12 Jul 2009

Spinoza's Foci for Deleuze's Contraction

by Corry Shores
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Spinoza's Foci

for Deleuze's Contraction

There is already extensive research and commentary on Spinoza’s philosophy. Somehow, it missed what might be one of the most important keys for unlocking his ideas. Spinoza’s day job was lens grinding. And so as well he was an expert in optics. We might keep this in mind and return anew to Spinoza’s fundamental concepts. In the very least we see a potent explanatory metaphor. We might consider for example our minds as lenses that could be polished to attain greater rational clarity. But the possibility remains that perhaps there is a more comprehensive way to view Spinoza’s philosophy: we might reexamine it under the light of his optical theories and craftsmanship. To do so would require extensive research, analysis, and commentary on these topics. Fortunately, all this original work has already been done. Kvond at Frames /sing indexes his many various treatments here. It is really incredible work, and it is changing the way I read and comprehend Spinoza.

And more recently kvond has begun work with analog and digital, relating it to Spinoza’s infinite. [See this entry and this one. Also see this entry on the topic of the infinite]. Kvond writes of Spinoza’s infinite:

The Infinite is Unbroken. And following this, modifications of the Infinite (how Spinoza defines the modes) do not break that unbroken state. (kvond, Spinoza on the Infinite, the Unbound: Part I)

We might call something analog if it has the property of being continuous, and digital if it is made of discrete parts. Because we may go on dividing some extending thing over and over and never come to ultimate divisions, it would seem to be continuous and analog. We might also think of it in terms of magnification. We look at a part of something under a magnifying glass. Then we grind a lens of a greater power, and see that there is still more parts within that one. And so on forever. No matter how powerful the lens, we can always (metaphorically) craft a stronger one, and see a smaller part of something. These divisions or magnifications that we make are really products of the imagination, imposed upon a substance that is fundamentally indivisible. Kvond expresses this another way when he writes:

One can see that there is a certain “faculative disorder” in the (digital) peak tracing of diagrammic representations, but, following Spinoza, these can only be analogical, which is to say continual, conjoinings. If Spinoza’s treatment of the infinite which disjoins the imaginarily discrete (mathematical) infinity from the real, expressive causal infinity, tells us anything, it is that diagrammic dis-organization and re-organization are imaginary processes which ever seek a continuity in the body itself, the body an infinite expression of magnitudes which press nestled upon each other. But unlike Deleuze’s pursuit of the chaotic elements (and this may only be an aesthetic difference), looking with the Intellect, as Spinoza would, is seeing-through these connections, not as bound, but as continually out-flowing and unitary. In this sense the ordering of numbers is a pale, imaginary imitation of the density of continuity in all things, a mechanism for our continual re-orientation. (kvond, Analog and Digital Intellect: Threshold Intensity, or Either/Or)

My aim here will be to explore Deleuze’s interpretation of Spinoza’s infinity, while building from some of kvond’s work on Spinoza’s optics and lens craftsmanship. However, my broader aim is to see if Deleuze’s version of Spinoza’s infinity is of a “contracted” series of discrete changes. I will begin first with kvond’s remarkable entries on Spinoza’s 39th and 40th letters. To really get a grasp of them in all their detail, see these entries by kvond:

Deciphering Spinoza’s Optical Letters

Some Observations on Spinoza’s Sight

A Diversity of Sight: Descartes vs. Spinoza

Spinoza: Letter 40 and Letter 39

Descartes’ Dioptrics 7th Discourse and Spinoza’s Letters 39 and 40

A Conflation of Spinoza Diagrams

Spherical Aberration: Descartes’ Solution

An origin of Spinoza’s “cones of rays” explanation, Letter 40

Spinoza’s Blunder and the Spherical Lens

Spinoza’s Letter 39: Descartes’ Silence

I would like to make use us of the circle diagram to help interpret Spinoza’s other circle diagram in his Letter on the Infinite.

The circle diagram depicts a circular lens.

Really, it is the ideal eyeball. The straight lines are rays of light.

To understand its properties, we make certain presumptions. We assume

1) this is a perfect circle

2) there is an exterior band of rays all parallel to line AB, and the spherical lens bends them so that they all converge at single point B.

Not every spherical lens will do this. But we might by some means construct one that could [see kvond's correction of the translation so that it is correctly formulated as a hypothetical, here and here]. Now, supposing that the spherical lens is perfectly circular, we can conclude that the same refraction-phenomenon will happen if a band of parallel rays shines from another axis, at a different angle. These are parallel to DC.

The light refracts the same way because “the circle, being everywhere the same, has everywhere the same properties.” (Spinoza, 215) No matter what the axis’s angle may be, the rays will converge on the opposite side of the lens. And they can be any of an infinite array of different axes; kvond writes

for Spinoza the Ideal Eye is one that in using the properties of a circle is able to focus rays parallel to a variety of axes (in fact, an infinity of axes). (Some Observations on Spinoza’s Sight)

Descartes has a diagram that might allow us to elaborate on how the circle is everywhere the same.

Descartes writes:

the surfaces of transparent bodies which are curved deflect the rays passing through each of their points in the same way as would the flat surfaces that we can imagine touching these bodies at the same points.

