Abstract
Medieval philosophers did not unequivocally support the Aristotelian doctrine of container-place, that is, that the place of a thing is the first immobile surface of what contains the thing. John Duns Scotus (d. 1308) famously developed a theory that tried to resolve the problems of container-place through an appeal to a notion of equivalence. Peter Auriol (d. 1322) took the radical step of reducing place to the category of position, understood with relation to the three-dimensional extension of the universe. Auriol called this “place according to metaphysical consideration” and contrasted it with “place according to physical consideration.” This division reflects one in another thinker, Nicholas Bonet (fl. 1333), who in his Philosophia naturalis distinguished between mathematical and natural senses of place. Rather than being influenced by Auriol, Bonet developed Scotus’ doctrine of equivalent place into a doctrine of mathematical place and time. To support his position, Bonet drew upon the Aristotelian notion of abstraction and selectively read Averroes as explicitly supporting his position.
All translations are the author’s, except where otherwise noted.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For an account of the Aristotelian notions of place and space, see Algra’s Chapter 2 in this volume.
- 2.
See Tiziana Suarez-Nani’s Chapter 4 in this volume.
- 3.
Piché-Lafleur 1999, 96 n49. As Algra’s Chapter 2 in this volume recalls, the first extant version of this thought experiment is found in the work of the Stoic philosopher Cleomedes . The same thought experiment plays a central role in the Leibniz-Clarke Correspondence; see Palmerino’s Chapter 12 in this volume.
- 4.
Cross 1998, 211.
- 5.
- 6.
Schabel 2000, 140 n65.
- 7.
Schabel 2011, 164: “But given that only one manuscript preserves what seems to be Auriol’s complete text, I now think it unlikely that Bonet had this text before his eyes, and that he was perhaps extrapolating from common statements on the mathematical vs. natural distinction going back to Aristotle himself.”
- 8.
The standard reference for Scotus’ doctrines of place and space is Cross 1998, 193–213.
- 9.
Ioannes Duns Scotus, Ordinatio II, d. 2, pars 2, q. 1–2 (1973, 256–257): “Dico igitur quod locus habet immobilitatem oppositam motui locali omnino, et incorruptibilitatem secundum aequivalentiam per comparationem ad motum localem. – Primum patet, quia si esset aliquo modo mobilis localiter, quantumcumque accipiatur per accidens, posset dici esse in loco et ei assignari posset alius et alius locus; sicut, licet similitudo moveatur quasi accidentaliter per accidens, scilicet quasi in quinto vel quarto gradu (quia primo corpus, et per hoc superficies, et per hoc albedo, et per hoc similitudo), tamen superficies vel similitudo vere est in alio et alio loco. – Similiter, tunc aliquid quiescens posset moveri localiter: nam quod habet alium et alium locum successive, localiter movetur; fixum autem posset habere alium et alium locum continentem, si locus moveretur per accidens.”
- 10.
Ioannes Duns Scotus, Ordinatio II, d. 2, pars 2, q. 1–2 (1973, 257–258): “Secundo probo, quia licet locus corrumpatur moto eius subiecto localiter, ita quod, moto aere localiter, non manet in eo eadem ratio loci quae prius (sicut patet ex iam probato), nec eadem ratio loci potest manere in aqua succedente, quia idem accidens numero non potest manere in duobus subiectis, − tamen illa ratio loci succedens (quae est alia a ratione praecedente) secundum veritatem est eadem praecedenti per aequivalentiam quantum ad motum localem, nam ita incompossibile est localem motum esse ab hoc loco in hunc locum sicut si esset omnino idem locus numero. Nullus autem motus localis potest esse ab uno ubi ad aliud ubi nisi quae duo ubi correspondent duobus locis differentibus specie, quia habentibus alium respectum – non tantum numero sed etiam specie – ad totum universum; ex hoc illi respectus qui sunt tantum alii numero videntur unus numero, quia ita sunt indistincti respectu motus localis sicut si tantum essent unus respectus.”
- 11.
Duba and Schabel 2017.
- 12.
Petrus Aureoli, In II Sententiarum, d. 2, pars 3, q. 1 (2000, 143–144): “Respondeo . Pono hic duas propositiones. Prima est quod locus per se et primo non est aliud quam positio, puta hic vel ibi. Secunda est quod per accidens locus est superficies corporis continentis.”
- 13.
Schabel 2000.
- 14.
