17 Dec 2008

Ideas Flowing through Boyle, Spinoza, and Deleuze: The Compatibility of Infinite Divisibility with Absolutely Simple Ultimate Parts

by Corry Shores
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[Text below exclusively is quotation, up to my commentary.]

Robert Boyle, The History of Fluidity and Firmness. The Fifth Part. Of Fluidity:

Section I:

Whether philosophers might not have done better in making fluidity and firmness rather states than qualities of bodies, we will not now examine. But under which soever of the two notions we look upon them, it is manifest enough, that they are to be reckoned amongst the most general affections of the conventions or associations of several particles of matter into bodies of any certain denomination, there being scarce any distinct portion of matter in the world, that is not either fluid, or else stable or consistent. And therefore, I presume, it may be well worth while to consider, what may be the general causes of these two states, qualities, or affections of matter, and to try, whether by associating chemical experiments to philosophical notions, there may not be given at least a more intelligible and more practical of both these subjects, than has been hitherto afforded us by the doctrine of the schools, which is wont to appear very unsatisfactory to discerning men, many of whom look upon what is wont to be taught by the Peripateticks, concerning fluidity and firmness, as well as other qualities, to be partly too general to teach us much, and partly too obscure to be understood. And that which at present invites us to this inquiry is, chiefly, that some circumstances of our author’s experiment, touching salt-petre, may afford us some useful assistance in our designed search. For though the chief phaenomena and circumstances of the experiment may be thought principally to respect fluidity; yet since that and firmness are contrary qualities, and since it is truly, as well as commonly, said, that contraries surveyed together serve to illustrate each other, it may reasonably be hoped, that the circumstances just now related may give to the nature of fluidity, may facilitate the knowledge of that of compactness: nevertheless, we shall often be obliged to treat of these two qualities together, because the experiments we are to produce do many of them relate to both.

(Boyle 240-241)

Response: Spinoza Letter VI:

Section 1. “It is quite manifest that they are to be reckoned among the most general states. . . etc.” In my view, notions which derive from popular usage, or which explicate Nature not as it is in itself but as it is related to human senses, should certainly not be regarded as concepts of the highest generality, nor should they be mixed (not to say confused) with notions that are pure and which explicate Nature as it is in itself. Of the latter kind are motion, rest, and their laws; of the former kind are visible, invisible, hot, cold, and to say it at once, also fluid, solid, etc.

(Spinoza 78)

Boyle, On Fluidity:

Section V

But instead of examining any further, how many bodies are or may be made visibly to appear fluid ones; let us now resume the consideration of what it is that makes bodies fluid: specifically, since having intimate some of the reasons, why we are unwilling to confine ourselves to the Epicurean notion, we hope it will the less be disliked that we thought fit to make such a description of a fluid substance, as may intimate, that we conceive the conditions of it to be chiefly these three.

The first is the littleness of the bodies that compose it: for in the big parcels of matter, besides the greater inequalities or roughnesses, that are usual upon their surfaces, and may hinder the easy sliding of those bodies along one another, and besides these things, I say, the bulk of it self is apt to make them to heavy, that they cannot be agitated by the power of those causes (whatever they be) that makes the minute parts of fluid bodies move so freely up and down among themselves: whereas it would scarce be believed, how much the smallness of parts may facilitate their being easily put into motion, and kept in it, if we were not able to confirm it by chymical experiments. But we see that lead, quicksilver, and even gold it self, though whilst they are of a sensible bulk, they will readily sink to the bottom of aqua regis, or any other such liquor; yet when the menstruum has corroded them, or fretted them asunder into very minute parts, those minute corpusels grow then so much more capable of agitation than before, that quitting the bottom of the liquor, they are carried freely every way, and to the top, with the associated parts of the liquor, without falling back again to the bottom. Nay, we see, that ponderous and mineral bodies divided into corpuscles small enough may be made to light and voluble, as to become ingredients even of distilled liquors; as we may learn by what some chymists call the butter, others (simply) the oil, and others the aleum glaciale of antimony; which, though it be after rectification a very limpid liquor, yet does in great part consist of the very body of the antinomy, as may appear (not to mention its weight) by this, that it is most easy to precipitate out of it with fair water store of a ponderous white calx, reducible by art to an antimonial glass. Nay, we make a menstruum, with which we can easily at the first or second distillation bring over gold enough, to make the distilled liquor appear and continue ennobled with a golden color.

(Boyle 242)

Response: Spinoza Letter VI:

Section 5.“The first is the littleness of the bodies that compose it, for in the larger bodies . . . etc.” Even though bodies are small, they have (or can have) surfaces that are uneven and rough. So if large bodies move in such a way that the ratio of their motion to their mass is that of minute bodies to their particular mass, then they too would have to be termed fluid, if the word ‘fluid’ did not signify something extrinsic and were not merely adapted from common usage to mean those moving bodies whose minuteness and intervening spaces escape detection by human senses. So to divide bodies into fluid and solid would be the same as to divide them into visible and invisible.

