Abstract
The aim of this paper is to show that John Wyclif’s theory of space is at once an interpretation of the Platonic theory of place and a Neopythagorean conception of magnitudes and numbers. The result is an original form of mathematical atomism in which atoms are point-like entities with a particular situation in space. If the core of this view comes from Boethius’ De arithmetica, John Wyclif is also influenced by Robert Grosseteste’s metaphysics, which includes the Boethian number theory within the Christian tale of the creation of the world ex nihilo. John Wyclif, however, adds some novelty to this theory concerning the epistemological status of this hypothetical description of the creation of the world out of atoms. First, according to Wyclif, whereas geometry is concerned with sensible and imaginable beings, arithmetic, which is purely intellectual, has access to the deep mathematical structure of the universe. He then suggests a subordination of geometry under arithmetic, which he considers the most solid basis for his metaphysics. As a result, with the attribution of numbers and units to every level of reality, it becomes possible to reform our natural imagination, so that it can imagine the atomic structure of matter and space.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
For an account of the Aristotelian notions of place and space see Algra’s Chapter 2 in this volume.
- 2.
Grant 1976, 138.
- 3.
On Walter Chatton see Robert 2012; on Guiral Ot see Bakker and De Boer 2009; on William Crathorn see Robert 2009 and Roques 2016; on Nicolas Bonet see Duba’s Chapter 5 in this volume; on Nicole Oresme see Kirschner 2000. We also find this kind of theory in the Arabic tradition, but I will limit this study to the Latin tradition.
- 4.
Albertson 2014.
- 5.
Robert 2017.
- 6.
Kretzmann 1986.
- 7.
Michael 2009.
- 8.
- 9.
- 10.
[Councils] 1973, 426.
- 11.
It is usually assumed that the third part of the Logica was written between 1360 and 1363, immediately after the first two parts, but the date 1383 is once mentioned in the text. Thomson (1983) considers the possibility that either a copyist changed the original date, or that it is a scribal mistake. Wyclif’s unedited commentary on the Physics survives in only one manuscript (Venice, Biblioteca Nazionale Marciana, MS Lat. VI. 173). Ivan Müller is presently preparing an edition of this text. The two important questions in which Wyclif deals with atomism and the nature of place are the following: Utrum omnia temporaliter existentia sunt in loco (ff. 38ra-40rb); Utrum omne tempus, magnitudo et motus diversificate se invicem consequuntur (ff. 52vb-55ra), and many arguments are similar to the ones developed in the Logica.
- 12.
- 13.
Michael 2009.
- 14.
Logica, I, ch. 4, in Wyclif 1893 13; see also De ente praedicamentali, 1, 3, in Wyclif 1891. The references to the Logica are from the nineteenth-century edition whenever the text is correct, but when it is not, in particular for the Logicae continuatio (henceforth LC), I will give my own transcription of MS Assisi, Biblioteca Communale 662 (now A), dating from c. 1385, which gives a better version.
- 15.
The edition has duratio but the MS A has dimensio.
- 16.
LC, III, ch. 9, in Wyclif 1899, 1–2.
- 17.
Kaluza 2003.
- 18.
LC, III, ch. 9, 2.
- 19.
LC, III, ch. 9, 3–4.
- 20.
LC, III, ch. 9, 11–29.
- 21.
LC, III, ch. 9, 30.
- 22.
See Annas 1975.
- 23.
LC, III, ch. 9, 49, 55, 59–61.
- 24.
The Latin has est (God is in as much place…), but in this context it is difficult to understand how God can be in this place.
- 25.
LC, III, ch. 9, 33–34, corrected with MS A, f. 72ra: “Similiter, ut credo, nullus theologus negaret quin Deus de potentia absoluta potest facere substantiam punctalem vel condensando vel noviter causando vel tertio faciendo spiritum esse in situ punctali et annihilando omnem aliam creaturam preter talem spiritum servatum immotum. Et tunc patet quod punctalitas vel punctus que est huiusmodi substantiam esse punctalem est accidens posterius illa substantia, sive sit separabile sive inseparabile. Punctus ergo potest esse. Nec dubium quin si Deus potest unum punctale producere potest et quodlibet iuxtaponere. Nec dubium quin situs essent correspondenter iuxtapositi, cum situs sit subiectum situari. Et ultra patet quod Deus potest ex talibus non quantis facere unum quantum […]. Creet substantiam ad omnem situm punctalem mundi unam substantiam punctalem et annihilet post omnem substantiam continuam servando punctales substantias immotas. Et patet quod Deus est adhuc per tantum locum sicut fuit in principio et per consequens est tantus locus vel saltem contingit tantum locum fieri ex illis punctalibus sicut prius. […] Posito ergo quocumque tali accidente continuo oportet ponere subiectum eius continuum et illud esset compositum ex punctalibus, quia illa forent eius principia intrinseca. Nec dubito quin admisso hoc pro possibili omnes philosophi mundi non haberent infallibilem evidentiam ad concludendum quod non est sic de facto.”
