,” in: C. LÜTHY – J.E. MURDOCH – W.R. NEWMAN (eds.), Late Medieval and Early Modern Corpuscular Matter Theories, Leiden 2001, pp. 91-131; C. TRIFOGLI, “Duns Scotus and the Medieval Debate about the Continuum,” in: Medioevo 29 (2004), pp. 233-266, esp. 248-266.

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corporeal forms. We can also see what motivates the tripartite order according to which Gregory discusses this topic. Clearly, all three issues that Gregory discusses in qq. 2-4 are relevant to the problem of intension and remission. It is important to notice, however, that only q. 4 deals exclusively with intension and remission (here Gregory advances his addition theory and disproves rivaling accounts). In qq. 2-3 Gregory discusses continuity and contrariety, and these topics exceed the narrow scope of intension and remission. Thus in q. 2 Gregory discusses the possibility of first instants and minima not just of gradual change, but of any change whatsoever. Moreover, a large part of q. 3 is dedicated to a discussion of the natural and supernatu- ral possibility of the co-presence of contraries in the same subject without even considering gradual change. Given that q. 4. is fully and qq. 2-3 are only partially relevant to the problem of intension and remission, I shall, in what follows, first discuss q. 4, (section 2), and then consider some of the issues raised in qq. 2-3 (section 3).

2. Gregory’s account of intension and remission In q. 4 Gregory defends an addition theory of intension and remission. On this theory, what happens in a process of intension is that an already existing qualitative form receives a new form of the same kind (eiusdem rationis49), such that these two forms constitute a novel total form (forma totalis50). The forms that constitute the total form remain fully actual once conjoined. Thus, the total form or ‘total quality’, as we may call it, consists of a “multitude” (multitudo) of equally actual partial forms or qualities.51 The assumption of the addition theory that a total quality results from two equally actual partial qualities seems to clash with Aristo- telian orthodoxy, specifically with the tenet that “if two things are in act, then not one thing in act results from them.”52 What Aristotle

49 Sent., 1, 17, 4, p. 380: “Dico […] quod forma quaelibet corporalis, quae intenditur, intenditur per acquisitionem novae formae totaliter vel partialiter eiusdem rationis.” 50 Sent., 1, 17, 4, p. 395: “[...] teneo […] quod forma praecedens remanet simul cum sequente et simul cum illa constituit formam totalem per se unam […].” 51 Sent., 1, 17, 4, p. 402. 52 See GILES OF ROME, In primum librum Sententiarum, Venice 1521 (repr. Frank- furt a.M. 1968), d.17, pars 2, princ. 1, q. 2, a. 1, fol. 96va. The relevant passage in Aristotle can be found in Metaphysics, VII, 13, 1039a3-5.

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means by this is that no two actual things could constitute the per se unity proper to a primary substance, but that this unity requires a potential and an actual part. John Duns Scotus, however, provided a famous counter-example to this claim, namely, the unity constituted by two quantities of water coming together.53 Add water to water, and what you get is not just an aggregate of unconnected ‘waters’, but actually one water that has per se unity. The two constituents of the new quantity of water are not related according to potency and act. Rather they are both actual.54 Yet, they constitute one quantity of water. Exploiting this analogy, Scotus argues (and Gregory follows him on this score) that something similar happens in intensive change. In a process of heating, for instance, the previous and the succeeding form of heat constitute one total heat that has per se unity. You do not just have an aggregate of ‘heats’. Nonetheless, the partial ‘heats’ constituting the total heat remain actual ‘heats’. On the Aristotelian account, only quantity is divisible into actual parts. Since Gregory holds, following Scotus, that total qualities consist of actual parts, we now need to see how he understands the relation between the concepts of quality and quantity. Are total qualities quantities of sorts? If so, what kind of quantities are they? All quantities are, on Aristotle’s view, either continuous or discrete. An example of a continuous quantity is a pebble. Every part of a pebble shares a common boundary with some other part of that pebble. A heap of pebbles, by contrast, is a discrete quantity. We can count the pebbles of the heap, but they do not share a common boundary. In the strict sense of ‘quantity’ (“sumendo quantitatem […] proprie”), Gregory says, total qualities are neither continuous nor discrete quantities.55 They are not discrete quantities because they have, as we have just seen, a per se unity, which discrete quantities such as heaps lack. Total qualities are not continuous quantities either, however. For, in a continuous quantity the parts have distinct spatial positions. For instance, as Gregory points out, in the human body, which is a

53 JOHN DUNS SCOTUS, Opera Omnia studio et cura Commissionis Scotisticae ad fidem codicum edita, vol. 5 (= Ordinatio, 1, dd. 11-25), Vatican 1959, d. 17, pars 2, q. 1, n. 229, p. 250, ll. 18-19. I shall refer to Scotus’s Ordinatio as Ord. 54 Sent., 1, 17, 4, p. 401. 55 Ibid.

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continuous quantity, the chest is not immediately adjacent to the head or the foot. Rather the chest is immediately adjacent to the abdomen.56 In a total quality, by contrast, the opposite is the case, Gregory thinks. Each partial ‘heat’ is immediately united to every other partial heat; the partial ‘heats’ do not have distinct spatial positions.57 Although total qualities are not quantities in the strict sense, on Gregory’s view, he thinks that we may nonetheless treat them as if they were quantities. To this end, Gregory suggests, we can take the term ‘discrete quantity’ in a broad sense (large) to denote any multitude constituting a non-continuous unity (not just a non-continuous multitude lacking per se unity). Since a total quality is constituted by a multitude of non-continuous partial qualities, it satisfies this description, and may thus be viewed as a discrete quantity in this broad sense.58 In view of this account of the constitution of total forms it is quite clear how Gregory solves the Aristotelian problem of the bearer of the ‘more or less’ of a quality: it is the quality itself that admits of more or less. Greater heat means a greater number of partial heats. Lesser heat means a smaller number of partial heats. For Gregory, then, the forms involved in intension and remission have a latitude, i.e., a range of intensive variation.59 Edith Sylla has shown that among the Oxford Calculators (includ- ing Burley) there were two conceptions of latitudes. On the one hand, there is Burley’s (and Roger Swineshead’s) abstract conception, which treats a latitude not so much as a physical reality, but rather as a “conceptual range determined by the various degrees of a species of quality that may be found in particular instances”.60 On this view, a

56 Ibid.: “[…] nec etiam (est quantitas) continua, quia quaelibet pars earum partium, de quibus loquimur, cuilibet immediate unitur; in quantitate autem continua non quaelibet immediate unitur cuilibet, sicut pectus non immediate unitur capiti neque pedi sed ventri.” 57 Sent., 1, 17, 4, p. 400 : “[…] quaelibet pars cuilibet alteri et eius etiam cuilibet parti immediate unitur et cum illa unam per se totalem formam constituit. Et loquor de partibus intensionalibus nequaquam situ distinctis.” 58 Sent., 1, 17, 4, pp. 402-403: “Si tamen large quis omnem multitudinem, tam constituentium aliquid per se unum quam non constituentium, velit appellare quantitatem discretam – quod tamen proprie non fit iuxta communem usum –, concedendum illi est quod talis forma est quantitas quaedam discreta.” 59 Sent., 1, 17, 2, 3, p. 308. 60 SYLLA, “Medieval Concepts,” p. 234.

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latitude such as a range of degrees of heat does not really exist. What really exists are indivisible degrees of heat, e.g., 31 or 32 degrees centigrade. However, on this view, we have a concept of all the possible degrees that heat may have with an upper and a lower limit, and this concept is the latitude. On the other hand, there are those Calculators like John Dumbleton and Richard Swineshead who have a realist conception of latitudes. On their view, latitudes are real because they are the same as the really existing maximum degrees, which, on their view, contain all the lower degrees.61 Since Gregory was influenced at least by the early Calculators (specifically by Burley and Kilvington), we may ask: can he be put in one of these two camps? Gregory does not provide an explicit account of how he understands latitudes. It is likely, however, that he adopts the realist conception, although the evidence is only indirect. For instance, when discussing a sophisma of Richard Kilvington’s, Greg- ory says that “after this instant there will not exist a maximal latitude of this form.” This suggests that on Gregory’s view, latitudes are real. For if latitudes were merely abstract ranges, then Gregory, being a nominalist, would not grant them existence. Furthermore, Gregory speaks interchangeably of “a maximal latitude or (vel) part of this form.”62 A form is a really existing entity, on Gregory’s view, a non- substantial individual. If latitudes were merely conceptual ranges, then the above identification of latitude with the part of a form would imply that a conceptual range could be a part of a really existing entity. It is not clear what that could possibly mean. This also suggests that Gregory adopts a realist conception of latitudes. This, then, is, in broad strokes, Gregory’s account of the ontology of the addition theory. The next question is: how does he defend it? Interestingly, Gregory does not provide any positive arguments for this theory and its corollary that qualities have latitudes. Instead he adopts this theory because he finds all the alternative theories unsatisfactory. What are these other theories, and what is wrong with them? Greg- ory considers three of them: the above-mentioned admixture theory defended by Simplicius, the esse theory of Giles of Rome, and the

61 Ibid, p. 252. 62 Sent. 1, 17, 4, p. 382.

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succession theory of Godfrey of Fontaines and Walter Burley. Maier has noted that many of the arguments, which Gregory deploys to refute these theories, can already be found in Ockham.63 As we shall see, however, Godfrey of Fontaines is also an important source. Moreover, some of Gregory’s arguments seem to be more original. Gregory first considers the admixture theory (q. 3). On this theory, defended by Simplicius, a process of intension occurs by a form’s becoming less mixed with its contrary. Gregory spends considerably less time refuting this theory than the other two. This could be explained by the fact that the admixture theory was no serious contender in the fourteenth century. No thinker, with the exception (perhaps) of Roger Swineshead,64 seemed to endorse it in that period. I shall nonetheless briefly present two arguments that Gregory adduces against the admixture theory. The first argument is that not all processes of intension involve contraries. Light can clearly be intensified. Yet, light has no contrary, but only a privation, namely, darkness.65 Similarly, sight may be intensified inasmuch as I can see more or less clearly, Gregory holds. Yet, there is no contrary to sight, but merely a privation, namely, blindness. A further argument that Gregory adduces is that if intension is no more than a quality’s being less mixed with its contrary, then the increase of that quality is not real,

63 MAIER, Zwei Grundprobleme, p. 74. To see how indebted Gregory was, here is (build- ing on some references provided by the editors of Gregory’s Lectura) a non-exhaustive list of some of the arguments which Gregory in d. 17, q. 4 (the question dedicated to defending the addition theory) takes from Ockham’s Ordinatio, d.17, qq. 4-7 (to which I shall refer as O). For Ockham’s Ordinatio, d. 17, I refer to the critical edition: Opera theologica, vol. 3, ed. G. ETZKORN (= Ordinatio), St. Bonaventure 1977. Five of Gregory’s six arguments “contra propriam responsionem” (arguments 1-5) (Sent., 1, 17, 4, pp. 381-382) are taken from Ockham (O, 1, 17, 6, pp. 513-514). Likewise, all of Gregory’s replies against these arguments (Sent., 17, 4, pp. 383-385) can be found with no or only minor variation in Ockham (O, 1, 17, 6, pp. 514-517). Two of the seven arguments that Gregory makes against Burley’s theory (Sent., 1, 17, 4, pp. 388-395) (arguments 1 and 6) are taken with no, or little, variation from Ockham (O, 1, 17, 5, pp. 491-492). Furthermore, at least five of Gregory’s twelve arguments “contra propriam conclusionem” (arguments 1-4, and 12) (Sent., 1, 17, 4, pp. 395-400) are taken from Ockham (O, 1, 17, 5, p. 487 and 17, 7, pp. 519-524). In addition, Gregory’s replies to arguments 1-4 (Sent., 1, 17, 4, pp. 401- 402) are very similar to those of Ockham’s (O, 1, 17, 7, p. 540 and p. 545). Finally, one of the initial arguments against the claim that in intension something new is acquired can also be found in Ockham (O, 1, 17, 5, p. 495). Gregory’s reply to it (Sent., 1, 17, 4, p. 414) is also the same as Ockham’s (O, 1, 17, 7, p. 519). 64 SYLLA, “Medieval Concepts,” p. 232. 65 Sent., 1, 17, 3, 1, p. 328.

