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SVOBODA and P. SOUSEDÍK A QUINAS’S CONCEPTION of number has attracted relatively little attention from historians dealing with medieval philosophy and mathematics. This lack of in terest is due to two reasons. The first has to do with the fact that Aquinas’s conception of number derives mainly from the Aristotelian Averroan tradition, whose representatives were not as concerned with mathematics and its philosophical problems as were, for example, the Pythagoreans or Plato.2 Aquinas therefore did not pursue this issue in a systematic manner, but only in the context of other, more immediate concerns. The other reason why this issue has not been very much explored thus far is closely related to the first one; as Aquinas worked out no systematic theory of number, we can only rely on marginal notes on this issue, making it difficult to arrive at a precise notion of his conception. Research in this field is by no means conclusive and presents a significant challenge to contemporary researchers of Aquinas’s thought. The present article attempts to respond to this challenge and to reconstruct 1 The work on this paper has been supported by the grant GAČR 13 08512S. See J. A. Aertsen, Studien und Texte zur Geistesgeschichte des Mittelalters 52 (Leiden, New York, Cologne: Brill, 1996), 201 4. 2 401 402 D. SVOBODA and P. SOUSEDÍK Aquinas’s conception of number, following up on the (neither very extensive nor intensive) investigation carried out so far.3 The most important accounts of the subject in Aquinas occur (1) in his commentaries on Aristotle’s and , where Aristotle’s conception of the one and the many is laid out; (2) in numerous passages of Aquinas’s theological works, where the nature of unity and multitude is discussed; and (3) in theological treatises on the Trinity and angels, in which Aquinas deals with the problems whether and in what sense numerical terms can be predicated of God and whether angels constitute a multitude.4 In these texts, two mutually related approaches toward the problem may be encountered. Their common point of depar ture is the logical ontological framework which Aquinas adopted from Aristotle. According to this framework, real created being divides into ten categories: the first category is being in its basic and primary meaning, that is, substance ( ); the other nine categories are being in its derived and 3 The fact that Aquinas did not work out a systematic conception of number probably explains the lack of secondary literature, which is as unsystematic as Aquinas’s reflections on this issue. On Aquinas’s conception of number, see D. Mercier, “L’unité et le nombre d’après Saint Thomas,” !" # . 8 (1901): 258 75; E. Bodewig, “Zahl und Kontinuum in der Philosophie des hl. Thomas”, $ 13 (1935): 55 77. We have also drawn valuable information on Aquinas’s conception of number from L. Oeing Hanhoff, % # & ' ' # , (Münster, 1953); T. O’Shaughnessy, “St. Thomas and Avicenna on the Nature of the One,” ' 41 (1960): 665 79; J. Owens, “Unity and Essence in St. Thomas Aquinas,” & 23 (1961): 240 59; P. C. Courtès, “L’un selon saint Thomas,” 68 (1968): 198 240; R. E. Houser, & , (Ph.D. diss., The University of Toronto, 1980); Aertsen, , 201 42; B. Blankenhorn, “Aquinas on the Transcendental One: An Overlooked Development in Doctrine,” 81 (2004): 615 37; A. Maurer, “Thomists and Thomas Aquinas on the Foundation of Mathematics,” ( 47 (1993): 43 61. 4 See, e.g., IV ., lect. 2 4; V ., lect. 7 8; X .; III # lect. 5.; VII ., lect. 8; I & ., d. 24, q. 1, aa. 2 3; & I, q. 11; & I, q. 30, a. 3 (“Utrum termini numerales ponant aliquid in divinis”); & I, q. 50, a. 3 (“De multitudine angelorum”); $ # q. 9, a. 7 (“Quomodo termini numerales praedicentur de divinis personis, utrum scilicet positive vel remotive tantum”). AQUINAS ON NUMBER 403 secondary meaning, that is, accidents ( ).5 To the ten categories are superordinated transcendental concepts ( ) ), which for example include, besides being, the concepts ‘one’, ‘true’, and ‘good’.6 The first approach to the problem of the mathematical one and number is related to Aristotle’s critical reaction to the con ception of the one and the many endorsed by the Pythagoreans and Plato. These thinkers, according to Aquinas, identified the transcendental one with the mathematical one, conceiving these unities—as well as the multitudes or numbers constituted by them—as subsisting separated substances.7 Rejecting this “sub stantial” conception of the one, Aquinas espoused the view that the mathematical one and number belong in the category of quantity.8 In this first approach, Aquinas therefore reflects on the mathematical one and number in relation to the category of quantity along with other concepts which also belong to this category. The point of departure of Aquinas’s second approach