Abstract
In the history of science perhaps the most influential Aristotelian division was that between mathematics and physics. From our modern perspective this seems like an unfortunate deviation from the Platonic unification of the two disciplines, which guided Kepler and Galileo towards the modern scientific revolution. By contrast, Aristotle’s sharp distinction between the disciplines seems to have led to a barren scholasticism in physics, together with an arid instrumentalism in Ptolemaic astronomy. On the positive side, however, astronomy was liberated from commonsense realism for the conceptual experiments of Aristarchus of Samos, whose heliocentric hypothesis was not adopted by later astronomers because it departed so much from the ancient cosmological consensus. It was only in the time of Newton that convincing physical arguments were able to overcome the legitimate objections against heliocentrism, which had looked like a mathematical hypothesis with no physical meaning.
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Notes
See W. Wians, “Scientific Examples in the Posterior Analytics” in Wians, ed.,Aristotle’s Philosophical Development. Problems and Prospects(Lanham: University of America Press, 1996 ), 131–150.
But see Posterior Analytics II.11 (94a28–31) where Aristotle puts a Euclidean proposition(Elements 1II.31) into syllogistic format.Aristotle’s Prior and Posterior Analytics ed. W.D. Ross (Oxford: Oxford University Press, 1949).[APst.]
Cf APst.71a15, 72a21, 75b2–5, 84a11–26, 88626, 90b33, 93b24.
Met.995b 13–18:Aristotle’s Metaphysics.trans. H.G. Apostle (Grinnell: Peripatetic Press, 1979).[Met].The parallel aporia in Met. XI. 1(1059a37 ff.) goes as follows: “In general, there is this problem, whether the science we now seek is concerned at all with sensible substances or not, but rather with some other substances. If with others, it would be either with the Forms or with Mathematical Objects.”
Cf. Alexander,in Metaph175.14–176.16. Syrianus(in Metaph.2.15 ff.) goes one better by combining three aporiaitogether, though he quotes and discusses each one separately.
Met. 996a12–15.
Perhaps it is by way of reaction against this assumption that the Neoplatonic commentator, Syrianus(in Metaph.12.25 ff.) adopts the strategy of simply asserting the Platonic order of priorities, beginning with the Forms of mathematical objects and concluding with their appearance in sensible things.
Thus Aristotle’s methodological attitude differs fundamentally from that of the ancient Sceptics who used the aporetic method as an end in itself within their philosophical inquiries.
In terms of his method, therefore, Aristotle owes something to “father Parmenides,” but his greatest methodological debt is to Plato’sParmenidesand its deliberately constructed antinomies. AtParm. 136c5 Parmenides recommends the gymnastic exercise of constructing antinomies as a way of seeing the truth more completely(re eíds)and better(Kvp(&s). See M. Schofield, “The Antinomies of Plato’s Parmenides,”Classical Quarterly27 (1977): 140–158.
In order to signpost this view as it is represented by Aristotle, I adopt the convention of italicizing the `in’ as follows: “chrw(133) mathematical objects in sensible things.”
In addition, Aristotle connects the argument “from the sciences” with the Parmenidean dictum that it is impossible to think or inquire about not-being; cf.Cael. III.1, 298b17–25.
Cf.In Metaph.725.4.
Met.1077a9–14.
Cf. T.L. Heath,The Thirteen Books of Euclid’s Elements.3 vols (Cambridge: Cambridge University Press, 1925), ii, 112 ff.
Cf. Heath, 138; W.R. Knorr The Evolution of the Euclidean Elements(Dordrecht-Boston: Reidel, 1975); D.R. Lachtennan The Ethics of Geometry. A Genealogy of Modernity(New York & London: Routledge, 1989), Ch. 2.
Met.1077a36-b11.
Cf. Cleary,Aristotle on the Many Senses of Priority (Carbondale: Southem Illinois University Press, 1988) for discussion of the different senses of priority in Aristotle.
ix npoo0éaews yap iti.) Aeuxw ò A.suxòs âvfpo mtos 7.éyetas — Met.1077b11.
Cf. Ps.-Alexander In Metaph. 733.23–24 & Syrianus, In Metaph. 93.22 ff.
D. D. Moukanos,Ontologie der ‘Mathematika’ in der Metaphysik des Aristoteles(Athens: Potamitis Press, 1981), 24 ff., claims that the conclusion of Mu 2–3 is that mathematics is about abstract objects, which exist through the separating reflection of mathematicians, but he fails to explain why the terminology of abstraction is conspicuously absent from Mu 3. For my explanation, see Cleary, “On the Terminology of Abstraction in Aristotle,”Phronesis30 (1985): 13–47.
