25 Dec 2008

Macherey's Hegel ou Spinoza, "Le fini et l'infini," summarized

by Corry Shores
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Pierre Macherey

Hegel ou Spinoza

IV. Omnis determinatio est negatio

Le finie et l'infinie

Hegel attributes to Spinoza the formulation omnis determinatio est negatio, (or as Spinza writes literally in Letter 50 to J. Jelles: determinatio negatio est). (158b)

[from Hegel's Lectures on the History of Philosophy:

in this nothing, falls negation generally, or in more concrete form, limitation, the finite, restriction: determinatio est negatio is Spinoza’s great saying.

from Hegel's Science of Logic:

From § 203

Determinateness is negation posited as affirmative and is the proposition of Spinoza: omnis determinatio est negatio. This proposition is infinitely important; only, negation as such is formless abstraction, However, speculative philosophy must not be charged with making negation or nothing an ultimate: negation is as little an ultimate for philosophy as reality is for it truth.

From § 204

Of this proposition that determinateness is negation, the unity of Spinoza's substance — or that there is only one substance — is the necessary consequence. Thought and being or extension, the two attributes, namely, which Spinoza had before him, he had of necessity to posit as one in this unity; for as determinate realities they are negations whose infinity is their unity. According to Spinoza's definition, of which more subsequently, the infinity of anything is its affirmation. He grasped them therefore as attributes, that is, as not having a separate existence, a self-subsistent being of their own, but only as sublated, as moments; or rather, since substance in its own self lacks any determination whatever, they are for him not even moments, and the attributes like the modes are distinctions made by an external intellect. Similarly, the substantiality of individuals cannot persist in the face of that proposition. The individual is a relation-to-self through its setting limits to everything else; but these limits are thereby also limits of itself, relations to an other, it does not possess its determinate being within itself. True, the individual is more than merely an entity bounded on all sides, but this more belongs to another sphere of the Notion; in the metaphysics of being, the individual is simply a determinate something, and in opposition to the independence and self-subsistence of such something, to the finite as such, determinateness effectively brings into play its essentially negative character, dragging what is finite into that same negative movement of the understanding which makes everything vanish in the abstract unity of substance.

From § 1179

Determinateness is negation-is the absolute principle of Spinoza's philosophy; this true and simple insight establishes the absolute unity of substance. But Spinoza stops short at negation as determinateness or quality; he does not advance to a cognition of negation as absolute, that is, self-negating, negation; thus his substance does not itself contain the absolute form, and cognition of it is not an immanent cognition.

From Hegel's Lectures on Logic:

Quite generally, in anything determinate we have a negation. Spinoza said "All determination is negation" [Omnis determinatio est negatio]. That is an important principle, which was especially important to Spinoza. Relative to [Spinoza's] One, everything else is determinate, and everything is a negation.
(Hegel 96)

From Hegel's The Encyclopaedia Logic:

The basis of all determinacy is negation (omnis determinatio est negatio, as Spinoza says). Unthinking opinion considers determinate things to be merely positive and holds them fast in the form of being.

Although L. Robinson claims that we cannot attribute the claim omnis determinatio est negatio to Spinoza, Macherey will not take such an extreme position. What is more important is Hegel's addition to Spinoza's phrase, the word "omnis," which confuses and changes what Spinoza originally meant. (Macherey 158c)

In the Letter 50 to J. Jelles, Spinoza does not address determination in general, although he discusses determination in relation to the case of a figure (158d).

The figure is neither an idea nor a thing; rather, it is a limit, so it is not physically real, but is solely a being of reason.

[Spinoza exemplifies beings of reason in his Short Treatise: "part and whole are not true or real entities, but only things of reason, and consequently there are in Nature neither whole nor parts."]

So to "perceive" a figure (which is not the same as perceiving some given thing), we conceive it as determined, which means we conceive it insofar as it is limited by something else. Thus the figure expresses nothing more than the reciprocal limitation found between determinate finite bodies; and moreover, the figure represents these bodies not according to their being, but according to what they are not. (159b.c)

Macherey elaborates Spinoza's idea by citing part of Letter 52 (The Letter on Blood):

On the question of wholes and parts, I consider things as parts of a whole to the extent that their natures adapt themselves to one another so that they are in the closest possible agreement. In so far as they are different from one another, to that extent each one forms in our mind a separate idea and is therefore considered as a whole, not a part.

