Clavius,Proclus,and the Limits of Interpretation: Snapshot-Idealization Versus Projectionism

A detailed biography can be found in Romano Antonella , La contre-réforme mathématique: Constitution et diffusion d'une culture mathématique jésuite à la Renaissance (Rome, 1999), 85–94, and in Smolarski Dennis , “The Jesuit Ratio Studiorum, Christopher Clavius, and the study of mathematical sciences in universities”, Science in context, xxv (2002), 2002–57, pp. 449–52.

Smolarski , op. cit. (ref. 1), 450.

For a detailed view of Clavius's literary production, see Knobloch Eberhard , “L'oeuvre de Clavius et ses sources scientifiques”, in Giard Luce (ed.), Les jésuites à la Renaissance: Système éducatif et production du savoir (Paris, 1995), 263–83. Most of these works were re-edited in Clavius Christopher , Opera mathematica (Mainz, 1611–12), literally his opus magnum, published in five volumes, containing almost 4000 pages.

See Romano , op. cit. (ref. 1), 89.

For example, Dear Peter , Discipline and experience: The mathematical way in the scientific revolution (Chicago, 1995); Kessler Eckhard , “Clavius entre Proclus et Descartes”, in Giard (ed.), op. cit. (ref. 3), 285–308; Feldhay Rivka , “The use and abuse of mathematical entities”, in MacHamer Peter (ed.), The Cambridge companion to Galileo (Cambridge, 1998), 80–145; Romano , op. cit. (ref. 1); and Sasaki Chikara , Descartes's mathematical thought (Dordrecht, Boston and London, 2003).

Kessler , op. cit. (ref. 5); Romano , op. cit. (ref. 1); and Sasaki , op. cit. (ref. 5).

Romano , op. cit. (ref. 1), 135, n. 8. Romano also names the example of Federico Commandino's preface to his Euclid edition of 1572.

Sorabji Richard , (ed.), Aristotle transformed: The ancient commentators and their influence (New York, 1990), 25.

Copenhaver Brian and Schmitt Charles , (eds), Renaissance philosophy (Oxford and New York, 1992), 69.

See Helbing Mario Otto , “La fortune des commentaires de Proclus sur le premier livre des Eléments d'Euclide à l'époque de Galilée”, in Bechtle Gerald and O'Meara Dominic (eds), La philosophie des mathématiques de l'Antiquité tardive (Fribourg, 2000), 173–93, pp. 175–180.

Piccolomini Alessandro , De certitudine mathematicarum (Rome, 1547), f. 69v: “… eousque comprimendam [sententiam meam] duxi, donec Proclum ipsum, hoc idem sentire cognoscens, maxima animi laetitia affectus, testem tam locupletem nactus, id ipsum dehinc clara voce frequenter asserui.”/ “I chose to withhold my opinion until I found Proclus himself, learning that he feels exactly the same. With my soul filled with enormous joy, for having found such a rich witness, I have declared it loudly since then”.

For example, Biancani Giuseppe , De natura mathematicarum (Bologna, 1615): “Sequatur tertio loco Procli ipsius auctoritas….”/ “Thirdly, let the authority of Proclus himself follow…”.

Piccolomini , op. cit. (ref. 11), f. 105r: “Plures etiam possem afferre rationes concludentes demonstrationes mathematicas, non esse potissimas. Sed haec sufficiant, praesertim cum authoritates habeamus mirabiles. Et primo Proclum virum illustrem in mathematicis.”/ “I could bring forward even more reasons that conclude that mathematical demonstrations are not potissimae. But these will be sufficient, especially because we have wonderful authorities. And first of all Proclus, an illustrious man in the field of mathematics”.

See Gilbert Neal , Renaissance concepts of method (New York and London, 1960), 30–36, and Copenhaver and Schmitt , op. cit. (ref. 9), 69.

Rose Paul Lawrence , The Italian renaissance of mathematics: Studies on humanists and mathematicians from Petrarch to Galileo (Geneva, 1975), 286; De Pace Anna , Le matematiche e il mondo: Ricerche su un dibattito in Italia nella seconda metà del Cinquecento (Milan, 1993), 11; and Sasaki , op. cit. (ref. 5), 42.

For the reoccurrence of the theme of concordance between Plato and Aristotle during the Renaissance, see De Pace , op. cit. (ref. 15), 263.

This is especially the case in the great success of the so-called mixed sciences, which applied mathematics to the study of nature. The recovery and spread of texts from Archimedes, Hero, and Pappus in mechanics, Euclid, Archimedes, and Appolonius in geometry, Ptolemy and Theodosius in astronomy, and so on, put the mathematicians of the Renaissance in possession of techniques far more sophisticated than those available in the Middle Ages, see Rose , op. cit. (ref. 15), 2.

