Abstract
In Posterior Analytics 1.13, Aristotle introduced a distinction between two kinds of demonstrations: of the fact (quia), and of the reasoned fact (propter quid). Both demonstrations take a syllogistic form, in which the middle term links either two facts (in the case of quia demonstrations) or a proximate cause and a fact (in the case of propter quid demonstrations). While Aristotle stated that all the terms of one demonstration must be taken from within the same subject matter, he admitted some exceptions in which the fact and the reasoned fact are instantiated by terms from different sciences, as when mathematics provides the reason and another science the empirical fact. This was the methodological foundation of the “mixed sciences”, a subject of varying interpretations in the thirteenth century. Roger Bacon (C. 1220–1292), adhering to Robert Grosseteste’s (C. 1168–1563) commentary on Posterior Analytics, presented a unique interpretation of this exception. He replaced propositional demonstrations with geometrical considerations and diagrams, thus producing geometrical arguments for theorems in natural philosophy. I focus on Bacon’s propter quid arguments, as applied in three case studies: (1) the heat caused by a body moving to its natural place; (2) the motion of the scale; (3) and the contraction of water. Based on an analysis of these demonstrations, I argue that Bacon’s interpretation of propter quid demonstration reflects his application of a scientific methodology that imbues geometrical objects with causal power over material bodies.
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Notes
Thomas Aquinas was not aware of this fact when he commented on the Posterior Analytics, since the Latin version of the Physics he used corrupted the key Physics II 2 passage, rendering it ‘the sciences which are more physical than mathematical.’ See Aquinas’ commentary on Physics, Book II, Lecture 3. For related discussions, See; Di Liscia (2009), p. 294; Distelzweig (2013), p. 90.
Sandoz refers to On the Heavens 290b1 for a claim about the permanence of celestial bodies, and to On the Soul II. 7–8 for a description of light and sound as incorporeal forms distinguished from sight and hearing.
This could explain why in On the Heavens Aristotle seems to breach his metabasis prohibition, undertaking a natural investigation of heaven, a breach indicated by Distelzweig (2013).
Note that the thirteenth-century interpretation—just like any other interpretation—does not necessarily conform with Aristotle’s original intention. Hussey (2002), Lear (1982), and Lennox (1986) argue that in Aristotle’s philosophy of mathematics, real mathematical properties are present within material terrestrial substances (but do not exhaust their essences). They deduce from this analysis a possible use of mathematical middle terms in natural philosophy. Lear, for example, argues that there is “evidence that he [Aristotle] thought that even in the sublunary world physical objects could perfectly instantiate geometrical properties” (1982, p. 177), and Lennox states that “There seems to be nothing in Galileo’s appeal in his new science to mathematical demonstration that Aristotle would not wholeheartedly endorse” (1986, p. 51). This, however, was not the accepted thirteenth-century interpretation.
van Dyke (2009) argues that this attempt was successful.
The broadening of the domain of optics in the Latin world owes its origin to Robert Kilwardby’s De ortu scientiarum. Kilwardby (C. 1215–1279) read Alhazan’s De aspectibus and defined the subject of optics as the “way of seeing” (modus videndi), and its end as the knowledge of the rules of vision and the perfection of our vision through such knowledge. He observed that although it is subalternate to geometry, perspectiva is more truly a natural science (De ortu scientiarum, pp. 36–50).
“Et necesse est omnia sciri per hanc scientiam, quia omnes actiones rerum fiunt secundum specierum et virtutum multiplicationem ab agentibus hujus mundi in materias patientes; et leges hujusmodi multiplicationum non sciuntur nisi a perspectiva, nec alibi sunt traditae”.
“res mathematica et naturalis non differant nisi penes consideracionem et non secundum esse et secundum rem ut Aristoteles dicit et demonstratum est in Methaphisicis, manifestum est quod id accidit mathematice et naturaliter secundum esse in rebus.”.
The diagram is taken from Vat. Lat. 4086, fol. 33r. This is a fourteenth-century manuscript, closely resembling (but not a copy of) Cotton Jul. D. v, which is the oldest manuscript of the Opus majus, held by the British Museum and partly destroyed by fire (Little, 1914, p. 379).
Jardine and Jardine (2010, p. 394) address the question of the purpose of a diagram and its relation to the text: is it mainly illustrative, or is it meant to convey information about geometrical configurations not explicitly present in the text? This question deserves, indeed, a separate discussion. However, it seems to me that Bacon would have argued that his demonstrations could not work without these diagrams, for to be convinced, one must see with his own eyes the difference between the straight and the diagonal, the vertical components, and the two arcs cut by the same cord (see the next two demonstrations).
Fisher and Unguru (1971, p. 360) note the tight link between mathematics and experience in Bacon. By drawing figures and counting, mathematics contains intrinsic certifying experiences. These ‘perceptible experiments’ make everything clear to the senses (rather than to reason) and preclude all possibility of doubting.
As Saito (2012) notes, it is often the case that medieval diagrams are schematic, oversimplified, and inaccurate. To communicate general mathematical relations, this is enough.
What we know about Jordanus de Nemore amounts to conjectures only. He taught in Paris (Hoyrup, 1988, p. 346), and was a capable mathematician, who composed treatises on algebra, arithmetic, geometry, and statics, all compiled before 1260. The works on statics ascribed to him exemplify a union of a philosophical Aristotelian approach with a more rigorous mathematical tradition of Euclid and Archimedes (Moody & Clagett, 1952, pp. 6–7). Some of the texts ascribed to him were likely written by members of his circle, which included, among others, Richard of Fournival (1201–?1260), Gerard of Brussels (fl. thirteenth century), and Campanus of Novara (C. 1220–1296). It has been suggested that Bacon and Jordanus may have met in the 1240s in Paris. On this suggestion, Bacon may have had close contact with Jordanus and his circle although he was not an actual member of this circle (Hoyrup, 1988, pp. 346–351). To be sure, Bacon provided several references to books associated with Jordanus, such as Liber philotegni, Liber de triangulis Jordani, and Liber Jordani de ponderibus, while expressing appreciation for his mathematical capabilities.
The diagram is taken from Vat. Lat. 4086, fol. 34r.
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Acknowledgements
I am grateful to Daniel Di Liscia and Giora Hon for their valuable comments on this paper, and to Jeremiah Hackett for his help with finding the manuscripts. This reaserch was supported by the Israel Science Foundation (Grant No. 2773/21).
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Kedar, Y. Propter quid demonstrations: Roger Bacon on geometrical causes in natural philosophy. Synthese 203, 18 (2024). https://doi.org/10.1007/s11229-023-04433-7
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DOI: https://doi.org/10.1007/s11229-023-04433-7