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but merely apparent.66 That, he thinks, is untenable, however, for the increase is real. Gregory’s refutation of the second contender, Giles’s esse theory, is much longer. On this theory, which goes back to Aquinas, a process of intension does not involve the addition of a new form. Rather the same form remains throughout a process of intension, but the inherence (inesse) or existence (esse) in the subject becomes greater.67 So, for instance, when heat is intensified or remitted there occurs, on Giles’s theory, no variation in the form of heat itself, but a variation in the object’s being hot. As Gregory puts it concisely, Giles denies that the form involved in intension and remission possesses any latitude in itself (secundum essentiam), but he accepts that there is a latitude with respect to the being of a given form (secundum esse) in a substance. Simply put, Giles holds that, e.g., heat itself has no degrees, but that heat may exist to a higher or lower degree in a given bearer. Scholars have already discussed Gregory’s arguments against Giles’s theory.68 I here consider one argument, which is taken, in large part, from Godfrey of Fontaines and clearly presents the difficulties of Giles’s theory.69 Gregory wonders how to understand the existence

66 Ibid. 67 GILES OF ROME, In 1 Sent., 17, 2, pars 2, princ.1, q.2, a.1, fol. 96vb : “[…] una et eadem accidentalis forma poterit magis et minus inesse. Et immo potest contingere quod sint in aliqua forma accidentali eiusdem speciei diversi gradus in esse absque eo quod sint diversi in essentia. Notandum quod accidentis esse est inesse: nam accidentia non sunt entia nisi quod sunt entis. Igitur diversi gradus secundum esse accidentis sunt diversi secundum inesse […] Unde una albedo non est maior alia: sed corpus unum est albior alio.” Giles’s theory is a Thomist theory. It should be noted, however, that Aquinas does not employ the esse/essentia distinction in his account of intension and remission. Aquinas’s account is that a quality such as charity is augmented secundum participationem. See THOMAS AQUINAS, Summa theologiae, II-II, 24, 4, in: Opera Omnia iussu Leonis XIII. P.M edita, vol. 8, Rome 1895, p. 177. 68 See GRASSI, “Gregorio da Rimini e l’agostinismo,” pp. 84-89; FIORENTINO, “Gre- gorio da Rimini a confronto,” pp. 24-35. Grassi claims (p. 84) that the (six) arguments which Gregory uses against Giles are analogous to the arguments that Ockham uses against Aquinas. I could only find one (O, 1, 17, 6, p. 501; Sent., 1, 17, 4, p. 385). I could, however, find two arguments that Gregory uses against Giles in PETER AURIOL, Commentariorum in Primum Librum Sententiarum, Pars Prima, Rome 1596, d. 17, pars 3, a.2, pp. 431 and 436. The argument on p. 431 can be found in GREGORY, Sent., 1, 17, 4, p. 378. Auriol’s argument on p. 436 can be found in GREGORY, Sent., 1, 17, 4, p. 376. 69 The arguments can be found in Godfrey’s Disputed Question 18. See J. CELEYRETTE – J.-L. SOLÈRE, “Édition de la question ordinaire n° 18, « De intensione virtutum », de Godefroid de Fontaines,” in: J. MEIRINHOS – O. WEIJERS (eds.), Florilegium Mediaevale.

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(esse) that according to Giles seems to admit of degrees, and he sets up a destructive dilemma. Either existence is understood as an additional entity (tertia entitas), and this entity admits of more or less, or existence is not so understood, and it is solely due to facts about the disposition of the bearer undergoing intension and remission that a form exists to a greater or lesser degree in a given bearer. The first horn of the dilemma might strike one as odd. Why think of existence as a third entity? It should be noted, however, that this account of existence is not at all implausible as applied to Giles because Giles takes existence to be a res superaddita.70 Gregory does not have much patience with the first horn. First, Gregory objects to thinking of existence as a third entity in standard nominalist fashion. He tells us that this entity “is posited superfluously and in vain” (“superflue et vane ponitur”).71 Gregory has a better argument than this appeal to Ockham’s razor, though. If there is this third entity, Gregory demands that we are given an account of its ontological status.72 Is this third entity of the same kind as the form or essence it belongs to or of a different kind? If it is of the same kind, i.e., if it is a form itself, then given that existence does admit of degrees for Giles, it follows that forms admit of degrees, which Giles expressly denies, as we have seen.73 If it is of a different kind then, Gregory points out, we will need to ascribe an essence to it in order to individuate it.74 If Giles grants that, then, Gregory continues, we can direct the very argument Giles uses to disprove the addition theory against Giles himself. The argument is that the addition theorist cannot explain

Études offertes à Jacqueline Hamesse, Turnhout 2009, pp. 83-107, esp. 99-101. Interest- ingly, Godfrey changed his mind on Giles’s theory. In his earlier Quodlibet II, q. 10 he seems to have in fact endorsed a version of it. For an analysis of Godfrey’s arguments against Giles in Disputed Question 18 and a comparison to Quodlibet II, q.10, see also J.F. WIPPEL, “Godfrey of Fontaines on Intension and Remission of Accidental Forms,” in: Franciscan Studies 39 (1979), pp. 316-355, esp. 328-336; J. CELEYRETTE – J.-L. SOLÈRE, “Godefroid de Fontaines et la théorie de la succession dans l’intensification des formes,” in: P. BAKKER (ed.), Chemins de la pensée médiévale. Études offertes à Zenon Kaluza, Turn- hout 2002, pp. 79-112, esp. 89. 70 GILES OF ROME, Theoremata de esse et essentia, ed. E. HOCEDEZ, Louvain 1930, Theorem 19, p. 129. 71 Sent., 1, 17, 4, p. 370. 72 Ibid., p. 372. 73 Ibid. 74 Ibid. : “[…] qualiscumque entitas sit, habebit propriam quidditatem et certitudi- nem […] ita, sicut et forma.”

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gradual change, which remains within the same species75, because he adds form to form, and thereby changes its species.76 In the background of this argument is, of course, Aristotle’s principle that species are like numbers because just as adding one number to another number yields a third number, so adding one form to a species yields another species.77 Now, Gregory seems to claim (without argument, however) that if existence has its own essence, then, Giles runs into the same problems as the addition theorist does, on Giles’s view. For if existence has its own essence, then, since existence admits of more or less, as Giles seems to think, the essence of existence should admit of more or less, too.78 But if the essence exists to a higher or lesser degree, for Giles, then, Gregory asks, would this not, by Aristotle’s principle, change its species, on Giles’s view? Taking the first horn of the dilemma is thus not a promising route for Giles. The second horn seems to Gregory “more subtle” (subtilior).79 If esse is not a separate entity, then Gregory suggests, Giles should be taken to claim that a form has more or less existence in a subject in the sense that the subject is more or less disposed to have this form (“propter maiorem vel minorem dispositionem”).80 Like the first horn, this second horn has its basis in Giles’s text. In fact, it seems to be the more appropriate interpretation of Giles’s. For Giles proposes exactly this sort of dispositional account in his discussion of intension and remission in his Sentences commentary, while he does not appeal at all to esse as a res superaddita or tertia entitas there.81 Ultimately, the dispositional account fails as well. Gregory observes that, on the dispositional account, the claim that a given quality Fness

75 Remember, gradual change is not a change, e.g., from white to black, but from being less white to being more white. 76 Sent., 1, 17, 4, pp. 367-368. 77 See ARISTOTLE, Metaph., VIII, 3, 1043b34-1044a2; Auctoritates Aristotelis, 1, 204, ed. J. HAMESSE, Louvain / Paris 1974, p. 132, ll. 45-48. 78 At least this is implicit in the following few lines (p. 372): “Et similiter talis entitas secundum se et suam essentiam erit sicut quidam numerus sicut et ipsa forma, et per consequens illae rationes aeque militant contra gradualitatem intrinsecam illius entitatis, quam vocant esse, sicut contra gradualitatem essentialem et intrinsecam formae.” 79 Ibid. 80 Ibid. 81 GILES, In 1 Sent., 17, 2, pars 2, princ. 1, q. 2, a. 1, fol. 96vb: “[…] sed inhaeren- tia in accidente maior et minor esse non potest nisi secundum maiorem et minorem dispositionem subiecti. Secundum ergo quod subiectum est magis et minus dispositum ad susceptionem alicuius formae, magis et minus suscipiet illam formam.”

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has more being in x always entails the claim that x is more disposed to being F. This implication, as Gregory remarks, is, however, false on empirical grounds (“patet ad sensum”).82 For instance, a less dense medium such as air is more disposed to being illuminated by light than a denser medium such as water. Yet, the sea is more strongly illuminated under sunlight than the air in a room that is only lit by a candle. Therefore, if Fness exists to a higher degree in x, this does not mean that x is more disposed to Fness than some y in which Fness exists to a lower degree. Having argued against the Simplician admixture and the Aegidian esse theory, there remains one more rival theory to consider for Gregory, namely, the succession theory. On this theory, a new form is gained in a process of intension or remission, but the preceding form does not remain. Rather it is corrupted and is replaced by the new advening form.83 The best-known advocates of this theory were Godfrey of Fon- taines and Walter Burley.84 The version of the theory that Gregory considers is Burley’s, as presented in Burley’s Tractatus Secundus. Gregory adduces some of the standard arguments against this theory. One such argument, developed by Scotus, is that the succession theory, if true, can only provide a strange explanation of remission.85 Suppose a hot thing remits the coldness in a hot object (i.e., the hot thing warms some cold(er) object). On the succession theory, the remission consists in a succession of ever lesser degrees of coldness, with none of the

82 Ibid. 83 This theory is expressed by two of Burley’s three famous conclusions from his Tractatus Secundus, 4, fols. 10va-11rb: “[…] pono tres conclusiones. Prima est quod in omni motu ad formam acquiritur aliquid novum quod est forma vel pars formae […]. Secunda est quod per omnem motum corrumpitur tota forma praecedens a qua est per se motus et acquiritur una forma totaliter nova cuius nihil praefuit.” 84 I take it that, despite arguments to the contrary, Stephen Dumont has shown that Godfrey did defend the succession theory. See S. DUMONT, “Godfrey of Fontaines and the Succession Theory of Forms at Paris in the Early Fourteenth Century,” in: S. BROWN – T. DEWENDER – T. KOBUSCH (eds.), Philosophical Debates at the University of Paris in the Early Fourteenth Century, Leiden 2009, pp. 39-125, esp. 78-93. Dumont has also drawn attention to some less well-known defenders of the succession theory at Paris in the fourteenth century (referred to as “multi alii” by Walter Burley in his Tractatus Secundus). First, Peter of St. Denys is reported to have held the succession theory. Further defenders were John of Pouilly and Thomas Bailly (who granted Burley his license in theology when Burley was writing his Tractatus Secundus). See DUMONT, “Godfrey,” pp. 94-96. See also B.F. CONOLLY, “Dietrich of Freiberg on the Succession of Forms in the Intensification of Qualities,” in: Recherches de Théologie et Philosophie médiévales 81 (2014), pp. 1-35, esp. 26-34. 85 Ord., 1, 17, 2, 1, 212, vol. 5, p. 243, ll. 8-14.

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preceding degrees remaining. It follows that a hot thing produces degrees of coldness, “which is implausible” (“quod est inconveniens”).86 Another argument that Gregory takes from his predecessors is that the succession theory cannot maintain the continuity of a change of intension.87 For, the succession theory has to conceive of it as a series of instants (or instantaneous acquisitions), which series, as we saw, can never yield a continuum on the Aristotelian conception of the continuum.88 Gregory develops this latter argument against the succession theory in an interesting way. Take two distinct degrees A and B of any arbitrary quality, where A

86 Sent., 1, 17, 4, p. 394. 87 The argument can be found, for instance, in AURIOL, Commentariorum, d. 17, pars 3, a. 2, pp. 436-437. 88 Sent., 1, 17, 4, pp. 391-392. 89 Sent., 1, 17, 4, p. 391. 90 TS, 4, fol. 11vb. 91 Sent., 1, 17, 4, p. 391: “[…] sequitur quod durante intensione continua per aliquod tempus mobile quievit et in nullo instanti illius temporis aliquid acquisivit; quod implicat contradictionem.”

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Perhaps, then, Burley should take the counter-intuitive second horn of the dilemma, viz. that t1 and t2 are identical. That, however, is a slippery slope, as Gregory points out. For once we admit that two distinct degrees are acquired in one instant, what prevents us from extending this reasoning to other degrees? Indeed, what prevents us from saying that all degrees are acquired in an instant? Every process of intension would then end up occurring instantaneously, which Burley, who wants to maintain the continuity of at least some intensive changes, would of course deny. Due to the failure of the three rivaling theories of intension and remission (the admixture, the esse, and the succession theory), the addition theory is, for Gregory, the only option that remains. Hence, he adopts it. Gregory, however, does not content himself with this defense of the addition theory by elimination. In the final section of q. 4 he also defends it against a series of counter-arguments. Most of these arguments can be found in Burley’s Tractatus Secundus, some of which I shall discuss shortly. First, however, I want to consider Greg- ory’s reply to two pre-Burleian arguments against the addition theory, one from Godfrey (repeated by Burley92), the other from Peter Auriol. Perhaps the best-known argument against the addition theory that Gregory considers is Godfrey’s so-called “termini-argument.”93 In Gregory’s (correct) presentation, it runs as follows: Term. 1) The terminus a quo and terminus ad quem of a change are opposites, and therefore incompatible. 2) The terminus a quo of an intension is the previous form, the terminus ad quem of an intension is the succeeding form. 3) Hence the previous form of an intension and the successive form are opposites, and therefore incompatible (1, 2).94

It is clear that this argument, if successful, rules out the truth of the addition theory. For, the addition theory holds that the previous and succeeding form constitute a unity, which of course presupposes that they are compatible.

