No CrossRef data available.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 21 October 2024
This contribution proposes an interpretation of Thomas Aquinas’s philosophy of mathematics. It is argued that Aquinas’s philosophy of mathematics is a coherent view whose main features enable us to understand it as a moderate realism according to which mathematical objects have an esse intentionale. This esse intentionale involves both mathematicians’ intellectual activity and natural things being knowable mathematically. It is shown that, in Aquinas’s view, mathematics’ constructive part does not conflict with mathematical realism. It is also held that mathematics’ imaginative reasoning is coherent with Aquinas’s doctrine of formal abstraction and his realistism. It focuses on some of Aquinas’s texts, which it places within their textual and doctrinal context and interprets them in the light of some historical elements.
1 David Svoboda and Prokop Sousedik, ‘Thomas Aquinas and Some Thomists on the Nature of Mathematics’, Review of Metaphysics, 73 (2020), 715–40 (p. 717).
2 Cf. David Svoboda and Prokop Sousedik (2020); David Svoboda, ‘Formal Abstraction and its Problems in Aquinas’, American Catholic Philosophical Quarterly, 96 (2022), 1–20.
3 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 1, co.
4 Ibid., q. 5, a. 1–3.
5 In this context, a ‘formal reason’ corresponds to what Aquinas designates in his Super Librum Boethii de Trinitate as ‘objects of speculation’, i.e., the proper objects of speculative sciences in virtue of which they are distinguished. This is how he refers to them in his De Anima, II, cap. 3, lect. 6, No. 307, and in his Summa Theologiae, Ia, q. 1, a. 3, co., where he speaks of objects of speculation or ‘subject-matters’ as ‘formal reasons’ (ratio formalis) through which an individual thing is known (audible, visible, intelligible, etc.).
6 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 1, ad. 3.
7 Cf. David Svoboda and Prokop Sousedik (2020), p. 724.
8 By ‘demarcation’, here, I am not referring to the demarcation problem in contemporary philosophy of science, but simply to medieval discussions on which of the newly discovered disciplines was to be legitimately considered as scientia.
9 Cf. Joan Cadden, ‘The Organization of Knowledge, Disciplines and Practices’, in The Cambridge History of Science, Vol. II, Medieval Science, ed. by David C. Lindberg and Michael H. Shank (New York: Cambridge University Press, 2013), pp. 240–67.
10 Cf. Olga Weijers, ‘The Organization and Content of Learning’, in A Scholar’s Paradise: Teaching and Debating in Medieval Paris, ed. by Olga Weijers (Turnhout: Brepols, 2015), pp. 45–58.
11 Cf. Olga Weijers (2015) and Joan Cadden (2013).
12 Cf. Joan Cadden (2013).
13 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 1, ad. 3.
14 Ibid., q. 5, a. 1, obj. 3.: ‘[…] Philosophy is commonly divided in seven liberal arts, which do not contain physics nor divine science, but only logic and mathematics. Therefore, physics and divine science should not be counted among speculative sciences.
15 Ibid., q. 5, a. 3, ad. 3.
16 Ibid., q. 5, a. 1, co.
17 Thomas Aquinas, In Aristotelis Libros Posteriorum Anayticorum, I, lect. 44.
18 Cf. Eleonore Stump, ‘Aquinas on the Foundations of Knowledge’, Canadian Journal of Philosophy, Supplementary Volume, 17 (1991), 125–58.
19 This is why he refers to Hugh of Saint Victor’s Didascalicon: ‘[…] seven arts are grouped [leaving out certain other ones] because those who wanted to learn philosophy were first instructed on them’. See Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 1, ad. 3, (italics mine).
20 One might think of Al-Fârâbî’s scheme of division of sciences, in which any discipline of the quadrivium was conceived to have an ‘active’ or practical part and a ‘speculative’ or theoretical part. Cf. e.g. Jens Hoyrup, ‘Jordanus de Nemore, 13th Century Mathematical Innovator: An Essay on Intellectual Context, Achievement, and Failure’, Archive for History of Exact Sciences, 38 (1998), 307–63. About Al-Fârâbî’s influence on Aquinas, see Claude Lafleur and Joanne Carrier, ‘Abstraction, séparation et tripartition de la philosophie théorétique: Quelques éléments de l’arrière fond farabien et artien de Thomas d’Aquin, Super Boethium « De Trinitate », question 5, article 3’, Recherches de Theologie et Philosophie Medievales, 67 (2000), 248–71.