- 26.
LC, III, ch. 9, 119, corrected with MS A, f. 88ra-b: “ymaginandum est igitur unam essentiam corpoream in principio productam, ex indivisibilibus compositam et occupare omnem locum possibilem, nec esse secundum eius partem aliquam corruptibilem nisi forte per divisionem et separationem unius partis a reliqua. Sed cum oportet illam totam essentiam habere quamcumque talem partem aliqualiter continuatam, patet quod illa essentia est simpliciter incorruptibilis et illa essentia primo confuse concipitur sub ratione qua [est] ens simpliciter, et nec ut ignis vel aer vel cuiuscumque alterius generis vel speciei […]. Sed philosophi ulterius considerantes quamlibet talem essentiam esse unum absolutum cui per se competit substare accidentibus attribuunt sibi substantialitatem. Et postmodum considerata eius extensione attribuunt sibi corporeitatem quam Lincolniensis vocat lucem. Et tertio formam generis proximi, ut animalitatem, lapiditatem vel aliud huiusmodi. Et quarto considerata ratione speciei attribuunt sibi formam specialissimam. Ideo dicunt philosophi quod substantiarum alia est materia, alia est forma, alia compositum ex hiis.”
- 27.
LC, III, ch. 9, 35.
- 28.
LC, III, ch. 9, 35–36.
- 29.
- 30.
For a general introduction to Nicomachus’s arithmetic, see Levin 1975.
- 31.
On Arithmetic, I, i, in Boethius 1983, 73.
- 32.
On Arithmetic, I, i, in Boethius 1983, 74.
- 33.
On Arithmetic, II, iv, in Boethius 1983, 130 (trans. slightly modified).
- 34.
Albertson 2014.
- 35.
Robert 2017.
- 36.
On this point see Mendell 1987.
- 37.
King 2004.
- 38.
- 39.
See fn. 1.
- 40.
See Robson 2008, 26–31.
- 41.
Trans. Lewis 2013.
- 42.
- 43.
On light, in Grosseteste 2013, 242.
- 44.
On light, Grosseteste 2013, 241–242.
- 45.
Lewis 2005, 160.
- 46.
Grosseteste, Commentarius in VIII libros physicorum, III, in Grosseteste 1963, 54.
- 47.
Grosseteste 1963, 54.
- 48.
Grosseteste 1963, 54.
- 49.
Grosseteste 1963, 55 (see also 93).
- 50.
Grosseteste 1963, 91–93.
- 51.
LC, III, ch. 9, 6.
- 52.
LC, III, ch. 9, 19.
- 53.
LC, III, ch. 9, 36, corrected with MS A, f. 72rb: “non interest tractare de partibus punctalibus in philosophia naturali in qua demonstratur causa per effectus demonstratione quia est., cuius principium est experientia vel sensus. Punctus autem non est sensibilis vel ymaginabilis, ideo tractatus eius non specialiter pertinet geometre qui solum de ymaginabilibus pertractat directe, sicut nec naturali philosopho. Sed illud conservandum est metaphysico et arithmetico.”
- 54.
LC, III, ch. 9, 45, corrected with MS A, f. 73va-b: “Ymaginatio autem non sufficit ista capere […]. Ideo oportet superius ascendere ad aciem intellectus in recte concipiendo compositionem continui ex non quantis. Quod grave est facere ex hoc quod ymaginatio coagit intellectum in apprehensione cuiuscumque ymaginabilis. Et cum in toto ambitu sui obiecti non reperit compositionem huiusmodi partium, non est mirabile si dissentit. Sed intellectus dicit sibi quod est dare partium indivisibilium compositionem aliam quam non est suum discutere.”
- 55.
LC, III, ch. 9, 40 and 58.
- 56.
Kretzmann 1986, 43–44.
- 57.
LC, III, ch. 9, 35–37.
- 58.
