15 Jul 2009

A Variation on Infinity, §11, Logic of Expression. Simon Duffy

[The following summarizes part of Simon Duffy's extraordinary book, The Logic of Expression: Quality, Quantity and Intensity in Spinoza, Hegel and Deleuze. My commentary is in brackets. Duffy's work is remarkable, so I highly recommend this book. If it costs too much, perhaps encourage your library to obtain a copy.]

A Variation on Infinity

Simon Duffy

The Logic of Expression:
Quality, Quantity and Intensity in Spinoza, Hegel andDeleuze

Chapter 1
"Spinoza from the point of view of an idealist or a materialist dialectic"

§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.

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