21 Thomas Aquinas, In Aristotelis Libros Posteriorum Analyticorum, I, lect. 25, cap. 13.
22 We may here think of what Robert Grosseteste has referred to as an ‘added condition’ in his Commentarius in Posteriorum Analyticorum Libros, 1.18, (ed. P. Rossi), Florence, 1981: ‘Scientia autem est subalternate alii cuius subjectum addit conditionem super subjectum subalternantis (…)’ (‘A science is subalternate to another when its object adds some condition to the subalternating object’), (italics mine). On Grosseteste’s treatment of subalternated sciences, see W. R. Laird’s, ‘Robert Grosseteste on the Subalternate Sciences’, Traditio, 43 (1987), 147–79, who gives a good overview on the topic and refer to relevant works as regards Grosseteste. As regards Thomas Aquinas, see C. A. Ribeiro do Nascimento, ‘Le statut épistémologique des “sciences intermédiaires” selon S. Thomas d’Aquin’, in La Science de la Nature: Théories et pratiques (Cahiers d’Études Médiévales 2, Montréal: Bellarmin, 1974), pp. 33–95; De Tomás de Aquino a Galileu (Campinas: IFCH, Unicamp, 1998–2ª ed), and W. A. Wallace, Causality and Scientific Explanation, 2 vols (Ann Arbor, 1972-74).
23 Thomas Aquinas, In Aristotelis Libros Posteriorum Analyticorum, I, lect. 25, cap. 13.
24 Such a ‘genus-species’ distinction has some explanatory value for, at least, some medieval mathematicians. An example is the preface to the so-called ‘Adelard III’ version of Euclid’s Elements, in which the author operates some divisions within mathematics according to such a genus-species model. Regarding the object of mathematics, for instance, he refers to ‘quantity’ as the genus, whose first species are ‘discrete quantity’ and ‘continuous quantity’ and on which the division between arithmetic and geometry is grounded. Speaking of geometry, he also says that regarding supposition (that is, the mathematical referents), the subject-matter of geometry is continuous quantity, which is a subspecies of the genus ‘quantity’. Similarly, regarding the content, he also says that the genus of geometry is ‘mathematics’ in so far as geometry is contained within mathematics simpliciter, which is about quantity simpliciter. He even suggests that Euclid himself operates a distinction within geometry itself ‘according to the parts of its matter’, that is, the species or specific instances of continuous quantity such as ‘line, surface, solid, and number, or according to the fifteenth distinctions made by [Euclid] himself (…) and called according to the distinction of principles and called by the mode of doing’. This appears to be a reference to the thirteen books of Euclid’s Elements and other available treatises of Euclid. See Johannes de Tinemue’s redaction of Euclid’s Elements, the So-called Adelard III version, vol. 45, ed. by H. L. L. Busard (Stuttgart: Franz Steiner Verlag, 2001), and The Commentary of Albertus Magnus on Book I of Euclid’s Elements of Geometry, vol. 3, ed. by A. Lo Bello (Boston: Brill Academic Publishers 2021) for the English translation.
25 Pascale Bermon’s approach gives some elements of Aquinas’s epistemology which may prove helpful to understand this division. Bermon refers to Aquinas’s theory of habits to treat the model of science he endorses as a large format (contrasting with those of other philosophers). To put it simply, Aquinas sees sciences as habits. Accordingly, the subordination of some sciences to another might be seen as a subordination of habits to other habits. Higher habits would be those whose perfecting objects are more general than those of lower habits. Since Aquinas characterises the objects of sciences as ‘formal reasons’ (see footnote No. 5), it could be said that objects of the quadrivial disciplines have a mathematical formal reason as their genus. Cf. Pascale Bermon, ‘Tot Scibilia quot Scientiae? Are There as Many Sciences as Objects of Science? The Format of Scientific Habits from Thomas Aquinas to Gregory of Rimini’, in The Ontology, Psychology, and Axiology of Habits (Habitus) in Medieval Philosophy, Historical-Analytical Studies on Nature, Mind, and Action, vol. 7, ed. by N. Faucher and M. Roques (Cham: Springer, 2018), pp. 301–19.
26 Cf. David Svoboda, Prokop Sousedik, (2020), p. 726: ‘[…] [Contrasted with formal abstraction], total abstraction is often characterized as an activity of the intellect, in which the universal is separated from the singular, but a similar characteristic of formal abstraction is not mentioned by Aquinas’.
27 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 3, obj. 3.
28 Ibid., q. 5, a. 1, co. By ‘connection with them’, Aquinas is referring to the degree of separation from matter and motion when not totally separated, which he adds at the end of the passage.
