On Ian Mueller on Aristotle on Abstraction, intelligible matter, and Geometrical Objects and Newton's 'De Gravitatione'
In an influential (1970) article, "Aristotle on Geometric Objects," my late teacher, Ian Mueller, noted that "Aristotle refers fairly frequently to mathematical objects as abstractions (εξ άφοαρέσεω$, εν αφαιρέσει, δι* αφαιρέσεως)." (p. 159) After running through a number of passages, Mueller concludes that for Aristotle, "abstracting involves eliminating something from consideration." (p. 160) An abstraction is a stripping away (of things to be bracketed) to create an object of enquiry. Crucially, and Mueller notes, "This is not a matter of collecting particulars and somehow arriving at a general idea, although abstraction is facilitated by seeing a number of different individuals." (emphasis added). That is, as Mueller helpfully remarks in an accompanying footnote (with a reference to Locke's Essay 2.11.9), collecting particulars in order to generate a general idea is the "notion of abstraction...which becomes crucial in British empiricism."+ So, on this account, for Aristotle abstraction is an epistemic tool to be used by the geometrician.
Now, in my own work on Newton, I have suggested that by “abstraction” Newton "means—in accord with his Scholastic and Platonic sources—something more akin to “focusing on some isolated qualities found within the empirical world without getting distracted by other empirical qualities." In context, I cite Mary Domski (2012) "Newton and proclus: geometry, imagination, and knowing space." The Southern Journal of Philosophy. In another paper on Newton, I claim something similar about abstraction, "which traditionally (in Scholasticism and Platonism) is used to isolate a particular feature of nature and make it amenable to analysis. Newton also uses this notion of “abstraction” in the scholium to the definitions in the Principia (“instead of absolute places and motions we use relatives ones…in [natural] philosophy abstraction from the senses is required” (Newton 1999: 410; see also De Gravitatione et Aequipondio Fluidorum, a manuscript unpublished in Newton’s time (generally known as “DeGrav,”) “we have an exceptionally clear idea of extension by abstracting the dispositions and properties of a body” Newton 2004: 22) In addition to Domski, I cite Gorham (2007) “Descartes on Time and Duration,” Early Science and Medicine. So, I have long held, without fully realizing it, that Newton uses Aristotle's account of abstraction, if Mueller is right, in his own work.*
Stephen Menn alerted me to the fact that the previous paragraph is a bit misleading. The medievals clearly think (I am now reporting Menn) of abstraction as dematerialization (which may come in stages and degrees—e.g. abstracting first from matter, then from the "conditions of matter"): it turns sensible species into intelligible species. This is not what I am attributing to Newton (or Buffon). Rather, what Newton’s use of ‘abstraction’ has in common with the earlier use is a certain epistemically useful bracketing/isolating operation.
Now, I went back to Mueller's essay because of the lovely summary of it in Stephen Menn's (2010) "In Memoriam" that I referenced last week while discussing Burnyeat's response to Strauss. And, in particular, I was very struck by Menn's treatment of Mueller's account of the following passage in Aristotle's Metaphysics. I quote from Mueller (1970: 164):
The mathematician theorizes about abstractions, for he theorizes having removed all sensibles such as weight and lightness, hardness and its opposite, heat and cold, and the other sensible opposites. He leaves only the quantitative and continuous in one, two, or three [dimensions] and the properties of these as quantitative and continuous. [Metaphysics, 1061a28—35]
In wider context [1061b] Aristotle is explaining what the philosopher does by way of analogy of what the mathematician does. So Aristotle takes what he is saying as relatively clear (and uncontroversial). Mueller uses this passage to illuminate another passage in Aristotle. Here's what Mueller says:
To say that the mathematician studies a man as solid is not to say that he studies a man at all. Rather, it is to say that he studies what is quantitative and continuous in three dimensions. And the mathematician comes to understand the quantitative and continuous by abstracting—i.e, ignoring—all the sensible properties of some sensible substance such as a man. There is, then, at least an initial plausibility in supposing Aristotle to have entertained a conception of mathematical objects, not as matterless properties, but as substance-like individuals with a special matter—intelligible matter. (p. 164)
I don't expect this to be fully clear because Mueller himself devotes the rest of the paper to making it more accessible in part by drawing on Philoponus. Going back to Mueller's (1970) essay was quite instructive because it made me see that Mueller was always willing to learn from ancient commentators, and cite them as authorities for his own readings--see also the very next paragraph. (Menn also emphasizes this in his In Memoriam, but I had not realized how entrenched this was in Mueller's practice from very early one.) At one point, Mueller summarizes his own explication as follows:
"Aristotle seems to have the idea of the purely dimensional underlying other properties. In part this is the idea of the three-dimensional underlying sensible properties in the physical world. But for Aristotle there is little if any difference between this idea and that of the one-, two-, or three-dimensional underlying geometric properties, which he calls intelligible matter." And then Mueller notes, "The author of the commentary on the Metaphysics which is attributed to Alexander of Aphrodisias refers to the intelligible matter in these passages as extension, and he is clearly right. For it is the extendedness of geometric objects, their continuity in one, two, or three dimensions, which makes them divisible. " (p. 166)
After dealing with some textual difficulties, Mueller notes
If Aristotle held the conception of geometric objects which I have developed here, it is easy to see how their exactitude is explained. For by abstraction one eliminates all sensible characteristics and arrives at the idea of pure extension. Pure extension does not seem to be sensible in the way that triangularity is, nor is it completely undifferentiated or purely potential in the way that prime matter seems to be. We cannot see a thing as just extended but only as extended so and so much with a certain shape. Simple extendedness we must grasp rationally. Geometric properties are imposed on this intelligible matter, but these properties are not the approximate properties of sensible substances precisely because they are imposed upon intelligible matter. (168; emphasis added)
Mueller then rewords the insight as follows, "Aristotle's treatment of extension as a kind of underlying stuff and as a very abstract notion, the genus of mathematical objects." (p. 171; Mueller also thinks that if we leave aside geometric points, the “distinction between genus and property is seen to be exactly the distinction between intelligible matter and form in the Metaphysics.” (p. 168))
So, what I suggest is that this is remarkably close to how Newton understands extension in De Gravitatione (a text that has been subject of intense scrutiny among philosophical Newton scholars, but is generally now thought to be a relatively early, pre-Principia work. [See here for a polemical version of that claim.]) In particular, what I have in mind are the following characteristics: (i) abstraction is used to bracket the topic of enquiry; this facilitates (ii) the rational or intuitive grasp of pure extension; (iii) pure extension is a special kind of intelligible matter; (iv) geometric properties have a special relationship with extension. Of these, above I have already shown that Newton holds (i). That is, I claim that both Aristotle in the Metaphysics (as interpreted by Mueller) and Newton in De Grav have an implied account of extension that shares non-trivial structural similarities (that is, i-iv).