to the problem is neither the category of quantity nor any other category, but the realm of transcendental concepts. The concept of the one convertible with being and the concept of the transcendental multitude play a crucial part.9 In this second approach, Aquinas reflects on the mathematical one and num ber in contrast to the transcendental one and many, and strives (mainly for theological reasons) to distinguish sharply between them. 5 V # lect. 9 (890 93); III # lect. 5. Parenthetical page number in references to the commentary on Aristotle’s refer to paragraphs numbers in Thomas Aquinas, & ed. M. R. Cathala and R. M. Spiazzi (Turin: Marietti, 1971). See also J. F. Wippel, “Thomas Aquinas’s Derivation of the Aristotelian Categories (Predicaments),” * + 25 (1987): 13 34. 6 See $ , ., q. 1, a. 1; q. 21, a. 1; I & #, d. 8, q. 1, a. 3. Cf. Aertsen, # 7 See, e.g., Aristotle, # 10.2.1054a13 19; Aquinas, X ., lect. 3 (1974 77). 8 See & I, q. 11, a. 1, ad 1. 9 See, e.g., $ , ., q. 1, a. 1; & I, q. 30, a. 3; $ #, q. 9, a. 7. 404 D. SVOBODA and P. SOUSEDÍK This distinction of two approaches provides the outline of the following exposition, which is divided into two main parts. First we focus on the relation of the mathematical one and number to the category of quantity, then on their relation to the transcendental one and multitude. I. THE MATHEMATICAL ONE AND NUMBER AS KINDS OF QUANTITY10 Quantity is an accident that determines only material substances, that is, substances composed of substantial form and prime matter ( ).11 Due to quantity the composite substance is extended and acquires a plurality of quantitative parts, which are at the same time ordered as to position, so that one part has being next to another. Aquinas accordingly defines quantity as “order of parts in a whole” ( ).12 This definition of quantity implies that every quantitative entity can be characterized by having a multitude of quantitative parts which are ordered as to position, next to one another. However, there are various kinds of quantity since there are various ways quantitative parts are ordered next to one another. These various ways of being ordered are determined by the relations among the quantitative parts, which can be twofold: either the parts are mutually connected, or they are not. If they are, we speak of (kinds of this quantity are line, surface, or body); if they are not, we speak of . Discrete quantity is also called—with a reservation to 10 Cf. D. P. Lang, “Aquinas and Suárez on the Essence of Continuous Physical Quantity,” . 3 (2002): 565 95. 11 See, e.g., & I, q. 77, a. 7, ad 2. As is well known, quantity is the first accident of material substance; all other (nonspiritual) accidents inhere in the substance by means of quantity. 12 See & I, q. 14, a. 12, ad 1: “De ratione quantitatis est ordo partium”; & ' IV, c. 65: “ … positio, quae est ordo partium in toto, in eius ratione includitur: est enim quantitas positionem habens.” Extensional quantity is not ordered as a relation, it is rather the “basis” of the relation of ordering: quantity is the basis of the “being next to one another” relation. AQUINAS ON NUMBER 405 be explained below— or (whose kinds are the numbers two, three, four, etc.).13 The specific difference that distinguishes continuous and discrete quantity is therefore the connectedness or dis connectedness of quantitative parts. This connectedness or disconnectedness can be characterized as follows: the parts of a quantitative being are connected if the end of one part is identical with the beginning of the other; they are disconnected if the end of one part is not identical with the beginning of the other. If the parts are disconnected, they are materially divided; if they are identical, they are not materially divided.14 Whether a quantitative being is or is not materially divided into many quantitative beings, therefore, depends on whether or not the beginnings and ends of its quantitative parts are identical.15 To summarize briefly the exposition so far, quantity is continuous if its parts are connected, and it is discrete if they are not. According to Aquinas, discrete quantity arises by material division of a continuous being and is constituted by a number of materially divided extended units.16 On the contrary, the or which Aquinas also calls “one as the principle of number,”17 is defined by the negation of material division. It is not possible to determine unequivocally from Aquinas’s texts whether or not he considers the concept of discrete quantity to be the genus of individual numbers (two, three, four, etc.), that is, whether or not he identifies discrete quantity with number as such. On the one hand, there are statements in his works that lead us to consider the concept of discrete quantity to be the immediate genus of individual numbers and 13 $ #, q. 9, a. 7: “duas species quantitatis, quae sunt discreta quantitas et continua.” 