Met.1077b17–22.
Syrianus(In Metaph.95.13–17) expresses some surprise at what he sees as Aristotle’s attempt to find a parallel in ontological status between universals and mathematical objects, since the former are logical entities belonging in the soul, whereas the latter are in sensibles and are also mental abstractions[étrt vol“ at 5 i’ cripotíoats] from sensibles. But such remarks cannot be taken to represent Aristotle’s views accurately, and they may even suggest that abstractionism was a product of commentators like Alexander, who proposed it as the official Aristotelian doctrine that was later opposed by Syrianus.
In his logical analysis of what he calls Aristotle’s theory of reduplication, Allan Back,On Reduplication. Logical Theories of Qualification (Leiden: Brill, 1996) points out that a qua proposition is actually a condensed demonstrative syllogism in which thequaterm functions as a middle term and as a cause; e.g., an isosceles triangle has this property because it is a triangle. He also argues that the qua phrase is attached to the predicate and does not change the reference of the subject term, which he takes to be a particular existent like this bronze triangle. He has objected (in personal communication) that my approach of making qua propositions fix our attention on the primary subject has the consequence of changing the reference of the subject term to some kind of Platonic entities about which it would be difficult to verify any knowledge claims. But I respond that the distinction between natural and logical priority in Aristotle separates the de ditto question of the primary logical subject from the de re question about the basic subject as substance.
Met.1077b22–30.
Met.1078a21–31.
F.A.J. de Haas, “Geometrical Objects in Aristotle,” (unpublished mss.) finds two major types of interpretations within the range given by scholars like I. Mueller, “Aristotle and the Quadrature of the Circle” in N. Kretzmann, ed.,Infinity and Continuity in Ancient and Medieval Thought(Ithaca: Cornell University Press, 1982), 146–64; J. Lear, “Aristotelian Infinity,”Proceedings of the Aristotelian Society1979–80: 188210; J. Barnes, “Aristotelian Arithmetic”,Revue de Philosophie Ancienne3 (1985): 97–133; M. Mignucci, “Aristotle’s Arithmetic”, in Graeser, ed.,Mathematics and Metaphysics in Aristotle, 175–211; J. Armas, “Die Gegenstände der Mathematik bei Aristoteles,” Graeser,Mathematics and Metaphysics in Aristotle, 131–147; Modrak, “Aristotle on the Difference between Mathematics and Physics and First Philosophy” in Penner & Kraut, eds.,Nature,Knowledge,and Virtue(Edmonton: Academic Printing, 1989), 121–139; E. Hussey, “Aristotle on Mathematical Objects” in Mueller, ed.,Peri Ton Mathematon (Apeiron24.4) (Edmonton: Academic Printing, 1991 ), 105–133.
Hussey (cited above) recognizes that Aristotle’s discussion in Metaphysics Mu 3 is incomplete on its own, but he fails to see the broader aporetic context within which one should understand the solutions given there. Although Barnes and Annas (both cited above) insist that the solution must be seen exclusively in terms of the inquiry at Mu 1–3, yet that context is surely too narrow.
Et its ròpi) Icextdptapévov Beítlxtapíoas—Met.1078a21–22.
If one accepts Frege’s analysis of number as a second-order property, one might still object that Aristotle is simply wrong to think of it as a first-order property of sensible things. But that would be a different objection from the one that I describe as missing the point.
Cf.APst.76a31–36, 76b3–11, 92b15–16, 93b21–28.
Cf. Met. 1025b3–18, 1059b14–21, Phy. 184b25–185a5.
Cf. P.F. Strawson, Individuals(London: Methuen, 1959), 167 ff.
In fact, R. Netz,The Shaping of Deduction in Greek Mathematics(Cambridge: Cambridge University Press, 1999), 214 claims that Aristotle should be regarded as a logicist in the modern sense but I think that Netz fails to take into account the different problem-situations that prevailed in the widely separated historical eras.
Lear, Aristotle: The Desire to Understand (Cambridge: Cambridge University Press,1988), 68n34.
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Cleary, J.J. (2002). Abstracting Aristotle’s Philosophy of Mathematics. In: Babich, B.E. (eds) Hermeneutic Philosophy of Science, Van Gogh’s Eyes, and God. Boston Studies in the Philosophy and History of Science, vol 225. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1767-0_13
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