For, after rejecting Descartes' account of extension, but keeping some of his terminology, Newton writes, "we have an exceptionally clear idea of extension by abstracting the dispositions and properties of a body so that there remains only the uniform and unlimited stretching out of space in length, breadth and depth." That we have an exceptionally clear idea of extension by abstracting is just to say that we grasp extension rationally through the Aristotelian process of abstraction. So, Newton holds something like (ii). In fact, Newton infers all kinds of properties of extension based on this clear idea (none of which fit well with the more familiar geometric empiricism of his later work).
And it turns that what we would call the content of this clear idea of extension just is, for Newton. a kind of underlying stuff. It's very important to Newton that space "has its own manner of existing which is proper to it and which fits neither substances nor accidents." (Newton's Philosophical Writings, revised edition, edited by Andrew Janiak, p. 35) In fact, for Newton "it is something more than an accident, and approaches more nearly to the nature of substance." (Janiak, p. 36; I don't mean to suggest that Aristotle and Newton have the same notion of substance.)
In fact, Newton writes:
there are everywhere all kinds of figures, everywhere spheres, cubes, triangles, straight lines, everywhere circular, elliptical, parabolical, and all other kinds of figures, and those of all shapes and sizes, even though they are not disclosed to sight. For the delineation of any material figure is not a new production of that figure with respect to space, but only a corporeal representation of it, so that what was formerly insensible in space now appears before the senses. (Janiak, p. 37, emphasis added)
So, it seems fair to say that while Newton isn't using Aristotle's categories and concepts, he is describing something like (iii), extension is like intelligible matter. And that, indeed Newton also holds (iv): geometric properties have a special relationship with extension.
Of course, in (iv) we also see a crucial difference between Mueller's Aristotle and Newton: that for Aristotle geometric figures are imposed on extension. (Menn thinks Mueller's right about this.) Whereas for Newton geometric figures inhere in extension in some sense and our delineation of them is not so much an imposition as a corporeal representation of that which was already there. (Newton explain this idea by the example of introducing color into a water and the color makes existing shapes visible.) So, I am not claiming that Newton and Aristotle have the exact same view of extension as a kind of intelligible matter. But rather that their positions have more than a family resemblance; they are structurally similar. (That I would think this is, alas, not wholly a coincidence of going back to Mueller because I already had an interpretive tendency, as my sometime co-author Zvi Biener noted to me when we we're discussing this hunch mine, to interpret Newton as a weird, Platonizing scholastic in De Grav.)
In fact, in his "In Memoriam," Menn (McGill) thinks it is not wholly obvious what it could mean to impose figures on extension. (p. 201) Menn, who is himself one of the leading contemporary experts on Aristotle's Metaphysics, proposes that for Aristotle, "geometric objects exist potentially within geometrical matter" (p. 201, or extension; emphasis in Menn.) Again, this is not quite the position of Newton, but it is awfully close. Newton seems to think geometrical objects exist within geometrical matter (and presumably visible to God), but only become visible to us by way of some operation (adding color, moving bodies, using measuring devices, etc).**
To the best of my knowledge Newton never read the relevant passages in Aristotle's Metaphysics. And it's possible he merely arrived at it by way of refuting Descartes (who himself is critically responding to various Scholastics) and creative reflection on Euclid (who is mentioned in DeGrav). But in correspondence, Niccolo Guicciardini suggested, not implausibly, that Barrow (Newton’s teacher) might well have been familiar with these passages. Anyway, it strikes me as useful topic for further research. In addition, what I hope to have made clear is that it's a mistake to read pre-Principia Newton as a proto-Lockean empiricist.
*I don't mean to suggest that after the rise of empiricism, this notion of abstraction fell into complete disuse. It's pretty clearly present in Buffon's Preliminary discourse. See Lyon, John, Buffon, and Georges Louis LeClerq. "The'Initial Discourse'to Buffon's" Histoire Naturelle": The First Complete English Translation." Journal of the History of Biology (1976): 176.
+I never talked about this 1980 article with Ian (who was reticent about pushing his own work on his students). I have seen no evidence that Howard Stein (one of my other teachers and a longtime colleague of Ian) drew on Ian's work on Aristotle in his interpretation of this material in Newton.
**By emphasizing that geometrical theorems are really about what could exist, Menn pushes the Aristotelian position to more familiar what I take to be the standard terrain. It's not controversial, I think, to claim that for Aristotle geometry is about possible objects of existence ([see 1078a28-31]—Menn alerted me to the fact that since Lear’s book it turns out that it is controversial, but I am confident this is right).)