92 TS, 1, fol. 4ra-va. 93 The term is Dumont’s. See DUMONT, “Godfrey,” p. 54. 94 Sent., 1, 17, 4, p. 399: “[…] termini motus sunt incompossibiles. Sed forma praecedens et subsequens sunt termini motus intensionis, igitur sunt incompossibiles.”

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Immediately after presenting the termini-argument, Gregory adds an interesting proviso, which he takes from Burley’s Tractatus Secun- dus: “and it cannot be said that the terminus a quo is the privation of the advening form, for intension is a motion, and motion is from what is affirmative to what is affirmative [...].”95 This remark is directed against Scotus and Ockham. These two thinkers react to the termini-argument by arguing that the termini of intension are not, as Term.3 maintains, the prior and the posterior form, but rather the privation of the form and the form itself.96 In his own reply to the termini-argument Gregory also rejects Term.3. Term.3 is false, as Gregory sees it, because the preceding and the succeeding form in intension and remission are not opposites. When whiteness is intensified, for instance, then the preceding and the succeeding terms are both forms of whiteness, rather than whiteness and blackness. But if the termini of intension and remission are not related as opposites, how are they related, on Gregory’s view? Gregory accepts the proviso just mentioned and opposes the Scotus- Ockham view that the termini of intension and remission are privation and form. Gregory rejects this view because, as he argues plausibly, when an intensification of a quality such as whiteness takes place the object “does not change from not being white to being white, for it is already white”.97 Gregory’s own suggestion is that we should take “more” and “less” (minus et magis) to be the termini of intension and remission, just as “greatness and smallness” (magnitudo et parvitas) are, for Aristotle98, the termini of the quantitative change of increase or decrease.99 Gregory thus replies to the termini-argument by con- struing intension and remission on the model of quantitative change.

95 Ibid.: “Nec potest dici quod terminus a quo sit privatio formae, nam intensio est motus, et motus est de affirmato in affirmatum […].” See TS, 2, 1, fol. 4ra. 96 Ord., 1, 17, 2, 1, 227, vol. 5, p. 249; O, 1, 17, 5, vol. 3, p. 493. The Scotus- Ockham view was also defended, e.g., by Francis of Marchia (who was Master of Theol- ogy in Paris when Gregory pursued his studium generale there in the 1320s). See Francis’s Commentarius in IV Libros Sententiarum Petri Lombardi, vol. 3 (= Distinctiones Primi Libri ab Undecima ad Vigesimam Octavam), ed. N. MARIANI, Grottaferrata 2007, d. 17, q. 3, 115, p. 137. 97 Sent., 1, 17, 4, 2: p. 412: “[…] quod fit magis album, non transmutatur ex non albo in album, cum ipsum iam actu sit album.” 98 Phys., VIII, 7, 261a35-6. 99 Sent., 1, 17, 4, p. 412.

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Again, Gregory treats qualities as if they were quantities, the more or less of quality being closely paralleled to the greatness and smallness of quantity. A similar move of modeling quality on quantity can be detected in Gregory’s reply to what could be called Peter Auriol’s unity-argument against the addition theory.100 The argument is that the addition theory is false because the addition theory claims that the total form has a per se unity, although the total form can, in fact, have none of the established forms of per se unity. At best the unity will be the weak one of an aggregate. Auriol lists three kinds of per se unity: (i) numer- ical unity, (ii) unity of composition, and (iii) the unity of a homogenous whole.101 It is clear that a total form cannot consist of numerically identical partial forms. Although the addition theorist would maintain that the partial qualities of a total form are immediately united to one another, he does not argue that they are numerically identical. A total form does not have the unity of composition either, on Auriol’s view. For, we can only find this kind of unity in sub- stances, and here the “parts” (i.e., matter and form) relate as potency to act, which the partial qualities do not, as we saw earlier. Given Scotus’s water analogy mentioned earlier, one might think that the unity of a homogenous whole is the unity that will do the trick. Is not, after all, every part of a total quality of heat, itself heat, just as every part of water is itself water? Auriol, however, does not think that total qualities have the unity of homogenous wholes. For,

100 Strictly speaking, the argument is directed against Scotus’s understanding of the addition theory. Auriol does not have a problem with the basic idea of the addition theory (Commentariorum, d. 17, pars 3, a. 1, p. 431). He has a problem with Scotus’s conception of the advening form. Auriol explains that, on Scotus’s theory, the advening form has a being of its own. It is a form praecise or a forma integra (ibid., a. 2, pp. 441-442). For instance, the form of charity added to the already existing charity is charity itself as a whole (charitas integra). On Auriol’s view, however, the advening form is a form imprae- cise. The advening charity is not charity itself as a whole, but “concharitas,” as Auriol calls it. (ibid.). For a detailed discussion of Auriol’s views, see C. SCHABEL, “Place, Space, and the Physics of Grace in Auriol’s Sentences Commentary,” in: Vivarium 38 (2000), pp. 117-161, esp. 122-126. 101 I cannot discuss here Auriol’s and Gregory’s views on ‘wholes’ and ‘parts’. The ‘wholes’ considered in this context are ‘continuous integral wholes’. On this topic, see A. ARLIG, “Is there a Medieval Mereology?,” in: M. CAMERON – J. MARENBON (eds.), Methods and Methodologies. Aristotelian Logic East and West, 500-1500, Leiden 2011, pp. 161-189, esp. 165, and ID., “Parts, Wholes, and Identity,” in: J. MARENBON (ed.), The Oxford Handbook of Medieval Philosophy, Oxford 2012, pp. 445-467.

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homogenous wholes are continua, and we have seen earlier that the parts of continua have distinct spatial positions, while the addition theorist’s parts of a total quality do not. Hence, (iv) the loose unity of an aggregate seems to be the only option left, which is not what the addition theorist wants for his total quality. Gregory replies to this argument as follows. He suggests that Auriol is in a sense right when he says that a total quality cannot have any of the per se unities in the above-defined sense. The addition theorist can react in two ways, however, as Gregory notes (following Ock- ham102). First, he could argue that Auriol’s argument is not problem- atic because the disjunction of the four kinds of unity is simply not exhaustive. That is, the addition theorist can argue that the unity of a total quality is, as Gregory puts it, “another type of unity, which you can call whatever you wish.”103 This is an ad hoc reply, and, fortunately, Gregory also proposes another more interesting reply. He suggests that we relax the conditions Auriol employs to define the per se unity of composition and the per se unity of a homogenous whole. Auriol claims that only parts that relate as potency to act to one another constitute a unity of composition. In other words, only the constituents of a substance can have such unity. Gregory observes, however, that our common linguistic practice (“communiter dicimus”) also allows us to say “that something is composed of its dimensional and quantitative parts.”104 We need not, then, Gregory suggests, restrict the unity of composition to the unity of a substance. We can broaden its meaning to also include the unity among quantitative parts. Once that is allowed, Gregory suggests, we can also consider a total quality to have a unity of composition, since this unity is at least as strong as the unity of a body’s quantitative parts. Gregory also thinks that we can relax the conditions that Auriol employs to define a homogenous whole. Auriol holds that homogenous wholes are necessarily continuous quantities, and this requires,

102 O, 1, 17, 7, vol. 3, p. 540. 103 Sent., 1, 17, 4, p. 402: “[…] faciunt unum […] quadam alia unitate seu modo unitatis, cui nomen imponas, ut tibi placeat.” 104 Ibid.: “[…] potest dici quod faciunt unum unitate compositionis, si quaelibet constituentia unum per se dicantur esse unum unitate compositionis et non solum ea, quorum unum aliud informat, quemadmodum communiter dicimus aliquid componi ex partibus suis dimensionalibus et quantitativis.”

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as we saw, that the parts of such a whole have distinct spatial positions. Gregory suggests, however, that we could drop this latter requirement and say, more generally, that all wholes consisting of parts of the same kind, whether they have distinct spatial positions or not, are homogenous wholes.105 On this broader understanding of ‘homogenous whole’, total qualities would, clearly, also count as homogenous wholes, since all total forms are of the same kind as the partial forms they include. Gregory does not repeat the move of appealing to common linguistic practice to make this point about the term ‘homogenous whole’. One could easily construe a linguistic argument on his behalf, however. One could argue that the term ‘homogeneity’ as such really only means that wholes and parts be of the same nature; it does not also include in its meaning that the parts have distinct spatial positions. Once again, we see how Gregory suggests extending the meaning of certain terms, which are commonly applied to quantities (here: ‘unity of homogeneity’) or quantitative parts (‘unity of composition’), to better explain tenets crucial to his doctrine of intension and remission. Gregory’s point seems to be that even though total qualities are distinct from quantitative wholes such as continua or unities of composition, we can nonetheless treat them as if they were such wholes. These were Gregory’s replies to the termini-argument and to Auri- ol’s unity-argument against the addition theory. I now turn to what constitutes one of the most interesting parts of Gregory’s discussion of intension and remission, namely, his replies to Walter Burley’s arguments against the addition theory. In his replies Gregory shows great technical rigor, and we can gain further insight into his quantitative approach to total qualities. Six of the twelve arguments against the addition theory that Greg- ory considers in q. 4 are taken from Burley’s Tractatus Secundus.106 I shall consider three of them, all of which Gregory reports quite accurately. The first two arguments that I shall consider rely on the

105 Ibid., p. 401: “Rursus dici potest quod sunt unum unitate homogeneitatis, si taliter dicantur esse unum quaecumque sunt eiusdem rationis et partes alicuius, quod est per se unum et eiusdem etiam rationis cum quolibet illorum.” 106 The arguments are the fifth to eleventh argument (ibid., pp. 396-401). Gregory’s replies can be found ibid., pp. 403-414.

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notion of the actual infinite (henceforth: infinity arguments). The third argument that I shall consider is based on observations concerning the relation between what changes and what is changed in a process of remission107 (henceforth: agent-patient108 argument). I discuss Gregory’s replies specifically to these arguments for two reasons. First, Gregory’s replies to the infinity-arguments are the longest he gives. Second, his reply to the agent-patient argument brings in an unexpected theological component. These replies are thus particularly interesting.109 Burley’s first infinity argument takes a natural phenomenon as its point of departure: a body’s being more and more lit as a light source approaches it. The argument seeks to show that if you want to explain this basic phenomenon by appeal to the addition theory you end up having to posit a form of light that has an actually infinite intensity, which is, of course, impossible on Aristotelian grounds. If we add one step to the argument that follows logically, but is not explicitly in Burley’s text, the argument can be put as follows:110 Inf.1 1) Suppose a light source x causes a certain amount of light in some body y in some instant t1 and that x approaches y after t1 with some continuous 111 motion M, which stops at t2.

107 See SYLLA, Oxford Calculators, p. 102. 108 For the sake of economy and for lack of a better term, I use the word ‘agent’ to indicate whatever changes something, and ‘patient’ to indicate whatever undergoes a change under the influence of the agent. 109 I shall not here adjudicate how ‘original’ Gregory’s replies are. Unfortunately, I have not been able to compare Gregory’s replies to earlier, still unprinted reactions to Burley’s Tractatus Secundus (e.g., in the Sentences commentary of Girald of Odo). I note here that Gregory’s replies to Burley’s second infinity argument and Burley’s agent-patient argument begin with the personal “nego” (Sent., 1, 17, 4, ad 7 and 8, pp. 409-410). Gregory’s reply to Burley’s first infinity argument, by contrast, begins with the impersonal “neganda est” (ibid., ad 5, p. 403). It is clear, however, that we cannot infer from this that Gregory is saying something original in the former two replies, but nothing original in the latter mentioned reply. 110 I shall do this in all of my more formal presentations of Burley’s arguments and Gregory’s replies. The absence of a footnote next to a step in the argument indicates that the step is my addition. Since my additions, as far as I can tell, follow from premises that are explicitly in the text, they are not arbitrary. 111 TS, 1, Prob. 1, fol. 2rab: “Ponatur quod aliquid corpus luminosum causet lumen in aliquo susceptivo luminis in instanti A et continue post A moveatur versus illud susceptivum luminis.”