So to perceive a figure is to conceive a thing insofar as it is limited by another thing opposing it, which is also to consider it as a whole distinguished from other things that do not belong to it.

But when we consider something as being acted upon by some exterior thing, we take it to be a part in relation to a greater whole proceeding from another determination. Thus in this case the representation of the figure is a matter of understanding something as removed from an infinite chain of singular things, and considering it as a whole.

The problem lies in the notion of determination. We see that in the 50th letter, the notion of determination does not apply to any type of reality. It does not concern the unlimited attributes whose essence involves no negation. (160d)

Moreover, it does not even seem that this meaning of determination can apply to modes of extension. Finite and determinate bodies are not determined in this negative sense. When we consider finite modes as discrete self-same unities apart from their chain of causation, we are abstractly imagining them in a manner that does not reflect the way they actually are in reality. (160-161b)

From Spinoza's 81st Letter to Tschirnhaus, we see that if we understand extension as Descartes did, we would also be conceiving figure negatively, because Descartes presupposes an exterior cause for the motion that separates bodies in extension. (161c)

Also, we see that if we begin by considering modal finitude, we will encounter contradiction when trying to then consider their infinity (that is, their infinite divisibility). Macherey quotes from Letter 12:

So it is nonsense, bordering on madness, to hold that extended Substance is composed of parts or bodies really distinct from one another. It is as if, by simply adding circle to circle and piling one on top of another, one were to attempt to construct a square or a triangle or any other figure of a completely different nature.
(Spinoza, Letters, 103a)

The way we imagine the infinite following from the finite is that we consider some part of extension being divided, and then each new divided part being continuously divided without end. But this only demonstrates our imagination's inability to see the infinite as anything other than being infinitely divisible, which is a negative and inadequate way to conceive it. In reality, the attribute Extension is indivisible, because quantity cannot be reduced to discrete parts, which are understood only by means of negation. (162b)

In §401 of his Science of Logic, Hegel refers to Spinoza's notion of pure quantity, quoting from the scholium to Proposition 15 of the Ethics. [Below is §401, and the underlined part is what Macherey quotes.]

It is the notion of pure quantity as opposed to the mere image of it that Spinoza, for whom it had especial importance, has in mind when he speaks of quantity as follows:

'Quantity is conceived by us in two manners, to wit, abstractly and superficially, as an offspring of imagination or as a substance, which is done by the intellect alone. If, then, we look at quantity as it is in the imagination, which we often and very easily do, it will be found to be finite, divisible, and composed of parts; but if we look at it as it is in the intellect and conceive it, in so far as it is a substance, which is done with great difficulty, then as we have already sufficiently shown, it will be found to be infinite, without like, and indivisible. This, to all who know how to distinguish between the imagination and the intellect, will be quite clear.'

So when we determine quantity in relation to an exterior cause, we prevent ourselves from positively understanding infinity's essence. (162c.d)

But Hegel's distinction between the bad infinite and the rational infinite does not correspond to Spinoza's distinction between the infinite in its kind and the absolute infinite (162d) [See Hegel's distinction between the mathematical infinite (bad infinite) and the Notion of the infinite (the rational infinite) in entries on §§538-566 and §§567-628.] Rather, the bad mathematical infinite corresponds to the imagination's tendency to understand everything as determinate. [Macherey's italics are here in underline.]

For there are many things that can in no way be apprehended by the imagination but only by the intellect, such as Substance, Eternity, and other things. If anyone tries to explicate such things by notions of this kind which are nothing more than aids to the imagination, he will meet with no more success than if he were deliberately to encourage his imagination to run made. Nor can the Modes of Substance ever be correctly understood if they are confused with such mental constructs (entia rationis) or aids to the imagination. For by doing so we are separating them from Substance and from the manner of their efflux from Eternity, and in such isolation they can never be correctly understood.

We know from Definition 2 of the Ethics that things are finite when they mutually limit each other, but our understanding finite modes is inadequate if they are considered in terms of their finitude. It is because we understand an effect from its cause (Axiom 4), that we must understand finite modes from their cause, the infinite substance [as Macherey quotes the axiom, "la connaissance de l'effet dépend de la connaissance de la cause et l'enveloppe."] But the imagination can only understand things in terms of finitude, namely, in terms of measure and number (163c.d) [see the Letter and Gueroult's commentary section XVI.]