The same section. “If we were not able to confirm it by chemical experiments.” One can never confirm it by chemical or any other experiments, but only by demonstration and by calculating. For it is by reason and calculation that we divide bodies to infinity, and consequently also the forces required to move them. We can never confirm this by experiments.

Editors’ footnote, 34:

Spinoza’s view is that the infinite divisibility of matter is not subject to experimental confirmation, and consequently that Boyle’s claim that effective forces can be indefinitely small is not experimentally confirmable either. He is not denying that the particulate structure of matter is confirmable, nor is he claiming, contrary to some of his commentators (such as the Halls), that experiments can have no demonstrative force.


Commentary, Deleuze, Expressionism in Philosophy:

The ultimate extensive parts are in fact the actual infinitely small parts of an infinity that is itself actual. Positing an actual infinity in Nature is no less important for Spinoza than for Leibniz: there is no contradiction between the idea of absolutely simple ultimate parts and the principle of infinite division, as long as this division is actually infinite.


[for more on actual infinity, see Spinoza's 12th Letter and Gueroult's commentary, Deleuze's Cours Vincennes: 10/03/1981.].

Spinoza et le problème de l'expression:

En vérité, les ultimes parties extensives sont les parties infiniment petites actuelles d’un infini lui-même actuel. La position d’un infini actuel dans la Nature n’a pas moins d’importance chez Spinoza que chez Leibniz : il n’y a aucune contradiction entre l’idée de parties ultimes absolument simples et le principe d’une division infinie, pour peu que cette division soit actuellement infinie.


Footnote 11:

I do not understand why, in his study of Spinoza’s physics, Rivaud saw here a contradiction: “How can one speak, in an extended space whose actual division is infinite, of completely simple bodies! Such bodies can be real only in relation to our perception” (“La Physique de Spinoza,” Chronicon Spinozanum IV.32). 1. There would be contradiction only between the idea of simple bodies and the principle of infinite divisibility. 2. The reality of simple bodies lies beyond any possible perception. For perception belongs only to composite modes with an infinity of parts, and itself grasps only such composites. Simple parts are not perceived, but apprehended by reason: cf. Letter 6 (to Oldenburg, III.21).


Nous ne comprenons pas pourquoi A. Rivaud, dans son étude sur la physique de Spinoza, voyait ici une contradiction : « Comment, dans une étendue où la division actuelle est infinie, parler de corps très simples ! De tels corps ne peuvent être réels qu’au regard de notre perception » (« La physique de Spinoza », Chronicon Spinozanum, IV, p. 32). 1) Il n’y aurait contradiction qu’entre l’idée de corps simples et le principe d’une divisibilité à l’infini. 2) Les corps simples ne sont réels qu’en deçà de toute perception possible. Car la perception n’appartient qu’à des modes composés d’une infinité de parties, et ne saisit elle-même qui de tels composés. Les parties simples ne sont pas perçues, mais appréhendées par le raisonnement : cf. Lettre 6, à Oldenburg (III, p. 21).


My commentary:

As Gueroult explains in his commentary on the Letter on Infinity, the actual infinite is not something we can imagine; rather, we can only conceive it in our understanding. So our imagination cannot imagine together two properties of the modally expressed actual infinite: 1) that it contains absolutely simple ultimate parts, and 2) that it is subject to infinite divisibility. However, our understanding is capable of conceiving together these two features of actual infinity by conceptualizing it as an idea. Deleuze suggests as concrete examples the infinitesimal and the limit concept: on the one hand, we regard the integrated differentials to be infinitely small, having no extensive magnitude, because they have been reduced down towards zero. And yet, despite this reduction, we do not regard the operation as impossible on account of Zeno’s paradox. Rather, we can still quantify the area of a finite region using integration. So just as in the history of calculus where geometrical imaginings of the infinitesimal hindered the calculus' theoretical grounding in the limit concept, so too do geometrical imaginings of extensivity prevent us from understanding that there can be infinite divisions as well as absolutely simple ultimate parts. The problem our imagination contributes is that it imagines quantity numerically, that is, as being constituted by discrete units. But the actual infinite is not constituted by discrete units, but rather by a continuum of differential relations.

Deleuze’s purpose for stressing this point is to show how intensive quantity matches with extensive quantity. When something has greater intensity, that means it has a greater infinity of ultimate parts, and vice versa.

Boyle, Robert. Works of the Honorable Robert Boyle: In Five Volumes. To Which is Prefixed the Life of the Author. Kessinger Publishing, 2003.

Limited preview available at Google Books:


Deleuze, Gilles. Spinoza et le problème de l'expression. Paris: Les Éditions de Minuit, 1968.

Deleuze, Gilles. Expressionism in Philosophy: Spinoza. Trans. Martin Joughin. New York: Zone Books, 1990.

Spinoza. The Letters. Transl Samuel Shirley. Cambridge: Hackett Publishing Company, Inc., 1995.

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