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ABSTRACT
This paper explores a specific problem within an important philosophical genre of the fourteenth century: the debates over the perfection of species. It investigates how the problem of defining limits for continuous magnitudes – a problem typical of Aristotelian physics – was integrated into these debates at the levels of genera, species, and individuals as these entities began to be conceptualized in quantitative terms. After explaining the emergence of this problem within fourteenth-century metaphysics, the paper examines the contributions of three philosophers – Hugolinus of Orvieto, John of Ripa, and Paul of Venice – who offered varying solutions to the challenge of defining limits between species. It demonstrates that two primary solutions arose, inspired by continuous and discrete mathematical objects. It is shown that whereas Hugolinus of Orvieto advocates for a continuist model, John of Ripa proposes a discrete one. The last part of the paper examines Paul of Venice's hybrid approach, which combines elements from both models, facilitating a more comprehensive treatment of species particularly difficult to analyse, namely geometric figures. The conclusions of this comparative study underscore how profoundly the metaphysical reflections of the Middle Ages contributed to the analysis of the structure of the continuum and the extension of the notion of quantity to various objects.
Ackowledgements
I am indebted to Yael Kedar, Elena Baltuta, and two anonymous referees for many comments that helped me improve this paper. I also thank Daniel A. Di Liscia for stimulating discussions and sharing his insights on this topic.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 On the general history of this problem, see Strobach, The Moment of Change; more specifically on its treatment in the Middle Ages, see Wilson, William Heytesbury, 29–56; Spade, “How to Start and Stop”; Trifogli, “Thomas Wylton”; Trifogli, “Walter Burley”; Nielsen and Trifogli, “Thomas Wylton’s Questions on Number”; Sylla, “Mathematics and Physics”; Di Liscia, “Walter Burley, Paulus Venetus”; De Libera, “La problématique de l’instant”; Pérez-Ilzarbe, “Socrates desinit esse”.
2 For the source of this expression, see Murdoch, “Propositional Analysis”, 118.
3 On natural minima, see Murdoch, “The Medieval and Renaissance Tradition”; Robert, “John of Jandun”; Trifogli, “Duns Scotus”. On the medieval debates over the limits of powers, usually called questions de maximo et minimo, see Wilson, William Heytesbury, 57–114; Longeway, William Heytesbury; Caroti, “Nuovi linguaggi”; Di Liscia, “A Tract De maximo et minimo”. For the logical backgrounds of these theories, see also Murdoch, “Scientia mediantibus vocibus”; Murdoch, “Propositional Analysis”.
4 The literature on this trend of fourteenth-century philosophy has generated a large number of studies. For up-to-date information, see Di Liscia and Sylla, Quantifying Aristotle, to which the title of the present article pays tribute; Roudaut, La mesure de l’être, and the bibliography therein.
5 See Murdoch, “Subtilitates Anglicanae”; Murdoch “Mathesis in philosophiam scholasticam introducta” and Mahoney, “Metaphysical Foundations”. On the geometric notions involved in this application to metaphysics, Di Liscia, ”Perfections and Latitudes”. On the conceptual transfer from physics to metaphysics, see Roudaut, La mesure de l’être, 321–73.
6 Maier, Zwei Grundprobleme, 1–109; Solère, “Plus ou moins”; Sylla, “Medieval Concepts”; Sylla, “Medieval Quantifications of Qualities”.
7 On this text, see Corvino, “Il ‘De perfectione specierum’”; Roudaut, “Hugolinus of Orvieto”. For an overview of Hugolinus’ thought, see Eckermann and Hucker, Hugolin von Orvieto.
8 Sub quolibet vero praedictorum generum sunt possibiles infinitae species vel imaginabiles, quarum nulla est alteri immediata. Sic igitur infimum genus incipit a non esse simpliciter seu a non gradu perfectionis exclusive. Dico autem ‘exclusive', tum quia est terminus extrinsecus totius latitudinis perfectionum tum secundo, quia quaelibet species distat a non gradu praedicto et nulla est infima species et immediata ad non esse.
9 On fourteenth-century accounts of points, see Celeyrette, “La problématique du point”.
10 In this passage, Hugolinus has just explained how to measure the essential perfection proper a given species, which he calls its “nobility”. The perfection of a species must be measured according to him by comparing it to a point of reference, which can be the lower or the upper limit of a genus. This point consequently leads him to determine what are the upper and lower limits of genera.
11 Ex ista ratione infero tria. Primum est, quod datur aliqua essentia infinitae modicae nobilitatis. Secundum est: Ordo essentialis est terminatus intrinsece ab inferiori puncto vel gradu. Tertium est, quod nullus gradus nobilitatis est immediatus ultimo gradui, quia ille immediatus non esset pura potentia nec actus, nisi infinite modice perfectus, quod non videtur dandum.
