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Spinoza and Mechanical Infinities

The Mechanically Bound Infinite

I want to respond to Corry Shores’ wonderful incorporation of my Spinoza Foci  research into his philosophical project (which has a declaimed Deleuzian/Bergsonian direction). It feels good to have one’s own ideas put in the service of another’s productive thoughts. You come to realize something more about what you were thinking. And to wade back through one’s arguments re-ordered is something like coming to your own house in a dream.

This being said, Corry’s reading of my material thrills, for he is, at least in evidentary fashion, one of the first persons to actually read it all closely. And the way that he fits it in with his own appreciation for Spinoza’s concepts of Infinity certainly open up new possibility for the Spinoza-as-lens-grinder, Spinoza-as-microscope-maker, Spinoza-as-technician interpretations of his thinking.

There is much to take up here, but I would like to begin at least with the way in which certain parallels Corry draws that change the way that I see what Spin0za was saying (or more exactly, what Spinoza was thinking of, and perhaps associating on), when talking about infinities. Key, as always, is coming to understand just what Spinoza had in mind when drawing his Bound Infinities diagram:

Corry points out in his analysis/summation of Letter 12, grafting from Gueroult’s commentary, that in order to understand the epistemic point (the status of mathematical figures, and what they can describe), one has to see that what Spinoza as in mind in writing to Meyer is a very similiar diagram found in his Principles of Cartesian Philosophy, of which Meyer was the active editor. There the diagram is not so Euclidean, but rather is mechanical, or, hydro-dynamical:

The diagram illustrates water moving at a constant rate (a “fixed ratio” one might say), but due to the nature of the tube it must be moving at point B, four time faster than at AC, and a full differential of speeds between. There you can see that any section of the intervening space between the two circles composed of “inequalities of distance”  in the Letter 12 diagram (AB/CD) is not really meant as an abstraction of lines and points as it would seem at first blush (the imaginary of mathematics), but rather real, mechanical differentials of speed and material change. The well-known passage

As, for instance, in the case of two circles, non-concentric, whereof one encloses the other, no number can express the inequalities of distance which exist between the two circles, nor all the variations which matter in motion in the intervening space may undergo. This conclusion is not based on the excessive size of the intervening space. However small a portion of it we take, the inequalities of this small portion will surpass all numerical expression. Nor, again, is the conclusion based on the fact, as in other cases, that we do not know the maximum and the minimum of the said space. It springs simply from the fact, that the nature of the space between two non-concentric circles cannot be expressed in number.

Letter 12

The Lathe Buried Under the Euclidean Figure

But, and this is where Corry Shores alerted me to something I did not formerly see, the relationship between the two diagrams is even further brought forth when we consider Spinoza’s daily preoccupation with lens-grinding and instrument making. It has been my intuition, in particular, that Spinoza’s work at the grinding lathe which required hours of patient and attentive toil, MUST have had a causal effect upon his conceptualizations; and the internal dynamics of the lathe (which fundamentally involve the frictioned interactions of two spherical forms under pressure – not to mention the knowing human eye and hand), must have been expressed by (or at least served as an experiential confirmation of) his resultant philosophy. If there was this heretofore under-evaluated structuring of his thought, it would see that it would make itself most known in his Natural Philosophy areas of concern, that is to say, where he most particularly engaged Descartes’s mechanics (and most explicitly where he refused aspects of his optics, in letters 39 and 40). And as we understand from Spinoza’s philosophy, Natural Philosophy and metaphysics necessarily coincide.

What Shores shows me is that Spinoza’s Bound infinities diagram (letter 12), his very conception of the circle, is intimately and “genetically”  linked to the kinds of motions that produce them. It is with great likelihood that Spinoza is thinking of his off-center circles, not only in terms of the hydrodynamics that circulate around them, but also in terms of Descartes’ tangents of Centrifugal force.

There is a tendency in Spinoza to conflate diagrams, and I cannot tell if this is unconscious (and thus a flaw in his reasoning process) or if he in his consummatephilosophy feels that all of these circular diagrams are describing the very same thing simply on different orders of description. But the connection between a tangential tendency to motion conception of the circle (which Corry makes beautifully explicit in terms of optics) and Spinoza’s consideration of bound Infinities in the letter 12 (which remains implicit in Corry’s organization of thoughts), unfolds the very picture of what Spinozahas in mind when he imagines two circles off-center to each other. Spinoza is thinking of is lens-grinding blank, and the spinning grinding form.