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2) In every instant within the time interval t1-t2 in which x approaches y, x 112 causes an equal amount of light in y as x caused at t1. (That is, there are as many amounts of lights as there are instants in between t1-t2.) 113 3) There are infinitely many instants in t1-t2. 4) There are infinitely many amounts of light produced during M. (2, 3) 5) The addition theory holds that the previous form remains together with the succeeding one in the intension of that form.114 6) The addition theory thus holds that infinitely many equal amounts of light produced during M remain in y at the end-point of M, i.e., at t2. (4, 5)115 7) Hence the addition theory holds that if x causes a certain amount of light in some body y at some instant t1 and that x approaches y after t1 with some continuous motion M, which stops at t2, then at t2 an infinitely intense light exists in y, which is impossible (1, 6).116 The argument appears to operate with two fairly unobjectionable assumptions, one involving continuous approximation (Inf.1.1), the other involving the causation of light (Inf.1.2). The two further premises, Inf.1.3 and Inf.1.5, are likewise unproblematic. Inf.1.3 is an Aristotelian mainstay, and Inf.1.5 is an accurate portrayal of the addition theory. So, how does Gregory react to Inf.1? He does not seem to contest its validity, but rather thinks that one of its premises is false.117 He says that what is wrong with Inf.1 is that it supposes that “in the continuous

112 Ibid.: “[…] corpus luminosum in quolibet instanti temporis mensurantis illum motum, seu illam appropinquationem causet tantum lumen in illo susceptivo quantum causavit in A.” 113 Ibid.: “[…] infinita sunt instantia in tempore illius appropinquationis.” 114 Ibid. : “[…] augmentum formae secundum istam positionem fit per additionem partis ad partem utraque parte remanente.” 115 Ibid.: “Sequitur quod infinita lumina aequalia in intensione erunt simul in illo susceptivo.” 116 Ibid.: “Illud totale lumen sit infinitum intensive, quod est impossibile.” 117 Maier argues that the argument is invalid (MAIER, Zwei Grundprobleme, p. 329). It is illicit, she argues, to infer from the claim that a multitude is composed of infinitely many parts (Inf.1.6) that the multitude constituted by these parts is infinite (Inf.1.7), i.e., exceeds any determinate measure. This entailment holds only if the multitude, or rather: series (here: the series of partial lights) is divergent, i.e., a series that does not have a finite limit (such as, e.g., the sum of positive integers 1+ 2 + 3...). But the entailment does not hold for a convergent series, which approaches a given number (such as, e.g., the sum of ½ + ¼ + ⅛…). Maier ignores, however, that Burley argues that equal amounts of light are added together. The sum of equal amounts of light, clearly, cannot approach a given number. It will have to be a divergent series, and this makes the argument valid.

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intension of light in each instant some light is first acquired, which is false”.118 The light that is “first acquired” in each instant is, I take it, an amount of light that has not been acquired before, i.e., a new amount of light. Thus, Gregory seems to object to what could be called the instantaneous acquisition claim: Inst.Acq. In any given instant of a continuous process of illumination a new amount of light is acquired. Inf.1 does not explicitly appeal to Inst.Acq. The opening premises of the argument, Inf.1.1 and Inf.1.2, seem to entail it, however. For in both premises Burley says that a light source causes a certain amount of light in some body in an instant. If some quality such as light is caused instantaneously in a body, then it seems plausible to say that the body instantaneously acquires that quality; and this vindicates attributing Inst.Acq. to Burley. But what is the problem with Inst. Acq.? Basically, Gregory’s argument is that Inst.Acq. involves an impossible view about spatial approximation. Here is Gregory’s argument: Red.Inst.Acq. 1) Suppose, as Inst.Acq. does, that a new amount of light is acquired by y in some instant t in between t1 and t2. 2) If in some instant t in the interval t1-t2 in which a continuous intension takes place a certain amount of light is acquired by y, then x, causing this light, approaches y at t.119 3) If x approaches y at t, x approaches y at t either by some divisible part of the path it traverses at t, or by some indivisible part of the path it traverses at t.120 4) x does not approach y by traversing some divisible part of the path at t. For t is an instant, and so x would traverse a divisible path instantaneously, which is impossible.121

118 Sent., 1, 17, 4, p. 403: “[…] supponit unum falsum, videlicet quod in intensione luminis continua in quolibet instanti aliquod lumen primo acquiratur, quod falsum est.” 119 Sent., 1, 17, 4, p. 403: “Si in casu dato in aliquo instanti temporis mensurantis intensionem luminis aliquod lumen acquiritur, susceptivum in illo instanti fit magis lucidum, […] igitur in illo instanti corpus causans lumen fit susceptivo magis propinquum. Sed hoc est falsum.” 120 Ibid.: “[…] aut fit sibi magis propinquum per divisibile spatii quod tunc acquirit […] aut fit sibi magis propinquum praecise per aliquod divisibile totum simul.” 121 Ibid.: “[…] et hoc est falsum, quia ipsum non acquirit aliquod divisibile totum simul.”

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5) x does not approach y by traversing some indivisible part of the path at t. For, it is impossible that x’s proximity to y become greater by adding an indivisible.122 6) Hence x does not approach y at t. (3-5) 7) Hence it is false that a certain amount of light is acquired by y in some instant t in between t1 and t2. (2, 6) (Contradiction 1, 7) Gregory here argues in Red.Inst.Acq.2, plausibly, that the acquisition of amounts of light in a continuous process of intension requires that the light source spatially approach the object that is lit. Gregory then claims that since the acquisition is instantaneous, the spatial approximation will be instantaneous too. If the light is received instantaneously, then the light source must come closer in an instant. Perhaps we would not take this claim to be plausible. It seems, however, plausible to construe Burley’s argument along these lines. For in Inf.1.2 Burley can be taken to correlate each new causation of light with the coming closer of the light source to the lit object. If it is legitimate to attribute to Burley something like instantaneous approximation, then Gregory can develop his argument that such an instantaneous approximation is an absurdity (Red.Inst.Acq.3 to 7). This type of approximation is absurd because we can only conceive of it in two ways, and both are impossible. The first way is that the approximation implies the traversing of a divisible distance (say, 3 centimeters), in which case a divisible path will be traversed instantaneously, and this is impossible, at least on the Aristotelian view (Red. Inst.Acq.4). The second way is that the approximation does not imply the traversing of a divisible path, but rather the traversing of an indivisible path (assuming such a thing exists). A traversal of an indivisible path, however, is impossible as well (Red.Inst.Acq.5) because, as Gregory sees it, approaching an object implies traversing an extended path. Now indivisibles have zero extension, for Gregory. So, a light source could never approach a lit object by traversing an indivisible path. The light source would never come closer. In sum, then, Gregory blocks Burley’s first infinity argument by showing that one of its implicit assumptions, namely, that there could

122 Ibid., pp. 401-402:“[…] si sic, sequitur quod indivisibile additum divisibili facit ipsum maius, et demptum facit minus, ex quo propinquitas est maior et distantia seu spatium quod intercipitur est minus propter solum indivisibile acquisitum.”

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be approximation in an instant, leads to absurdities. Burley, however, has another infinity argument to marshal against the addition theory. It turns crucially on the theory of ratios, i.e., the theory of relations between two quantities. Here is the argument: Inf.2 1) According to the addition theory individual qualities are infinitely divisible into individual quality parts.123 2) A whole whiteness w is more perfect than any of its parts, and the perfection can be measured according to some ratio (proportio)124 (E.g., w is twice as perfect as its half, four times as perfect as its fourth, and so forth in infinitum.125) 3) Therefore, according to the addition theory, w is infinitely divisible into infinitely many less perfect ‘whitenesses’ according to some ratio, such that w is twice as perfect as its half, four times as perfect as its fourth, and so forth in infinitum (1, 2). 4) It is impossible that two individuals of two distinct species are of equal perfection.126 Any given individual of one species will always be more perfect than any given individual of another species (e.g., any individual bit of whiteness is more perfect than any individual bit of blackness). 5) Suppose there is an individual whiteness w and an individual blackness b.127 6) Any whiteness that is a part of w is more perfect than b (4, 5).128 7) What exceeds the more perfect exceeds the less perfect.129 8) Therefore, according to the addition theory, the whole whiteness w exceeds b beyond any ratio (ultra proportionem) (3, 6, 7).130 9) If x exceeds y beyond any ratio, x exceeds y infinitely.131

123 Ibid.: “[…] a habet infinitas partes qualitativas sive formales secundum istam opinionem.” 124 In medieval geometry, proportio translates Euclid’s λόγος (Elements, V, def. 3), now commonly rendered as ‘ratio’ in English, while proportionalitas translates ἀναλογία (Elements, V, def. 5), now commonly rendered as ‘proportion’. See the relevant passages in the Latin translation of the Elements: EUCLID, Elementa Geometriae per Erhardum Ratdolt. Trans. Adelard of Bath, ed. J. Campanus, Venice 1482. 125 TS, 1, 4, fol. 3rb: “[…] a excedit aliquam sui partem in duplo; et aliam in triplo [...] et sic in infinitum.” 126 Ibid.: “[…] impossibile est duas species esse aeque perfectas in universo. Nec est possibile duo individua diversarum specierum esse aeque perfecta.” 127 Ibid.: “Sint a et b duo individua specierum contrariarum. Et sit a individuum albedinis. Et sit b individuum nigredinis.” 128 Ibid.: “[…] quaelibet pars ipsius a est perfectior in entitate quam ipsum b.” 129 Ibid.: “[…] quod excedit perfectius excedit minus perfectum.” This step is miss- ing in Gregory’s presentation. Sent., 1, 17, 4, pp. 398-399. 130 TS, 1, 4, 3rb: “Et per consequens a excedit b in perfectione sine omni proportione.” 131 Ibid.: “Huic rationi forte diceretur quod a excedit b in perfectione sine omni proportione numerali. Nec plus concludit argumentum nec ex hoc sequitur quod a sit

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10) If x exceeds y infinitely, then x is infinitely perfect.132 11) Hence, according to the addition theory, w is infinitely perfect in terms of intensity (8-10).133 The purport of this argument is to show that if the addition theorist’s account of qualities admitting of more or less (such as whiteness) as constituted by partial qualities (such as whiteness parts) is correct, then these qualities will necessarily be of infinite intensity, which is absurd. The first seven steps are fairly unproblematic. Inf.2.1 accurately represents a tenet of the addition theory. Inf.2.2 and Inf.2.3 show that if the defender of the addition theory accepts the infinite division of total forms according to ratios, then he must commit himself to a gradation of perfection between partial and total forms. The prima facie odd Inf.2.4 states a principle that Burley’s contemporaries would have accepted. Inf.2.5 and Inf.2.6 show what follows from this principle if we assume the existence of an individual whiteness and an individual blackness, and Inf.2.7 states an obvious truth. What the reader is likely going to take issue with is the derivation of Inf.2.8 via Inf.2.3, Inf.2.6 and Inf.2.7. This claim arguably constitutes the heart of the argument. Burley’s derivation of Inf.2.8 is not implausible, however. Inf.2.3 tells us that w is more perfect than any of the parts it is divisible into according to some ratio (proportio).134 W is either twice, four times, and so on in infinitum, as perfect as

infinitae perfectionis. Sed ista responsio non valet.” This passage implies that, for Burley, if x exceeds y beyond any ratio then x is of infinite perfection. Notice, however, that Burley does not directly infer that x is of infinite perfection from the claim that x exceeds y beyond any ratio. As n. 131 below shows, Burley first reasons: if x exceeds y beyond any ratio, then x exceeds y infinitely, and then deduces that if x exceeds y infinitely, then x must be of infinite perfection. 132 Ibid.: “[…] si a excedat b in perfectione excedit finite aut infinite: si infinite: ergo est infinitae perfectionis: quia nullus finitus excedit aliud finitum in infinitum.” 133 Ibid.: “[…] forma excedens esset infinitae perfectionis intensive.” 134 On the medieval conception of ratios and their Greek and Arabic sources, see J.E. MURDOCH, “The Medieval Language of Proportions,” in: A.C. CROMBIE (ed.), Scien- tific Change. Historical Studies in the Intellectual, Social and Technical Conditions for Scien tific Discovery and Technical Invention, from Antiquity to the Present, London 1963, pp. 237- 271, esp. 251-261; H.L.L BUSARD, “Die Traktate De Proportionibus von Jordanus Nemorarius und Campanus,” in: Centaurus 15 (1970), pp. 193-227; E.D. SYLLA, “Com- pounding Ratios: Bradwardine, Oresme, and the First Edition of Newton’s Principia,” in: E. MENDELSSOHN (ed.), Transformation and Tradition in the Sciences. Essays in Honor of I. Bernard Cohen, Cambridge 1984, pp. 1-44, esp. 17-26.

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some part of it. The in infinitum-part is important. Burley, I take it, wants to pair off any given ratio with some relation between the whole of w and a part of w. Take any ratio you want, and there will be a relation between w and a part of w that instantiates that ratio. Now if you accept that, and if it is also true that any part of w exceeds b in perfection, as Inf.2.6 claims, then, given that what exceeds the more perfect also exceeds the less perfect Inf.2.7, there is no ratio according to which w is greater than b. In other words, Inf.2.8 is true. To see why this is so suppose you take the ratio of being a million times greater. This will be a ratio relating w and some part of w. Since any part of w is greater in perfection than b, w will not be a million times greater than b, but even more than a million times greater. Now take the ratio of being a billion times greater. This will, again, be a ratio relating w and some part of w, and so w will also be more than a billion times greater than b. You can reiterate this procedure ad infinitum: you will not find a ratio relating w and b. For you can only relate w and parts of w according to a ratio, and any part of w will exceed b. Hence, w exceeds b beyond any ratio (ultra proportionem). This, then, is the strategy underlying Burley’s second infinity argument. How does Gregory react to it? Interestingly, Gregory fully accepts Burley’s strictly quantitative treatment of the relation between total and partial quality in terms of ratios. What he attacks are Inf.2.8 and Inf.2.9. First he observes that the phrase “x exceeds y beyond ratio” (“excedere aliud sine proportione”135) can be understood in two ways: Exc.rat.