29 As indicated by Svoboda and Sousedik, (2020), p. 727: ‘[…] [That Aquinas speaks of formal abstraction as concerned by the universal is difficult to defend]. This is attested to by Aquinas’s reaction to the objection according to which a mathematician works with geometrical objects as if they were singulars […]. Aquinas does not say that mathematics does not deal with singular objects because it is a theoretical science. He chooses a different strategy, according to which, even when quantity is abstracted “from sensibly available matter, it is still possible to imagine numerically different beings of the same kind” […]. Thus, geometry deals with singular, not universal, mathematical objects because objects obtained by formal abstraction are numerically different and can even be imagined’.
30 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 3, ad. 3.
31 Cf. Svoboda and Sousedik, (2020), p. 726–27: ‘[Regarding formal abstraction, Aquinas] mostly speaks about “abstraction of form from sensibly available matter”, but he does not add, as we would expect, that an object grasped in this way is universal. But the form of a circle of a certain radius ought to be universal because it is really present in many singular ten-crown coins and can be predicated of them’ (italics mine).
32 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 3, co, (italics mine).
33 Ibid., q. 5, a. 3, co.
34 Ibid., q. 5, a. 3, co.
35 Ibid., q. 5, a. 3, co: ‘This is the abstraction of a whole, in which we consider a nature absolutely, according to its essential character’.
36 Ibid., q. 5, a. 3, co.
37 Thomas Aquinas, De Ente et Essentia, cap. 3.
38 Thomas Aquinas, De Ente et Essentia, cap. 6.
39 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 1.
40 Ibid., q. 5, a. 1.
41 Ibid., q. 5, a. 1.
42 Thomas Aquinas, In Aristotelis Libros Posteriorum Anayticorum, I, Lect. 10, cap. 4.
43 Ibid., I, Lect. 5: ‘Another type of position is the one which does not signify existence or non-existence: in this way a definition is a position. For the definition of “one” is laid down in arithmetic as a principle, namely, that “one is the quantitatively indivisible”’.
44 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 3, ad. 4, (italics mine).
45 On Aquinas’s fourfold distinction of matter as regards ‘mathematical abstractions’, see Summa Theologicae, q. 85, a. 1, ad. 2, where he distinguishes ‘individual sensible matter’, ‘common sensible matter’, ‘individual intelligible matter’, and ‘common intelligible matter’.
46 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 3, co.
47 Thomas Aquinas, In Duodecim Libros Metaphysicorum Aristotelis Expositio, VII, lect. 10, No. 1495.
48 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 6, a. 2, co.
49 According to Aquinas, formal abstraction is a purely intellectual operation (Expositio Super Librum Boethii de Trinitate, q. 5, a. 3, co.) whereas imagination is, at best, an internal power which stores traces of sensory acts.
50 Thomas Aquinas, In Duodecim Libros Metaphysicorum Aristotelis Expositio, VII, lect 10, No. 1494-1495.
51 In commenting on Aristotle’s Metaphysics VII, Aquinas suggests that being individual, and then, material, does not entail being sensible. See Thomas Aquinas, In Duodecim Libros Metaphysicorum Aristotelis Exposition, VII, lect. 11, No. 1521: ‘[Aristotle] answers that it makes no difference to his thesis whether the material parts [of mathematical objects] are sensible or not, because there is intelligible matter even in things which are not sensible’. The parts at stake are not parts of the species. These are called ‘matter’ as they are the principle of individuation of an individual (this line) whose being is not identical to its species (line). In other words, Aquinas’s point is that ‘anything sensible must be material whereas anything material need not be sensible’.
52 More should be said about Aquinas’s view on the status of imagination in mathematics. However, I have limited myself, here, to an explanation of the passage in question which serves my purpose without extending the content of the paper, since I deal with this question in an article based on a talk given at the ‘Nancy-Liège Workshop on Mathematical Intuition’, hosted by Andrew Arana, Yacin Hamami, Gerhard Heinzmann and Bruno Leclercq in May and November 2023. Cf. Daniel E. Usma Gomez, ‘Mathematical Intuition as Imagination: The Case of Aquinas Philosophy of Mathematics’. The volume is currently being edited by the journal Logique et Analyse and scheduled for publication in 2025.
