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Category Archives: Descartes

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.

Spherical Aberration: Descartes’ Solution

Philosophical Context: Optics

For those unclear about what spherical aberration is, and attempting to follow Spinoza’s comments about Descartes below, it is the tendency of rays at the edge of a spherical lens to refract at a focal length shorter than those near the axis:

This tendency is corrected by a hyperbolic lens, as theorized by Rene Descartes, which in its progressive sloping, sends the rays to one mathematical point.

It was Spinoza’s thought, probably working from the theories of mathematician Johannes Hudde, that light rays gathered at what he called a “mechanical point” and not a mathematical point. So spherical aberration was considered inconsequential, a fact of working with lenses and light. (A practical mid-century address of spherical aberration in telescopes was the use of much larger, flatter lenses, with longer focal lengths, making telescopes sometimes reach lengths over 30 ft.)

Unfortunately, for the greater part of the 17th century this spherical aberration effect was confused with, and assumed to be the cause of, chromatic aberration which produced a bluish halo at the edges of images, and which is, in most cases, more disruptive than spherical blurring. Not until Newton discovered the spectrum components of light from 1670 – 72 was it theoretically realized that the one was not the other. Even Christiaan Huygens, as he worked to resolve spherical aberration solely with spherical lenses, probably thought that he was resolving what was latter understood to be chromatic, as he was somewhat deflated by Newton’s discovery:

Descartes as a Scotist

I post here the second chapter of professor Roger Ariew’s Descartes and the Last Scholastics. Importantly it presents the historical and argumentative support for Descartes haveng significant debt to the Scottist strain of Scholasticism which was dominant in Paris at the time of his education and early life. This adds somewhat to the context of Deleuze’s conclusion that Spinoza was Scotist in thought, and gives added significance to Behan’s and Yolton’s suggestion that due to the scholastic conception of “sign” a representationalist reading of Descartes’ notion of “idea” is not complete.

Descartes and the Scotists

Gilson’s Index

To date, the most substantial works on the intellectual relations between Descartes and his predecessors have been Etienne Gilson’s masterful studies.1 In the Index scolasticocartésien, Gilson catalogued various concepts in Descartes and matching ones in his scholastic predecessors. Gilson’s choice of antecedents was carefully chosen. He compared Descartes’ works with those of Thomas Aquinas, the Jesuits of the University of Coimbra, Francisco Suarez, Franciscus Toletus, Antonius Rubius, and Eustachius a Sancto Paulo.2 As Gilson indicated in his introduction to the Index scolastico-cartésien, the teaching at Descartes’ Jesuit college, La Flèche, was based on Saint Thomas, and Descartes continued to consult Thomas throughout his life. Further, Descartes became acquainted at La Flèche with the works of the Coimbran Jesuits, Toletus, and Rubius. Gilson defended the choice of Suarez by indicating that Descartes was familiar with his work-that Suarez’s Disputationes metaphysicae was basically the handbook in metaphysics for Descartes’ teachers. Read more of this post

Slightly, Re-evaluating Descartes

The Flexed Lens of Hyperbolic Doubt, as it Imaginatively Focuses the MInd


Instead of seeing Descartes as the harbinger of the tremendous severing of the Body and the Mind, as philosophy can be thought to have suffered over the centuries that followed, there are more subtle readings that grasp the cohesive project that Descartes attempting, one in which the imagination is seen to play a role in rational understanding. Such a take remains critical, but at a level which is more nuanced in the historical contexts of his ideas, while understanding the breadth of Descartes’ vision of how things cohere.

An important if prospective conclusion in concert with Descartes’ re-evaluation is reached by Graham Burnett, as he places Descartes’ pursuit of hyperbolic lenses in tentative relation to the use of hyperbolic doubt (I find this connection to be brilliant):

Descartes’ greatest philosophical success lay, from his perspective, in a systematic investigation of the human mind and the perfection of cognitive operations those investigations promised; that human mind received, via natural light of reason, an instantaneous, clear, and distinct illumination, but only by means of interposition of another hyperbolic focusing device – hyperbolic doubt…

…In Gaukroger’s reconstruction of Descartes’ psychology [Descartes, an intellectual biography ], a quite elaborate extension of the hyperbolic (lens)/hyperbolic (doubt) analogy would be possible. In Gaukroger’s reading, the imagination mediates between the pure intellect and the realm of the senses, and the experience of cognition inheres in this intermediate faculty, which represents the content of the intellect and the content of the senses both as “imagination.” Where these two map onto each other the experience is that of “perceptual cognition.” As the project of hyperbolic doubt is abundantly imaginative, and as Descartes has insisted that the natural light of reason does not stream down from God but is within our intellects, it would be possible to argue that the imagination plays the role of the focusing of the hyperbolic lens, and receives the light emanating from the intellect, which normally enters the imagination confusedly, quickly distorted by the “blinding” profusion of imagery from the senses.