1) x exceeds y beyond ratio =def (i) x exceeds y, (ii) not according to any numerable ratio (proportio numeralis), but (iii) in such a way that any part of x relates to y according to some ratio, and (iv) one part of x is equal to y (i.e., x and y relate as 1:1) 2) x exceeds y beyond ratio =def (i) x exceeds y, (ii) not according to any numerable ratio, and (iii) no part of x exceeds y according to a numerable ratio, and (iv) no part of x is equal to y.136

135 I think it is legitimate to translate “sine proportione” here as “beyond proportion” because Gregory is here making reference to Burley’s argument, which employs “ultra proportionem.” If excess “sine proportione” meant something different from excess “ultra proportionem,” Gregory’s counter-argument could not get off the ground, since it would employ a conception of excess that Burley need not accept. 136 Sent., 1, 17, 4, 2: pp. 410-411: “[…] dico quod excedere aliud sine proportione potest dupliciter intelligi. Uno modo quod excedat ultra omnem proportionem numeralem,

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What does this distinction amount to? First we need to understand the notion of ‘numerable ratio’. It is useful here to turn to Brad- wardine’s Geometria speculativa (written in the 1320s), which presents in summary fashion the basic tenets of (late) medieval geometry. Bradwardine distinguishes between two kinds of ratio: rational and irrational ones (proportio rationalis/irrationalis).137 Rational ratios are for instance: 2:3, 3:4, 5:6, etc. The two quantities that are related according to a rational ratio have in common a quantity that “numbers” (quantitas numerans) them. For instance, the quantity that 2 and 3 have in common is 1. Two times 1 yields 2, and three times 1 yields three. Hence, the rational ratio 2:3 is, as Bradwardine tells us, numerable (“proportio est […] numeralis”).138 In irrational ratios the quantities do not have a quantity in common (at least not one that can be expressed by a positive integer). Bradwardine gives as an example the relation between the side and the diagonal of a square. There is no common element that, when multiplied, will yield both the side and the diagonal. Here the ratio is not numerable.139 It seems plausible to take Gregory’s distinction between two types of excess to be a distinction between excess according to a rational (= numerable) ratio (Exc.rat.1) and excess that is not according to rational ratio (Exc.rat.2) (which includes, but is not exhausted by excess according to an irrational ratio). According to Exc.rat.1, then, any given part of x is related to y according to a rational ratio. One rational ratio will be that of 1:1, which is why Gregory says that one part of x will be equal to y. Notice that here only the parts of x are related according to ratios to y. The whole of x is not so related to y. This is why x, according to Exc.rat.1, will exceed y beyond numerable ratio, simply by virtue of exceeding any of its parts. According to Exc.rat.2, neither x nor any part of x is related to y according to

sic tamen quod in qualibet proportione numerali aliqua pars eius excedat aliud, et per consequens etiam aliqua pars eius se habet ad aliud in proportione aequalitatis. Alio modo quod excedat sine proportione, id est excessu improportionali seu non se habente in aliqua proportione numerali, et cum hoc etiam nulla pars eius excedat aliud secundum aliquam proportionem numeralem, nulla etiam se habeat ad illud in proportione aequalitatis.” 137 THOMAS BRADWARDINE, Geometria speculativa, Latin text and English translation with an introduction and commentary by G. MOLLAND, Stuttgart 1989, Tertia Pars, Prop. 3.1, p. 88. 138 Ibid. 139 Ibid.

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some numerable ratio. This can mean that there is no ratio whatsoever between parts of x and y. It can also mean, however, that the ratio between x and y is irrational, i.e., that there is no common ‘number- ing’ quantity, as in the case of the diagonal of the square. Having made this distinction, Gregory can launch his attack on Inf.2.8 and Inf.2.9. He argues that if Burley’s argument employs “x exceeds y beyond ratio” in the first sense, then Inf.2.8 is false, and that if Burley’s argument uses “x exceeds y beyond ratio” in the second sense, then, Inf.2.9 is false. Here is Gregory’s argument as to why Inf.2.8 is false if we understand “x exceeds y beyond ratio” in the first sense. Taken in the first sense, x exceeds y beyond ratio only if one part of x is equal to y (related according to the ratio 1:1). Burley, however, states in Inf.2.6 that there is never such a relation between an individual whiteness and an individual blackness. An individual whiteness will always be more perfect than an individual blackness. Hence, if we read Inf.2.8 according to the first sense of “x exceeds y beyond ratio,” Inf.2.8 will be false. If Burley’s argument employs “x exceeds y beyond ratio” in the second sense, then it is Inf.2.9 that will turn out false. We rewrite Inf.2.9 using Exc.rat.2: Inf.2.9’ If (i) x exceeds y, (ii) not according to any numerable ratio, and (iii) no part of x exceeds y according to a numerable ratio, and (iv) no part of x is equal to y, then x exceeds y infinitely. To see why Inf.2.9’ is false, on Gregory’s view, we need to consider a restriction that Gregory places on what it means to infinitely exceed something. The restriction is that x can infinitely exceed y only if x, in fact, has parts equal to y, but according to an infinite quantity, while y has these parts only according to a finite quantity.140 To understand what Gregory wants to convey with this restriction, consider the number 2 and the divergent series 1+1+1+…etc. Both the number 2 and the series consist of 1s. Thus, they have equal parts. However, 2 consists of finitely many 1s, namely, two, while the series consists of infinitely many 1s. Hence, the series infinitely exceeds 2, but according to parts equal to parts of 2, viz. 1s.

140 Sent., 1, 17, 4, p. 411.

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It is in light of this restriction that Gregory’s attack on Inf.2.9’ becomes intelligible. The antecedent of Inf.2.9’ rules out that there is any equality between a part of x and y. If that is so, then the antecedent cannot entail that x exceeds y infinitely in the sense of having an infinite quantity of parts equal to y. Hence Inf.2.9’, as a whole, is false, for Gregory. At first blush, Gregory’s restriction on infinite excess in terms of equal parts seems arbitrary. True, if the restriction is accepted, Inf.2.9’ is false. But why should we accept the restriction in the first place? Gregory provides an indirect argument to motivate this restriction, one appealing to a mathematical analogy having to do with certain types of angles. The argument runs as follows. Suppose you do not accept the above-mentioned restriction. That is, suppose you accept that if x exceeds y in such a way that no part of x is equal to y, then x exceeds y infinitely. If you accept this, Gregory argues, then “you could also prove” the false claim “that any rectilinear angle is infinite or of infinite magnitude”. Here is why. Gregory has us consider a horn angle (angulus contingentiae). This is the angle formed by a cir- cle and a straight line tangent to it. As the scholastics knew from Euclid’s Elements (which was translated several times into Latin from the twelfth century onwards141), this angle had the property of being smaller than any other acute rectilinear angle.142 In his discussion of the horn angle, Campanus (1220-1296), the influential editor and com- mentator of Euclid’s Elements in the Middle Ages, moreover, observed that no matter how often you divide a given rectilinear angle, you will never arrive at an angle equal to the horn angle.143

141 The Elements were translated from Arabic into Latin before they were translated from Greek into Latin. The first translations were made by Gerard of Cremona, Herman of Carinthia, and Adelard of Bath. See M. CLAGETT, “The Medieval Latin Translations from the Arabic of the Elements of Euclid with Special Emphasis on the Versions of Adelard of Bath,” in: Isis 44 (1953), pp. 16-42; H.L.L. BUSARD: “Über die Überlieferung der Elemente Euklids über die Länder des Nahen Ostens nach West-Europa,” in: Historia Mathematica 3 (1976), pp. 279-290. 142 The locus classicus for the discussion of this kind of angle is Elements, III, 16. 143 As John Murdoch has shown, Campanus saw that this property ruled out that the size of the horn-angle could ever have a Euclidian ratio to the size of any rectilinear angle. For, the definition of ‘ratio’ given by Euclid in V, def. 4, states, in modern terms, that A and B have a ratio, just in case (i) A < B, and (ii) there exists some positive integer n such that nA>B. There is no such ratio between the horn angle and any rectilinear angle. See MURDOCH, “The Medieval Language,” pp. 248-249. For more on the use of the

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This latter observation proved to be influential in the later Middle Ages, as John Murdoch has shown. Bradwardine picks it up, and Gregory likewise follows this path.144 Gregory states that, according to Euclid, we may divide the rectilinear angle into infinitely many rectilinear angles.145 This is of course supposed to parallel Burley’s move in Inf.2.3 of dividing the individual whiteness into infinitely many less perfect parts. Since the horn angle is smaller than any rectilinear angle, Gregory argues that none of the infinitely many rectilinear angles will ever be equal to the horn angle, just as no part of w, according to Burley (Inf.2.6), will ever be equal to b. Any given rectilinear angle thus exceeds the horn angle beyond any ratio, as Gregory concludes. Should we therefore infer that any rectilinear angle infinitely exceeds the horn angle, and that it is therefore of infinite perfection? Does a rectilinear angle of, e.g., 30 degrees have an infinite perfection? We would have to accept this absurd conclusion, Gregory argues, if we did not accept his restriction that x infinitely exceeds y only if x has (infinitely many) parts equal to y. To avoid this consequence, we had better accept the restriction. Therefore, Gregory’s rejection of Inf.2.9’ is vindicated. Thus, Burley’s second infinity argument Inf.2 turns out to fail, just as his first one Inf.1 did. Gregory’s replies to Burley’s infinity arguments share a common feature. Gregory fully accepts Burley’s presupposition that qualities

example of the horn angle in the later Middle Ages, see J. MURDOCH, “Mathesis in Philo- sophiam,” p. 243, and ID., “Beyond Aristotle: Indivisibles and Infinite Divisibility in the Later Middle Ages,” in: C. GRELLARD – A. ROBERT (eds.), Atomism in Late Medieval Philosophy and Theology, Leiden 2009, pp. 15-38, esp. 35-37. Murdoch, in the two latter papers, discusses Peter Ceffons, who in his Sentences commentary (late 1340s) used the example of the horn angle to discuss, like Gregory, diverse senses of ‘exceeding’. I could also find the same horn angle argument that Gregory uses in Oresme’s discussion of intension and remission in his commentary on the Physics (terminus ante quem 1347). See S. KIRSCHNER (ed.), Nicole Oresmes Kommentar zur Physik des Aristoteles. Kommentar mit Edition der Quaestionen zu Buch 3 und 4 der Aristotelischen Physik sowie von vier Quaes- tionen zu Buch 5, Stuttgart 1997, Liber 5, q. 7, p. 402, ll. 79-84. For the dating of this latter commentary see Stefan Kirschner, “Oresme on Intension and Remission of Qualities in His Commentary on Aristotle’s Physics,” in: Vivarium 38 (2000), pp. 255-274, esp. 269. 144 See, e.g., his Geometria speculativa, 2.306, p. 66. Bradwardine also uses this claim as a premise in an argument against an indivisibilist account of the continuum. See Thomas Bradwardine, Tractatus de Continuo, in: J.E. MURDOCH, “Geometry and the Continuum in the Fourteenth Century: A Philosophical Analysis of Thomas Bradwardine’s Tractatus de Continuo” (PhD Dissertation, University of Wisconsin, 1957), Conclusio 79, p. 417. 145 Elements, I, 9.

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admit of a mathematical treatment as if they were no more than quantities. What he does to refute Burley’s arguments is to show that they lead to logical impossibilities, such as instantaneous traversals of divisible paths, or mathematical absurdities, such as the claim that adding an indivisible to an extended amount will make that amount greater, or that acute angles are of infinite perfection. Nowhere in his replies does Gregory appeal to any of his own (notorious) views about the infinite, to wit, that there can be, and indeed are, actual infinities.146 His replies to Burley are thus elegant inasmuch as Gregory does not require us to adopt any of his own (more idiosyncratic and contentious) views. The same elegance cannot be found in Gregory’s reply to the third (and last) of Burley’s arguments that we shall consider here. This argument, which I dubbed the agent-patient argument, turns crucially on the relation between agent and patient. It can be laid out as follows: Ag.Pat. 1) If no form is remitted by the subtraction of a part from a part, then no form is intended by the addition of a part to a part.147 2) If a remission of a form occurs, then (i) one half (medietas), h1, of that form is as close to the remitting agent as the other half, h2, (ii) and h1 has the same resistance to the remitting agent as h2 (the same holds for all fourths (medietates medietatum) of that form, and any proportional parts).148 3) If h1 is as close to the remitting agent as h2, and h1 has the same resistance to the remitting agent as h2, then there is no reason why the remitting 149 agent would destroy h1 rather than h2. 4) If there is no reason why the remitting agent would destroy h1 rather than 150 h2, then it destroys both halves (i.e., the whole form).

