## 14 Jul 2009

### Good Infinities Gone Bad, §9, 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.]

Simon Duffy

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

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

§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.