14 We will focus on the concept of (material) division more closely below. 15 & I, q. 30, a. 3: “Est autem duplex divisio. Una materialis, quae fit secundum divisionem continui …” 16 Ibid. 17 II & ., d. 3, q. 1, a. 3, ad 1: “unum quod est principium numeri, qui est discreta quantitas, causatus ex divisione materiae vel continui.” 406 D. SVOBODA and P. SOUSEDÍK to identify discrete quantity with number as such; on the other hand, it seems, especially in contexts where he explicitly defines number, that number as such cannot be identified with discrete quantity and that it is rather a species subordinate to discrete quantity, insofar as he defines the categorial number as an “aggregate of (materially divided) units.”18 However, if number is an aggregate of units, there seems to be a difference between a multitude that is aggregated (number as such) and a multitude that is not aggregated (discrete quantity). Aquinas mostly de fines number as “[quantitative] multitude measured by the one.”19 This definition makes clear that “number” is a species, “quantitative multitude” is a genus, and “to be measured by the one” is the specific difference. Number differs from discrete quantity, that is, from quantitative multitude, by being its further specification. The above difficulty concerning the relationship of number and quantitative multitude (discrete quantity) plays no essential part in our considerations. In what follows we will therefore espouse the view that according to Aquinas number as such is a species of discrete quantity. Number as such is further divided into the numbers two, three, four, etc. From this point of view, number as such is a genus and the individual numbers are its species. Of course, the different species of number as such differ by specific differences. According to Aquinas, the specific difference of the individual numbers is the last unit of the number, which defines its species,20 since the last unit is what makes the number three different from the number two, the number four from the number three, etc. From this it is apparent, incidentally, why Aquinas does not hold that the mathematical one is a number 18 VII ., lect. 8: “numerus . . . est . . . aggregatio unitatum.” I & #, d. 24, q. 1, a. 2, obj. 4: “numerus est multitudo mensurata per unum”; see I & ., d. 24, q. 1, a. 3, ad 3; & I, q. 7, a. 4; & I, q. 11, a. 2; $ # q. 9, a. 5; and elsewhere. 20 VIII ., lect. 3 (1725): “ultima unitas dat numero speciem.” 19 AQUINAS ON NUMBER 407 but rather the principle of number,21 since the mathematical one is not a multitude, but that which constitutes a multitude. Since the numbers two, three, etc. are generated by an abstractive act of the intellect from a particular number of items, these numbers exist only as objects of the intellect. Aquinas speaks of numbers generated by an abstractive act of the intellect as numbers in an absolute sense ( ) ). Particular numbers ( ) of items then fall under these absolute numbers—for example, all pairs (two persons, two horses, etc.) fall under the (absolute) number two.22 Of course, unlike an absolute number, the particular number, being a special type of accident, has real existence. From the above division of the category of quantity one can see what number is and how it relates to the other concepts within the category of quantity. The particular number is a cate gorial accident, specific in that its subject is not a substance but an aggregate of substances.23 For example, the subject of the number of the Evangelists is not one particular substance (e.g., John), but the whole aggregate of the Evangelists—Matthew, Mark, Luke, and John. A particular number relates to the given aggregate as an accident to its subject in such a manner that, as a categorial accident, it really differs not only from each and every individual substance, but also from the aggregate as a whole.24 Another specificity of the categorial number may be the fact that, unlike other generic concepts, there are potentially infinitely many different species of number (2, 3, 4, 21 $ , cap. III: “unitatem numeri, quam constat non esse numerum”; Aquinas specifies this claim in $ #, q. 7, a. 3, ad 7: “per reductionem . . . unitas in genere numeri.” 22 See & I, q. 30, a. 1 ad 4: “numerus est duplex, scilicet numerus simplex vel absolutus, ut duo et tria et quatuor; et numerus qui est in rebus numeratis, ut duo homines et duo equi. Si accipiatur numerus absolute sive abstracte . . . sic non est nisi in acceptione intellectus nostri; non enim numerus absolutus a rebus numeratis est nisi in intellectu.” 23 / # XI, q. 1, ad 1. 