146 On this topic, see the articles mentioned in n. 8 above. 147 TS, 1, 3, fol. 3vb: “[…] illud quod non diminuitur per ablationem partis a parte non augetur per additionem partis ad partem.” 148 Ibid.: “Accipio […] duas medietates qualitativas formae corrumpendae; et sint a et b […] aequaliter distant ab agente habente potestatem diminuendi et corrumpendi illam formam, et istae medietates sunt aequales in virtute ad resistendum actioni agentis […] Et eodem modo arguitur de medietatibus medietatum.” 149 Ibid., fol. 3rb: “[…] nec est aliquod impedimentum quare magis agit in unam illarum potius quam in reliquam […] Ergo in eadem mensura in qua destruit unam medietatem destruit reliquam.” 150 Ibid.: “Ergo tota forma simul diminuitur et simul corrumpitur.” It is interesting to note that Gregory presents an expanded version of the argument. On Gregory’s version, from the contention that there is no reason why one half rather than the other is corrupted, Burley does not immediately infer that both halves are corrupted. Rather he offers a

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5) If a remission occurs, then both halves of the remitted form are destroyed (2-4). 6) If it is the case that a remission occurs in which both halves of the remitted form are destroyed, then remission does not occur by the subtraction of a part from a part.151 7) Hence no form is remitted by the subtraction of a part from a part (5, 6). 8) Hence no form is intended by the addition of a part to a part (1, 7).

The first four premises require some comments. The argument begins with the then commonly accepted premise Ag.Pat.1 that if remission occurs in a particular way (addition of part to part), then intension must occur in the inverse way (subtraction of part from part). The reason for this is that remission is the contrary of intension, and vice versa. Contraries are, on Aristotle’s view, in the same genus.152 Hence the same generic features are applicable to them. The second premise may seem puzzling at first. To better understand what Burley is saying here consider the process of heating a pot of cold water on a stove. When the pot is being heated the water’s form of coldness is remitted. Let us say the coldness consists of a temperature of 10 degrees centigrade. The addition theory would hold that this form of coldness is composed of parts. Thus, Burley takes it that we can divide it into halves, i.e., into two parts of 5 degrees each. Now, as we have seen, these degrees do not have distinct spatial positions (“nequaquam situ distinctae”) according to the addition theory (at least according to the version that Gregory defends). Remember that, for Gregory, each part of a total form is “immediately united” (“immediate unitur”) to every other part of the form. Hence, any part of the water that contains one half of heat also contains the other half. Thus, these halves are, for the addition theorist, equally close to the remitting agent, which is precisely what Burley claims. Since these parts are of the same degree, Burley argues, they also have the same resistance to the remitting agent. Ag.Pat.2, then, is not an implausible claim about remission as it is understood according to the addition theory.

disjunction: if there is no reason why one half rather than the other is corrupted, then either no half or both halves are corrupted. Of course, something is corrupted in remission. Therefore, both halves must be corrupted. See Sent., 1, 17, 4, p. 396. 151 TS, 1, 3, fol. 3rb: “Ergo non per ablationem partis a parte.” 152 Metaph., X, 3, 1055a1.

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Ag.Pat.3 is fairly unobjectionable, given Ag.Pat.2. There is nothing intrinsic about these 5 degrees that makes them more or less corruptible than those 5 degrees. However, Ag.Pat.4, which is the nervus probandi, is, as Maier pointed out, clearly false.153 Maier argued that there is of course a good reason as to why the remitting agent cannot destroy both halves if it can destroy one. The reason is that when the agent destroys one half, then the patient (i.e., the half) reacts to the action of the agent, and thereby limits the power of the agent, thus preventing it from destroying the other half.154 So it is because a remitting agent only has a limited amount of power (due to the patient’s reaction to the agent) that it can only destroy one rather than both halves. Some fourteenth-century authors responded to Ag.Pat.4, like Maier, by drawing attention to the limits of the agent’s power through reaction. For instance, Gregory’s confrere, Alphonsus Vargas Toletanus, who was Gregory’s successor in the Parisian Augustinian chair of theology in 1344-45,155 argued that the remitting agent cannot destroy both h1 and h2 because it does not “have so much power that it could destroy both”.156 While Alphonsus does not explicitly develop this point in light of the problem of reaction, it is plausible to assume that this is what he is getting at here. Gregory also rejects Ag.Pat.4. Surprisingly, though, he does not, like Alphonsus and Maier, rebut it by reference to the limited power of the agent (due to the patient’s reaction). Instead he appeals to God. He argues that the remitting agent does not corrupt both halves at once, but rather one part after another “because the first agent, namely, God, […] freely determines the natural agent so that this part be corrupted prior to that one.”157

153 MAIER, Zwei Grundprobleme, p. 330. 154 She also draws attention to the fact that Burley, in other works, was aware of this feature (ibid.). 155 See BERMON, “La Lectura,” p. 268. 156 ALPHONSUS VARGAS TOLETANUS, Lectura super primo Sententiarum. ed. Th. DE SPILIMBERGO, Venice 1490, d. 17, q. 3, a. 2: “[…] non sequitur: forma diminuitur per ablationem partis a parte, ergo tota simul perditur. Ed ad probationem illius consequentiae, quando dicitur: quandocumque aliquid agens est aequaliter approximatum, tunc dico: quod illa maior non est vera si illud agens non sit tantae virtutis quod simul possit destruere. Et sic est in proposito, quia quamvis medietates qualificativae formae corrumpendae sint aequalis intensionis et aequales […] ad resistendum actioni agentis, citius tamen destruitur una quam altera, quia agens non est tantae virtutis quod possit omnes simul destruere.” 157 Sent., 1, 17, 4, pp. 409-410: “[…] causa, propter quam una pars prius corrumpitur quam altera, est quia primum agens, scilicet, deus […] determinat libere agens naturale, ut haec pars prius quam illa corrumpatur.”

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Gregory recognizes that this reply to Burley’s objection could be perceived as weak. As he explains, however, his reply “must not seem as an ad hoc solution on account of not knowing how to deal with the argument.”158 This is so, Gregory thinks, because when it comes to sense perception there is a similar case where “everyone is forced to maintain this very solution.”159 Gregory asks us to consider the situation in which someone looks at one visible object, say, a painting, for some time. While looking at it, the perceiver’s eyes open and close. The painting thus successively causes several visual perceptions in the perceiver. These impressions are of the same kind (eiusdem rationis). They are perceptions of a certain color, shape, etc. Now, Gregory argues that, before actually occurring, these impressions were equally disposed to occur (“aequaliter in potentia ut fiant”). There is no essential order (ordo essentialis) determining which perception comes first and which second. At t2, the perceiver could have had the perception p1 of the painting that he actually had at t1, and at t1 he could have had the perception p2 that he had at t2. Why, Gregory asks, did the perceiver have p1 at t1 rather than at t2, and why did he have p2 at t2 rather than at t1? Why is the order the way it is? Gregory thinks that we cannot determine a “precise cause” (causa praecisa) for this in the created world. The painting itself is in active potency to produce any of these perceptions at any time. The eye and the power of sight are in passive potency to any of these perceptions at any time. Thus, Gregory concludes that the order can only be due to God’s will.160 Gregory seems to move rather quickly to invoking God as cause. At first blush, it seems there could be other causes (apart from the painting, the eye, and sight) that explain the order of perceptions. For instance, I might have p1 at t1 whilst sitting, and p2 at t2 whilst standing. What explains the order of the perceptions is that I was first sitting and then chose to stand up. Or again, I might have p1 at t1 with my head in a straight position, and p2 at t2 with my head slightly tilted. The reason why I had the perceptions in this rather than that order is

158 Ibid., p. 410: “[…] nec hoc alicui videri debet fuga quaedam propter ignorantiam solutionis argumenti.” 159 Ibid.: “[…] hoc ipsum in simili quilibet cogitur dicere.” 160 Ibid.: “[…] nulla omnino huius ordinis causa poterit assignari nisi libera deter- minatio agentis primi.”

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that I moved my head. It is very likely, however, that, in his example from perception, Gregory means to rule out precisely such variations. We are, it seems, meant to think of a series of perceptions that are exactly identical except with respect to their temporal positions. Likely, such perceptions do not exist, although their assumption involves no contradiction. If they did exist, one could simply argue, however, contra Gregory that it is brute fact that the perceptions have the temporal positions they have. There is no need to invoke God here just as there is no need to invoke him to explain remission.

3. Continuity and contraries in intension and remission according to Gregory We now have a picture of the addition theory, as Gregory understands it. To summarize: on this theory, a qualitative form increases in intensity by the addition of a new form of the same kind. The old form is not corrupted in this process. Instead it remains, and, together with the new form, constitutes a total form that has per se unity. We have seen how Gregory conceives of total qualities as being in important ways similar to quantities. Furthermore, we have seen how Gregory refutes other theories (notably, Giles’s esse and Burley’s succession theory), and rebuts various counter-arguments (especially those advanced by Burley) against the addition theory. In doing so, he sometimes appeals to conceptual considerations, sometimes to mathematical considerations. Finally, God enters into the picture, too. To complete my presentation of Gregory’s account of intension and remission, I would like to briefly examine the two remaining questions of Gregory’s discussion, qq. 2-3. There, Gregory explores the two other main issues pertaining to intensive change that the scholastics inherited from Aristotle: the problem of the continuity of intension and remission, and the problem of the co-presence of contraries. Let us first consider the problem of continuity. Gregory argues for a tripartite distinction between types of intension and remission: some intensions/remissions are continuous, others are instantaneous, yet others can be mixed, having an instantaneous beginning (by creation) and from then on proceeding continuously.161 Gregory gives

161 Sent., 1, 17, 2, 2, p. 260.

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most attention to continuous intensions. I shall thus leave aside the other two kinds here. Why think that intensions are continuous at all? To show that some intensions are continuous, Gregory appeals to empirical considerations.162 He asks us to consider a light source and a medium that can be lit. Take some part P in this medium that is at a certain distance from the light source, and let the light source approach P during a certain time interval. It is observable, Gregory argues, that P will become more luminous as the light source approaches. To claim the opposite would be, as Gregory puts it, “contra sensum.”163 Hence some intensions are continuous. If (at least some) intensions are continuous, this means that at least some qualitative changes are continuous. But how is this compatible with what seems to be Aristotle’s view, namely, that all qualities are indivisible and therefore acquired immediately in a first instant? Gregory suggests that in order to harmonize the continuity of intensive changes with Aristotelian first instants we first need to introduce a distinction between two senses of ‘having a first instant’. First, a quality brought about by change can be said to have a first instant of existence (i) in the sense that it exists at this instant and before this instant no part of it (“nihil eius”) existed. There can, however, also be a first instant in a different sense according to Gregory, namely, (ii) in the sense that a quality exists at this instant and that it did not exist before this instant, although parts of it did.164 Now, Gregory shows that first instants in sense (i) are incompatible with the continuity of intensive changes, while first instants of type (ii) are compatible. Furthermore, he provides a reason for thinking that intensive changes actually have first instants of type (ii).165 Let us consider his arguments in turn. Gregory establishes that first instants in sense (i) are incompatible with the continuity of intensions as follows. Suppose a continuous

162 Ibid., p. 252. 163 Ibid., p. 253. 164 Ibid., p. 262: “Nam aliquid habere primum instans esse potest dupliciter intelligi: Uno modo quod ipsum sit in aliquo instanti et ante instans illud nihil eius fuit; alio modo quod sit in instanti aliquo et ante illud instans ipsum non fuit, quamvis aliquid eius fuerit.” 165 The arguments can be found in ibid., pp. 323-325.

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intension has an instant of type (i). This would mean that there is a point when the intensive quality, say, a certain degree of heat, is acquired such that no part of this quality had been acquired before. Now this violates the Aristotelian theorem (introduced in section 1 above), accepted by Gregory, that if x is in continuous motion and acquires a form Fness at t, then x was moving toward Fness at some t’ prior to t. It violates the theorem because if the theorem is true then if x is moving toward Fness at some t’ prior to t, it has already acquired at t’ some part of the form that it would acquire at t. For instance, if x acquires a certain amount of heat at t then x has already a bit of heat at t’. But this is exactly what positing a first instant of type (i) rules out. Therefore, a first instant in sense (i) is incompatible with the continuity of intension and remission (at least on the Aris- totelian model of continuity). Gregory also shows that, in contrast, instants of type (ii) are compatible with the continuity of intensions. Here is his argument. A first instant of type (ii) is only a first instant of a quality’s complete existence. It is not, however, the first instant of existence simpliciter, since the quality may have existed before in a partial mode. Thus first instants in sense (ii) do not violate the theorem that if x is in continuous motion to Fness and acquires a form Fness at t, then x was moving toward Fness (and thus acquired a part of Fness) at some t’ prior to t. For, first instants of type (ii) allow for the partial existence of Fness prior to the total existence of Fness. Thus, Gregory has shown that while continuous intensive changes cannot have first instants of type (i), they can have first instants of type (ii). But why should we think that intensive changes actually have first instants of type (ii)? To establish this claim Gregory argues as follows. Although parts of a given form, e.g., a degree of heat, can be acquired continuously, the form is acquired “primo et tota” in an instant. Gregory provides an analogy from locomotion to drive his point home. It makes sense to say that a person completely gets to a place “primo perfecte et totus” in a certain instant, although the ‘getting there’ (acquisitio) was continuous and successive.166 To use an example, when a runner gets to the finishing line, he continuously moves

