Frames /sing


Spinoza’s Circle and the Interior

Some Ruminations on Spinoza’s “Simplest” Thought

A single figure keeps returning to me as I contemplate Spinoza’s optical concepts and metaphysics, in view of the prevailing Cartesian science of the age, that elementary figure that Spinoza provides to help explain how he conceives of the reality of the modes, even the modes of ideas for non-existent things, in relationship to the ultimate reality of Substance, God or Nature. As Spinoza explains, the figure stands for something that has not parallel, not example, because it is a fundamental relationship which is unique and totalizing.

Ethics part 2, prop 8 schol. — The nature of a circle is such that if any number of straight lines intersect within it, the rectangles formed by their segments will be equal to one another; thus, infinite equal rectangles are contained in a circle. Yet none of these rectangles can be said to exist, except in so far as the circle exists; nor can the idea of any of these rectangles be said to exist, except in so far as they are comprehended in the idea of the circle. Let us grant that, from this infinite number of rectangles, two only exist. The ideas of these two not only exist, in so far as they are contained in the idea of the circle, but also as they involve the existence of those rectangles; wherefore they are distinguished from the remaining ideas of the remaining rectangles. (Elwes trans)

This has always been a convenient image for the explanation how the modes dependently are an expression of the totality which is all of what there is. One practically sees how the modes, while causally interacting with each other to produce current states of being, immanently rely upon a larger comprehesivity. And if one wants to even leave off from the narrowness of Spinoza’s explicit thought, the chords D and E could be read as vectors which produce an intersecting point (unlabeled), the modal expression of two lines of force, a point of focus, this seems at least arguable when considering much of the rest of Spinoza’s view.

But two other realizations come to me as well. The first is that the rectilinearitythat Spinoza presents within the borders of the circle very well may for Spinoza have represented the linear analyses of motion provided by Descartes. Part of the challenge presented to Spinoza, as he tried to heal a perceived breech between Body and Soul, Thought and Extension, was how to bridge mathematical descriptions with affective realities. In a certain sense the rectilinearity with the circle of Substance for Spinoza represents the ideational crispness of mathematical description understood as a partial, yet sufficient expression of substance. The mathematics is to some degree subsumed. Just as Huygens and others struggled to assume the advantages of the Perspectiva  tradition in optics, yet still fully explain the properties of light’s propagation (which Huygens would show to be that of a wave or a pulse), Spinoza too sought to preserve the mechanized sureties of Descartes mathematizations, yet with a view that could account for the affective propagations living bodies (as early as Kepler’s Paralipomena light is described as an “affectus”).

Secondly though is the realization that the circle does not only represent Substance as a whole, but also any conatus expression of Body (defined by Spinoza as a preservation of a ratio of motion and rest over time). The circle above can also be read as the recursive coherence of a body, and the internal causal relationships of which it is composed, immanent to its essence. 

In other words, a body, in the ordinary as well as in the Cartesian sense, preserves its physical integrity in just the same way the whole universe preserves it: they differ only with regard to the degree of probability that each can ‘survive’ externally caused modifications. Only the unique individual of level-ω can be modified in infinite ways without giving up its identity.

The Physics of Spinoza’s Ethics, David Lachterman

What level-ω is in Lachterman’s nicely conceived reduction is the last border of an entire body. Level-2 is the most elementary kind of body for Spinoza, a hypothetical composition of what one supposes is at least two of the most simple bodies “corpora simplicissima”. The human body, and all such bodies we regularly recognize as such are what Lachterman regards as level-k bodies (2 < k < ω). Following these descriptions, the circle above from the Ethics could be read not only as the schema of the level-ω Body, but also of any typical level-k Body. It is only the degree of power, freedom and being that preserves such a circle (ratio) that distinguishes it from the ultimate whole. Spinoza world can be seen as one of circles within such circles. The modal causal interactions are immanent expressions not only the totality of Substance or Nature, but also are expressions of the ratio of motion and rest of which a said body (level-k) is composed. The existential modality of ideas and extensions which is interior to the circle (level-k) can be read as contingent to the essence of that Body taken as a whole, as it is determined to exist.

While it is fair and best to understand the diagram exactly in the way that Spinoza meant it, I think it is important to realize that in a figure such as this, representing something for which Spinoza says there is no parallel, the image is likely overdetermined by several other Spinoza concerns and conceptions. The circle and its interior floats through Spinoza’s thinking process (additionally). The diagram can be read across at least these two layers: the affective subsumption of Cartesian mathematicization of Nature in terms of propogating, interacting bodies; and, the immanence of constituent modes of bodies under gradations of Being and power. And I would end that the suggestive recursivity of this body (level-k) opens itself perhaps to Autopoietic descriptions, at least for those bodies considered to be alive, and Autopoiesis itself to Spinozist re-description.


2 responses to “Spinoza’s Circle and the Interior

  1. visblog September 26, 2008 at 5:30 pm


    I really liked browsing through your blog. I am going to read in more detail. It’s so interesting how you like to illustrate thoughts and philosophy.

    I am going to add you on my blog roll. I also have written couple of things in my about page.

    Anyways, I was just going to tell you that, interestingly I remember this circle example set as two non-concentric circles, one being inside of the other. and the distances between these two. Do you remember that example? It could even be more interesting to illustrate.

    Just a thought and thanks for preparing such an awesome blog-site.

  2. kvond September 26, 2008 at 6:04 pm

    Thanks for the very good words. I believe that the illustration you have in mind is from letter 12 to Lodewijk, which interestingly Hegel himself was attracted to in his Lectures on the History of Philsophy:

    This indeed would be another elemental diagram to discuss. I’m glad to have the site added to your blogroll, best of thoughts to your own site.

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