Piccolomini , op. cit. (ref. 11), f. 69r: “Mathematicas demonstrationes, in primo esse ordine certitudinis … testatur Averroes 2. Metaph. com. 16. super illis verbis Aristotelis…. Quam quidem Averrois authoritatem, omnes fere latini, quos ego viderim, veluti ex antiquioribus, Divus Albertus, Divus Thomas, Marsilius, et Egidius; ex recenterioribus vero, Zimarra, Suessanus, Acciaiolus, et plerique alii … ita interpretati sunt.”/ “Mathematical demonstrations are in the first order of certainty … so Averroës attests on Aristotle's words in 2. Metaph. com. 16…. Indeed, nearly all Latin [commentators], from the ancient ones St Albertus, St Thomas, Marsilius and Egidius; from the recent ones Zimarra, Suessanus, Acciaiolus and many more … have interpreted that authority of Averroës as such”.

For a more extensive treatment of this technical matter, see Jardine Nicholas , “Epistemology of the sciences”, in Schmitt Charles and Skinner Quentin (eds), The Cambridge history of Renaissance philosophy (Cambridge, 1988), 685–712, p. 687.

Mathematical demonstrations do not always provide the true and only cause, since there may be different mathematical demonstrations of the same property, as Piccolomini displays with examples from Euclid, see Piccolomini , op. cit. (ref. 11), ff. 100r–105v. For a more detailed view on Piccolomini's technical arguments, see Mancosu Paolo , Philosophy of mathematics and mathematical practice in the seventeenth century (New York and Oxford, 1996), 12, and Jardine , op. cit. (ref. 19), 693–4.

Piccolomini , op. cit. (ref. 11), f. 95r: “Concludit ergo Proclus ex Platone, quod res ipsae mathematicae, de quibus fiunt demonstrationes, nec omnino in subiecto sensibiles sunt, nec penitus ab ipso liberatae, sed in phantasia ipsa reperiuntur figurae illae mathematicae, habita tamen occasione a quantitatibus in materia sensibili repertis…. Materia ergo harum scientiarum, erit quantum ipsum, hoc modo, ut ita dicam, phantasiatum…”.

Piccolomini refers to quantitas as “omnium sensatissimorum sensatissimum”, ibid., f. 106v.

Feldhay , op. cit. (ref. 5), 83–84.

Van Dyck Maarten , An archaeology of Galileo's science of motion (Gent, 2006), 45.

Piccolomini , op. cit. (ref. 11), f. 97r: “cum non detur inter quanta naturalia, quod trinam dimensionem non recipiat, nec vera aequalitas, nec tandem quaevis figura, circularis, triangularis, aut quaevis alia, exacta penitus & absoluta.”/ “since among natural quantities nothing occurs that does not receive three dimensions, nor true equality is found, nor any figure, circular, triangular, or whatever, that is truly exact and absolute”.

Dyck Van , op. cit. (ref. 24), 46.

De Pace , op. cit. (ref. 15), 71.

For a comprehensive account of imagination and mathematics in Proclus, see Nikulin Dmitri , “Imagination and mathematics in Proclus”, Ancient philosophy, xxviii (2008), 153–72.

Proclus , In primum Euclidis elementorum librum commentarii, p. 1 [I follow the pagination of Friedlein's edition (Leipzig, 1873)].

Proclus. A commentary on the first book of Euclid's Elements, translated, with introduction and notes, by Morrow Glenn R. and with a new foreword by Mueller Ian (Princeton, 1992), p. xviii.

Proclus , op. cit. (ref. 29), 18.

Ibid., 12–18 and 48–57.

Ibid., 12.

See Morrow (transl.), op. cit. (ref. 30), 12–13.

Proclus , op. cit. (ref. 29), 12–13.

Ibid., 13–14.

Ibid., 14–15.

Mueller , op. cit. (ref. 30), p. xxvi.

De Pace , op. cit. (ref. 15), 65.

See Mueller Ian , “Aristotle's doctrine of abstraction in the commentators”, in Sorabji (ed.), op. cit. (ref. 8), 463–80.

E.g. Proclus , op. cit. (ref. 29), 54: “”/ “Therefore, when geometry says something about the circle or its diameter and about its accidental properties, such as tangents, segments and the like, let us not say that it is instructing us about sensible things for it attempts to separate from them — Nor about the Form in the understanding”.