166 Sent., 1, 17, 4, p. 323: “Nam aliquando, licet forma in aliquo instanti sit primo tota et perfecta acquisita, ipsa tamen prius partibiliter et continue acquirebatur, sicut, licet

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over the line. Yet, there is a determinate first instant when the runner has completely crossed the finishing line. The same distinction, Greg- ory thinks, applies to qualities that admit of more or less. When water is being heated to, say, 36 degrees, it continuously approaches 36 degrees. Yet, there is a determinate instant when water first acquires the exact temperature of 36 degrees. In sum, by arguing that intensive changes have first instants in sense (ii) Gregory seeks to save the continuity of intensive change as well as preserve a qualified version of the Aristotelian idea that qualities are acquired instantaneously. Two things should be noted here. Notice first that, for Gregory, only in the natural course of things, continuous intensions do not have first instants in sense (i).167 Since the instantaneous existence of a normally divisible entity does not involve a contradiction God can of course produce a divisible thing that “can exist for a unique instant such that no part of it was before this instant nor after it.”168 Second, notice that Gregory’s solution of the problem of first instants seems to presuppose the truth of the addition theory of intensive change. For, Gregory’s claim that there are first instants in the sense that a quality exists at this instant and that it did not exist before this instant, although parts of it did, assumes that intensive change occurs by a part-by-part acquisition. This, evidently, weakens Gregory’s pro- posal. Let us now turn to Gregory’s discussion of another relevant issue pertaining to the problem of the continuity of intensive change: the question as to whether intensive minima, i.e., smallest possible degrees of a form, exist. Remember, the problem posed by these intensive minima was that, if they existed, they could not be acquired continuously

in aliquo instanti locus aliquis sit primo perfecte et totus acquisitus, eius tamen acquisitio fuit successiva et continua.” 167 Sent., 1, 17, 2, 2, p. 262: “Item, cum quaeritur, utrum sit dare ultimum vel primum instans etc, hoc potest intelligi dupliciter, scilicet an de facto hoc vel illud con- tingat seu naturaliter possit contingere, et an absolute per quamcumque potentiam natu- ralem vel supernaturalem seu divinam.” 168 Ibid., p. 266: “[…] possibile est per divinam potentiam aliquam rem esse per unicum instans sic, quod nihil eius fuerit ante illud instans nec aliquid eius erit post […] nulla contradictoria ponuntur ex eo, quod aliquid affirmatur esse et negatur fuisse et fore. Confirmatur, quia haec non magis implicant circa unam rem impartibilem [...] quam circa rem habentem partes.”

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over the course of a period of time, but only in an instant. They could not be acquired over a period of time because that would require that parts of the minimum would be acquired in the intervals into which the time period is divisible. Minima, however, have no parts. In order to answer the question as to whether minima exist, Gregory first introduces a distinction.169 As Gregory explains, something may be an intensive minimum (i) in the sense that there can be no part of the form (“nihil formae”) smaller than it, which still belongs to the same species as the minimum.170 Something may, however, also be an intensive minimum (ii) in the sense that it is the smallest possible amount of that form which can exist separately.171 Gregory’s illustration of this distinction relies on a rather compli- cated example involving Aristotelian physiology.172 Here is another example that I hope will more easily illustrate the distinction. Con- sider a form of heat. Assume that this form is on the threshold of coldness and heat, i.e., that any form below this form counts as a form of coldness. Now, one way to look at this threshold form of heat is to say that it does not consist of any heat parts because a part is necessarily lesser than its whole, and whatever is lesser than the form in question counts as cold (as we stipulated). Taken in this way, the heat form would be taken as an intensive minimum in sense (i). One could also view the heat form differently, however. One could hold that the heat form is constituted by heat parts, but that these parts (due to their being below the threshold) cannot exist as heats on their own, but only, as Gregory puts, “as inexistent and in conjunction” (“inexistens et coniunctum”).173 Taken in this way, the heat form would be taken as an intensive minimum in sense (ii). It is not denied here that the heat form has parts, but only that these parts have separate existence.

169 Gregory also discusses extensive minima, i.e., smallest possible quantities in which a given form can inhere. See Sent., 1, 17, 3, p. 289. Here I shall only consider intensive minima. For extensive minima in Gregory, see SYLLA, “Disputationes,” pp. 403-404. 170 Sent., 1, 17, 3, p. 289: “[…] illud dicatur minimum, quo nihil formae est minus nec potest esse minus in specie illa.” 171 Ibid.: “[…] illud potest vocari minimum, quod potest per se actu separatum subsistere, et nihil eiusdem speciei minus eo posset per se separatum subsistere remanens in specie illa.” 172 Sent., 1, 17, 2, p. 304 and p. 310. 173 Ibid., p. 325.

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Now do intensive minima of type (i) exist, on Gregory’s view? Or are there perhaps minima of type (ii)? Gregory first argues that there are no intensive minima in sense (i).174 He has two arguments for this claim. The first is that if we were to accept such minima we could not maintain that any intensions are continuous. We could not maintain this, Gregory thinks, because on the assumption of minima the whole process of intension would have to be ultimately analyzed into instantaneous acquisitions of minima. We have seen, however, that an Aristotelian continuum cannot consist of such instants. Therefore, no intension would be continuous. Gregory’s second argument for the claim that there are no intensive minima in sense (i) is weak. The argument is that if there were such minima, then nothing would ever become more intense. For, minima are extensionless or indivisible, and adding indivisibles to indivisibles yields no greater amount. This reasoning is weak because it clearly seems to presuppose the truth of the addition theory, which a defender of minima need not accept. What does Gregory have to say about intensive minima in sense (ii)? He advocates a cautious position. He argues that some forms do not have an intensive minimum in sense (ii),175 and he considers it likely (probabiliter) that his reasoning could be extended to any corporeal form capable of intension and remission.176 To show that there are cases of intension in which there are no intensive minima, Gregory once again appeals to an empirical scenario involving light.177 Consider a light source that illuminates the air. Gregory first makes the plausible observation that whatever is closer to the light source is more luminous, and whatever is farther away is less luminous. Then Gregory argues that there is no ‘last’ part of the air.178 By this he means that the air, being a continuous magnitude, is infinitely divisible into actual (rather than merely potential179) parts such that there is no last part that is farthest away from the light source. Indeed, given the infinite divisibility of a

174 Ibid., p. 290. 175 Sent., 1, 17, 3, pp. 295-296. 176 Ibid., p. 296. 177 Ibid., p. 295. 178 Ibid.: “[…] non est dare ultimam partem simpliciter.” 179 Sent., 1, 24, 1, 1, vol. 3, p. 18. On infinite divisibility into actual parts in Gregory, see CROSS, “Infinity, Continuity,” p. 97.

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continuous magnitude, it follows that for any part x of the air, which is at some distance d from the light source, there is always some further part y, which is at a greater distance than d from the light source. Y, being farther away from the light source than x, is therefore also less luminous. There is, then, no part of the air that has the least degree of luminosity. Since every part of the continuum exists actually (and is thus capable of separate existence), there are, in this case, no intensive minima understood as the smallest possible amounts of form, which can exist separately. There is always a smaller one. Thus, Gregory (like Burley180) rejects the existence of intensive minima in sense (i), and (probably) also in sense (ii). He constructs, however, an ingenious objection to this ‘anti-minimalist’ view of his on the basis of Richard Kilvington’s sophisma 8 (which does not explicitly deal with intensive minima). The sophisma reads: “Socrates will be precisely as white as Plato will be white in any of these (instants).”181 To see how Gregory can use this sophisma to concoct an argument for minima, consider the hypothesis of the sophisma. The hypothesis is this. Suppose Socrates and Plato are both not white and both begin to become white at the same time, their whiteness being continuously increased at the same rate until they reach some instant A. At A Plato dies, while Socrates stays alive. Is Socrates whiter than Plato at A or not? Here is how a plausible answer might look like, and, as we shall see shortly, it seems to force us to posit minima. First, it is clear that Socrates is not less white than Plato at A. For, both started becoming white at the same time, and were changing at the same rate. Neither is Socrates as white as Plato at A. For, Plato ceased to be at A, and so did not acquire any whiteness at A. Hence, it seems to follow that Socrates is whiter than Plato at A. If that is granted, then the defender of intensive minima can ask: by virtue of what is Socrates whiter than Plato at A? Is this because of some divisible amount of whiteness, or some indivisible part, i.e., an intensive minimum? Socrates cannot be whiter because of a divisible amount of whiteness. For, he would then have to have acquired

180 SYLLA, “Disputationes,” p. 405. 181 Sent., 1, 17, 3, pp. 298-299. For Kilvington’s Sophisma 8, see N. KRETZMANN – B. ENSIGN KRETZMANN, The Sophismata of Richard Kilvington, repr. New York 2011 (11990), pp. 16-17 (for the sophisma), and pp. 177-180 (for commentary).

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a divisible amount of whiteness in an indivisible instant of time, viz. A, which (on the Aristotelian view) is impossible. Hence he must be whiter by an indivisible amount of whiteness. This amount would also have been acquired instantaneously, and that is unproblematic, since indivisibles can be acquired in an instant, on the Aristotelian view. Now, the defender of minima can ask: what else can this indivisible amount of whiteness be than an intensive minimum? Richard Kilvington’s solution to the sophisma is that Socrates is whiter than Plato at A, but that Socrates is not whiter by a determinate bit of whiteness. Gregory follows suit, and thereby finds a way to rebut the argument for minima.182 In support of his denial that Socrates is whiter by a determinate bit of whiteness Gregory provides a very dense explanation. Socrates is not whiter by a determinate bit of whiteness […] because there is not some maximal degree of whiteness or a maximal latitude of a maximal whiteness by which Plato will be white and because, in no instant, will Plato be so white that he could not be whiter because in this case there is no last instant in which he will be white.183 It is clear why Gregory claims that Plato does not have a maximal degree of whiteness. The reason is that, ex hypothesi, he is exceeded in his whiteness by Socrates. Why does Gregory say that Plato has no last instant of being white, though? Here we need to introduce a distinction between what scholars of medieval natural philosophy call the “intrinsic” and the “extrinsic limit” of a motion.184 Roughly speaking, the intrinsic limit of a motion is the last instant in which the thing in motion is in motion, while the extrinsic limit of a motion is the first instant in which the moving thing is no longer in motion.185

182 Sent., 1, 17, 2, 3, p. 307: “[…] per nihil tamen albedinis praecise erit albior quam Plato […] cum comparatur albedo Socratis, quam habebit in instanti A, ad albedinem Platonis, nullus est determinatus excessus quo illam excedet praecise.” 183 Ibid.: “[…] causa est, quia non erit aliquis maximus gradus albedinis vel maxima latitudo seu maxima albedo per quam Plato erit albus, et quia in nullo instanti Plato erit tantum albus, quin post illud erit albior, quia secundum casum nullum ultimum instans erit in quo erit albus.” 184 For this distinction see C. WILSON, William Heytesbury. Medieval Logic and the Rise of Mathematical Physics, Madison 1956, pp. 29-30; KRETZMANN, “Incipit/Desinit,” pp. 101-103. For the application of this distinction to Richard’s Sophisma 8, see SYLLA, “Infinite Indivisibles,” p. 252; KRETZMANN – KRETZMANN, The Sophismata, p. 180. 185 I am here only considering limits as end points. Of course intrinsic or extrinsic limits may also be starting points of motion. Thus there may be intrinsic beginnings, i.e.,

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In the case at hand, Plato’s whitening has an extrinsic limit, namely, A. At A Plato has first ceased to become white. But Gregory thinks that Plato’s whitening has no intrinsic limit, i.e., no last instant in which he is white. Why? Recall here Aristotle’s theorem, espoused by Gregory (and Richard), that no two instants can “immediately succeed” one another. This is logically impossible because there is between any two instants a further instant. Now Aristotle’s theorem would be violated if there were an intrinsic limit to Plato’s whitening. For this intrinsic limit would be Plato’s last instant of being white and it would be immediately succeeded by the extrinsic limit to Plato’s whitening, i.e., the first instant in which Plato is no longer whitening. Since this theorem must not be violated, there cannot be a last a moment when Plato was white. Instead there are infinitely many instants of Plato’s becoming white, which ‘approach’ the extrinsic limit, such that whichever instant tn prior to the extrinsic limit you choose, there will be another instant tm(m>n) closer to the limit. Once we grant that there is no unique last instant of Plato’s being white, and therefore no last degree of his whiteness, we can also see why Gregory thinks that Socrates does not exceed Plato by a determinate amount of whiteness. There is just no determinate last degree of Plato’s being white to which we could compare Socrates’ whiteness at A. Hence there is, as Edith Sylla puts it (here interpreting Richard Kilvington), no “determinable amount” according to which Socrates’ whiteness at A exceeds Plato’s whiteness prior to A.186 If there is no determinable amount of excess we cannot posit a specific entity that accounts for Socrates’ whiteness being greater than Plato’s. The defender of minima who claims that Socrates’ whiteness exceeds Plato’s by an indivisible degree of whiteness posits just such an entity, however. So his view must be rejected, Gregory concludes. This then is Gregory’s account of two crucial issues connected with the problem of the continuity of the process of intension and remission. We saw, first, that he denies first instants of intension in the sense in which these preclude the prior existence of parts of the intensified quality. Moreover, we saw that he likewise (cautiously) denies

first instants in which the thing in motion is in motion, or extrinsic beginnings, i.e., last instants in which a thing in motion was no longer stationary. 186 SYLLA, “Infinite Indivisibles,” p. 253.

