Lawrence of Lindores on the Royal Road to Geometry

In his questions on Aristotle's Physics, written in Paris at around the turn of the 15th century, the Scottish arts master Lawrence of Lindores asked whether one could arrive at knowledge of effects from knowledge of their causes (I.4, utrum ex cognitione causarum contingat devenire in cognitionem effectuum, ed. Dewender 2002).

One of the arguments under consideration involved the claim that, if knowledge of a cause was sufficient for knowledge of its effect,

sequeretur quod cognito aliquo statim cognoscerentur simul omnia possibilia cognosci ex illo.  Consequens falsum, quia tunc sequeretur quod cognitis principiis geometriae statim cognoscerentur omnes conclusiones eius.  Probatur, quia notitia principiorum geometriae est causa sufficiens ad habendum notitiam primae conclusionis, et notitiae principiorum una cum notitia primae conclusionis esset causa sufficiens ad habendum notitiam secundae conclusionis, et sic ulterius de tertia, quarta et quinta, et sic de aliis, ergo propositum.

‘it would follow that, once something was known, all that could be known from it would immediately be known at the same time.  The consequent is false, because in that case it would follow that once the principles of geometry were known, all the conclusions of geometry would be known immediately.  Proof: knowledge of the principles of geometry is a sufficient cause for having knowledge of the first conclusion, and knowledge of the principles together with knowledge of the first conclusion would be a sufficient cause for having knowledge of the second conclusion, and so on for the third, fourth and fifth, and so for the others, QED.’

Lindores was more down to earth in his response:

Et si dicatur "Ponatur quod Socrates habeat notitiam omnium principiorum geometriae et velit agere toto conatu suo ad disponendum illa principia in debito modo et figura ad inferendum omnes conclusiones geometriae, et non habeat impedimentum extrinsecum nec ex parte famis nec sitis aut frigoris vel quocumque alio extrinseco, tunc in isto casu, ex quo velit agere, sequitur quod statim cognosceret omnes conclusiones geometriae", respondetur admisso casu negando consequentiam.  Et causa est ista, quia, quamvis Socrates non haberet impedimentum extrinsecum, tamen haberet intrinsecum, quia actualis consideratio circa illationem unius conclusionis impediret actualem considerationem circa illationem alterius conclusionis.  Quo dato poneret magnum tempus ad inferendum duas conclusiones, ut pateret experientia, igitur et cetera.

‘And if someone says "Suppose Socrates had knowledge of all the principles of geometry, and wanted to put all of his effort into setting those principles in the mode and figure necessary for inferring all the conclusions of geometry, and had no extrinsic impediment from hunger, thirst, cold, or any other extrinsic thing – then in that case, from his wanting to do this, it follows that he would at once know all the conclusions of geometry", the response is to allow the case and deny the consequence.  And the reason is as follows: although Socrates would not have an extrinsic impediment, he would still have an intrinsic one, because actual consideration concerning the deduction of one conclusion would impede actual consideration concerning the deduction of another.  Given which, it would take a long time to infer two conclusions, as should be clear from experience; therefore etc.’

If only knowledge was closed under implication!

Editions of 14th-century Sentences commentaries

I'm compiling a list of sizeable critical editions of 14th-century commentaries on Lombard's Sentences. In case it's of use to anyone else, here it is. At present the roll-call is: Scotus, Auriol, Marchia, Ockham, Chatton, Wodeham, Crathorn, Roseth, Langeley, Rimini, and Marsilius of Inghen.

'Contingit' and 'accidit' in Boethius and Moerbeke

Albrecht Becker-Freyseng complained in his classic study Die Vorgeschichte des philosophischen Terminus 'contingens' (1938) that Boethius used 'contingit' as a dual-purpose translation of Aristotle's endechetai (it may be) and sumbainei (it happens) despite the availability of 'licet' for the former and 'accidit' for the latter.

Funnily enough, William of Moerbeke remedied this defect by rendering sumbainei as 'accidit' in his De interpretatione translation of 1268, but this version never gained currency – so much so that it still hadn't been printed by the time Becker-Freyseng was writing.

Becker-Freyseng would not, however, have been happy to see that Moerbeke followed the editio composita in using 'contingit' – rather than, say, 'accidit' – in his translation of Aristotle's phrase hopoteron/hopoter' etuchen (whichever/however it chances).