One can see the fundamental dynamic of the lathe from Van Gutschoven’s 1663 letter to Christiaan Huygens, illustrating techniques for grinding and polishing small lenses,

And it is my presumption that Spinoza worked at a Springpole lathe, much like one used by Hevelius, Spinoza’s Grinding Lathe: An Extended Hypothesis, the dynamics of which are shown here:

In any case, when one considers Spinoza’s Bound Infinity diagram, under the auspices of tangential motion tendencies, and the hydrodynamic model of concentric motions, I believe one cannot help but also see that the inner circle BC which is off-center from the first, is representationallythe lens-blank, and the larger circle AD, is potentially the grinding form. And the reason why Spinoza is so interested in the differenitals of speed (and inequalities of distance) between two, is that daily, in his hand he felt the lived, craftsman consequence of these off-center disequilibria. To put it one way sympathetic to Corry’s thinking, one could feel them analogically, with the hand, though one could not know them digitally, with math. The human body’s material (extensional) engagements with those differentials (that ratio, to those ratios), is what produced the near perfectly spherical lens; and the Intellect intuitionally – and not mathematically – understands the relationship, in a clear and distinct fashion, a fashion aided by mathematics and figure illustration, which are products of the imagination.

What is compelling about this view is that what at first stands as a cold, abstract figure of simply Euclidean relationships, suddenly takes on a certain flesh when considering Spinoza’s own physical experiences at lens-grinding. Coming to the fore in such a juxtaposition is not only a richer understanding of the associations that helped produce it, but also the very nature of Spinoza’s objection to the sufficiency of mathematical knowledge itself. For him the magnitudes of size, speed and intensity that are buried between any two limits are not just abstract divisions of line and figure, or number to number. They are felt  differentials of real material force and powers of interaction, in which, of which, the body itself necessarily participates. The infinities within (and determinatively outside of) any bound limits, are mechanical, analogical, felt and rational.

Corry raises some very interesting relationship question between the Spinoza Bound Infinities Diagram and the Diagram of the Ideal Eye from letter 39. They are things I might have to think on. The image of the ideal eye is most interesting because it represents (as it did for Descartes) a difficult body/world shore that duplicates itself in the experiential/mathematical dichotomy. Much as our reading of the duplicity of the Bound Infinity Diagram which shows mathematical knowledge to be a product of the imaginary, the diagram of the ideal eye, also exposes a vital nexus point between maths, world and experience.

From Mechanics to Optics (to Perception)

It should be worthy to note that Spinoza’s take on the impossibility of maths to distinguish any of the bound infinities (aside from imposing the bounds themselves), bears some homology to Spinoza’s pragmatic dismissal of the problem of spherical aberration which drove Descartes to champion the hyperbolic lens. When one considers Spinoza’s ideal eye and sees the focusing of pencils of light upon the back at the retina (focusingswhich as drawn do not include the spherical aberration which Spinoza was well-aware of), one understands Spinoza’s appreciation of the approximate nature of perceptual and even mathematical knowledge. This is to say, as these rays gather in soft focus near the back of the eye (an effect over-stated, as Spinoza found it to be via Hudde’s Specilla circularia), we encounter once again that infinite grade of differential relations, something to be traced mathematically, but resultantlyexperienced under the pragmatic effects of the body itself. “The eye is not so perfectly constructed” Spinoza says, knowing as well that even if it were a perfect sphere there as yet would be gradations of focus from the continumof rays of light so refracted by the circular lens. What Spinoza has in mind, one strongly suspects, and that I have argued at length, is that the Intellect, with its comprehensive rational in-struction from the whole, ultimately Substance/God, in intuitional and almost anagogic fashion, is the very best instrument for grasping and acting through the nature of Nature, something that neither bodily perception, or mathematical analysis may grasp. Indeed, as Corry Shores suggests in his piece, it is the very continuum of expressional variability of Substance (real infinities within infinities) which defies the sufficiency of mathematical description, but it is the holistic, rational cohesion of expression which defies experiential clusterings of the imagination: the two, mathematics and imaginary perception, forming a related pair.

In the end I suspect that there is much more to mine from the interelationship between Spinoza’s various circular diagrams, in particular these three: that of the relationship of the modes to Substance (EIIps), that of the the hydrodynamics of circulating water (PCP, implicit in the Letter 12 diagram of Bound Infinites), and the Ideal eye (letter 39), each of these to be seen in the light of the fundamental dynamics of the lens-grinding lathe to which Spinoza applied himself for so many years, and at which he achieved European renown expertise.