[and he continues:

As, for example, the refraction of the rays AB, AC, AD, which, coming from the flame A, fall on the curved surface of the crystal ball BCD, must be considered in the same way as if AB fell on the flat surface EBF, and AC on GCH, and AD on IDK, and so with the others. From which you can see that these rays can be variously gathered or dispersed, according as they fall on surfaces which are differently curved. And it is time that I start to describe the structure of the eye to you, in order that you will be able to understand how the rays which enter within it are so disposed there as to cause the sensation of sight. (Descartes, Optics, Second Discourse, p.83b.d)]

What I note is that Descartes seems to draw tangents along the circle as being those surfaces which refract the light. We can also think of tangents this way: we spin a ball on a rope. If we cut the string at any time, the ball flies off at a ninety degree angle to the string. This is also the tangent to the circle.

Align Centre

In fact, Spinoza says that the definition for a circle, like all other definitions, should be genetic, in that the definition produces the thing defined. This is the first requirement for definitions:

Rule I:

[96] (1) The definition must contain the proximate cause of the thing. [96] (2) So for example, we should define the circle as the figure traced by the end of a moving line rotating around its other endpoint, which is fixed in place, like how we draw a circle with a compass. (Spinoza, Improvement of the Understanding)

In fact, as kvond points out, Spinoza’s work on a lathe could have continually reminded him of a circle formed by motion:

It is no secret that Spinoza had great love for the circle as a diagramed exemplar of the relationship between the modes and Substance (Ethics 2p8s, pictured below), but also as an ideal of vision and the actions of optical focus (Letter 39 to Jelles, March 3rd 1667). Here, for hours on end Spinoza would stare determinitively as a rotating circular form which remained both fixed (stable in its ratio), and changing, expressing in both the consummation of the vectors of the room’s actions. (Spinoza’s “Spring Pole” Lathe: Experience to Metaphysics and Back)

(image obtained gratefully from kvond’s entry:

Spinoza’s “Spring Pole” Lathe: Experience to Metaphysics and Back)

So one way to conceive a circle is in terms of it being made of points all tending 90 degrees from their respective radius. That means each point tends a different direction. But they all tend the same degree away from the center point. So in this sense, the circle is everywhere tending to change direction at the same "rate."

Now we will turn to Spinoza’s diagram from his 12th letter, the “Letter on Infinity.”

[It is more thoroughly explained in this entry.]

The inner circle does not share the same center with the larger one. Let’s consider first if it did. The smaller circle would be tending to change direction continuously at the same rate. The larger one will as well. Because they both change at the same rate relative to the same center point, the distances between them will remain the same. But if the center circle is off-set, then the two will no longer be continuously coordinated. Rather, they will be continuously divergent. So the circles remain everywhere the same, but the differences between them are everywhere different.

Because the differing of the inter-mediate space is continuous, there will be infinitely many differences between them. Extensive space fills that continuously differing gap. We may divide that space up over-and-over again, and never arrive upon a finite set of small parts.

This is where we might find grounds for Deleuze’s mixture of analog and digital, or continuous and discrete, by means of a contraction rather than a flowing continuum.

there are last terms: the simplest bodies‚ for Spinoza. These are the ultimate terms, these are the terms which are last, which you can no longer divide. But, these terms are infinitely small. They are the infinitely small, and this is the actual infinite. (Deleuze, Cours Vincennes 10/03/1981)

Because they are infinitely small, these terms do not have extensive measure. They are "vanishing" quantities. In the case of our diagram, they would be like the infinite series of limits found throughout the middle space. Look at these diagrams to see that as we add more rectangles under the curve, they become narrower each time.

(Image from Edwards & Penney Calculus)

Imagine if we did something similar with Spinoza’s off-set circles. As the number of rectangle-like forms nears infinity, the spaces between will approach zero. We want to consider them once they are smaller than any finite extent, but still not yet zero. So they are vanishing, but not yet entirely vanished. There will be an infinity of limits in the circle’s middle space. But that does not mean it has somehow become like a solid of limits. One limit does not overlap with another to make a flowing continuum. They are still discretely different, even though their number is infinite. [I need a mathematician's help to determine if this is correct, and what is the better way to express it. The relevant calculus entries are this one and this one.]

So we may infinitely divide the space between the circles, and arrive upon these discrete inextensive simple bodies, which are limits. The change of tendency from one to another can have a certain degree of intensity. Deleuze writes:

An intensive quantity is inseparable from a threshold, that is an intensive quantity is fundamentally, in itself, already a difference. The intensive quantity is made of differences. (Deleuze, Cours Vincennes 20-01-1981)

We may consider any one limit and determine its tendency toward changing, by using differential calculus. So to go from one discrete limit to the one on its border there is not an extensive difference, but a difference of degree (of change in tendency). We might consider the sum of all the differential tendencies. If the circles were concentric, then there would not be a series of such differences. Each circle’s tendencies to change would continuously correspond to the other’s. But because Spinoza’s circles are offset, there is a sort of sum of internal differentiation.

We would consider such a collection of intensive differences an ‘individual.’

an individual is not a simple body, an individual, whatever it is, and however small it is, an individual has an infinity of simple bodies, an individual has an infinite collection of the infinitely small. (Deleuze, Cours Vincennes 10/03/1981)

So the region between the circles is infinitely divisible into an infinity of infinitely-small discrete “spaces” (although, none extend in space; however, they do not equal zero either). We may consider any one of them alone, and in that sense they are discrete. But, there is no extensive space between them either. So in that way they are continuous. But they are not continuous in a flowing way where each part overlaps with its neighbors. Rather, the discrete parts have no extensive space between them, so they are contracted together. However, the changes from one to the next can have a certain intensity. When later discussing Bergson’s duration, we will look at this contraction that creates an intense series of discrete consciousness states.

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

Edwards & Penney. Calculus. New Jersey: Prentice Hall, 2002.

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