Descartes and the Hyperbolic Quest

Compare this conclusion to Augustine’s own Neo-Platonic conception of our own self-knowledge, Augustine whose “Si fallor sum” preceded Descartes “Cogito ergo sum”, and we can see the legs of this approach in relation to an pervasive conception of the divine:

For we exist and we know that we exist, and we take delight in our existence and our knowledge of it. Moreover, in respect of these three things of which I speak [a trinity of being, knowing and loving], no falsehood which only resembles the truth troubles us. For we do not make contact with these things by means of our bodily senses, as we do in the case of things extrinsic to ourselves…[in] these cases it is the images resembling the sensible objects, but not the corporeal objects themselves, which we perceive in the mind and retain in the memory, and which excite us to desire the objects…

City of God against the Pagans, Book XI, Chapter 24

The much defamed “doubt” of Descartes really is not so much a doubt for skeptical doubt’s sake, or even a doubt played as a pretense for the foundation of an augment; it rather acts as a kind of imaginary corrective to sense images and experience itself, a use of the imagination upon the imagination, something that focuses the mind on just what is most sure, under a conception that reason is something that both resides and connects. When seen in this way, the division of mind and the body becomes not only ludicrously joined by the pituitary gland, but also by the imagination itself.

Descartes Sans Homunculus!?

One might productively add to this John Yolton’s reconfiguration of Descartes’ project to be one of an immediate Realism, and Natural Philosophy. Here, the scholastic division of the sign’s two parts, that of its signfication, and that of its representation, promises to free up the cliched reading of Descartes as harboring the perverse theoretical imp of an infinity of homunculi buried inside the head. As David Behan points out, scholastic formal signs (ideas) can be read by minds entirely without awareness. Representation, per se, no longer becomes the standard for Descartes’ notion of knowledge. A Few Selections…

The being of an object of the mind is epistemic; it is (in a phrase that I picked up from Norman Wells) the being of being known. The epistemic rendering of “being in the mind” is an important shift from an attempted ontic transfer of an objects reality to a cognitive transfer. The explication of “existence in the mind” does not only occur in Descartes. Behan calls attention to a passage from William of Auvernue which employs the same language, “What it does mean is that it is in the soul according the mode of the being of the soul, which is cognitive”…

David Behan interprets Descartes’ brain motions as formal signs. In support of this interpretation, he refers to the scholastic tradition just behind Descartes, a tradition to which Descartes must have been exposed. As Behan explains, formal signs in that tradition are not themselves known, they signify without without or being aware of them. If we read Descartes’ suggestion of brain motions as signs in this way, the supposed need, which commentators are fond of insisting upon, for a code-reader or, as Wolf-Devine repeatedly says, a homunculus, does not arise…

As a mode of mind, an idea does not…make ‘something other than itself come into the mind’. If an idea represents or if,…the act of cognizing by means of ideas does the representing (the combination of act and idea), in that function, ideas are not ideas as such. That is, in that representing function, they are not modes of mind. I do not suppose that there are any ideas on Descartes’ account that are only modes of mind in the narrow sense I am suggesting. I simply want to distinguish their nature as modes of mind from their nature of function as objectively real. It is this objective reality that is in some cases (e.g. the idea of God, some physical objects) caused by something other than the mind. Ideas as objectively real (or the combination of act and idea) do not play a sign role: they simply are the objects, that which is known.

There is an interesting similarity in Descartes’ account of brain motions and ideas: both play two roles or have two function. Brain motions are both physical events and signs carrying meaning. The motions become something other than motion. Ideas are ideas and objects, modes of mind and the object known. In this secondary role, ideas are something other than ideas. Brain motions become signs to a mind. Signs must refer beyond themselves. Ideas as objects do not really refer beyond themselves on Descartes’ account: they are the objects known. Thus the relation or function of representation is not a signifying relation, signifying differs from representing. Both are necessary for knowledge and perceptual awareness. To represent is to be that which is represented. The combination of signifying and representing ‘gets the object into the mind’, that is, makes the object known.

John Yolton, “Response to Fellow Symposiasts” found in, Descartes’ Natural Philosophy

I think sometimes we moderns, (even we post-moderns), are in the habit of setting up our grand narratives. And in our story about the errors of our historic ways Descartes has come to play the conveniently villanous role that makes any good story worth telling. He plays this role in a curious way though, in particular in the form of the rather easily used and ubiquitous adjective “Cartesian”. We should watch just how satisfying this word is, how simplifying. It is tossed about in Philosophy of Mind and in so many other fields with remarkable reassurance. In regards to it, there is supposed to be a neat and tidy error – some want to call it a irrefutable sounding “categorical error” – which is consistently present in Descartes’ program, and ferreting out this error (or even defending it obstinately) wherever we may find it makes up a very good portion of our philosophical endeavors.

In such a perspective Spinoza can be of very good use for he represents a turning point just before Idealism took up and swallowed the Cartesian poisoned pill. Descartes severed the Mind from the Body, but Spinoza just would not let him. I do think that there is much to be said for such a broad brushing of philosophical history on the West, and even for the very useful distinctions which underpin it. But I also suspect that Descartes’ thought holds within itself much more subtlety and tension that is otherwise granted. Representation simply does not hold such a privledged, and pristine place in Descartes’ thinking about knowing. And in this way, Spinoza’s thought, in relation to Descartes, is perhaps more complex and sympathetic than we otherwise might suspect.

This does not make Spinoza’s thinking “Cartesian” – that adjective again ! how it works something like the words “Communist” or “Racist” – for I am not even sure how frequently we can be assured that Descartes is entirely “Cartesian”; but it does make the connections between the two thinkers more imbricated than a simple comparison of a position of Attributes affords. I suspect that in the grey penumbra of Descartes, somewhere in Descartes’ conception of the blind man’s cane, for instance, in the folds of his treatment of the Imagination, and in the signifying, homunculus-defying aspects of Idea, there are sweet-spots of affinity between the two that may be good to trace.