De Pace , op. cit. (ref. 15), 64.

Procli Diadochi Lycii in primum Euclidis Elementorum commentariorum libri iv, transl. by Barozzi Francesco (Padua, 1560).

Van Dyck , op. cit. (ref. 24), 47.

See, for example, Barozzi Francesco , Lectiones in Procli commentarios, 22, ed. by De Pace as an appendix: De Pace, op. cit. (ref. 15), 339–430: “Naturales autem probabiles atque debilissimae sunt, tum propter instabilitatem sensibilis materiae, tum quoniam ab opinione et sensu diiudicantur, qui facile decipi fallique potest.”/ “Natural [objects] are probable and feeble, both due to the instability of sensible matter and to the fact that they are judged by opinion and the senses, which can easily be misled and deceived”.

For example, the first chapter of the Prolegomena seems to be an explicit refutation of the arguments given by Pereira, see Feldhay , op. cit. (ref. 5), 94.

E.g. Kessler , op. cit. (ref. 5); Romano , op. cit. (ref. 1), 158; and Sasaki , op. cit. (ref. 5), 55. The exception here is Peter Dear, to whom we shall return.

Clavius , op. cit. (ref. 3), i, 3: “Quam quidem divisionem … ferme ad verbum ex Proclo iuxta interpretationem Francisci Barocii Patricii Veneti excerptam his subiicere statui.”/ “I chose to annex this division almost literally from Proclus, hereby following the transcribed interpretation of Francesco Barozzi, Venetian nobleman.” See also Romano , op. cit. (ref. 1), 158, and Sasaki , op. cit. (ref. 5), 50–51. For Clavius's knowledge of the Greek editio princeps, see Kessler , op. cit. (ref. 5), 295ff.

For Clavius's acknowledgement of Proclus's authority, compare: Clavius , op. cit. (ref. 3), i, 126: “qui enim negat maiorem esse auctoritatem, meliora argumenta Procli quam Peletarii?”/ “After all, who denies that Proclus's authority is greater, that Proclus's arguments are better than those of Peletarius?” Here Clavius appeals to Proclus in the technical discussion concerning the angulus contactus; see ref. 10.

Clavius , op. cit. (ref. 3), i, 5: “Quoniam disciplinae mathematicae de rebus agunt, quae absque ulla materia sensibili considerantur, quamvis reipsa materiae sint immersae, perspicuum est eas medium inter metaphysicum, & naturalem scientiam obtinere locum, si subiectum earum consideremus, ut recte a Proclo probatur, metaphysices etenim subiectum ab omni est materia seiunctum, & re, & ratione: Physices vero subiectum & re, & ratione materiae sensibili est coniunctum: Unde cum subiectum mathematicarum disciplinarum extra omnem materiam consideretur, quamvis re ipsa in ea reperiatur, liquido constat hoc medium esse inter alia duo. Si vero nobilitas atque praestantia scientiae ex certitudine demonstrationum, quibus utitur, sit iudicanda: Haud dubie mathematicae disciplinae inter caeteras omnes principem habebunt locum”.

Dear , op. cit. (ref. 5), 37.

Ibid., 36–37; Feldhay , op. cit. (ref. 5), 94; and Smolarski , op. cit. (ref. 1).

Clavius , op. cit. (ref. 3), i, 5: “ab Aristotele, veluti rami ex trunco aliquo, exortae, adeo & inter se, & nonnumquam a fonte ipso Aristotle diffident.”/ “From Aristotle, like branches from a trunk, they originate, and to such a degree they disagree with each other and sometimes with Aristotle, the source himself.” This argument is repeated more extensively in Clavius's In Sphaeram Ioannis de Sacro Bosco Commentarius of 1570 (Clavius, op. cit. (ref. 3), iii, 4) and is also found in Justus Velsius's Oratio de mathematicarum disciplinarum vario usu dignitateque (Leuven, 1544); see Vanden Broecke Steven , “Humanism, philosophy and the teaching of Euclid at a northern university: The Oration on the various uses and dignities of the mathematical disciplines, 1544, of Justus Velsius”, Lias, xxv (1998), 43–68, pp. 54–55.

Clavius , op. cit. (ref. 3), i, 5: “Neque enim ad metaphysicam, ut eleganter ostendit Proclus, ulli patet aditus, nisi per mathematicas disciplinas…. Non parum etiam conducunt hae artes ad philosophiam naturalem, moralem, dialecticam, & ad reliquas id genus doctrinas … ut perspicue docet Proclus.”/ “There is not any entrance open to metaphysics, as Proclus elegantly shows, but through the mathematical disciplines…. These arts also contribute to natural, moral, dialectical philosophy, and other doctrines of that sort … as Proclus clearly demonstrates”.