Loveland on Auriol on Foreknowledge of the Sunrise

Jeff Loveland's article ‘Buffon, the Certainty of Sunrise, and the Probabilistic Reductio ad Absurdum’ (2001) contains a tantalizing reference to Auriol's take on foreknowledge of the sunrise in S I.30:

The authors Robert Holcot … and Peter de Rivo … – the latter perhaps influenced by Petrus Aureolus (Aureolus, 1312–20 [sic], pt. 1, p. 673) – were among those investigating the status of the assertion sol orietur cras – "the sun will rise tomorrow."  (p. 468)

Alas, if you follow this up in the 1596 edition, you'll find no such thing.  What's worse, the faulty reference doesn't look like a mere typo.  Here's something you will find on that page:

Quarta demum propositio est, quod relationi repugnat in communi ex sua ratione, quod sit res in natura existens.  Dicitur enim de ea, quod acquiritur, et oritur in fundamento sine sui mutatione, sola mutatione facta in termino.  Illud ergo non est res, quod acquiritur in subiecto nullo agente ipsum attingente; sed manifestum est, quod albo existente in oriente, si fiat aliud album in occidente, similitudo intelligitur in esse [inesse] albo, quod praeteriit in oriente, absque hoc, quod aliquod agens ipsum attingat, praeterquam intellectus, qui ipsum comparat ad album aliud, quod de novo producitur in occidente.  Non enim dealbans in occidente actionem suam protendit usque in oriens, et [ut] ibi similitudinem imprimit [imprimat], nec coelum, aut aliquid aliud potest impremere eam, nisi solus intellectus.  Ergo poni non potest, quod aliqua talis res fuerit acquista, et per consequens relationi ex sui natura repugnat habere existentiam in natura. (p. 673, col. 2, A–C)
(The variants are from Henninger's transcription of MS Borghese 329 – which wrongly has 'ullo' for 'nullo' – in his Relations, p. 154, n. 13.)

The context is the ontological status of relations, which Auriol takes to be mental constructs.  Here we find him defending his view with a thought experiment: if there is already a white thing in the east, and a white thing appears in the west, then the former acquires a similarity to the latter without any contact between the two.  (You might say that the pre-existing white thing undergoes a 'Cambridge change'.)

Now, someone could conceivably misconstrue this passage as having something to do with the sunrise.  But to do so he would have to read sola and solus as 'the sun' (rather than 'only') and oriens and occidens as 'rising' and 'setting' (rather than 'the east' and 'the west'); the former is indefensible, and the latter would make no sense in the context.

I hope this isn't the explanation – not just for academia's sake, but because I'd love to know what Auriol thinks about the necessity of the sunrise – but its ready availability would be a striking coincidence.

Bradwardine and Augustine, Pagans and Pelagians

It is well known that Bradwardine's magnum opus De causa Dei was heavily influenced by Augustine's De civitate Dei, but scholars rarely if ever mention the obvious parallel between the two titles.  From a quick survey, the best I can find is an incidental comment in Minnis and Johnson's introduction to The Cambridge History of Literary Criticism vol. 2 (2005): ‘the prosimetric elegance of Thomas Bradwardine's De causa Dei (written in proud imitation of Augustine's De civitate Dei)’.

The full title of Augustine's work is De civitate Dei contra Paganos. Bradwardine's work is usually cited as De causa Dei contra Pelagium, but sometimes as De causa Dei contra Pelagianos.  If Bradwardine's title is a homage to Augustine, shouldn't that count in favour of the second version?  Just a thought.

Rimini as Torturer of Infants (II)

Oreste Delucca's biographical contribution to Gregorio da Rimini filosofo (2003) makes the origins of the nickname Tortor Infantium seem even murkier:

‘Quanto al casato di Gregorio, taluno lo dice appartenere alla famiglia Tortorini, o Tortorucci, o Tortorici; ma non si hanno prove documentarie al riguardo, per cui l'affermazione viene accolta con prudenza, ignorata o addirittura rigettata da molti storici.’ (p. 46)

‘As for Gregory's surname, some say he belongs to the Tortorini, Tortorucci, or Tortorici family; but there is no documentary evidence in this regard, because of which the statement is greeted with caution, ignored, or even rejected by many historians.’

Delucca cites Battaglini (1794), L. Tonini (1880), C. Tonini (1884), and Perini (1929) as being of the former persuasion; earlier we saw Trapp (1980) using yet another variant, 'Tortoricci'.

I'd like to follow up Delucca's references, but the Bodleian – which doesn't even stretch to Gregorio da Rimini filosofo – only has the Perini. The others are all at the Warburg (ENH 360, HNB 174, 175).

Leff on Future Contingents

Gordon Leff's Gregory of Rimini: Tradition and Innovation in Fourteenth Century Thought (1961) might be expected to shed light on Rimini's nickname Tortor Infantium, but I'm afraid it doesn't.

I was pleased to read of Auriol's position as ‘somewhat anachronistically described by Michalski as 'three-value logic'’; Leff's wariness is echoed by Schabel in Theology at Paris, p. 3. But the following raised an eyebrow:

‘These two postulates [omniscience and bivalence] are the foundation of Gregory's view of the future; and they constitute an uncompromising repudiation of the two leading ideas among many of Gregory's confrères: that God cannot know future contingents necessarily, and that, as the future is itself undetermined, propositions about it are neutral.’ (p. 115)

Auriol is the only example Leff gives of the "many of Gregory's confrères" who think that propositions about future contingents are neither true nor false. Whether or not a Franciscan and an Augustinian can be said to be confrères, who are the others?