Petrus Aureoli, In II Sententiarum, d. 2, pars 3, q. 1 (2000, 151–152): “Ratio loci aliter accipitur secundum considerationem metaphysicam, aliter secundum considerationem physicam. Physicus enim definit per materiam, non quidem per materiam quae est pars altera compositi, quia hoc modo metaphysicus definit per materiam, sed debet hic accipi “materia” pro omni eo quod est extra rationem quidditatis. Secundum hoc igitur differt consideratio metaphysici a consideratione physici, quia metaphysicus tantummodo accipit illud quod intrinsece pertinet ad quidditatem, sed physicus accipit materialia et accidentia ac extranea quidditati. Unde concernit in sua consideratione qualitates sensibiles secundum quas res ipsa est subiecta motui, actioni, et passioni. – Tunc ad propositum, dico quod de per se quidditate loci non est aliud quam ratio ipsius ubi, et ideo locus quidditative est in praedicamento ubi. Quapropter locus, secundum considerationem metaphysicam, non est aliud quam ipsum ubi sive positio. Et nota quod immediatum subiectum est quantitas continua. Unde nihil est situabile primo et per se nisi quantum, ut alibi apparere poterit. Haec est intentio Commentatoris expressa, V Metaphysicae, capitulo de quantitate, ubi reddit rationem quare Aristoteles ibi non numerat locum inter species quantitatis sicut in Praedicamentis. Et dicit ‘et forte dimisit hic locum quia apud ipsum locus est de passionibus quantitatis.’ Igitur quidditative locus non est quantitas, sed aliquid quod accidit quantitati. Illud autem est situs ipse, sive est ubi, ut dictum est.”
- 15.
Petrus Aureoli, In II Sententiarum, d. 2, pars 3, q. 1 (2000, 152–153): “Sed secundum physicam considerationem, locus ultra ipsum ubi et situm dicit aliquid materiale, puta ultimum continentis, et sic locus physice non est aliquod unum ens per se, sed est unum per accidens ex duobus praedicamentis aggregatum. Aut si non placet quod sit ens per accidens, ita quod includat res duorum generum in recto et in principali significato, oportet saltem dicere quod aliquid includat in recto, puta quidditatem ipsius loci; et in obliquo per modum connotati includit illud quod est materiale ipsi loco, scilicet ultimum continentis. – Et idcirco, ubi Aristoteles definivit locum physice, dixit quod est ultimum continentis immobile primum, non quod formaliter sit ultimum continentis et quidditative, quia sic esset quantitas secundum substantiam et non esset passio quantitatis, quod improbat Commentator ubi supra. Unde 4° Physicorum, definiens locum, capit illud quod est formale in loco in hoc quod dicit ‘immobile primum.’ Non enim locus est immobilis nisi quia situs vel ubi est immobile. Capit etiam materiale cum dicit ‘ultimum continentis.’”
- 16.
Schabel 2000, 156. Schabel implicitly presents the second passage as authentic, since, after the part cited above, it continues by addressing objections that would otherwise be left open. This passage appears in five manuscripts, including Paris, Bibliothèque nationale de France, Latin 3066, which Florian Wöller has recently argued is a witness to Auriol’s final revision of book II (Wöller forthcoming). With the reference to the Commentator, the second passage explicitly refers to the first, arguing for its authenticity. Most likely, Peter Auriol had these two passages written in the margin or on easily overlooked cedulae, sometime after the text had begun to circulate. This would explain why the copyist of the only manuscript witness to the first passage, immediately after the first passage, began copying the next question before finding the second passage. That is, in the manuscript Firenze, Biblioteca Nazionale Centrale, Conv. Sopp. B.6.121, f. 21rb, the copyist finished with “ut dictum est.,” the last words of the first passage; he then began copying the next question (“Utrum angeli sint creati in celo empireo sicut in loco”) before abandoning it (on f. 21va). At that point, the copyist crossed out the text of the new question, then went back and wrote after the “ut dictum est” the beginning of the second passage, namely “Sed secundum phisicam consideracionem,” extending into the margin.
- 17.
- 18.
In what follows, therefore, in addition to the 1505 edition, the following manuscripts are used: Paris, Bibliothèque nationale de France, Latin 6678 (=P, Metaphysica and Physica); Paris, Bibliothèque nationale de France, Latin 16132 (=S, Metaphysica and Physica, the copy from the library of the Sorbonne); Città del Vaticano, Bibliotheca Apostolica Vaticana, Vat. lat. 3040 (=V1, Metaphysica) and Vat. lat. 3039 (=V2, Physica, the copies used by Francesco della Rovere, later Pope Sixtus IV).
- 19.
- 20.
- 21.
Grellard 2004, 189, citing Nicholas Bonetus, Physica IV, c. 2 (see below, n26).