24 Ibid.: “Praeterea, unus numerus, licet sit in omnibus numeratis sicut unica essentia, non tamen est in qualibet parte; quia non quaelibet pars numeratur eodem numero#” 408 D. SVOBODA and P. SOUSEDÍK ). However, leaving these differences aside, the categorial number has the same characteristics as other cate gorial accidents. Just as an ordinary accident determines a particular substance, a categorial number determines a particu lar aggregate of individuals; just as an accident can be predi cated of a particular substance in the absolute sense, (e.g., Socrates is wise) an absolute number can be predicated of a particular aggregate of individuals in the absolute sense (e.g., the Evangelists are four).25 II. NUMBER AND THE MATHEMATICAL ONE VERSUS TRANSCENDENTAL ONE AND MULTITUDE The above exposition conceals a difficulty. When we predicate a number of an aggregate of individuals, we also predicate its whole conceptual “content.” The content of the concept of a number obviously includes all concepts superior to the given number, including the concept of discrete quantity. Thus, when we predicate a number of an aggregate of individuals, we also predicate of them discrete quantity, by which we say that it is an aggregate of materially divided units. When we say, for example, “the Evangelists are four,” we obviously claim that Matthew, Mark, Luke, and John are materially divided individuals. 25 A question widely discussed among medieval and later Scholastics was whether number is or is not one absolutely ( ), since only beings which have absolute unity fall in this category. Thus, if number is a real species of quantity, it must have absolute unity. Aquinas (unlike Duns Scotus and others) defended a positive reply to this question. See VIII ., lect. 3 (1725): “Est enim per se unum numerus, inquantum ultima unitas dat numero speciem et unitatem; sicut etiam in rebus compositis ex materia et forma, per formam est aliquid unum, et unitatem et speciem sortitur.” See also & I, q. 11, a. 1. For Scotus’s position, see Duns Scotus, //# # 0,, q. 2, n. 100 (B. Ioannis Duns Scoti, / , Libri IV, ed. G. Etzkorn, R. Andrews, G. Gál, R. Green, F. Kelly, G. Marcil, T. Noone, and R. Wood, 1 3 [St. Bonaventure, N.Y.: The Franciscan Institute, 1997], 344); J. A. Aertsen, “Being and One, The Doctrine of the Convertible Transcendentals in Duns Scotus,” in % + ' % 2 3 4 , ed. T. Noone and G. A. Wilson, 3 & 56 (1998): 13 26. AQUINAS ON NUMBER 409 This reveals the difficulty with the exposition provided above. We commonly predicate numbers, not only of aggregates of materially divided units, but also of such entities that are not, and cannot be, materially divided. We speak, for example, of a number of ideas or the number of Aristotle’s categories. In this context Aquinas, being a theologian, focuses mostly on state ments concerning God. According to Christian doctrine, God is not only one, but a Trinity of persons who share the divine essence—the Father, the Son, and the Holy Spirit. One can therefore say that there are three persons in God. By saying that, however, we also apparently claim that the divine persons are an aggregate of materially divided entities—a conclusion that, of course, was totally unacceptable for Aquinas.26 This motivated Aquinas to supplement and develop the doctrine of number presented so far, especially when dealing with numerical terms predicated of God.27 This is the context in which the second of the two approaches described above most often occurs. In this context Aquinas defines the quantitative one and many in contrast to the transcendental one and multitude and tries to distinguish them as sharply as possible. In what follows, we shall therefore attempt briefly to summarize some of Aquinas’s thoughts on the transcendental one and mul titude. This will allow us better to understand their relationship to the mathematical one and number. It needs to be said, however, that Aquinas’s conception of transcendental concepts (and thus also of the transcendental one and many) is very complex and has been the focus of considerable attention in recent years.28 The complexity of the matter precludes us from presenting it here in detail. In what follows, we will therefore 26 See & I, q. 30, a. 3. See I & ., d. 24, q. 1, aa. 2 3; & I, q. 30, a. 3; $ ., q. 9, a. 7. 28 Aquinas speaks of “transcendental” unity and multitude because they are not limited to one category but occur in every category and thus “transcend” all categories by being common to all beings. See X ., lect. 3 (1981): “unum quod convertitur cum ente circuit omnia entia”; & I, q. 50, a. 3 ad 1: “multitudo est de transcendentibus.” Cf. H. J. Tallon, “Does Thomism Neglect Multitude?”, ! ( & 37 (1963): 267 92; Aertsen, , 223 31. 