Henninger on Fiorentino on Rimini

Francesco Fiorentino's book Gregorio da Rimini: Contingenza, futuro e scienza nel pensiero tardo-medievale (2004) is riddled with errors.  For example, here is n. 16 on p. 9 of the introduction:

‘C. Normore, Peter Aureoli and his contemporaries on future contingents and the excluded middle, "Synthèse", 96 (1993), pp. 83-92.  Nel 1999 C. Normore è ritornato sul tema secondo una prospettiva più ampia; cf. C. Normore, Contingenti futuri, in La logica nel Medioevo, The Cambridge History of Later Medieval Philosophy, a cura di N. Kretzmann, A. Kenny, J. Pinborg, Milano 1999.’

An attentive reader working on future contingents will immediately spot three mistakes in this one footnote.  The first two are comparatively insignificant: the title of Normore's article begins with ‘Petrus’ and does not contain the definite article.  The third is more worrying: the CHLMP was published in 1982, so Normore's 1993 article appeared after his CHLMP chapter, but Fiorentino (presumably using a 1999 translation) tells us that in the latter Normore ‘returned to the theme from a wider perspective’.  The worrying thing is that the CHLMP has been the standard handbook of mediaeval philosophy since its appearance over 25 years ago.

Both author and publisher deserve a rap on the knuckles, so I was pleased to discover that Mark Henninger had already reviewed the book in Gregorianum.  Pleased, that is, until I read the review.

Henninger is grateful for ‘certain logical principles he wisely provides in the introduction [actually Chapter 1]: the principle of bivalence, i.e., in whatever statement, either its affirmation or its negation is true; the law of contradictories, i.e., for every affirmative statement there exists its contradictory, and vice versa’.  Unfortunately, both of these are more or less obvious howlers on Fiorentino's part.

If the meaning of ‘bivalence’ in a logical context is not clear enough from the etymology, even a standard dictionary will define it as the existence of only two truth-values, the corresponding principle being that every statement is either true or false.  Worse, Fiorentino goes on (p. 24 n. 23) to cite Łukasiewicz's insistence on distinguishing between the principle of bivalence and the law of excluded middle.  A reader following this up will be astonished, given Fiorentino's formulation, to read on p. 82 of Łukasiewicz, Aristotle's Syllogistic: ‘…the so-called principle of bivalence, which states that every proposition is either true or false, i.e. that it has one and only one of two possible truth-values: truth and falsity.  This principle must not be mixed up with the law of the excluded middle, according to which of two contradictory propositions one must be true.’

As for the law of contradictories, Fiorentino refers only to Aristotle, De int. 9, 18a28-31.  The reader following this up will find a discussion of the truth or falsity of an affirmation and its corresponding negation – not the mere assertion that there is a corresponding negation.  And I think the reader checking in Rimini and Auriol will find that the law of contradictories is that if one part of a contradictory pair is false then the other part is true (Rimini), or vice versa (Auriol).

Henninger's verdict is that ‘Fiorentino provides a well-researched, thorough and balanced investigation of the teaching of Gregory of Rimini … In the introduction, Fiorentino sets the context with a helpful historiography of the problem of future contingents and divine foreknowledge as treated mostly by researchers in the last century, as well as a survey of the literature on Gregory of Rimini. … This is a fine work by a fine scholar.’

Sadly, then, Fiorentino's book still awaits a properly critical review.

Rimini as Torturer of Infants

Historians of philosophy like to point out that Gregory of Rimini's views on baptism and salvation earned him the nickname Tortor Infantium, but they are remarkably coy about supporting this with references. The nickname was supposedly derived from Rimini's family name, Tortoricci (Trapp, "Notes on the Tübingen Edition of Gregory of Rimini II", 1980). But when did it first appear in print?

A preliminary search puts an upper bound on the date: 1709, when Leibniz wrote in his Essais de Théodicée (§92): ‘Grégoire de Rimini, général des Augustins, avec peu d'autres, a suivi saint Augustin contre l'opinion reçue des écoles de son temps, et pour cela était appelé le bourreau des enfants, tortor infantum.’

But surely it can't be hard to find something that antedates Leibniz.

Frustra fit per plura (III)

Here's another of Auriol's statements of ontological parsimony:

‘multitudo ponenda non est nisi ratio evidens necessaria illud probet aliter per pauciora salvari non posse.  Deus enim et natura nihil faciunt frustra’ (S II.14.1.ii)

(Mark Henninger, Relations: Medieval Theories (1989), p. 156 n. 25.)