 

 

The Infinities Beneath the Microscope

I would like to leave, if only for Corry Shores’ consideration, one more element to this story about Real Infinities (and I have mentioned it in passing before on my blog). There is an extraordinary historical invocation of something very much like Spinoza’s Bound Infinities in the annals of anatomical debates that were occurring in last decade of Spinoza’s life. I would like to treat this in a separate post and analysis, but it is enough to say that with the coming of the microscope what was revealed about the nature of the human body actually produced more confusion than understandings in what it revealed, at least for several decades. Only recently was even the basic fact of the circulation of blood in the body, something we take for granted, grasped. And in the 1670s the overall structure or system of human anatomy was quite contested, contradictory evidence from the microscope being called in support one theory or another. Among these debators was Theodore Kerckring, who was weighing in against the theory that the human body was primarily a system of “glands” (and not ducts). Kerckring’s  connection to Spinoza is most interesting, much of it brought to light in Wim Klever’s inferential and quite compelling treatment of the relationship of Van den Enden  and Spinoza. In any case Kerckring  is in possession of a microscope made by Spinoza (the only record of its kind), and by virtue of its powers of clarity he is exploring the structure of ducts and lymph nodes. Yet he has skepticism for what is found in the still oft-clouded microscope glass leads him to muse about the very nature of perception and magnification, after he tells of the swarming of tiny animals he has seen covering the viscera of the cadaver, (what might be the first human sighting of bacteria). He writes of the way in which even if we see things clearly, unless we understand all the relationships between things, from the greatest breadth to the smallest, we simply cannot fully know what is happening, if it is destruction or preservation:

On this account by my wondrous instrument’s clear power I detected something seen that is even more wondrous: the intestines plainly, the liver, and other organs of the viscera to swarm with infinitely minute animalcules, which whether by their perpetual motion they corrupt or preserve one would be in doubt, for something is considered to flourish and shine as a home while it is lived in, just the same, a habitation is exhausted by continuous cultivation. Marvelous is nature in her arts, and more marvelous still is Nature’s Lord, how as he brought forth bodies, thus to the infinite itself one after another by magnitude they having withdrawn so that no intellect is able to follow whether it is, which it is, or where is the end of their magnitude; thus if in diminishments you would descend, never will you discover where you would be able to stand.

Spicilegium Anatomicum 1670

Several things are going on here (and in the surrounding context), but what seems most striking given our topic, we once again get a glimpse into the material, and indeed historical matterings of what bound, mechanical infinities might be. (As a point of reference, at the time of Kerckring’s  publishing Spinoza had just moved to the Hague and published his Theological-Political Treatise, having taken a respite from his Ethics approximately half done, and he will have died seven years later.) Kerckring  in a remarkable sense of historical conflation looks on real retreating infinities with Spinoza’s own microscope, and exacts much of the same ultimate skepticism toward human scientific knowledge, as per these infinities, as Spinoza  does in his letter to Meyer. This does not mean that we cannot know things through observation, or that imaginary products are not of use to us, but only that there is ultimately for Spinoza and Kerckring  a higher, rational power of interpretation, the comprehensiveness of what abounds. Neither measurement or calculation is disqualified, in fact Spinoza in his letters and experiments and instrument making showed himself to be quite attentive to each. It is rather that the very nature of human engagement requires both attention to the bodily interaction with devices and the measured thing, and also a sensitivity to anagogic, rational clarity, something found in the very unbroken nature of Substance’s Infinity. What Kerckring’s description does is perform the very consequence of conception in scientific observation itself, almost in Spinoza’s stead (expressing very simililar  sentiments as Spinoza does in Letter 32 to Oldenburg on lymph and blood, and the figure of the worm in blood,

Let us imagine, with your permission, a little worm, living in the blood¹, able to distinguish by sight the particles of blood, lymph, &c., and to reflect on the manner in which each particle, on meeting with another particle, either is repulsed, or communicates a portion of its own motion. This little worm would live in the blood, in the same way as we live in a part of the universe, and would consider each particle of blood, not as a part, but as a whole. He would be unable to determine, how all the parts are modified by the general nature of blood, and are compelled by it to adapt themselves, so as to stand in a fixed relation to one another.

