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Some Observations on Spinoza’s Sight

How The Two Philosophers “See”

I feel that there are some important things to say about my recent post, A Diversity of Sight: Descartes vs. Spinoza , but I am still undecided just how deep the influence of these thoughts run. So pervasive is the metaphor of vision and light within Western metaphysics, any identification of an ocular appropriation into the field of metaphysics, and the questioning of its radical truth or application, may have far reaching interpretive effects.

What may prove the advantage of this analysis is that it promotes a simplification. Like all simplifications it is misleading to take this as the whole story, but it does help us identify a core element of disagreement between the two Natural Philosophers. The difference between Descartes and Spinoza cannot be reduced to these two diagrams, of course. But there is an essential divergence in the thinking about vision as a metaphor for thought that is expressed in them. 

Descartes' Ur Image: The Hyperbola

Spinoza's Ideal Optical Eye

The first of these, for those uninterested in the optics under question in Spinoza’s letters 39 and 40, shows the capacity of a hyperbolic lens to focus any rays that are parallel to its central axis to a point along that axis. What the hyperbola provides is a schema for thinking about vision and clarity, the analogy of imagining that a focused image of the world that is “clear and distinct” is one where all the rays of a kind are brought to a mathematical ideal, poured into a point. We are not dealing here with all the details of lenses, and how they interact with the human eye and light in the fullness of their variety, but rather with a guiding diagram of what a lens should do – focus rays of light to a center point – and what that means for the experience of vision. For this reason, it is best to understand that this image for Descartes is likely intuitive of directions for investigation, steering both his theories and empirical observations.

The second of these is from Spinoza’s Letter 39, and works as a vivid contrast to Descartes’ Hyperbola. Instead of imagined parallel rays focusing down into an ideal point in the very center of the eye (which in some ways Descartes will conflate with the free Will), 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). Rays coming from all directions are hoped to be focused across the back of the eye. And Spinoza sees the human eye (insofar as it does not have a spherical lens), as failing to achieve this kind of vision. Ideal mental vision, instead of being modeled upon a central point of focus, Spinoza conceives of as panoptical; that is, one “sees” as best as a human mind can the cross-section of rays as they converge from every direction upon the human being.

As admitted, this is truly a vast over-simplification, for much unites these two philosophers, and the kinds of radical divergences that Spinoza makes are must more diverse than this simple diagram comparison. But really there is something suggestively profound in this contrast. For one, in that Descartes’ hyperbola inheritance may be traced to Kepler’s Paralipomena its conceptual framework should be viewed as grafted from that Neo-Platonic Ideal, opening up the question of what aporias arise under such a graft (for instance, a point of focus in a Neo-Platonic realm, does not operate with the same powers or meanings as a point of focus does within a Will-driven conception of the soul). Additionally, Spinoza’s rejection of the naturalization of the hyperbola, and the analogy of center-focused human vision, has far-reaching consequences for the reading of the place of the Self in his philosophy of power and affect. If Ideal vision occurs across a field of foci, the periphery has no less a “truth” than any center. The margin does not merely, as Kepler says, “serve” the axis – so goes the critique in so many postmodern attacks on a philosophy of Presence – hence the margin is the very place where a search for truth is made, whether it be the margin of society or a comprehensive Totality of Being.

It is my hope that these two sketches of focus, one by Descartes and one by Spinoza, can help draw out the more refined differences of both philosophers, along an analogy of sight.

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

[addendum: in addition to these thoughts, the influence of a more recent source, James Gregory’s Optica Promota (1663) has to be considered]

Kepler and How Spinoza Viewed the Eye and Light

As a point of reference it is important to locate the origin of Spinoza’s phrase “cones of rays” found in his letter 40, since implicit in this phrase is likely the conception of light and refraction which would help us make sense of his objection to Descartes. This phrase has a history of what seems a bit of interpretive confusion, for instance, that expressed by Alan Gabbey in his Cambridge Companion to Spinoza article, “Spinoza’s Natural Science and Methodology”. Here professor Gabbey quotes the phrase as if it embodies the locus of Spinoza’s befuddlement:

Spinoza explained that light rays from a relatively distant object are in fact only approximently parallel, since they arrive as “cones of rays” from different points on the object. Yet he maintains the same property of the circle in the case of ray cones, apparently unaware of the importance of the “[other] figures” [the famous “Ovals of Descartes”] that Descartes had constructed in Book 2 of La Géométrie to provide a general solution to the problem of spherical aberration [Ep 40].

I have already pointed out that Spinoza indeed was not “unaware” of the “importance” of Descartes’ figures (since he was intimate with the debate over that importance), and that part of Gabbey’s difficulty may stem from a weakness in translation, or not taking into account Spinoza’s familiarity with Hudde’s Specilla circularia: here. Spinoza, all the same, is constructing an argument that seems to shift parameters. In Letter 39 he speaks of the capacity of spherical lenses to focus parallel rays to an (approximate) point of focus opposite, along an infinity of axes, and now he tells Jelles that this capacity is to be understood not for parallel rays, but for “cones of rays”, which is more accurate to what is actually occurring. Where does Spinoza get his conception of “cones of rays”?