To better explain this "implication" or "involvement" (enveloppement), Spinoza draws from his circle diagram in the 12th Letter , which Hegel examines specifically in two texts: 1) in his Lectures on the History of Philosophy when discussing Definition 6 of Spinoza's Ethics, and 2) §566 of his Science of Logic.

So to be clear on this example, Macherey displays the circle diagram and provides the following quote from Letter 12 [The quotation comes from the English translation, but is modified to better resemble the French version Macherey cites in his text]:

All the inequalities of the space (inegalitates spatii) lying between the two circles, AB and CD, and all the variations of speed of matter moving through that area, exceed any number. Now this conclusion is not reached because of the excessive magnitude of the intervening space; for however small a portion of it we take, the inequalities of this small portion will still be beyond any numerical expression. Nor again is this conclusion reached, as happens in other cases, because we do not know the maximum and minimum; in our example we know them both, the maximum being AB and the minimum CD. Our conclusion is not reached because number is not applicable to the nature of the space between two non-concentric circles. Therefore, if anyone sought to express all those inequalities by a definite number, he would also have to bring it about that a circle should not be a circle.
(Spinoza Letters 105b.d)

Here, the "intervening space" between the circles contains all the inequalites between that separate their circumferences; and, the "inequalities of space" is the totality of differences between unequal distances, that is, the totality of their variation. We cannot reduce this totality to any number, because it is a continuous variation resulting from the off-set circularities of the figures (Macherey 164d) [for more on this continuous variation, see Deleuze's Cours Vincennes: 24/01/1978 and 20/01/1981.]

The indefiniteness does not result from there being an "excessive magnitude," because the space is enclosed between limits (164-165). The indefiniteness persists even if we take only a portion of this space, because no matter how small the portion, there will still always be a quantity of spatial differences which cannot be given a number, a point Spinoza emphasizes in his Letter 81 to Tschirnhaus (165a.b).

The imagination's tendency to represent and analyze magnitude in terms of number leads to insoluble paradoxes. But mathematicians recognize this and do not let themselves be hindered. Macherey quotes again from Letter 12 [where Spinoza refers to mathematicians who deal with infinite magnitudes]:

For not only have they come upon many things inexpressible by any number (which clearly reveals the inadequacy of number to determine all things) but they also have many instances which cannot be equated with any number, and exceed any possible number. Yet they do not draw the conclusion that it is because of the multitude of parts that such things exceed all number; rather, it is because the nature of the thing is such that number is inapplicable to it without manifest contradiction.
(Spinoza Letters 105a)

Certain magnitudes cannot be assigned a number because the "movement" constituting them is continuous and hence indivisible. For the imagination, continuity is paradoxical; but for the understanding, the concept of continuity is clear and distinct. (Macherey 165d)

Macherey then cites at length Hegel's two commentaries on Spinoza's circle diagrams, the one from Science of Logic that Hegel himself wrote, but the one in the Lectures on the History of Philosophy is reconstituted from students' notes.

Macherey quotes from §566 of his Science of Logic:

The mathematical example with which he illustrates the true infinite is a space between two unequal circles which are not concentric, one of which lies inside the other without touching it. It seems that he thought highly of this figure and of the concept which it was used to illustrate, making it the motto of his Ethics. 'Mathematicians conclude', he says, 'that the inequalities possible in such a space are infinite, not from the infinite amount of parts, for its size is fixed and limited and one can assume larger and smaller such spaces, but because the nature of the fact surpasses every determinateness.' It is evident that Spinoza rejects that conception of the infinite which represents it as an amount or as a series which is not completed, and he points out that here, in the space of his example, the infinite is not beyond, but actually present and complete; this space is bounded, but it is infinite 'because the nature of the fact surpasses every determinateness', because the determination of magnitude contained in it cannot at the same time be represented as a quantum, or in Kant's words already quoted, the synthesis cannot be completed to form a (discrete) quantum. How in general the opposition of continuous and discrete quantum leads to the infinite, will be shown in detail in a later Remark. Spinoza calls the infinite of a series the infinite of the imagination; on the other hand, the infinite as self-relation he calls the infinite of thought, or infinitum actu. It is, namely, actu, actually infinite because it is complete and present within itself.