27 410 D. SVOBODA and P. SOUSEDÍK limit ourselves to those aspects of the topic that are essential for the subject at hand. The point of departure of Aquinas’s reflections on tran scendental unity and multitude is the concept of division, since multitude arises by division and the negation of division constitutes unity.29 Aquinas distinguishes between division absolutely ( ) and division in a certain respect ( ).30 Division in a certain respect is division as to some accidental properties of the thing; thus, for example, a man is divided in a certain respect, if he is white in one respect and black in another. We shall not pay further attention to this type of division.31 Division absolutely is either division as to essence or division as to quantity. Aquinas calls the division as to essence and the division as to quantity, which we treated above, .32 All beings (including quantitative things) are formally divided by their essences insofar as one being is not identical with another being ( ).33 Only quantitative, spatially separated, discrete bodies (e.g., two lines or bodies) are divided materially. We may now take a closer look at the concept of formal division, which is crucial for defining transcendental unity and multitude.34 As the above characteristic shows, formal division—unlike material division—is not limited to quantitative beings. According to Aquinas, all beings are divided by this type of division, including spiritual beings and the divine persons of 29 $ ., q. 9, a. 7: “divisio multitudinem causet, indivisio vero unitatem, oportet secundum rationem divisionis de uno et multo iudicium sumi.” 30 See I & ., d. 24, a. 2. In a few places Aquinas speaks of so called “essential division”; by that he means the potential division of an actually undivided being into its essential and entitative parts, that is, substantial form and prime matter, act of and essence. See I & ., d. 24, a. 1. 31 I & ., d. 24, a. 2. 32 See & I, q. 30, a. 3. 33 X #, lect. 4 (1997): “hoc ens et illud, dicuntur diversa, ex eo quo hoc non est illud.” 34 We have dealt with the concept of formal division in detail elsewhere. See D. Svoboda, “The RATIO of Unity: Positive or Negative? The Case of Thomas Aquinas,” / 86 (2012): 47 70, esp. 48 53. AQUINAS ON NUMBER 411 the Trinity.35 Since formal division transcends the borders of all categories, the negation of this division is included in the concept of the transcendental one: the negation of formal division of being constitutes the concept of this one.36 The principle of formal division is the opposition between affirmation and negation, that is, the principle of con tradiction.37 As is well known, the principle of contradiction states that the same cannot be simultaneously affirmed and negated.38 According to Aquinas, the principle of contradiction is “based” on the concept of being ( ) and nonbeing ( ) ), since the same cannot be simultaneously affirmed and denied of any being.39 Hence it is clear that the formal indivision of a being being ( ) consists in that the being is not divided into being and nonbeing. The formal indivision of a being being (transcendental unity) therefore manifests itself in that the predicate “being ( )” and its negation cannot be simultaneously truthfully predicated of any real being. Aquinas defines the transcendental one as undivided being ( ).40 With respect to the above stated exposition, this definition may be further specified: the transcendental one 35 $ ., q. 9, a. 7; see, & I, q. 30, a. 3. / # X, q. 1, a. 1, ad 3: “huius divisionis negatio constituit rationem unius#” 37 IV ., lect. 3 (566): “5divisio> formalis quae fit per opposita, cuius prima radix est oppositio affirmationis et negationis”; X ., lect. 4 (1997): “divisio quam causat contradictio”; I & ., d. 24, a. 3: “prima autem ratio divisionis, secundum quam aliquid ab aliquo distinquitur, est in affirmatione et negatione.” 38 & I–II, q. 94, a. 2: “primum principium indemonstrabile est quod non est simul affirmare et negare, quod fundatur supra rationem entis et non entis” (“the first indemonstrable principle is based on the concept of being and non being: the same cannot be simultaneously affirmed and denied”). 39 Ibid. Cf. J. A. Aertsen, “Natural Law in the Light of the Doctrine of Transcendentals,” in Leo J. Elders and K. Hedwig, eds., - 4 3 - ( &# , Studi Tomistici 30 (Vatican City: Libreria editrice vaticana, 1987), 99–112. 40 IV ., lect. 2 (553): “Est enim unum ens indivisum”; see also X ., lect. 1 (1932); V ., lect. 7 8 (848 84). 