Quis castigabit ipsum castigatorem?

In his handbook chapter on ‘Mathematics in Fourteenth-Century Theology’ (2009), which introduces historians of mathematics to the surprising relevance of Sentences commentaries, Mark Thakkar focusses on a stand-off between the positions of Thomas Bradwardine and Gregory of Rimini on infinite multitudes.  ‘Unfortunately,’ he writes, ‘[Bradwardine's] 'Sentences' commentary, which would have been written in around 1332, has not come down to us.’

Now, anyone familiar with the literature on Bradwardine will know that Jean-François Genest and Katherine Tachau claimed to have identified part of his Sentences commentary as long ago as 1990.  Still, in some contexts Thakkar's remark might have been seen as an excusable simplification.  What is embarrassing is the appearance in a collection that he himself cites of a chapter by Genest, ‘Les premiers écrits théologiques de Bradwardine: textes inédits et découvertes récentes’ (2002), that includes a 2½-page discussion of a question on infinity (c.1333) that is probably from Bradwardine's Sentences commentary and would have served his purposes admirably.

What he chooses to look at instead is De causa Dei, Bradwardine's ‘sprawling magnum opus’ of 1344.  Readers may be interested to learn that, almost four hundred years after Henry Savile's lavish editio princeps (1618), the Austrian Science Fund has approved a project led by Edit Anna Lukács to study the manuscript tradition in Vienna (home to 11 of the 50-odd extant MSS) and to establish a ‘reception-specific’ critical edition.  Hopefully this will pave the way for the full edition that John Marenbon recently highlighted as a desideratum for the British Academy's Auctores Britannici Medii Aevi series.

Gregory of Rimini on the Eternity of the World

Gregory of Rimini's treatment of the paradoxes of infinity (S I.42-44.4) allowed him a swift response in S II.1.2.i to an Aristotelian argument that an endless world could not have had a beginning.

Here is the core of the Aristotelian argument:
‘Probatur minor, quia si detur oppositum, sequitur quod erit infinitum maius infinito. Nam tempus aeternum ex utraque parte super utrumque aeternum ex altera tantum addit reliquum tempus aeternum ex reliqua parte, ut super aeternum in praeterito tantum addit tempus aeternum in futuro tantum, et econverso. Cum igitur omne aeternum in futuro sit ens tempore infinito, sequitur quod fuit etiam ens infinito tempore praeterito.’

And here is Gregory's response:
‘Ad secundam rationem principalem nego minorem, et dico quod probatio aeque militat contra Philosophum, qui posuit tempus aeternum in praeterito et in futuro, cum tamen aeternum in praeterito sit infinitum, et similiter aeternum in futuro, et aeternum ex utraque parte utrumque comprehendat. Quia tamen, secundum quod inferius dicam, possibile fuit tempus esse aeternum, quamvis de facto habuit initium, ac per hoc fuit possibile infinitum comprehendere in se plura infinita, dico quod hoc non est inconveniens, sicut in primo libro distinctione 44 declaratum est. Et pro nunc sufficiat exemplum de aliquo continuo, cuius utraque medietas continet infinitas partes proportionales alias ab his, quas reliqua continet.’

The tu quoque here is a nice move.