Spinoza has us recall his non-concentric circle diagram, which demonstrated that within an extensive space there are an infinity of inequalities of difference, but within half that space, there are half as many such inequalities, even though it is still an infinite quantity. This was because what makes something infinite is its indivisibility and unlimitedness. So when we take modes to be discrete units that can be subdivided, we are using our imaginations to think abstractly about how substance is modified; so substance cannot be divided, and when our imagination tries to divide it by considering modal divisions, we find that our divisions can go on to infinity, because the substance underlying Extension is indivisible. But, when thought of as pure relations of difference, that is, as relations considered without terms of relation, then we have the simple bodies, which Deleuze associates with calculus differentials, pure quantitative relations without quantities being related. So when we consider the infinities of differential relations, we find that greater extensive space can have more of an infinity of simple bodies, even though we could keep dividing that space and always have an infinity of bodies making up any smallest piece of extensive space. The simple bodies, then, are intensive magnitudes, because they do not extend in space, but vary in terms of degrees of relative difference.
So because this diagram demonstrated that we can have different sizes of infinities, it cannot be that we can say something is infinite merely because it has more parts than can be given number; what makes something infinite is if it is unlimited and indivisible.
Spinoza's second point in this letter to Tschirnhaus is that it is impossible to demonstrate the existence of bodies if we presuppose Extension as Descartes envisioned, that is, as inert mass.
For matter at rest, as far as in it lies, will continue to be at rest, and will not be set in motion except by a more powerful external cause.
So if all we presume is matter at rest, we just have an indeterminate mass. We must then presume an exterior force in addition, namely God, in order for their to be the forces separating parts away from each other in this mass at rest. Hence we cannot demonstrate the existence of bodies if we only presume that Extension is matter at rest. Spinoza, on the other hand, presupposes Extension to be a qualitative expression of infinite substance, which is self-caused, so no additional thesis is needed to conceive of bodies as self-modifications of substance.
The letter's text:
LETTER LXX. (LXXXI.)
SPINOZA TO * * * * * 1
[Spinoza explains his view of the infinite.]
Distinguished Sir,—My statement concerning the infinite, that an infinity of parts cannot be inferred from a multitude of parts, is plain when we consider that, if such a conclusion could be drawn from a multitude of parts, we should not be able to imagine a greater multitude of parts; the first-named multitude, whatever it was, would have to be the greater, which is contrary to fact. For in the whole space between two non-concentric circles we conceive a greater multitude of parts than in half that space, yet the number of parts in the half, as in the whole of the space, exceeds any assignable number. Again, from extension, as Descartes conceives it, to wit, a quiescent mass, it is not only difficult, as you say, but absolutely impossible to prove the existence of bodies. For matter at rest, as it is in itself, will continue at rest, and will only be determined to motion by some more powerful external cause; for this reason I have not hesitated on a former occasion to affirm, that the Cartesian principles of natural things are useless, not to say absurd.
The Hague, 5 May, 1676.
Spinoza. The Letters. Transl Samuel Shirley. Cambridge: Hackett Publishing Company, Inc., 1995, p.352.
Text reproduction from R. H. M. Elwes translation, available online at:
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