36 412 D. SVOBODA and P. SOUSEDÍK is being which is not divided into being and nonbeing.41 Transcendental unity is a concomitant property of every being being.42 From this it further follows that the transcendental one adds only the negation of formal division to being, that is, only something intentional, not a new real nature, which would contract it to a certain categorial mode of being. Therefore, the transcendental one and being do not really differ ( ); the difference between them is only intentional ( ). It is important to state that according to Aquinas the transcendental one can be analogously predicated of God and other spiritual beings.43 The one convertible with being does not exclude the existence of other beings, which together constitute a tran scendental multitude. The innovative feature of Aquinas’s thought on the one and the many is that he considers “multitude” to be a transcendental concept. The starting point of his reflections on multitude is again the concept of formal division. According to Aquinas, transcendental multitude is constituted by formal division44 insofar as we understand that a being is divided from another and each is one in itself.45 If the concept of transcendental one adds one negation to the concept of being—being as undivided being—the concept of multitude adds two negations to the concept of being—being as undivided 41 I & ., d. 24, a. 3, ad 2: “ens et non ens . . . sufficiunt ad definitionem unius, secundum quod intelligimus unum esse ens, in quo non est distinctio per ens et non ens.” 42 Incidentally, these conclusions are closely related to Aquinas’s account of the origin of the so called first concepts of human reason, which, however, cannot be addressed here. See X ., lect. 4 (1998); IV ., lect. 3 (566); I & ., d. 24, a. 1; $ ., q. 9, a. 7, ad 15; & I, q. 11, a. 2, ad 4; cf. Aertsen, , 221 23; J. F. Wippel, “Thomas Aquinas on the Distinction and Derivation of the Many from the One: A Dialectic between Being and NonBeing,” ( 30 (1985): 563 90. 43 $ ., q. 9, a. 7; Aquinas discusses analogy in many places, e.g., & I, q. 13, a. 5; & ' II c. 34; I & ., d. 19, q. 5. 44 & I, q. 30, a. 3: “divisio formalis, quae . . . sequitur multitudo quae non est in aliquo genere, sed est de transcendentibus.” 45 $ #, q. 9, a. 7: “sequitur intellectus multitudinis, prout scilicet hoc ens intelligitur divisum ab alio, et utrumque ipsorum esse in se unum#” AQUINAS ON NUMBER 413 in itself and simultaneously divided from another.46 Since the concept of multitude adds only negations, that is, intentional beings, to the concept of being, it does not add anything real to it and therefore does not differ from being but only intentionally. Multitude thus conceived is, according to Aquinas, not limited to one category, but occurs in every category, which is why he ranks it among the transcendentals.47 Aquinas holds that transcendental multitude, just as the transcendental one, can be analogously predicated of God and other spiritual beings. In light of the preceding exposition we may now attempt to clarify how in Aquinas’s thinking the transcendental one and many differs from the mathematical one and number.48 Tran scendental concepts have the same extension as being being; they differ mutually only in their conceptual defini being is internally undivided, that is, tions.49 Every being one, and therefore the one convertible with being occurs in every category. The transcendental one, as we have seen, does not add any material determination to the concept of being, but only a conceptual determination, the privation or negation of internal division.50 46 Ibid.: “Et sic, cum unum addat supra ens unam negationem, secundum quod aliquid est indivisum in se, multitudo addit duas negationes, prout scilicet aliquid est in se indivisum, et prout est ab alio divisum. Quod quidem dividi est unum eorum non esse alterum.” 47 In $ , # q. 1, a. 1 Aquinas does not present multitude as a transcendental concept; instead, he mentions “something” ( ). The transcendental term expresses an essential characteristic of multitude, that is, that one being is divided from another. There is a difficulty with classifying multitude under transcendental concepts, since other transcendental concepts are convertible with being, while multitude relates to , not to being. This is probably why Aquinas in $ , # q. 1, a. 1 prefers the term to multitude. 48 IV ., lect. 2 (559). 49 I & #, d. 8, q. 1, a. 3: “bonum, verum et unum, addunt super ens, non quidem, naturam aliquam, sed rationem.” 50 $ , #, q. 1, a. 1: “Negatio autem consequens omne ens absolute, est indivisio; et hanc exprimit hoc nomen unum: nihil aliud enim est unum quam ens indivisum.” 414 D. SVOBODA and P. SOUSEDÍK In this lack of material determination, the fundamental difference between the transcendental and the mathematical can be seen. One as the principle of number does not have the same extension as being but instead occurs only in the category of quantity; it adds a real determination to the concept of being by which being is contracted to a certain essence, that is, to a categorial quantitative mode of being.51 A question may arise: in what exactly does the nature of the mathematical one consist and what real characteristic does it add to being? Aquinas’s reply to this question derives mainly from Aristotle’s analysis of the concept of one ( # 10.1) and from Averroës’ later interpretation of the problem. When Aristotle