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the existence of intensive minima. In sum, his account of the continuity of intension and remission is thoroughly divisibilist. On his view, we need not alter the Aristotelian understanding of the continuum as applied to quantitative and local change to account for the continuity of continuous gradual change. The exact same story that we can tell about the continuity of quantity and local change can be told of qualities. Again, we find Gregory treating qualities (specifically: intensities) as if they were quantities. Aristotle’s cautionary note that alterations are only divisible per accidens is abandoned in favor of a unified story of the continuity of qualitative, quantitative, and local change. We now need to consider one last issue in Gregory’s account of intension and remission, namely, his account of the contraries involved in intension and remission (q. 3). Scotus presented two well-known arguments for the claim that contraries co-exist in the same subject during a process of intension and/or remission. The first argument is that the remission of a quality cannot occur unless an agent introduces another quality incompatible with the quality to be remitted, which entails that contrary qualities will both be in the changing subject.187 For instance, a hot object cannot become cold unless an agent introduces coldness into the object. So, there will be heat and coldness together in the object. The second argument is that when it is true to say that water gains heat, then it is also true to say that it loses coldness.188 In other words, Scotus maintained that the intension of a form and the remission of its contrary occur in tandem. Thinkers like Chatton and Wodeham followed Scotus on this score.189 Thus it is not surpris- ing that Gregory can speak of “multi” defending this view.190 While these multi clearly state that contraries co-exist in the same subject, they do not explicitly say that contraries co-exist in the same subject according to the same part or respect. Much hinges on this qualification, however. If they accept this qualification then their view is that, for instance, one and the same piece of wood is both wet and dry in the same part or respect, which is a very contentious position. If they do not accept this qualification, then their view is that, e.g.,

187 Ord., 2, 2, 2, 5, vol. 7, p. 330, ll. 2-6. 188 Ibid., p. 330, ll. 6-7. 189 See n. 45 above. 190 Sent., 1, 17, 3, 2, p. 331.

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the same piece of wood can be both wet and dry according to distinct parts, which is innocuous. After all, who will deny that, e.g., a stick can be partly submersed in water, and partly dry? Gregory (following Ockham and Burley191) understands the doctrine according to its contentious version, and he emphatically rejects it.192 To counter the doctrine, Gregory develops some of the arguments that Burley already used against the co-presence of contraries in the same subject. One such argument is that the defender of the co-presence of contraries in the same subject would have to grant that one and the same object could be both hot and cold.193 Gregory argues, however, that we experience that whenever a hot and a cold object are in physical proximity (and no impediment is present), one or the other quality is destroyed. When fire approaches ice, for instance, ice melts and so loses its coldness. When water approaches fire, fire is extinguished, and so loses its heat. Now if one contrary expels the other in a spatially proximate entity, Gregory argues, then, a fortiori one contrary would have to expel another in the same entity. Hence hot and cold, and contraries in general, cannot co-exist in the same substance. Another argument that Gregory employs against the co-existence of contraries (which, in outline, can also be found in Burley194) is that the acceptance of this co-presence would entail the simultaneous truth of contradictories. Suppose an object has the contrary qualifica- tions of being hard and soft at the same time. Aristotle defines the

191 TS, 2, fol. 10vb; OCKHAM, Quaestiones in Libros Physicorum Aristotelis, q. 150, in: ID., Opera philosophica, vol. 6, ed. S. BROWN (= Brevis Summa Libri Physicorum, Summa Philosophiae Naturalis et Quaestiones in Libros Physicorum Aristotelis), St. Bonaven- ture, N.Y. 1984, pp. 806-808. 192 Sent., 1, 17, 3, 2, p. 331: “[…] dico quod impossibile est naturaliter sive virtute agentium secundariorum contraria in aliquo gradu intenso vel remisso simul esse in eodem subiecto primo.” Gregory’s views on whether this is possible by God’s absolute power are somewhat difficult to determine. On the one hand, he tells us that “dubium pro certo grande est,” which does not mean that it is impossible (ibid., p. 338). On the other hand, he then presents a series of objections against what he calls his “secundam conclusionem.” All of the objections argue that God can make contraries co-exist in the same subject, which suggests that Gregory’s second conclusion stated the opposite, i.e., that God cannot make contraries co-exist (ibid., p. 346). The conclusion cannot be found, however, in the text. The editors of Gregory’s Lectura state: “non exstat” (ibid.). 193 Sent., 1, 17, 3, 2, pp. 331-332; TS, 2, fol. 6va. 194 TS, 2, fol. 5rb.

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hard as “what does not yield into itself, and the soft as what does yield into itself”.195 It follows that if an object at the same time had the contraries of being hard and soft it would yield and not yield into itself, which is a contradiction.196 This is all well and good, but how does Gregory react to the Scotist arguments in favor of the co-presence of contraries? Gregory’s reply to the argument that remission can only occur by the introduction of an incompatible form is unsatisfactory. He merely asserts that he thinks it is false.197 His reply to the argument that intension and remission occur in tandem, by contrast, is more interesting. Gregory writes: […] never in any part of the water is there any heat, as long as there remains in this part (ibi) an uncorrupted form of coldness, although the water becomes continuously less cold, until the whole coldness is corrupted and the form of heat induced. But according to diverse parts of the water this can obtain: that in one thing there is the form of heat and in the other the form of coldness.198 Gregory here makes two points. First, he accepts Burley’s account of the transition from one contrary to another according to which one contrary is first completely remitted, and the other contrary then begins to be acquired.199 Second, however, he adopts the view that two contraries can co-exist in the same subject during a continuous change, but according to spatially distinct parts. Thus, Gregory can maintain that in some sense the intension of one form and the remission of its contrary occur in tandem. As he makes clear, however, it is not possible that one and the same part of the body gains one quality and remits its opposite.

195 See ARISTOTLE, Meteorologica, IV, 382a11-13. Strictly speaking, the definition of soft is “what does not yield into itself without interchanging place” (emphasis mine). 196 Sent., 1, 17, 3, 2, p. 335. 197 Ibid., p. 347: “[…] dicendum quod falsum assumit, scilicet quod agens non remittit formam nisi causando aliquid incompossibile formae in eadem parte subiecti.” 198 Ibid., p. 349: “[…] numquam in aliqua parte aquae inducitur aliquid caloris, id est aliquis calor, quamdiu manet ibi incorrupta forma frigiditatis, quamvis continue fiat minus frigida, quousque frigiditas tota sit corrupta et inducatur forma caloris. Secundum diversas tamen partes aquae hoc potest contingere, quod in una sit forma caloris et in alia sit forma frigiditatis.” See also Sent., 1, 17, 2, 3, p. 310: “Et si tunc dicatur quod tunc subiectum simul esset sub formis contrariis, dicendum est quod verum est secundum aliam et aliam eius partem.” 199 SYLLA, “Disputationes,” p. 406.

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Conclusion It would be difficult to summarize all the details of Gregory’s intricate account of the problem of the intension and remission of forms in one conclusion. However, as I pointed out in the introduction, there is, to my mind, a common theme running through the various points Gregory is making, namely, what I called his ‘quantitative approach’ to qualities. This approach consists in the view that although qualities (specifically intensities) are not quantities, they may be treated as if they were quantities. That is to say, while qualities differ ontologically from quantities, on Gregory’s view, they are, nonetheless, amenable to an analysis in quantitative terms. This is so because qualities and quantities are in important respects alike. Here in summary fashion are some of the similarities Gregory sees: 1) Qualities (specifically: intensities) are infinitely divisible into parts in the same way as quantities are. 2) A total quality constituted by intension and remission is like a homogenous quantitative whole because all parts of the total quality are of the same kind as the total quality. 3) A total quality is a totality whose parts form a non-continuous unity, which is why it can be treated as if it were a discrete quantity (this being the type of quantity opposed to continuous quantity). 4) A total quality has a unity of composition just like the quantitative parts of a body. 5) The termini of intension and remission, viz. the more and the less, are like the termini of the quantitative changes of increase and decrease, viz. greatness and smallness. 6) The relation between total and partial quality can be expressed mathematically just like a whole-part relation of quantities, namely, by appeal to ratios and the notion of excess. 7) Continuous intensions and remissions are infinitely divisible in the same way (not per accidens) as quantitative and local changes are (and, therefore, there are probably no intensive minima). A quantitative approach to qualities is a general feature of fourteenth- century natural philosophy, as is widely known.200 Scotus’s theory of

200 E.J. DIJKSTERHUIS, The Mechanization of the World Picture. Pythagoras to Newton, trans. C. Dikshoorn, Princeton 1986, p. 164; M. CLAGETT, The Science of Mechanics in

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intension and remission is often viewed as being crucial for this development. Gregory takes up this approach, but he takes it one step further. For, unlike Scotus (and Ockham), he thinks, for instance, that the termini of intension and remission are the more and the less (see [5]). Moreover, like Burley, he discusses intension and remission in the mathematical terms of ‘ratio’ and ‘excess’ (see [6]), which Scotus and Ockham, to my knowledge, do not do. None of this is to say that Gregory endorses anything like the early modern quantitative approach to intensities. The central features of this modern approach is that intensities are in fact viewed as quantities (not just as as-if quantities), and that something can be said about them on the basis of experiments and certain laws that are formulated in mathematical terms.201 All of these aspects are absent in Gregory. For him intensities are simply in many important ways like quantities (and thus, as-if quantities). Moreover, he appeals almost exclusively to imaginary and hypothetical scenarios in his ‘empirical arguments’. Finally, he never uses mathematics to the extent that he actually for- mulates any laws of motion. Nonetheless, I think Gregory’s view that qualities are fully amenable to a quantitative treatment has a certain proximity to the basically quantitative approach of early modern science. The extent to which medieval science actually influenced early modern science is a matter of considerable debate.202 There is, however, on my view, no denying

the Middle Ages, Madison 1961, p. 206; E.D. SYLLA, “Medieval Quantification of Qualities: The ‘Merton’ School,” in: Archive for the History of Exact Sciences 8 (1971), pp. 9-39; E. GRANT, The Foundations of Modern Science in the Middle Ages. Their Religious, Institu- tional, and Intellectual Contexts, repr. Cambridge 2011 (11996), p. 194. 201 These are considered central features of (early) modern science by historians of science like E.J. Dijksterhuis or Peter Dear. See DIJKSTERHUIS, The Mechanization, p. 3; P. D EAR, Mersenne and the Learning of the Schools, Ithaca, N.Y. 1988, p. 1. 202 The debate in the twentieth century was to a large extent a reply to Pierre Duhem’s (now obsolete) contention that the Scientific Revolution of the seventeenth century was anticipated, in its essentials, by Parisian fourteenth-century thought. See P. D UHEM, Études sur Léonard de Vinci, vol. 3. (=Les précurseurs parisiens de Galilée), Paris 1913, pp. v-vi. Koyré famously opposed this view by denying that early modern science was in any way anticipated by fourteenth-century science. Maier and Clagett argued that fourteenth-century thinkers did not anticipate early modern science, but that they at least prepared certain ideas in important ways (a position that I am inclined to). Nowadays, the picture is a mixed one, with John Murdoch arguing for a strong discontinuity between medieval and early modern science, Edward Grant arguing for a weak influence, and Amos Funkenstein and David Lindberg suggesting a stronger continuity. On the various

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that a variety of the quantitative approach to qualities crucial to the (early) modern scientific view that the book of nature is written in the language of mathematics took shape in the fourteenth century. In this development Gregory of Rimini occupies an important position.203

Can L. LOEWE FWO Aspirant Institute of Philosophy – University of Leuven Kard. Mercierplein 2 B-3000 Leuven [email protected]

reactions to Duhem from Koyré to Clagett, as well as a statement of his own view, see J.E. MURDOCH, “Pierre Duhem and the History of Late Medieval Science and Philosophy in the Latin West,” in: R. IMBACH – A. MAIERÙ (eds.), Gli studi di filosofia medievale fra Otto e Novecento: Contributo a un bilancio storiografico, Rome 1991, pp. 253-302, esp. 272-283; For Duhem, see also R. ARIEW, “Pierre Duhem,” in: Stanford Encyclopedia of Philosophy (2014): http://plato.stanford.edu/entries/duhem/. For the remaining views discussed above, see GRANT, The Foundations of Medieval Science, pp. 171-205; A. FUNKEN- STEIN, Theology and the Scientific Imagination, From the Middle Ages to the Seventeenth Century, Princeton 1986, pp. 12-18; D.C. LINDBERG, The Beginnings of Western Science. The European Scientific Tradition in Philosophical, Religious, and Institutional Context, Prehistory to A.D. 1450, 2nd ed., Chicago 2007, pp. 357-367. 203 I thank the referees of RTPM and Russ Friedman for their comments on previous versions of this paper.

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