- 22.
Duba 2014, 480–484.
- 23.
Nicholaus Bonetus, Metaphysica VIII, c. 2 (1505, ff. 42vb-43ra; P, f. 102r-v; S, f. 77va; V1, f. 78v): “Post hec autem de separatione mathematica est dicendum, et primo de separatione mathematicorum a sensibilibus, et primo quantum ad magnitudines, postmodum autem de numeris fiet sermo. Separatio autem mathematicorum potest intelligi tripliciter. Prima separatio magnitudinis singularis. Secunda universalis magnitudinis a singularibus. Tertia universalis magnitudinis ab omni subiecto, et in quolibet ordine potest intelligi separatio fieri vel apud intellectum in esse cognito, vel extra intellectum et in esse reali. – Dicamus igitur in primis de separatione mathematicorum in esse cognito quod in isto triplici ordine separatio possibilis est; potest namque intellectus abstrahere magnitudinem particularem ab omni subiecto obiective, quia quantitas potest capere esse cognitum absque hoc quod subiectum in quo est capiat esse cognitum. Et sic abstrahere nihil aliud est nisi considerare hoc preter hoc. Et talium abstrahentium non est mendacium, et talis abstractio quantitatis ab omni subiecto et materia sensibili proprie est mathematica. Considerant namque mathematici quantitates corporum non curantes in qua materia existant.”
- 24.
Cf. Aristoteles, Physica II, c. 2 (193b 34–35); Auctoritates Aristotelis (ed. Hamesse 1972, 145 n57): “Abstrahentium non est mendacium.”
- 25.
Nicholaus Bonetus, Physica IV, c. 2 (1505, f. 60rb; P, f. 142va; S, f. 111rb; V2, f. 76r-v): “Habet autem dubitationem circa ista dicta de realitate motus, quoniam sic dicendo est dimittere Aristotelem in contradictione, ut patet VI Physicorum per totum. Fertur quod locutus est Aristoteles de motu, tempore, et loco mathematice, non autem physice, vel tantum de motu in esse conceptibili, non autem in esse reali. Concessum est autem quod in esse conceptibili motus habet continuitatem, divisibilitatem, et forte infinitatem, et sic de aliis proprietatibus que VIII Physicorum scribuntur, non autem habet illa in esse reali.”
- 26.
Nicholaus Bonetus, Physica VI, c. 1 (1505, f. 67rb; P, f. 161r; S, f. 127ra-b; V2, f. 54v): “Et si arguas ‘ergo in eodem nunc in quo comedo alius non vivit, et in eodem in quo loquor, Secana non currit,’ respondeo: non in eodem realiter, sed in eodem equivalenter, per quod intelligo nunc mensurans extrinsece et non intrinsece. Debes diligenter advertere quod nunc secundum considerationem physicam non est idem; secundum considerationem tamen mathematicam est idem nunc numero, sicut dicetur inferius de tempore. – Et si dicas: ‘contradicis progenitoribus tuis,’ respondeo: illi loquti sunt in hac materia de tempore et de nunc mathematice, abstrahendo nunc per intellectum ab isto mutato esse in hoc motu et in illo, et ut sic idem nunc est saltem equivalenter. Progenitores etiam nostri habuerunt simpliciter pro impossibili plures esse mundos et per consequens plures esse motus eque primos, ergo plura tempora simul et plura nunc; nos autem ab eis deviamus in hoc principio, ergo et oportet deviare in conclusione necessario sequente ex illo principio.”
- 27.
Duhem 1956, 431: “En affirmant que Dieu peut, s’il lui plaît, créer plusieurs Mondes, Étienne Tempier a ruiné le fondement qui portait la théorie peripatéticienne du temps; de même en affirmant que Dieu peut imposer à l’Univers un mouvement de translation, il avait privé de base la théorie péripatéticienne du lieu.” The condemnation Duhem has in mind is of the doctrine that God cannot move the universe rectilinearly; see Piché-Lafleur 1999, 96 n49.
- 28.
Nicholaus Bonetus, Physica VI, c. 2 (1505, f. 67vb; P, f. 162r-v; S, f. 128rb-va; V2, f. 55r): “De simultate autem partium temporis est inquirendum et de eius unitate. Ad cuius evidentiam est intelligendum quod consideratio de tempore est duplex: una naturalis et alia mathematica. Ideo aliter est dicendum de simultate et unitate temporis secundum esse naturale (naturale] nature mss.), aliter secundum esse mathematicum, et hoc est quod dicit Commentator ille Averroys, 4 (4] 8 P, 2 V2) Physicorum, capitulo de tempore, commento 131, dicens quod tempus eundem modum dicitur habere esse extra animam simile loco (eundem ... loco] secundum hunc modum dicitur inesse extra animam simile perfecto etsi non sit perfectum 1505), et addit: ‘ista consideratio de tempore magis est mathematica quam naturalis.’ Una littera habet ‘mathematica,’ alia ‘divina,’ alia habet ‘philosophica.’”