has, in his typical manner, summarized the four meanings of the term “one” and consequently established the proper nature of the one (“that, which is indivisible”), he immediately states a necessary property of the one, which is that the one is a measure: “[T]o be one” means “to be indivisible,” being essentially one means a “this” and capable of being isolated either in place, or in form or thought; or perhaps “to be whole and indivisible”; but it means especially “to be the first measure ( ) of a kind,” and most strictly of quantity; for it is from this that it has been extended to the other categories. For measure is that by which quantity is known; and quantity qua quantity is known either by a “one” ( ) or by a number ( ), and all number is known by a “one.” Therefore all quantity qua quantity is known by the one, and that by which quantities are primarily known is the one itself; and so the one is the starting point of number qua number.52 Aristotle distinguishes the property “to be indivisible,” as the of unity, from its proper attribute, “to be a measure.” That which is indivisible in some way is a measure in every category. Nevertheless, measure primarily and prevailingly occurs in the category of quantity. Unity as the indivisibility or undividedness of being occurs primarily in the category of 51 & I, q. 5, a. 3, ad 1: “substantia, quantitas et qualitas, et ea quae sub eis continentur, contrahunt ens applicando ad aliquam quidditatem seu naturam.” 52 Aristotle, . 10.1.1052b16 25 (trans. W. D. Ross; available online at http://classics.mit.edu/index.html [accessed October 29, 2014]). AQUINAS ON NUMBER 415 substance, whereas unity as measure belongs primarily in the category of quantity, since quantity is known by means of measure. We know quantity by means of the one and number: by means of the one, as when we say one meter or one mile; by means of number, as when we say three meters or three miles. Nevertheless, we know every number by means of the one, and in this sense the one is the principle of number as such. It is not our task to follow Aristotle’s argumentation and its historical context further; for our purposes here it suffices to note that Aristotle seems not to have distinguished between the tran scendental one and the one as the principle of number as two different “kinds” of unity.53 It was Aristotle’s Arabic commentator, Averroës, who explicitly distinguished between the one convertible with being and the one as the principle of number as two different “kinds” of unity, as he strove to respond to the doctrine of his predecessor Avicenna. On Averroës’ interpretation, Avicenna identified the one convertible with being with the one as the principle of number: Avicenna allegedly held that unity adds a positive accidental determination to being, ‘one’ signifying a certain accidental being.54 Averroës, however, regarded this opinion as false and held, on the contrary, that each thing is one in itself.55 His argument is that if a being (substance) becomes 53 H. G. Apostle, 6 (Chicago: University of Chicago Press, 1952), 30 40. 54 Averroës, IV ., comm. 3, 1 , vol. 8 (Venice, 1562 74), fol 67 B; D E: “Avicenna autem peccavit multum in hoc, quod existimavit, quod unum et ens significant dispositiones additas essentiae rei. . . . Et etiam, quia existimavit, quod unum dictum de omnibus praedicamentis, est illud unum quod est principium numerorum. Numerus autem est accidens. Unde opinatus fuit iste, quod hoc nomen unum significat accidens in entibus.” Cf. A. de Libera, “D’Avicenne à Averroès, et retour: Sur les sources arabes de la théorie scolastique de l’un transcendental,” & 4 (1994): 141 79. 55 Recent research shows, however, that Averroës’ (and Aquinas’s) interpretation of Avicenna’s conception of the one is not fully appropriate. See Houser, , 120 98; O’Shaughnessy, “St. Thomas and Avicenna on the Nature of the One,” 665 79; Aertsen, , 213 15. According to these 416 D. SVOBODA and P. SOUSEDÍK one due to an accidental “addition” ( ), we could say that this addition is a being. But since the concepts of being and one are convertible, this “addition” must also be a one. And then the same problem returns: either this addition is one in itself, or it is one due to some other “addition”; but then we could proceed . It is therefore necessary to stop the infinite regress in the first case and maintain that a substantial being is one in itself.56 To avoid Avicenna’s mistake while remaining loyal to Aristotle’s legacy, Averroës distinguished between two kinds of unity: transcendental unity and quantitative unity. What distinguishes them, according to Averroes, is that as the principle of number unity falls into the category of quantity and adds a real determination to the concept of being, while unity convertible with being occurs in all categories and does not add any real determination to being, only the negation of internal division.57 Aquinas espouses Averroes’s distinction of the two kinds of unity and interprets the of the mathematical one as well as its relation to the transcendental one in an innovative manner.58 According to Aquinas, the one as the principle of number adds a real determination to the concept of being and it belongs in the category of quantity. Its consists in being the measure of scholars, Avicenna in fact did not identify the one convertible with being with the one as the principle of number. 