53 Cf. Clelia Crialesi, ‘The Status of Mathematics in Boethius: Remarks in the Light of his Commentaries on the Isagoge’, in The Sustainability of Thought: An Itinerary Through the History of Philosophy, ed. by Lorenzo Giovannetti (Napoli: Bibliopolis, 2020), pp. 95–124, who provides an overview of this phenomenon as regards Boethius’s philosophy of mathematics. Cf. also Alain de Libéra, L’art des généralités: théories de l’abstraction (Paris: Aubier, 1999).
54 Cf. Allan Bäck, Aristotle’s Theory of Abstraction (Cham: Springer, 2014). Cf. also John J. Cleary, ‘On the Terminology of “Abstraction” in Aristotle’, Phronesis, 30 (1985), 13–45.
55 Admitting that would have led him to a sort of dualism. Indeed, Aquinas’s epistemology would have consisted of an account of knowledge through intellectual powers independent of an account of knowledge through an inner-sense power, as imagination.
56 Cf. David Svoboda and Prokop Sousedik (2020).
57 For a general overview, cf. Ralph W. Clark, ‘Saint Thomas Aquinas’s Theory of Universals’, The Monist, 58 (1974), 163–72; Gabriele Galluzzo, ‘Aquinas on Common Nature and Universals’, Recherches de Theologie Et Philosophie Medievales, 71 (2004), 131–71; Jeffrey E. Brower, ‘Aquinas on the Problem of Universals’, Philosophy and Phenomenological Research, 92 (2016), pp. 715–35; Luiz Marcos da Silva Filho, ‘Fundamento do universal no singular em Tomás de Aquino: Natureza Comum, Similitude e/ou Ideia?’, Dois Pontos, 18 (2021), 86–112.
58 Thomas Aquinas, Scriptum Super Libros Sententiarum Magistri Petri Lombardi Episcopi Parisiensis, I, d. 2, q. 1, a. 3; In Doudecem Libros Metaphysicorum Aristotelis expositio, VII, 13, No. 1570.
59 Thomas Aquinas, Espositio Super Librum Boethii de Trinitate, q. 5, a. 2, co.
60 Ibid., q. 5, a. 3, co, (italics mine).
61 Ibid., q. 5, a. 3, co.
62 I take the terms ‘hylomorphic union’ and ‘mereological union’ from Claude Lafleur and Joanne Carrier, ‘Abstraction et séparation: de Thomas d’Aquin aux néoscolastiques, avec retour à Aristote et aux artiens’, Laval Théologique et Philosophique, 66 (2010), 105–26. The term ‘hylomorphic’ refers to ‘form-matter’ compounds, where ‘mereological’ refers to ‘whole-part’ relations. In this case, since Aquinas sees the union of singulars to universals as a ‘part-whole’ relation, it could be named a ‘mereological union’.
63 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 2, ad. 4.
64 Cf. Gabriele Galluzzo (2004).
65 Cf. Thomas Aquinas, In Duodecem Libros Metaphysicorum Aristotelis Expositio, XI, lect. 1, No. 2161.
66 Cf. David Svoboda and Prokop Sousedik, (2020), p. 729. The authors do refer to the passage of In Duodecem Libros Metaphysicorum Aristotelis Expositio, XI, lect. 1, No. 2161, in which Aquinas states that some mathematical objects do not exist in the reality as the mathematician investigates them and, at best, that mathematical objects do not consist of physical properties. I deal below with this passage.
67 Ibid., (2020), p. 733.
68 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 6, a. 2, co.
69 I take the expression ‘essential dependence’ from Ross Inman, ‘Essential Dependence, Truthmaking, and Mereology: Then and Now’, in Metaphysics: Aristotelian, Scholastic, Analytic, ed. by Lukás Novák, Daniel D. Novotný, Prokop Sousedik, David Svoboda (Heusenstamm: Ontos Verlag, 2012), pp. 73–90.
70 I take the expression ‘intelligible dependence’ from Thomas C. Anderson, ‘Intelligible Matter and the Objects of Mathematics in Aquinas’, New Scholasticism, 43 (1969), pp. 555–76.
71 Thomas Aquinas, Expositio Super Librum Boethii de Trinitate, q. 5, a. 3, co.
72 Ibid., q. 5, a. 3.
73 Ibid., q. 5, a. 3.
74 Ibid., q. 6, a. 2, co.
75 Armand Maurer, ‘A Neglected Thomistic Text on the Foundation of Mathematics’, Pontifical Institute of Mediaeval Studies 21, (1959), 185–92; ‘Thomists and Thomas Aquinas on the Foundation of Mathematics’, Review of Metaphysics, 47 (1993), 43–61.