Sasaki , op. cit. (ref. 5), 56.

Barozzi Francesco , Opusculum, in quo una oratio, et duae quaestiones: Altera de certitudine, et altera de medietate mathematicarum, continentur (Padua, 1560), f. 40 r, quoted in Sasaki, op. cit. (ref. 5), 56: “Naturalem esse primam ordine, quoad nos, Mathematicam secundam, Divinam tertiam: Ordine vero naturae divinam esse primam, secundam mathematicam, postremam naturalem, ita ut Mathematica re vera media semper sit inter naturalem, & divinam…”.

Ammonius , In quinque Porphirii Voces Commentarii, f. 6r. See De Pace, op. cit. (ref. 15), 132–3.

Barozzi , op. cit. (ref. 56), f. 38r, quoted in De Pace, op. cit. (ref. 15), 129–30: “Plato et Aristoteles discrepare videntur. Plato enim eiusque sectatores mathematicam naturali praeposuere. Aristoteles autem omnesque peripatetici naturalem mathematica priorem esse dixerunt.”/ “Plato and Aristotle seem to disagree. After all, Plato and his followers preferred mathematics to natural philosophy. Yet Aristotle and all the Peripatetics said that natural philosophy is prior to mathematics”.

Sasaki , op. cit. (ref. 5), 56. For the relation between Barozzi and Clavius, see also Romano , op. cit. (ref. 1), 158.

Barozzi Francesco , Oratio ad philosophiam virtutemque ipsam adhortatoria, habita Patavii in Academia Potentium die 25 Novembri 1557 (Padua, 1558).

Ibid., ff. 9r–9v: “Alia immersa penitus in materia sunt, ita ut ab ea nec secundum esse, neque secundum rationem separari possint, quae naturalia appellantur. Alia autem a materia, et secundum esse, et secundum rationem formalem seiuncta sunt, quae divina esse dicuntur. Alia vero, quae secundum rationem formalem quidem separari, secundum vero esse haud separari a materia sensibili possunt, quae Mathematica nuncupantur”.

Barozzi , op. cit. (ref. 45), 15: “verum animadvertum est quod, dum in materia sensibili haec [sc. the mathematical objects] consistunt, res naturales esse dicuntur; cum autem ab omni materia sensibili separata[e] fuerint, tunc mathematicae essentiae vere appellantur, quae quidem nullibi nisi in phantasia et in anima nostra subsistunt”.

De Pace , op. cit. (ref. 15), 137.

Barozzi , op. cit. (ref. 45), 11: “alia autem (ut Proclus noster passim asseverat, utque in sequentibus melius nobis perspicuum erit) cum secundum esse, tum secundum rationem formalem a materia sensibili separata sunt in intelligibilique materia subsistunt, quae quidem mathematica dicuntur et a mathematica contemplatricis philosophiae parte considerantur.” Compare also with Barozzi, op. cit. (ref. 56), f. 37v: “quae autem mathematica esse dicuntur, et a materia sensibili quidem omino separata, intelligibili autem inseparabilia sunt, medium inter haec locum sibi vendicarunt.”/ “Those which are called mathematical objects, and are indeed fully separated from sensible matter, but are inseparable from intelligible [matter], have claimed the place intermediate between them” (my emphasis).

De Pace , op. cit. (ref. 15), 138.

Cf. Barozzi , Lettera di F. Barozzi ad Alvise Buonrizzo, f. 13r, quoted in De Pace, op. cit. (ref. 15): “non vol stare sopra la scorza delle parole, ma penetrar piu dentro alla midolla della cosa, ritrovera, che tutti duo questi filosofi sono stati d'accordo ma differenti solamente in parole”.

“Unde cum subiectum mathematicarum disciplinarum extra omnem materiam consideretur, quamvis re ipsa in ea reperiatur …” should be translated as “then, as the subject of the mathematical disciplines is considered free from all matter, although in reality it [sc. the subject] is found in it [sc. matter]”, this in contradistinction to the translation “although it [i.e. matter] is found in the thing itself” ( Feldhay Rivka , Galileo and the Church: Political inquisition or critical dialogue? (Cambridge, 1995), 217), “even though it [matter] is found in the thing itself” (Feldhay, op. cit. (ref. 5), 96), “although it [i.e., matter] is found in the thing itself” (Dear, op. cit. (ref. 5), 37), and “although matter is really found in it” ( Sasaki , op. cit. (ref. 5), 56).

