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"if you think childlike, you'll stay young. If you keep your energy going, and do everything with a little flair, you're gunna stay young. But most people do things without energy, and they atrophy their mind as well as their body. you have to think young, you have to laugh a lot, and you have to have good feelings for everyone in the world, because if you don't, it's going to come inside, your own poison, and it's over" Jerry Lewis "I don’t believe in the irreversibility of situations" Deleuze
The numerical citations refer to page number. The source's text-space (including footnote region) is divided into four equal portions, a, b, c, d. If the citation is found in one such section, then for example it would be cited p.15c. If the cited text lies at a boundary, then it would be for example p.16cd. If it spans from one section to another, it is rendered either for example p.15a.d or p.15a-d. If it goes from a 'd' section and/or arrives at an 'a' section, the letters are omitted: p.15-16.
§11 The adequate and the inadequate idea of the actual infinite
Macherey follows Hegel’s translation of Spinoza’s geometrical example in the 12th letter, “The Letter on the Infinite.” There is an irregular distribution of space between lines AB and CD.
Thus there are many “inequalities of space” between the two lines.
Macherey considers them in terms of the variation that might appear if there was motion from end to end:
The ‘inequalities of the space’ should then be understood to refer to ‘the set of the differences between these unequal distances’, or, what Macherey emphasizes as ‘the variation’ of these ‘differences’, which is determined by the rotation of the segments from AB towards CD, ‘in the sense of hands of a watch’. This set, which is ‘the sum of the inequalities of distance included in this ... total space’, is a continuous and therefore infinite variation. (27bc, emphasis mine)
For Hegel, the differences are limited by the maximum, AB, and the minimum, CD. But for Macherey, the variations are limited by how much they may vary. So the difference between the length of AB and CD is the margin that limits the range of variation.
According to Hegel’s interpretation, it would not matter if the circles were concentric or not. He merely considered infinity as the infinite divisibility of continuous magnitudes. But for Macherey, the actual infinite is the infinite variation of differences in the middle-space. Substance expresses itself through an infinity of different qualities or attributes. Extension is one of them. Finite extended things, then, would be modes or modifications of substance as expressed extensionally. When we use our reason to understand the limited expanse in the geometrical example, we see that it is caused by infinite substance, and for that reason the mode itself is infinite by force of its cause. However, if we use our imagination to conceive the infinity between the limits, then we begin to imagine it being divided into smaller and smaller parts on to infinity. This would be to inadequately understand it as unlimited or indefinite. (27-28)
According to Spinoza, some things are indefinite because we cannot describe them in terms of numbers. But even though we cannot give them a number, we can still know that certain ones are larger or smaller than certain other ones. (28a)
When we conceive of the geometrical example with our imagination, we can consider it as being divisible into an infinite number of variations. Ones with a greater range of variation could then be seen as having a greater infinity of differences. Hegel thinks that this would be the bad infinite. For, to say that one infinity is larger is to indicate that it has a greater number of parts. But number does not apply to the actual infinite, so we cannot say that one actual infinite is greater or lesser than another, says Hegel. (28b.c)
According to Macherey, we encounter such contradictions when we conceive the infinite by means of the imagination, “which wants to represent everything by numbers.” (28d) However, reason can clearly and distinctly conceive the notion of the continuous without encountering paradoxes. (28-29)
Macherey disagrees with Hegel’s interpretation. For Hegel, Spinoza’s geometrical example first indicates an infinity that is a negation of the finite: when we infinitely divide something, there is always something beyond or more than the finite parts that division produces. But this is the bad infinite. When we negate this notion, we realize that the actual infinite is such because no number applies to it in the first place. So Spinoza’s geometrical example illustrates the dialectical mediation leading to the actual infinity by means of the negation of negation. [again, see this entry for more.] But for Macherey, the example depicts both infinites at the same time. For when we use our imagination, we inadequately conceive it as indefinite and unlimited. And when we use our reason, we adequately understand it as infinite by force of its cause. (29b)
So no negation is involved for Macherey. We adequately and positively know the infinite as resulting from the infiniteness of its immanent cause: substance. Hence Hegel’s formulation omnis determinatio est negatio does not apply [for more see this entry and this one.]
We only partially understand the infinite when we use our imagination. But when using reason, we may have “knowledge of the third kind.” (29-30)
Duffy, Simon. The Logic of Expression: Quality, Quantity and Intensity in Spinoza, Hegel and Deleuze. Aldershot: Ashgate Publishing, 2006.
§10 The problem of the ‘bad infinite’
Previously we discussed Hegel’s misinterpretation of Spinoza’s infinite. It is not merely the infinite divisibility of finite extensions. Conceiving it so is often the doing of our imaginations rather than our more able rational faculties.