Then Macherey quotes the following portion of Hegel's Lectures on the History of Philosophy:

Spinoza here also employs geometrical figures as illustrations of the Notion of infinity. In his Opera postuma, preceding his Ethics, and also in the letter quoted above, he has two circles, one of which lies within the other, which have not, however, a common centre.
“The inequalities of the space between A B and C D exceed every number; and yet the space which lies between is not so very great.” That is to say, if I wish to determine them all, I must enter upon an infinite series. This “beyond” always, however, remains defective, is always affected with negation; and yet this false infinite is there to hand, circumscribed, affirmative, actual and present in that plane as a complete space between the two circles. Or a finite line consists of an infinite number of points; and yet the line is present here and determined; the “beyond” of the infinite number of points, which are not complete, is in it complete and called back into unity. The infinite should be represented as actually present, and this comes to pass in the Notion of the cause of itself, which is therefore the true infinity. As soon as the cause has something else opposed to it — the effect — finitude is present; but here this something else is at the same time abrogated and it becomes once more the cause itself.

Macherey notes that Hegel's interpretation misconstrues Spinoza's example, which Gueroult addresses in his Spinoza text [see Gueroult's commentary on the Letter, section XIII, for Hegel's mis-translation of "inequalities of distance."] We know that unlike Hegel's interpretation, the diagram's inequalities of difference result from the continuity of variation between the off-set circles, which is why the differences cannot be given any number.

According to Macherey, Hegel on the contrary seems to suggest that Spinoza means that there is an infinite series of different magnitudes between circumferences, and their sum cannot be assigned a number, despite all those magnitudes falling within delimited space [see section XIII of Gueroult's commentary for more on this incorrect manner of interpreting the diagrams.] But we know this cannot be Spinoza's intent, because he could have used concentric circles: the sum of their distances likewise could not be given any number. Hegel, then, misses what is essential to Spinoza's illustration: a continuum of differences, even one spanning between determinate bounds, cannot be assigned a determinate value. (168b.d)

Spinoza distinguishes different senses for the term infinite:

A1) infinite by nature or definition
A2) infinite by force of its cause (and not its essence)

B1) infinite because it is without limits
B2) infinite because it cannot be determined by a number
(Macherey 169a)

[see the beginning of the Letter, and sections I through VII in Gueroult's commentary for more on these distinctions.]

The intractable paradoxes normally associated with the infinite may be traced back to confusions between these very distinct senses for the term infinite (169b.c). For example, substance is infinite by nature, and is indivisible. Modes are infinite because they are caused by infinite substance. But our imagination is able to conceive of modes as finite, which leads to Zeno's paradox. Also, Zeno's paradox can be avoided if we see that there are infinities bound between limits that nonetheless cannot be determined by a number.

Thus the geometrical example illustrates this infinite that cannot be assigned a number despite its magnitude being confined between a minimum and a maximum limit. The values between the limits alter continuously, and thus cannot be divided into discrete units, not even an infinity of them, which is why it cannot be determined numerically. (169d) [for Hegel's distinction between continuous and discrete quantities, see his Science of Logic §429-§431, summary, and §434-§435.]

To account for the imagination's error, Macherey cites Definition 2 from Part I of the Ethics, which says that something is finite if it is limited by something else. The space of the geometrical example has an upper and lower limit, because the right side of the circular area is limited by the left side. Thus the imagination should take this space as indivisible, thinks Macherey, when we are regarding it as a finite limited entity. The paradoxes result when we try to give a number to its internal variations rather than to the its finite limits. (169-170.ab)

But infinite and finite are not so distinct from each other, because the infinite produces the finite, and the finite involves (enveloppe) the infinite. The variation between the non-concentric circles is infinite insofar as it is an affection of infinite substance. (170b.c)

We turn then to Hegel for clarifying this subtlety, even though in other ways he misinterprets the example. On the one hand, Hegel realizes that the example illustrates the causal relation between substance and its affections. But on the other hand, he contributes his own distinction to describe the relation, because he speaks here in terms of the infinite Notion and the actual infinite (the infinite in act). Spinoza, in fact, speaks of the actual infinite in the paragraph before the one discussing the geometrical example. He writes in regard to Number, Measure, and Time:

Hence one can easily see why many people, confusing these three concepts with reality because of their ignorance of the true nature of reality, have denied the actual existence of the infinite.