I believe it is found in Kepler’s Paralipomena to  Witelo (1604), a work I am beginning to suspect holds some of Spinoza’s resistance to Descartes. Descartes called Kepler his “first teacher” in optics, so when there is a divergence between the two, Kepler and Descartes, one may perhaps look to Kepler as a source for other resistance to Descartes’ conclusions. (It is a mistake to assume that solely in terms of temporal advancement, all of Descartes deviations from Kepler are corrections, for in some ways Kepler held views antecedent to our better conceptions on the nature of light.) In letters 39 and 40 Spinoza is critiquing Descartes explanation of how image size is produced in telescopes, and he finds in Descartes’ explanation some delinquencies which give undue favor to the hyperbola. Where Spinoza likely draws his conception of “cones of rays” is where Kepler is discussing the manner in which images are formed in the human eye:

Now in order to approach closer to the way this picturing happens, and to prepare myself gradually for the demonstration, I say that this picturing consists of as many pairs of cones as there are points in an object seen, the pairs always being on the same base, the breadth of the crystalline humor, or making use of a small part of it, so that one of the cones is set up with its vertex at the point seen and its base at the crystalline (though it is altered somewhat by refraction in entering the cornea), the other, with the base at the crystalline, common with the former, the vertex at some point of the picture, reaches to the surface of the retina, this too undergoing refraction in departing from the crystalline. And all the outside cones come together at the opening of the uvea [pupil], at which space the intersection of the cones takes place, and right becomes left..

…[now speaking of a single cone of those cones of rays] Thus those rays which previously were spreading out in their progress through the air, are gathered together now that they have encountered in to the cornea, so much so that any great circle described by those rays upon the cornea, which in their decent touch the edges of the opening is wider than the circle of the opening of the uvea; however, these rays, all the way to the opening of the uvea, are so strongly gathered together through such a small depth of the aqueous humor, that now the edges of that opening are trimmed of by the extremes, and by that decent they have made illuminous a portion on the surface of the crystalline humor, if indeed they all have first arisen at a point at a certain and proportionate distance (which is pecular to each eye, and not just the same for all), they fall approximately perpendicularly, because of the similar convexity of the cornea and the crystalline humor. (trans. Donahoe, 170)  

Included in this reference is also the obvious fact that for an object to be seen, light from all its points must be gathered. It is part of Kepler’s picture:

Spinoza writes: “…in order to see an entire object, we need not only rays coming from a single point but also all the other rays that come from all the other points. And therefore it is also necessary that, on passing through the glass, they should come together in as many other foci.”

Because Spinoza is arguing that the hyperbolic lens – designed to receive rays solely parallel to its one axis – is insufficient for the variety of angles at which light arrives, the question of parallel or coned rays does not seem germane to his argument. His emphasis in the original description seems meant to be in terms of axes, assuming a “mechanical point” of focus definition. (Whether it is ultimately germane to contemporary telescope construction is another question.)

It must be noted, though here is both a most significant implication of the cone of light having a spherical (wave?)front, something ungrasped by Descartes but captured later by Huygens, in the text that follows as Kepler closely describes this action of cones of rays in the eye, he emphasizes the “hyperbolic posterior surface of the crystalline” (171), possibly disturbing the cohesion of Spinoza’s purely spherical ideal of light refraction. If indeed Spinoza is taking Kepler’s description as his source, this gives us to consider how Spinoza might mean the inexactness of the construction of the eye (letter 40). In what way can the eye be considered imperfect, and is there a Kepler source for this notion?

Spinoza writes: “And although the eye is not so exactly constructed that all the rays coming from different points of an object come together in just so many foci at the back of the eye, yet it is certain that the figures that can bring this about are to be preferred above all others.”

There is an antecedent to this in Keplers’ description of the action of rays as they come from cones at angles oblique to the axis of the cornea:

All the lines of the direct cone [a cone whose axis is the same as the axis of the cornea and crystalline] are approximately perpendicular to the crystalline, none of those of the oblique cones are, The direct cone is cut equally by the anterior surface of the crystalline; the oblique cones are cut very unequally, because where the anterior surface of the crystalline is more inclined [aspherical], it cuts the oblique cone more deeply. The direct cone cuts the hyberbolic surface of the crystalline, or the boss, circularly and equally; the oblique cone cuts its unequally. All the rays of the direct cone are gathered together at one point in the retina, which is the chief thing in the process; the lines of the oblique cones cannot quite be gathered together, because of the causes previously mentioned here, as a result, the picture is more confused. The direct cone aims the middle ray at center of the retina; the oblique cones aim the rays to the side

…so the sides of the retina use their measure of sense not for its own sake, but whatever they can do they carry over to the perfection of the direct vision. That is we see an object perfectly when at last we perceive it with all the surroundings of the hemisphere. On this account, oblique vision is least satisfying to the soul, but only invites one to turn the eyes thither so that they may be seen directly (174). 

