Are mathematical suppositions of physical theories determined uniquely according to Aristotle and Plato?

Question

Does mathematics apply to physics in one way or multiple ways? What do Aristotle and Plato think?

• It would seem that Aristotle thinks mathematics can be applied to physics in one way only because, for him, mathematics is abstracted from physical objects, and isn't abstraction unique? Cf. this. However, In II De cælo lect. 17 n. 451, St. Thomas Aquinas's commentary on Aristotle's On the Heavens bk. 2, portrays Aristotle as thinking that mathematics could be applied to physics (or mathematicals abstracted from physical beings) in many ways. St. Thomas writes that astronomers have

  tried to reduce the irregularities [of the planets] to a right order by assigning diverse motions to the planets…Yet it is not necessary that the various suppositions which they hit upon be true—for although these suppositions save the appearances, we are nevertheless not obliged to say that these suppositions are true, because perhaps there is some other way men have not yet grasped by which the things which appear as to the stars are saved. Aristotle nevertheless uses suppositions of this kind, in what regards the quality of the motions, as true.

  The "suppositions" mentioned are mathematical-physical theories, like Ptolemy's theory or, more recently, Copernicus's.

  ^{Ampère also thought physical theories, expressed in terms of mathematics, are uniquely (uniquement) deduced from experiment/experience, evidenced by the title of his famous electrodynamics work: Mémoire sur la théorie mathématique des phénomènes électrodynamiques uniquement déduite de l’expérience (Memoir on the mathematical theory of electrodynamic phenomena uniquely deduced from experiment).}

• Plato would seem to think mathematics can be applied to physics in multiple ways because he invented the phrase "to save the phenomena" or "σώζειν τὰ φαινόμενα" to describe how various theories can "save" or "account for" the same sensible appearances (cf. scientific formalism).

Look at this thread and these comments for background. This question also relates to the debate on "Structural Realism vs. Scientific Formalism" (=instrumentalism?).

Answer 1

I am not sure that saving phenomena can be used to argue that Plato and Aristotle admitted or did not admit that different suppositions might be consistent with them. At the time Plato posed the problem of reconciling apparent motions of planets with the Pythagorean ideal of uniform circular motions not only wasn't Ptolemy's system around, but no such theory existed at all. It wasn't clear that it could be done, and there definitely was no notion of mathematized physical theories, or of the scientific method for verifying them. It took the genius of Eudoxus to produce a model that accounted for backward planetary motions at least qualitatively, in response to Plato's challenge. It was extremely clever and counterintuitive, this cleverness might have even suggested to many that it was the only right track.

This doesn't tell us that Plato believed that multiple suppositions could be used for the task. It's likely that he believed it possible on philosophical grounds, but also believed that there was a unique "true" way to do it, and that discovering it would assert the glory of the ideal. The same goes for Aristotle, at his time the only offer on the table was still a Eudoxian type model, only with many more spheres than originally, to account for more details. Aristarchus's heliocentrism couldn't account for the apparent absence of the parallax, and so didn't "save the phenomena". Who knows what Aristotle would have thought if he was presented with Ptolemy's system as an alternative. Church theologists presented by Copernicus with a similar situation chose to admit multiple suppositions rather than give up their preferred cosmology.

Also, there is a difference between abstraction, which he may have believed was unique, and trying to guess at the unseen from the visible, which by common daily experiences can't be done uniquely. As Aristotle says in Metsaphysics 1010b:"And as concerning reality, that not every appearance is real, we shall say, first, that indeed the perception, at least of the proper object of a sense, is not false, but the impression we get of it is not the same as the perception". When Aristotle was inferring his theory of natural and forced motions from pulled carts and falling rock and feather he was abstracting, but when Eudoxus and Calippus were attaching planets to homocentric spheres they were just speculating about a mechanism behind the visible motions. It's unlikely that Aristotle believed that specific arrangements of inclination angles and rotation speeds they came up with, which were still refined and contested in his time, were uniquely suitable. Aristotle made his own additions to the arrangement to connect spheres for different planets into a single chain driven by his unmoved mover, which required adding counter-spheres in between to prevent planet specific motions from being transferred. In other words, he was aware that mathematics can be altered to fit a theoretical goal without affecting the phenomena.

We know that soon after Aristotle's time Epicurus criticized Eudoxean models exactly by pointing out underdetermination of intrinsic properties by observable ones, see Sedley's Epicurus and the Mathematicians of Cyzicus, presumably such criticisms were familiar to him already. Here is Epicurus arguing that Eudoxean planetaria, built based on "mathematical suppositions", reflected the wrong properties:

"All that this leaves is a pretence and a diehard dogma that the indications on the instrument create an analogy that corresponds with what we see in the heavens. For our friend must, it seems to me, make the distinction: (a) that when he argues about the cosmos and what we see in the cosmos he is arguing about a certain image arising from certain accidental properties of things passed through the medium of vision into a thought-process or into a memory-process permanently preserved by the mind itself, quantities, qualities; but (b) that when he argues about the indications on his instrument he is arguing about the intrinsic properties of an object".

In other words, intrinsic properties of planetaria reflected only accidental properties of the cosmos, so according to Epicurus not only are "mathematical suppositions" for saving astronomical phenomena not unique, but they do not even stand a chance of matching the reality. It is likely that Aristotle would not go that far.