There is great conceptual proximity in these two descriptions, suggesting I imagine that Spinoza used his microscopes as well, for observation, not to mention that Kerckring and Spinoza come from a kind of school of thought on scientific observation of human anatomy, perhaps inspired by or orchestrated by Van den Enden, as argued by Klever. Just the same, at the very least, Kerckring  presents greater context of just what kinds of retreating infinities Spinoza  had in mind in his letter 12 diagram, not simply a differential of motions, but also a differential of microscopic magnitudes, each of which were an expression of an ultimate destruction/preservation analysis, something that falls to the very nature of what is body is. Spinoza not only ground lenses, but also made both telescopes and microscopes, gazing through each at the world, this at a time when the microcosmic and macrocosmic, nested infinities were just presenting themselves to human beings. And as such his critique of scientific observation and mathematical calculation preserves a valuable potentiality for our (postish) modern distancings and embrace of the sciences.

Huygens’s Criticism of Descartes’ La Dioptrique

In order to understand Spinoza’s dissatisfaction with and objection to Descartes’ La Dioptrique  (found in letters 39 and 40 linked below), one has to understand the opinions of those contemporary to Spinoza. Below I post a selection from Fokko Jan Dijksterhuis comprehensive book on Christian Huygens, who is well-noted for having been Spinoza’s neighbor in Voorburg.  

La Dioptrique was written, Descartes said in the opening discourse, for the benefit of craftsmen who would have to grind and apply his elliptic and hyperbolic lenses. Therefore the mathematical content was kept to a minimum. Apparently this implied that Descartes need not elaborate a theory of the dioptrical properties of lenses. Descartes adopted the term Kepler had coined for the mathematical study of lenses. He had not, however, adopted the spirit of Kepler’s study. Dioptrice and La Dioptrique approached the telescope from opposite directions. Kepler had discussed actual telescopes and drudged on properties of lenses that did not fit mathematics so neatly. Descartes prescribed what the telescope should be according to mathematical theory. The telescope, having been invented and thus far cultivated by experience and fortune, could not reach a state of perfection by explaining its difficulties. Huygens was harsh in his judgement of La Dioptrique. In 1693, he wrote:

“Monsieur Descartes did not know what would be the effect of his hyperbolic telescopes, and assumed incomparably more about it than he should have. He did not understand sufficiently the theory of dioptrics, as his poor build-up demonstration of the telescope reveals” (OC10 402-403)

We can say that Descartes, according the Huygens, had failed to develop a theory of the telescope. He had ignored the questions that really mattered according the Huygens: an exact theory of the dioptrical properties of lenses and their configurations. La Dioptrique glanced over the telescope that existed only in ideal world of mathematics (37).

Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century, Fokko Jan Dijksterhuis

Here we get a sense of Spinoza’s complaint that Descartes may have avoided certain questions due to mathematical complexity, and that Descartes is not dealing with actual telescopes. It is significant to follow that indeed the analogical form of arguments in the La Dioptrique was designed to be in favor of the craftsmen and artisans who were meant to grind Descartes’ miraculous hyperbolic lens, while in life (and theory), it was the great tension with this human and technical reality that that produced the Cartesian telescope failure.

A line by line interpretation of Spinoza’s Optical letters: Deciphering Spinoza’s Optical Letters

Jerry Rothman’s website dedicated to Kepler’s Dioptrice: here

 

The Hole at the “Center of Vision”

Spinoza as Seer

In a sense, if we are to understand Spinoza’s optical influences we have to come to at least consider what seeing, or more helpfully, perceiving meant for Spinoza, for behind any optical conceptions Spinoza had lies the very act of actively engaging the world. Much as Descartes worked from definitive values of what clear perception was, wrestling with both empirical experiment and mathematical analysis, so too Spinoza held core positions on what clear perception involved, and these factors into the nature of Spinoza’s break with his precursor. It has become my running thought in this research that if we can generalize, Descartes’ model of clear perception involved the hyperbola’s capacity to refract rays come from a single point on an object, to another co-ordinate point on the surface of the back of the eye, and that importantly this point fell upon the central axis of the hyperbola, a mathematical line which expressed, or was the locus of, the human freedom of Will. This point of focus was – at least in the accounts of vision where Descartesis in praise of the hyperbola and the remarkable representational accuracy of the eye – the fulcrum of a naturalized embrace of narrow focus, frontal clarity.