56 See Averroës, IV ., comm. 3, fol 67 F G; Aquinas, IV ., lect. 2 (555): “Quod autem ens et unum praedicentur de substantia cuiuslibet rei per se et non secundum accidens, sic potest probari. Si enim praedicarentur de substantia cuiuslibet rei per aliquod ens ei additum, de illo iterum necesse est praedicari ens, quia unumquodque est unum et ens. Aut ergo iterum de hoc praedicatur per se, aut per aliquid aliud additum. Si per aliquid aliud, iterum esset quaestio de illo addito, et sic erit procedere usque ad infinitum. Hoc autem est impossibile: ergo necesse est stare in primo, scilicet quod substantia rei sit una et ens per seipsam, et non per aliquid additum.” 57 See Averroës, IV ., comm. 3; cf. also Houser, , 237 39. 58 I & ., d. 24, a. 3. AQUINAS ON NUMBER 417 quantitative multitude.59 Aquinas seems to have attained the distinction between these two unities by sharply distinguishing the of the transcendental one and of the mathematical one.60 Yet this does not mean, as Aertsen points out, that the mathematical one stands outside the transcendental one, since the latter extends to all modes of being one.61 There is therefore an analogy between these two kinds of unity, that is, a resemblance between their “ #” The of the one convertible with being, as we know, is given by the negation of formal division of an entity while the of mathematical one results from a particular form of indivision, that is, the negation of the material or quantitative division. That is why, according to Aquinas, there is the order of between the two kinds of unity. The mathematical one is a categorical specification of the transcendental one.62 What has been said about the relationship between the transcendental one and the mathematical one applies— —to the relationship between the transcendental multitude and categorical number. Unlike transcendental multitude, number adds real categorial determination to being ( ) according to which it is contracted to a quantitative multitude of materially divided units, the measure of which is the mathematical one. Categorial number is therefore con stituted by a multitude of continuous beings, each of which is materially undivided (i.e., a quantitative one). And the measure of this multitude is the one as the principle of number. Categorial number is, therefore, a quantitative multitude measured by the one. 59 Ibid.: “unum, secundum quod est principium numeri, ponit aliquid additum ad esse, scilicet esse mensurae, cujus ratio primo invenitur in unitate, et deinde consequenter in aliis numeris et deinceps in quantitatibus continuis.” 60 V ., lect. 8 (875); X ., lect. 2 (1938). 61 See Aertsen, , 218. 62 V # lect. 8 (875): “Haec autem ratio mensurae consequitur rationem indivisionis. . . . Et ideo unum non omnino aequivoce dicitur de eo quod convertitur cum ente, et de eo quod est principium numeri; sed secundum prius et posterius.” 418 D. SVOBODA and P. SOUSEDÍK CONCLUSION The above exposition has shown why Aquinas holds that categorial number and the mathematical one can in the proper sense of the word be predicated only of aggregates of quantitative beings, not of abstract or spiritual entities. If number is a kind of quantity, it can only be univocally predicated of quantitative beings. We predicate number and the mathematical one of abstract and spiritual objects only meta phorically, since spiritual entities are not quantitative, have no common measure, and are not composed of units of the same nature.63 Only the transcendental one and many can be analogically predicated of abstract and spiritual objects, including the divine persons. But this is a hard and challenging topic which cannot be addressed here. Aquinas views number as a kind of quantity. In subsequent centuries this thesis was defended especially by the Thomistic philosophical school (Thomas de Vio, John Poinsot, Antoine Goudin, and others) and more recently it was also in a way endorsed by John Stuart Mill. However, as outlined above, the conception implies certain problems. These problems form the basis of criticisms issued by many rivals of the Thomistic conception (John Duns Scotus, and later the entire Scotistic school, William Ockham, Francisco Suárez and most Jesuit theologians, etc.). 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