Macherey builds from Gueroult’s commentary to argue that Hegel misses Spinoza’s point: finite things are not infinite by force of their cause. Finitely-extended modes are nonetheless infinite insofar as they express the infinite of substance, which is their immanent cause. (25c)
Macherey evokes the distinction between
1) what is infinite by its nature. Substance is absolutely infinite; and
2) what is infinite by force of its cause. The attributes or modes of substance are infinite in this way.
He further distinguishes
1) what is infinite because it has no limits; and
2) what is infinite because it cannot be determined by numbers. It is unlimited or indefinite despite the fact that it is bound within a maximum and minimum.
When we use reason, we may adequately understand things. But our imagination is inadequate for this purpose. Substance is absolutely infinite. We can only conceive it through reason, and not through imagination. We can, however, imagine substance’s affections. But when we do so, we understand them inadequately as unlimited and indefinite. Only reason can adequately understand affections as infinite by force of their cause. (26a)
Unless we maintain these distinctions, we will run into contradictions when conceiving the infinite.
According to Macherey, the bad infinite is the unlimited or indefinite. It is the same thing as the actual infinite. When using reason, we understand it adequately as the actual infinite, but when using our imagination, we inadequately conceive it as the bad infinite.
For Macherey, we adequately understand something when we grasp it affirmatively in terms of its cause. Hegelian dialectic views the infinite in terms of negations, and hence does not understand it adequately.
According to Macherey, the actual infinite, as apprehended adequately in the mode, ‘is not different to that infinity constitutive of substance, but is formally the same’. The actual infinite is therefore the immanent expression in the affections, or the finite modes, of infinite substance, which is their cause. (26-27)
§9 Infinite actu and the ‘bad infinite’
Hegel discusses Spinoza’s bounded infinite in Science of Logic and the History of Philosophy. And he refers specifically to Spinoza’s geometrical example in the 12th Letter.
Hegel interprets Spinoza as saying that the space between AB and CD is filled with an infinity of “inequalities of space.” (21d) One might think then that the infinite for Spinoza results when we try to count all the infinitely many unequal spaces. Just as soon as we think we counted them all, more pop-up in-between the ones we just counted. Hence this would be an infinity resulting from a continuously incomplete series. However, Hegel claims that this is not how Spinoza actually viewed infinity. For Hegel, a continuum is by nature not divisible into any number of parts. Hence, we misrepresented a continuous value when we regard it as being made-up of a determinate number of discrete elements. And there is nothing incomplete about Spinoza’s infinity, because it is all entirely there between the two boundaries. Such is the “actual infinite.” (22b.c)
So the actual infinite can be found within the finite. We would instead be dealing with the “bad infinite” if we were to think of infinity as more finite parts than can be counted. (22c)
The bad infinite is a negation of the finite, (because it says that the finite is never enough, and the infinite is always something more). The actual infinite is a negation of the bad infinite, (because it says that the infinite is found within bounds, and need not always be more more more.) Thus the actual infinite is “the negation of negation.” (22-23)
Hegel portrays mathematicians as incorrectly conceiving the infinite in the “bad” way. Spinoza thinks instead that mathematicians conceive the infinite correctly, however. He says that something is infinite for them not on account of the number of its parts, but rather because it is not expressible by any number. So we need not consider the actual infinite in terms of a contradiction or negation. It is not something that exceeds number. It is merely something that cannot be expressed by numbers. Hegel does not recognize this of mathematicians. He wants to contrast the mathematical (bad) infinite with the philosophical (good/actual) infinite. This fits his dialectical logic. The bad mathematical infinite is the first negation, and the philosophical infinite is the negation of negation.
Hence we see that Hegel misrepresents Spinoza’s characterizing the mathematical infinite as being the same as the actual infinite. Instead for Hegel, the mathematical infinite is a sort of dialectical stepping-stone leading to the philosophical infinite:
The infinite, when opposed to the finite, is conceived as the bad infinite, which is then sublated and subsumed in the actual infinite, that is, the finite realizes itself as actually infinite. This is how Hegel resolves the relation of the infinite to the finite from the point of view of his interpretation of Spinoza. (24a)
Yet Hegel’s interpretation of Spinoza’s geometrical example misses its subtle but important peculiarities. [See Macherey’s commentary.] The circles are off-set. Hegel’s rendition could also be expressed if the example were of concentric circles. For he also thinks that the infinity of points on a line also exemplifies the actual infinite. In his view, all finite things contain the philosophical infinite in this way. (24c)
Duffy, Simon. The Logic of Expression: Quality, Quantity and Intensity in Spinoza, Hegel and Deleuze. Aldershot: Ashgate Publishing, 2006.
part and whole are not true or real entities, but only things of reason, and consequently there are in Nature neither whole nor parts.