Feldhay , op. cit. (ref. 67), 216. For the ‘abstract’ character of mathematical objects, see: Clavius, op. cit. (ref. 3), i, 5: “Quam ob rem, antequam a rebus physicis, quae materiae sensibus obnoxiae sunt coniunctae, ad res metaphysicas, quae sunt ab eadem maxime avulsae, intellectus ascendat, necesse est, ne harum claritate offundatur, prius eum assuefieri rebus minus abstractis, quales a mathematicis considerantur, ut facilius illas possit comprehendere.”/ “Therefore, before the intellect ascends from physical objects, which are joined to that matter subject to the senses, to metaphysical entities, which are the most torn away from it [sc. sensible matter], it is necessary, in order that it would not be damaged by the latter's clarity, that it first grows familiar with things that are less abstract, which are considered by mathematics, so it is able to better understand them [sc. metaphysical objects]” (my emphasis).

Aristotle , Metaphysics, E, I, 1025b26–1026a16.

Dear , op. cit. (ref. 5), 37.

Iamblichus, Syrianus, and Proclus put forward the projectionist point of view, see Mueller , op. cit. (ref. 40).

Alexander of Aphrodisias's interpretation seems to oscillate between a constructive abstractionist-reading and a view at first glance similar to Clavius's. See Cattanei Elisabetta , “Gli enti matematici ‘per astrazione’ secondo Alessandro di Afrodisia e lo Pseudo-Alessandro”, in Movia Giancarlo (ed.), Alessandro di Afrodisia e la “Metafisica” di Aristotele (Milan, 2003), 255–76, pp. 260–72. Nevertheless, I did not find any evidence of Clavius's having been acquainted first hand with Alexander's interpretation.

Mueller , op. cit. (ref. 40), 480. See also Cattanei, op. cit. (ref. 72), for a critical reading of Mueller's rather rigid division.

Pereira Benedictus , De communibus omnium rerum naturalium principiis (Rome, 1576), ff. 374–5.

Aristotle , op. cit. (ref. 69), M, 3, 1077b24–30: [25] [30] . My translation is based on the translation of Jonathan Barnes (“Metaphysics”, in Barnes Jonathan (ed.), The Cambridge companion to Aristotle (Cambridge, 1995), 66–108, p. 85), with some modifications.

Lear Jonathan , “Aristotle's philosophy of mathematics”, The philosophical review, xci (1982), 161–92, pp. 168–9.

Aristotle , op. cit. (ref. 69), M, 3, 1077b18–1078a31.

Barnes , op. cit. (ref. 75), 85.

Ibid., 86.

Hussey Edward , “Aristotle on mathematical objects”, in Mueller Ian (ed.), ПEPI TΩN MAHMATΩN (Edmonton, 1991), 105–34, p. 116.

From this perspective, one could consider Barozzi's “derivative being” as “a being assumed as a helpful fiction”.

Aristotle , op. cit. (ref. 69), M, 3, 1078a28.

Lear , op. cit. (ref. 76), 175.

Aristotle , Physics, B, 2, 193b23.

Clavius , op. cit. (ref. 3), i, 5: “rebus minus abstractis, quales a mathematicis considerantur.”/ “less abstract entities which are considered by mathematics”.

Clavius, op. cit. (ref. 3), i, 5: “Cum igitur disciplinae mathematicae veritatem adeo expectant, adamant, excolantque, ut non solum nihil, quod sit falsum, verum etiam nihil, quod tantum probabile existat, nihil denique admittant, quod certissimis demonstrationibus non confirment, corroborentque, dubium esse non potest, quin eis primus locus inter alias scientias omnes sit concedendus”.

See Mueller Ian , “Aristotle on geometrical objects”, in Barnes Jonathan Schofield Malcolm and Sorabji Richard (eds), Articles on Aristotle. 3: Metaphysics (London, 1979), 96–107; Lear , op. cit. (ref. 76); Mueller , op. cit. (ref. 40); Hussey , op. cit. (ref. 80); Barnes , op. cit. (ref. 75); and Cleary John , Aristotle and mathematics: Aporetic method in cosmology and metaphysics (Leiden, New York and Cologne, 1995).

Feldhay , op. cit. (ref. 5), 97.

See, for example, Clavius , op. cit. (ref. 3), i, 5–6, the chapter “Various uses of the mathematical disciplines”.

See Biagioli Mario , “The social status of Italian mathematicians, 1450–1600”, History of science, xxvii (1989), 41–95, p. 54.