Here Kepler seems to be making the exact same point as Spinoza, with an additional hint towards the necessity of the oblique in Spinoza’s concern. The construction of the eye, in so far as its lenses are aspherical, it is retarded its capacity to handle the focus of cones of rays oblique to its single axis. This first calls our attention to the limits of human vision (in individuals and in plan), and then suggests that Spinoza’s point is one of practical application in terms of lenses: that in aiding human vision and constructing telescopes, the symmetry of spherical lenses is preferred for magnification, handling a greater variety of angles of incidence through its infinity of axes.

This does not of course establish the veracity of Spinoza’s argument, but in locating a likely origin for Spinoza’s conception, we at least place Spinoza’s argument within the context of a larger view, to be weighed with all other anti-hyperbolic (Cartesian) positions  of his day (Hudde, Huygens). As I have said, it is my sense that Spinoza derives more than this from Kepler’s account of light. More posts to follow.

A Conflation of Spinoza Diagrams

How Spinoza Thought of the Eye, the Lens and The Modes

Perhaps this is an irresponsible and trite comparison, but sometimes the mind indeed works visually, even in authors as exacting and deductive as Spinoza attempts to be. It is striking that Spinoza uses two very similar diagrams to illustrate on the hand, the powers of spherical lenses to most ideally focus rays across an infinity of axes, (the manifestation of which is subject to the properties of real lenses):

Text not available

Letter 39 to Jelles, March 3rd 1667
Benedicti de Spinoza opera quotquot reperta sunt quotquot reperta sunt By Benedictus de Spinoza, Baruch Spinoza, Johannes van Vloten, Jan Pieter Nicolaas Land

Depicted above are the hypothetical intersection of rays, in two sets taken to be parallel, as they arrive at the surface of a spherical lens. Such rays are taken to be then focused at the back of the circumfrance, as the would be at the back of the eye, or as part of the refractions of a lens.

 

In this diagram, Spinoza illustrates how each contingently expressive mode – what is usually taken to have come into existence and then will pass away – are implied by, that is caused by as immanent to, the Idea of an infinity of points that make up a circle. In this way, the rectangles that are immanent to a circle’s circumfence are by analogy seen to be dependent upon that circle. The rectangles come and go, the circle remains eternal. As explained in Ethics IIp8s:

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.

There is the simple coincidence of using a circle to diagram both physical effects, and metaphysical effects (which for Spinoza are of course commensurate). But if one allows a conflation, one that may have occurred within Spinoza’s thinking, in the first we have the effects what occur within the eye, as it interacts with events outside of it, and in the second, we have the effects (modes) as expressed immanent to the circle that contains them.

Because Adequate Ideas are understood by Spinoza to be Ideas uncaused by something external to them, I don’t think it is too big of a leap to understand that when Spinoza is diagramming the effects of light with the eye (and for a lens, post-angle of incidence), he is thinkingof the second diagram. It is perhaps for this reason that Spinoza is not obsessed with the crystality of vision that occupied Descartes in his quest for the hyperbolic lens. The sharpness of an image is but a part played in an assemblage of knowledge. However clearly one’s eye, or lenses work, this simply is not clear thinking. Of course Descartes understood this as well, but there is something to how Descartes and Spinoza each responded to spherical aberration which reveals a difference of emphasis in the very project of mental and physical liberation. I believe in this co-incidence of diagrams, a profound conflation is being accomplished in Spinoza’s process of thinking.

I see hear as well an interesting graphic subsumption of the scattering of rays that occur with spherical aberration, as in being focused they tend about a “mechanical point” [Johannes Hudde]. Much as rays are never entirely focused to a mathematical point (even with real, hyperbolic lenses), so too we never possess wholly adequate ideas. The focus rays as seen in the first diagram (again, if we allow an analogical thought), appear to enact indices found in the second diagram. Is Spinoza at some level conceiving of rays of focus as being parallel to the adequacy of ideas? And is Spinoza’s theoretical acceptance of spherical aberration [a la Hudde] a product of his acceptance of the fundamentally inadequate nature of ideas we hold? Is his mechanical project of lens focusing analogous to a mechanical – that is, pragmatic, rational and crafted – construction of human freedom? These are large and obscured questions.

This certainly does not make up an argument either for Spinoza’s position, or for an interpretation of Spinoza’s position. It is really more an intuition into the kinds of thought processes Spinoza may have been engaged in, in part elicited by the diagrams he used to make things clear. Meant is a direction for future analysis.