76 Thomas Aquinas, Scriptum Super Libros Sententiarum Magistri Petri Lombardi Episcopi Parisiensis, ed. by R. P. Mandonnet (Paris: Lethielleux, 1929), I, d. 2, q. 1, a. 3, co.
77 Cf. Armand Maurer (1993).
78 Thomas Aquinas, In Duodecem Libros Metaphysicorum Aristotelis Expositio, XI, lect. 16, No. 2161.
79 Ibid., XI, lect. 16, No. 2161.
80 Ibid., XI, lect. 16, No. 2161 (italics mine).
81 Ibid., XI, lect. 16, No. 2162 (italics mine).
82 Cf. Armand Maurer (1959).
83 Cf. Armand Maurer (1993).
84 Thomas Aquinas, Scriptum Super Libros Sententiarum Magistri Petri Lombardi Episcopi Parisiensis, I, d. 2, q. 1, a. 3, co.
85 Cf. Antoine Dondaine, ‘Saint Thomas et la dispute des attributs divins (I Sent., d. 2, a. 3): authenticité et origine’, Archivum Fratrum Praedicatorum, (1938), pp. 253–62. Maurer refers to this paper in his two articles.
86 Thomas Aquinas, Scriptum Super Libros Sententiarum Magistri Petri Lombardi Episcopi Parisiensis, I, d. 2, q. 1, a. 3, co, (italics mine).
87 David Svoboda and Prokop Sousedik (2020), p. 737: ‘Both absolute numbers and geometrical objects are thus understood as sui generis fictions, as constructs of our minds (…). It seems that according to Aquinas, even whole numbers are invented by human beings’.
88 Cf. Armand Maurer (1959), p. 189; (1993), p. 53.
89 Thomas Aquinas, De Ente et Essentia, cap. 3.
90 Ibid., cap. 3, (italics mine).
91 Ibid., cap. 3.
92 The ‘mathematical’ formal reason ‘being intellectually separated from sensible matter’.
93 Thomas Aquinas. In Aristotelis Libros Posteriorum Analyticorum, I, lect. 2.
94 Cf. Ibid., I, lect. 5, where Aquinas asserts that, sometimes, specifically in mathematics, a definition can be laid down as a (first) principle of a syllogism (positio). As such, definitions can be, at least virtually, truly predicated on a subject in the conclusions.
95 Cf. Jean W. Rioux, Thomas Aquinas’ Mathematical Realism (Cham: Springer Verlag, 2023), chapters 6–7.
96 I develop further elsewhere the interaction between intellectual powers and imagination in Aquinas’s view on mathematical activity. See footnote 52.
97 Thomas Aquinas. In IX Metaphys., lect. 10, No. 1888 and ff, (italics mine).
98 Broadly speaking, ‘fictionalism’ is the nominalist idea that mathematical concepts are like fictional terms since they refer to nothing in the real world. Mathematical objects would be fictional or imaginary entities like fictional characters. The best-known contemporary source on this subject is Hartry Field (1946–present). To avoid any anachronism, I would like to note that in the context of a discussion of Aquinas’s philosophy of mathematics, the term ‘fictionalism’ has a purely philosophical meaning (just as ‘mathematical Platonism’ does not necessarily refer to Plato’s own position).
99 Cf. Jean W. Rioux, Thomas Aquinas’ Mathematical Realism (Cham: Springer Verlag, 2023), chapters 6–7. I recommend considering Rioux’s use of this line of interpretation to shed some new light on the well-known debate ‘classical-intuitionist’ mathematicians about the law of the excluded-middle.
100 The existence of the quantitative essences may be seen as ‘intentional’ in so far as the intellectual grasping of them through (formal) abstraction may be described in terms of reception of a form by its appropriate powers (in this case, quantitative forms), which, according to some prominent commentators, is what intentionality is about. See John Haldane, ‘Brentano’s Problem’, Grazer Philosophische Studien, 35 (1989), 1–32; Anthony J. Lisska, ‘Axioms of Intentionality in Aquinas’s Theory of Knowledge’, International Philosophical Quarterly, 16 (1976), 305–22; Aquinas’s Theory of Perception: An Analytic Reconstruction (New York: Oxford University Press, 2016); Anthony Kenny, Aquinas on Mind (New York: Routledge, 1993); Roger Pouivet, Après Wittgenstein, Saint Thomas (Paris: Librairie Philosophique Vrin, 2014), to find out more about debates on Aquinas’s notion of ‘intentionality’. The constructions can be seen, in turn, as ‘second intentions’ since they result from the intellectual reflection upon a previously abstracted notion.