LC, III, ch. 9, 109, corrected with MS A, f. 86ra-rb: “Omnes, ut dictum est., intelliguntur de lineis angulis et figuris ymaginacioni subiectis, nos autem loquimur de illis que a solo intellectu cognosci possunt, ut docet Augustinus in De quantitate animae. Quod si aliquis dicat quod eque verificantur omnes conclusiones geometrice de puris intelligibilibus, sicut de ymaginabilibus, leve verbum est et sine probacionis efficacia eructatum. Ideo non credetur nisi efficaciter comprobetur. Quod si dicatur Campanum et multos alios expositores Euclidis illud asserere, revera multi expositores, ut Pitagoras, Democritus, Plato et inter moderniores Lincolniensis cum aliis sequentibus tramitem veritatis constanter asserunt oppositum. Ideo tales topice raciones in materia doctrinali deficientes demonstraciones adducte indicant defectum garulum argumentorum. Et sic dico quod nulla est conclusio demonstrabilis in continuis quin sit demonstrabilis in numeris, sed forte<non> econtra propter amplitudinem obiecti prioris. Et patet quod conclusiones arismetice non demonstrant cum precisione, sed cum exclusione erroris sensibilis, ut dictum est de divisione cuiuscumque date linee vel dati anguli in duo [A: f. 86rb] equalia. […] incertum est cuilibet geometro de quantitate et proporcione intelligibilis dyametri, sicut, secundum Licolniensem, incognitus est sibi numerus punctalium componencium. Et de dyametro sensibili habet coniecturam probabilem vel veram vel veritati propinquam a sensu incorrigibilem. In numeris autem cognitis a beatis consistit certitudo sciencie et in sensibilem langor erroneus et confusus.”
- 59.
LC, III, ch. 9, 57.
- 60.
LC, III, ch. 9, 55–58.
- 61.
Proclus 1970, 30.
- 62.
Proclus 1970, XX.
- 63.
Proclus 1970, 78.
- 64.
Proclus 1970, 79–80.
- 65.
Proclus 1970, 216–217.
- 66.
Proclus 1970, 48.
References
Albertson, David. 2014. Mathematical Theologies: Nicholas of Cusa and the Legacy of Thierry of Chartres. Oxford: Oxford University Press.
Annas, Julia E. 1975. Aristotle, Number and Time. The Philosophical Quarterly 25: 97–113.
Bakker Paul, J.J.M., and Sander W. De Boer. 2009. Locus est spatium: On Gerald Odonis’ Quaestio de loco. Vivarium 47: 295–330.
Boethius. 1983. On Arithmetic. Trans. Michael M. In Boethian Number Theory: A Translation of the De institutione arithmetica. Amsterdam: Rodopi.
Burkert, Walter. 1972. Lore and Science in Ancient Pythagoreanism. Harvard: Harvard University Press.
Cesalli, Laurent. 2005. Le ‘pan-propositionnalisme’ de Jean Wyclif. Vivarium 43: 124–155.
Conti, Alessandro. 2006. Wyclif’s Logic and Metaphysics. In A Companion to John Wyclif, ed. Ian C. Levy, 67–125. Leiden: Brill.
Cornelli, Gabriele. 2013. In Search of Pythagoreanism: Pythagoreanism as an Historiographical Category. Berlin: Walter de Gruyter.
[Councils]. 1973. Conciliorum oecumenicorum decreta, ed. Giuseppe Alberigo et al., 3rd ed. Bologna: Istituto per le scienze religiose.
Grant, Edward. 1976. Place and Space in Medieval Physical Thought. In Motion and Time, Space and Matter, ed. Peter K. Machamer and Robert G. Turnbull, 137–167. Columbus: Ohio State University Press.
Horky, Phillip S. 2013. Plato and Pythagoreanism. Oxford: Oxford University Press.
John Wyclif. 1869. In Trialogus, ed. Gotthard V. Lechler. Oxford: Clarendon Press.
———. 1891. In De ente praedicamentali, ed. Rudolf Beer. London: Trübner for the Wyclif Society.
———. 1893–1899. Tractatus de logica. 3 vols., ed. Michael H. Dziewicki. London: Trübner for the Wyclif Society.
———. 2012. Trialogus. Trans Stephen Lahey. Cambridge, MA: Cambridge University Press.
Kaluza, Zénon. 2003. La notion de matière et son évolution dans la doctrine wyclifienne. In John Wyclif: Logica, politica, teologia, ed. Mariateresa Fumagalli Beonio Brocchieri and Stefano Simonetta, 113–151. Florence: Edizioni del Galluzzo.
King, Peter. 2004. The metaphysics of Peter Abelard. In The Cambridge companion to Abelard, ed. Jeff Brower and Kevin Guilfoy, 65–125. Cambridge: Cambridge University Press.
Kirschner, Stefan. 2000. Oresme’s Concepts of Place, Space, and Time in His Commentary on Aristotle’s Physics. Oriens – Occidens: Sciences, mathématiques et philosophie de l’antiquité à l’âge classique 3: 145–179.