The actual infinite is not given in an unlimited series, because, as Hegel says, it is "complete and present" within its limits. What is important for Spinoza is that the infinite is in the finite on account of the causal act of substance that makes it actually infinite (170-171d) [for more on actual infinity, see Spinoza's 12th Letter and Gueroult's commentary, Deleuze's Cours Vincennes: 10/03/1981. and Deleuze's commentary on Spinoza's critique of Boyle's notion of divisibility].

Hegel's inaccurate interpretation is helpful in another way, because Spinoza says in Proposition 18 of Ethics Part I that God is the immanent cause of everything. Hegel's understanding of the actual infinite as complete and present within its bounds captures the immanent causality of the infinite. (171d)

If Spinoza only wanted to explain infinity contained within a finite extent, he need not have overcomplicated his circle illustration by making the circles non-concentric. The reason he does is because he wants to articulate a more particular kind of infinity. A variation of difference cannot be given a magnitude, even if it is within finite bounds, because it passes through more differences than can be given a number. So the number of differences does not result from infinite division, because that goes on forever. The differences are pure differential relations, of the sort in differential calculus. The sort of infinity here is not a matter of extensity, but rather of intensity, because the differential relations vary in changing degrees, not extents. This, says Macherey, is how Deleuze comes to distinguish two further types of infinities, extensive and intensive, at the beginning of the 13th chapter of Spinoza et le Problème de l'expression (Expressionism in Philosophy: Spinoza). (172b.d)

The intensive infinite is the expression of the immanent relation between substance and its affections, because in the intensive infinite are the varying degrees of powers making up the continuum of essences whose proper modes temporarily come into existence (172d).

So because substance expresses itself in a continuum of essences that cannot be divided, and because these essences become modally determined, to remove but one part of matter will annihilate the whole (173bc) [Macherey here quotes Letter 4. See the end of the entry for an alternative explanation based on Deleuze's theory of simple bodies.] Likewise for the ideas of Thought. So the unity and infinity of substance expresses itself in the intensive continuum of essences, and knowing this actual infinity of substance constitutes the knowledge of the third kind, which is the "intellectual love of God." (173c)

This knowledge is affirmative; for, it does not begin first from finite modes and proceed regressively to substance, because this leads to us rejecting the infinite. Rather, it begins in substance and proceeds to its affections, moving from cause to effect. In this way, we make a necessary and continuous logical progress that never resorts to notions of the possible or of negation. So in a way, we might borrow from Hegel and say that it is in a sense a doing away with negation. (173-174)

Thus we know that Hegel is incorrect to impute to Spinoza the principle that "all determination is negation;" because the actual infinite for Spinoza lies within determinate bounds, but is not the product of negation, nor can it be conceived in terms of negation (174a). To determine something negatively would be to consider it as a discrete thing distinct from God, which happens when we use our imaginations to abstractly consider modal existences. When instead we use our understanding to conceive something positively, we regard it as being produced by infinite substance and hence to be essentially infinite and eternal.

Thus we see from Spinoza's infinite that there are not two distinct realities or worlds, one infinite and the other finite. For Spinoza, there is only one "order" that is concrete and physically real.

Hegel. The Encyclopaedia Logic. with the Zusätze: Part I of the Encyclopaedia of Philosophical Sciences with the Zusätze. Transl. Garaets, Suchting, and Harris. Hacket Publishing, 1991.
Limited view of text available at:

Hegel. Lectures on the History of Philosophy. Transl. E.S. Haldane.
See electronic text site for further publication information, and for the entirety of the text itself:

Hegel. Lectures on Logic. Transl. Clark Butler. Indiana Univerisity Press, 2008.
Limited view of text available at:

Hegel. Science of Logic. Transl. A.V. Miller. George Allen & Unwin, 1969.
Text available online at:

Macherey, Pierre. Hegel ou Spinoza. Paris: Libraire François Maspero, 1979.

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

Spinoza. Short Treatise on God, Man, and His Well-Being Benedict de Spinoza. Transl. A. Wolf.
Text available online at:

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