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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.

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Spinoza, On The Immortality of the Soul

Controversially, the question of the immortality of the soul/mind arises in Spinoza’s writings, and with it the definition of personal identity. At one point he takes up thoughts about a poet who has, in an Altzheimer’s way, has lost contact with his person. In what sense is the poet still himself? To answer this, Spinoza argues for the existent essence of non-existent modes, a position which Deleuze sums as such:

“A mode’s essence is not a logical possibility, nor a mathematical structure, nor a metaphysical entity, but a physical reality, a res physica. Spinoza means that the essence, qua essence, as an existence. A modal essence has an existence distinct from that of the corresponding mode.” Expression in Philosophy (192)

Despite Deleuze’s assurance that this reality is not mathematical, Spinoza does take recourse to mathematical analogy to make clear his meaning, for instance (cited below), the existence of essence of an infinity of equal rectangles within the essence of a circle (Theorem 35, Euclid) which exist even if only one or even none exist modally.

 

So the essence of a mind is said to exist within the mind of God, eternally, despite its own limited duration. What this does is give the human mind a kind of eternity, an existence outside of the brief flicker of expression, but what this also does is place that eternal existence in relation to all other essences, of all other things, animate and inanimate, which are also produced by God/Nature. The human mind is eternal in essence as all other things are eternal in essence. But further, (as is shown in the note to EIV39 below), identity itself, our preservation of ourselves as ourselves in duration, is also not guaranteed, and is in fact likely an illusion of perspective. Just as his Spanish poet has died to himself, despite the continuity of his body, unable to recognize even his own writings, we too would only be an infinite series of eternal essences – slight modifications of a rectangle within its circle – defined only by our momentary consonance of parts – both ideational and extended. It is not so much that Spinoza has awarded undue eternity to the human mind, but rather has radically (categorically) undermined the basis upon which the human mind privileges itself to be unique among things in this world, given eternal life, but a life fused with all other things, capable as alien to its own “past” as akin to another thing. I list below relevant passages and definitions to this thinking:

EV29- The human mind cannot be absolutely destroyed with the body, but there remains of it something which is eternal.

(Proof) There is necessarily in God a concept or idea which expresses the essence of the human body, which, therefore is necessarily something appertaining to the essence of the human mind. But we have not assigned to the human mind any duration, definable by time, except insofar as it expresses the actual existence of the body, which is explained through duration, and may be defined by time – that is we do not assign to it duration, except while the body endures. Yet there is something, notwithstanding, which is conceived by a certain eternal necessity through the very essence of God; this something, which appertains to the essence of the mind, will necessarily be eternal.

EIV39note – …But here it should be noted that I understand the Body to die when its parts are so disposed that they acquire a different ratio of motion and rest to one another. For I dare not deny that – even though the circulation of the blood is maintained, as well as the other [signs] on account of which the Body is thought to be alive – the human Body can nevertheless be changed into another nature completely different from its own. For no reason forces me to think that the Body does not die unless it is changed into a corpse. And, indeed, experience seems to urge the opposite conclusion. Sometimes a man undergoes such changes that I should hardly believe that he was the same man. For example, I have heard tell of a Spanish poet who was struck by an illness; though he recovered, he remained so oblivious to his past life that he did not believe the tales and tragedies he had written were his own. He might have been taken for a grown-up infant had he also forgotten his native tongue.

EIp8 – By eternity, I mean existence itself, insofar as it is conceived necessarily to follow solely from the definition of that which is eternal. (Explanation) – Existence of this kind is conceived as an eternal truth, like the essence of a thing, and, therefore, cannot be explained by means of continuance or time, though continuance may be conceived without a beginning or end.

EIp24- The essence of things produced by God does not involve existence. (Corollary)… God must be the sole cause, inasmuch as to him alone existence appertain.

EIp25 – God is the efficient cause not only of the existence of things, but also of their essence.

EIIp8 – The ideas of particular things, or of modes, that do not exist, must be comprehended in the infinite idea of God, in the same way as the formal essences of particular things or modes are contained in the attributes of God. Note – If anyone desires an example to throw more light on this question, I shall, I fear, not be able to give him any, which adequately explains the thing of which I here speak, inasmuch as it is unique; however, I will endeavour to illustrate it as far as possible. 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.