- 29.
Averroes , Physica IV, comm. 131 (1562, f. 202vaH): “Secundum igitur hunc modum dicitur tempus habere esse extra animam simile perfecto, etsi non sit perfectum. Et ista perscrutatio de tempore magis est philosophica quam naturalis.”
- 30.
Of the texts I consulted, only the 1505 edition has the correct reading for the first part of the text (simile perfecto), and this may be the corrective work of the editor. Cf. Duhem 1956, 432: “C’est ce que dit le Commentateur d’Aristote au commentaire 131 sur le huitième livre des Physiques: il remarque que la façon dont le temps se comporte hors de l’esprit est analogue à celle dont se comporte le lieu.”
- 31.
Nicholaus Bonetus, Physica VIII, c. 4 (1505, f. 75rb; P, f. 183r; S, f. 145vb; V2, f. 80r): “Ultimum autem dictum nostrorum progenitorum de immobilitate loci est istud: quoniam de loco est duplex speculatio, una mathematica et alia naturalis. Consideratio autem mathematica de loco fuit apud Aristotelem cum diffinit locum dicens quod est superficies corporis continentis immobilis primum. Consideratio autem naturalis de loco sic diffiniret locum: quod est superficies corporis continentis mobilis primum. Naturalis enim ut naturalis est non considerat proprie rationem loci, sed rationem vasis; omnis autem locus qui habet rationem vasis mobilis est.”
- 32.
Nicholaus Bonetus, Physica VIII, c. 4 (1505, f. 75rb; P, f. 183r; S, ff. 144vb-145ra; V2, f. 80r): “Debes igitur diligenter advertere quod mathematica consideratio de loco est consideratio de superficiei corporis continentis primum, absque hoc quod consideretur corpus naturale cuius est illa superficies. Unde mathematicus considerat superficiem aeris te ambientem et continentem immediate, non curando in quo corpore existat, sive sit aer vel aliquid aliud, sed precise considerat illam superficiem, et, ut sit abstracta ab omni corpore naturali, immobilis est, quia omnia mathematicalia (]mathematica mss.) sunt immobilia, quia abstrahunt a motu et a materia sensibili. Nec in tali abstractione est mendacium, quia consideratur hoc preter hoc, non autem hoc sine hoc. Ideo locus, ut de eo mathematicus considerat est omnino immobilis. Unde superficies aeris te ambiens et continens considerata quasi separata ab aere et ab alio corpore per variationem corporis naturalis continentis numquam mutabitur. Superficies enim ambiens ut mathematice considerata, quoniam semper consideratur ut una, ideo locus est omnino immobilis secundum considerationem mathematicam; secundum autem considerationem naturalem, mobilis est et tantum habet rationem vasis, quia consideratur locus naturalis, ut scilicet superficies est in corpore naturali isto et illo, et illa superficies bene est mobilis et subiective et obiective, sicut et corpus naturale cuius est, sicut partes fluvii continue variantur secundum esse naturale fluvii, et per consequens superficies illorum partium ut sub esse naturali considerantur, non autem ut mathematice considerantur.”
- 33.
Nicholaus Bonetus , Physica VIII, c. 4 (1505, f. 75rb; P, f. 183r-v; S, f. 145ra, V2, f. 80r): “Palam autem secundum Commentatorem, IV Physicorum, commento 131, quoniam consideratio de loco est magis mathematica quam naturalis. – Concludamus ergo quod ista fuit intentio progenitorum nostrorum de loco immobilitate, et qui ista triplici immobilitate predicta non fuerit contentus, querat aliam.”
References
Averroes. 1562. Quartum Volumen Aristotelis De physico auditu libri octo cum Averrois Cordubensis variis in eosdem commentariis, Quae omnia, a summis huius aetatis Philosophis a mendis quamplurimis expurgata cernuntur. Marci Antonij Zimarae Contradictionum in eosdem Libros Solutiones. Venetiis apud Iunctas (USTC 810959, v. 4 = Physica).
Cross, Richard. 1998. The Physics of Duns Scotus. Oxford: Oxford University Press.