12 De terminis meddi ordinis, scilicet generis, ponuntur tres conclusiones. Prima: Ordo essentialium perfectionum primi generis terminatur infra exclusive ad materiam primam, sed supra terminatur exclusive ad genus immediatum. Secunda: Ordo essentialium perfectionum generis medii sub et supra terminatur exclusive ad genus immediatum.
13 For Facinus’ view on limits between genera, see 370.
14 Pro isto dubio sciendum primo, quod non quaeritur hic de speciebus connotativis, quae non denotant distinctas essentias, sed potius denotant aliqualiter esse, ut numeri et figurae, linea, superficies et cetera.
15 For interesting comments on John of Ripa’s theory, see Nannini, “Introduzione”, V-XXIII. See also Nannini, Giovanni da Ripa. A good introduction to John’s doctrinal framework can be found in Combes, “L’intensité des formes”. On his sources, see also Coleman, “Jean de Ripa”.
16 Dicit igitur ista opinio quod in latitudine entis quaelibet species est indivisibilis secundum gradum specificum, ita quod nulla species claudit infra se aliquam latitudinem. […] Sed quomodo cum indivisibilitate gradus specifici stet infinita latitudo numeralis in eadem specie?
17 Nullus gradus specificus in latitudine entis simpliciter est indivisibilis perfectionaliter. […] Nam da oppositum: sequitur quod ex talibus gradibus non potest constitui aliqua latitudo sicut nec ex punctis aliqua linea; et ideo punctus non est gradus lineae.
18 Sexta conclusio: quacumque specie finita infra latitudinem entis data quae non sit infinita, sub tali est species immediate inferior vel superior in huiusmodi latitudine, ita quod sicut numeri sunt immediati, ita et species in latitudine entis. Et in hoc species se habent ad invicem per modum discreti et non continui.
19 Quocumque enim genere dato sub quo sit aliqua species finita, et data ratione a qua sumitur tale genus, huiusmodi genus non est potentiale ad totam latitudinem huiusmodi rationis, sed aliquis est maximus gradus finitus ad quem terminatur tota potentialitas dicti generis, ita quod ultra non potest progredi sed ibi necessario incipit exclusive aliquod aliud genus.
20 This point exceeds the scope of this study; see Nannini, “Introduzione”.
21 Omnia ista exempla mathematica de figuris et excessibus angulorum sunt multo deridenda si quis cupit attingere ad indaginem veritatis.
22 Some aspects of Paul’s theory of the perfection of species are studied in Di Liscia, “Perfections and Latitudes,” 314–7. See also Di Liscia, “Walter Burley, Paulus Venetus” for a study of his position on temporal limits. Many works have now been devoted to Paul’s philosophy. For an overview, see Conti, Esistenza e verità; for some recent work on his metaphysics, see Majcherek, “Paul of Venice,” and the bibliography therein. On the role of the quantifying trend typical of the fourteenth century on his philosophy, see Roudaut, “Paul of Venice’s Theory of Quantification”.
23 […] Notandum quod sicut unus numerus immediate incipit ab alio ut quaternarius a ternario, et ternarius a binario, et binarius ab unitate quae dicitur esse non gradus latitudinis numerorum, ita in latitudine entis una species incipit immediate incipit ab alia […]. Tertia conclusio. Quaelibet species praeter primam et ultimam in latitudine entis habet aliquam immediate superiorem et aliquam immediate inferiorem ad modum existentiae numerorum.
24 Paul seems elsewhere to concede that species are infinite with respect to some particular aspect. The species of figures, for instance, are infinite in the sense that the number of angles of polygons can increase up to infinity; see Summa naturalium, 157rb.
25 Notando primo quod harum latitudinum quaedam est discontinua omnino ut latitudo entis, latitudo figurarum et latitudo numerorum; quaedam continua omnino, ut latitudo numeralis, scilicet hominis vel equi, caliditatis vel frigiditatis, albedinis vel nigredinis; quaedam vero partim continua et partim discontinua, ut latitudo angulorum.
26 Prima pars est angulorum acutorum habens gradus intensiores et remissiores. Secunda pars est angulorum rectorum non habens gradus intensiores et remissiores, quia omnes anguli recti sunt aequales. Sed est quidam gradus imaginabilis longus terminans exclusive totam longitudinem angulorum acutorum, in qua longitudine sunt imaginabiles infniti anguli recti, sicut et infinita puncta. Tertia est pars angulorum obtusorum incipiens exclusive a longitudine angulorum rectorum, et habet latitudinem graduum intensiorum et remissiorum.