Kretzmann, Norman. 1986. Continua, Indivisibles, and Change in Wyclif’s Logic of Scripture. In Wyclif in his Times, ed. Anthony Kenny, 31–65. Oxford: Clarendon Press.
Lahey, S.E. 2009. John Wyclif. Oxford: Oxford University Press.
Levin, Flora R. 1975. The Harmonics of Nicomachus and the Pythagorean Tradition. University Park: American Philological Association.
Levy, Ian C. 2003. John Wyclif: Scriptural Logic, Real Presence, and the Parameters of Orthodoxy. Milwaukee: Marquette University Press.
Lewis, Neil. 2005. Robert Grosseteste and the Continuum. In Albertus Magnus and the Beginnings of the Medieval Reception of Aristotle in the Latin West, ed. Ludger Honnefelder, Rega Wood, Mechthild Dreyer, and Marc-Aeilko Aris, 159–187. Münster: Aschendorff.
Lewis, Neil. 2013. Robert Grosseteste’s on light: An English translation. In Robert Grosseteste and his intellectual milieu, ed. John Flood, James R. Ginther, and Joseph W. Goering, 239–247. Toronto: Pontifical Institute of Mediaeval Studies.
Maier, Anneliese. 1949. Kontinuum, Minimum und aktuell Unendliches. Die Vorläufer Galileis im 14. Jahrhundert, 155–215. Rome: Edizioni di Storia e Letteratura.
Mendell, Henry. 1987. Topoi on Topos: The Development of Aristotle’s Theory of Place. Phronesis 32: 206–231.
Michael, Emily. 2009. John Wyclif’s Atomism. In Atomism in Late Medieval Philosophy and Theology, ed. Christophe Grellard and Aurélien Robert, 183–220. Leiden: Brill.
Murdoch, John E. 1974. Naissance et développement de l’atomisme au bas moyen âge latin. In Cahiers d’études médiévales, 2: La science de la nature: Théories et pratiques, 11–32. Montreal: Bellarmine.
———. 1982. Infinity and Continuity. In The Cambridge History of Later Medieval Philosophy, ed. Norman Kretzmann, Anthony Kenny, and Jan Pinborg, 564–591. Cambridge, MA: Cambridge University Press.
Panti, Cecilia. 2012. The Evolution of the Idea of Corporeity in Robert Grosseteste’s Writings. In Robert Grosseteste His Thought and Its Impact, ed. Jack P. Cunningham, 111–139. Toronto: Pontifical Institute of Mediaeval Studies.
Philip, James A. 1966. The ‘Pythagorean’ Theory of the Derivation of Magnitudes. Phoenix 20: 32–50.
Proclus. 1970. A Commentary on the First Book of Euclid’s Elements. Trans. Glenn Morrow, with a foreword by Ian Mueller. Princeton: Princeton University Press.
Robert, Aurélien. 2009. William Crathorn’s Mereotopological Atomism. In Atomism in Late Medieval Philosophy and Theology, ed. Christophe Grellard and Aurélien Robert, 127–162. Leiden: Brill.
———. 2012. Le vide, le lieu et l’espace chez quelques atomistes du XIVe siècle. In La nature et le vide dans la physique médiévale: Etudes dédiées à Edward Grant, ed. J. Biard and Sabine Rommevaux, 67–98. Turnhout: Brepols.
———. 2017. Atomisme pythagoricien et espace géométrique au moyen âge. In Lieu, espace, mouvement: Physique, métaphysique et cosmologie (XII e–XVI esiècles), ed. Tiziana Suarez-Nani, Olivier Ribordy, and Antonio Petagine, 181–206. Turnhout: Brepols.
Robert Grosseteste. 1963. Commentarius in VIII libros physicorum Aristotelis, ed. Richard C. Dales. Boulder: University of Colorado Press.
Robson, John A. 2008. Wyclif and the Oxford Schools: The Relation of the Summa de ente to Scholastic Debates at Oxford in the Later Fourteenth Century. Cambridge, MA: Cambridge University Press.
Roques, Magali. 2016. Crathorn on Extension. Recherches de Théologie et Philosophie médiévales 83 (2): 423–467.
Thomson, Williell R. 1983. The Latin Writings of John Wyclif. Toronto: The Pontifical Institute of Medieval Studies.
Zhmud, Leonid J. 2012. Pythagoras and the Early Pythagoreans. Oxford: Oxford University Press.
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
Robert, A. (2018). Space, Imagination, and Numbers in John Wyclif’s Mathematical Theology. 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_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-02765-0_6
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)