Duba, William. 2014. Three Franciscan Metaphysicians After Scotus: Antonius Andreae, Francis of Marchia, and Nicholas Bonet. In A Companion to the Latin Medieval Commentaries on Aristotle’s Metaphysics, ed. Fabrizio Amerini and Gabriele Galluzzo, 413–493. Leiden: Brill.
Duba, William, and Chris Schabel. 2017. Remigio, Auriol, Scotus, and the Myth of the Two-Year Sentences Lecture at Paris. Recherches de Théologie et Philosophie Médiévales 84: 143–179.
Duhem, Pierre. 1956. Le système du monde: Histoire des doctrines cosmologiques de Platon à Copernic. Vol. 7. Paris: Hermann.
———. 1985. Medieval Cosmology. Theories of Infinity, Place, Time, Void, and The Plurality of Worlds, trans. Roger Ariew. Chicago: University of Chicago Press.
Goris, Wouter. 2015. Transzendentale Einheit. Leiden: Brill.
Grellard, Christophe. 2004. Les présupposés méthodologiques de l’atomisme: La théorie du contenu de Nicolas d’Autrécourt et Nicolas Bonet. In Méthodes et statut des sciences à la fin du Moyen-Âge, ed. Christophe Grellard, 181–199. Paris: Presses Universitaires du Septentrion.
Grellard, Christophe, and Aurélien Robert, eds. 2009. Atomism in Late-Medieval Philosophy and Theology. Leiden: Brill.
Hamesse, Jacqueline. 1972. Les Auctoritates Aristotelis: Un florilège médiéval. Étude historique et édition critique. Louvain: Université Catholique de Louvain.
Ioannes Duns Scotus. 1973. Ordinatio. Liber Secundus. Distinctiones 1–3, ed. Carolus Balič. Vatican City: Typis Polyglottis Vaticanis.
Maier, Anneliese. 1949. Die Vorläufer Galileis im 14. Jahrhundert. Rome: Storia e Letteratura.
———. 1968. Zwei Grundprobleme der Scholastischen Naturphilosophie. 3rd ed. Rome: Storia e Letteratura.
Murdoch, John. 1984. Atomism and Motion in the Fourteenth Century. In Transformation and Tradition in the Sciences: Essays in Honor of I. Bernard Cohen, ed. Everett Mendelsohn, 45–66. Cambridge: Cambridge University Press.
Nicholaus Bonetus. 1505. Habes Nicholai Bonetti viri perspicacissimi quattuor volumina: Metaphysicam videlicet naturalem phylosophiam praedicamenta necnon theologiam naturalem, Venice, (USTC 816175 = Philosophia naturalis (Metaphysica, Physica, Praedicamenta, et Theologia naturalis).
Petrus Aureoli. 2000. In II Sententiarum, d. 2, pars 3, q. 1, ed. Chris Schabel. In Chris Schabel, Place, Space, and the Physics of Grace in Auriol’s Sentences Commentary. Vivarium 38:117–161, at 143–154.
Piché, David, and Claude Lafleur. 1999. La condamnation parisienne de 1277: Nouvelle édition du texte latin, traduction, introduction, et commentaire. Paris: J. Vrin.
Robert, Aurélien. 2012. Le vide, le lieu et l’espace chez quelques atomistes du XIVe siècle. In La nature et le vide dans la physique médiévale, ed. Joël Biard and Sabine Rommevaux, 67–98. Turnhout: Brepols.
Schabel, Chris. 2000. Place, Space, and the Physics of Grace in Auriol’s Sentences Commentary. Vivarium 38: 117–161.
———. 2011. The Reception of Peter Auriol’s Doctrine of Place, with Editions of Questions by Landulph Caracciolo and Gerard of Siena. In Représentations et conceptions de l’espace dans la culture médiévale: Colloque Fribourgeois 2009, ed. Tiziana Suarez-Nani and Martin Rohde, 147–192. Berlin: De Gruyter.
Wöller, Florian. Forthcoming. Inaugural Speeches by Bachelors of Theology: Principial Collationes and their Transmission (1317–1319). In Principia on the Sentences, eds. Monica Brînzei and William Duba. Turnhout: Brepols.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Duba, W.O. (2018). Mathematical and Metaphysical Space in the Early Fourteenth Century. In: Bakker, F., Bellis, D., Palmerino, C. (eds) Space, Imagination and the Cosmos from Antiquity to the Early Modern Period. Studies in History and Philosophy of Science, vol 48. Springer, Cham. https://doi.org/10.1007/978-3-030-02765-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-02765-0_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02764-3
Online ISBN: 978-3-030-02765-0
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)