The hyperbolas central point of focus as a model of clarity

The hyperbola's central point of focus as a model of clarity

This seems to be contrasted in Spinoza with an emphasis upon the multiplicity of visual axes that a spherical lens affords (Spinoza’s optical letters are talked about here: Deciphering Spinoza’s Optical Letters ). Spinoza was privileged enough to be familiar with thinkers who argued that spherical aberrationwas over-emphasized as a problem, and he seemed to hold that there was also a problem of “hyperbolic aberration” (my term), that is the inability of such lenses to focus rays cleanly to any points peripheral to the exact center-line of vision. Whatever one is to make of the impact such aberrationwould have had on telescope construction, it is plain that Spinoza’s view of a model of vision was panoramic, that is, anything that had clarity in the center, was clear due to its place within the context of the clarity of all that surrounded. Instead of a vague and confused border of “confused images” which only “serve” the central crispness (Kepler), because Spinoza felt that we looked with the Mind and not the eyes – something that Descartes also argued but withthe burden of theological-theoretical commitments to a free faculty of Will – Spinoza holds that ideal vision embraces the tableau, the scope of things. The hyperbola’s acute focus, as Spinoza understood it, just did not provide the convincing analogical force of what clarity would mean. I think it safe to say, neither thinker, Spinoza or Descartes, had a sure enough idea of what exact effect spherial aberration had on telescopes nor how refractionproduced its images, and it was their different notions of mental claritywhich governed their arguments for ideal lense shapes, filling in the blanks of what was known.

a modified diagram from Spinozas letter 39 designed to bring out the capacity of spherical lenses to focus peripheral rays

a modified diagram from Spinoza's letter 39 designed to bring out the difference between spherical and hyperbolic lens focus as it pertains to peripheral rays

Within this overview of differences, it is worthwhile to consider my guiding assumption of this research: that Spinoza’s experiences as a lens-grinder and instrument maker (not to mention his social standing having come from an artisan class) decisively gave him a craftsman’s appreciation of perception, one that reflects itself in his metaphysis. To get a firmer grasp on what a “craftsman’s appreciation of perception” is, I turned to Richard Sennett’s book on the subject, The Craftsman. There he writes adroitly on the nature of craftsman perceptions, thinking processes and environments, in particular the relationship to tools and on-site difficulties. This has been of great value. In his sum of craftsman perceptions he turns to “cognitive dissonance’ theory to help explain how the craft perception functions. This strikes me to be of use in pointing out just where Spinoza and Descartes seem to optically diverge. Below he discusses the nature of “focal attention” (he mentions two examples he has discussed previously, the house the philosopher Wittgenstein designed and had built, in which he infamously had the ceiling height of a room changed 3 cm, just as the worksite was being cleaned up; and Gehry’s explorations into the processes of forming titanium, designed for the rippling skin of his Bilbao project).

The capacity to localize names the power to specify where something important is happening…Localizing can result from sensory stimulation, when in a dissection a scalpel unexpected hard matter; at this moment the anatomist’s hand movements become slower and smaller. Localization can also occur when the sensory stimulation is of something missing, absent or ambiguous. An abscess in the body, sending the physical signal of a loss of tension, will localize the hand movement…

In cognitive studies, localizing is sometimes called “focal attention.” Gregory Bateson and Leon Festinger suppose that human beings focus on the difficulties and contradictions they call “cognitive dissonances.” Wittgenstein’s obsession with the precise height of a celing in one room of his house [citing that the philosopher had a ceiling lowered 3 centimeters in a house he had designed just the construction was being completed] derives from what he perceivedj as a cognitive dissonance in his rules of proportion. Localization can also occur when something works successfully. Once Frank Gehry could make titanium quilting work [citing the design of a material of specific relfective and textural capacities], he became more focused on the possibilities of the material. These complicated experiences of cognitive dissonance trace directly, as Festinger has argued, from animal behavior; the behavior consists in an animal’s capacity to attend to “here” or “this.” Parallel processing in the brain activates different neural circuits to establish the attention. In human beings, particularly in people practicing a craft, this animal thinking locates specifically where a material, a practice, or a problem matters.

The capacity to question is no less or more a matter of investigating the locale. Neurologists who follow the cognitive dissonance model believe the brain does something like image in sequence the fact that all the doors in a mental room are locked. There is then no longer doubt, but curiosity remains, the brain asking if different keys have locked them and, if so, why.  Questioning can also occur through operational success…This is explained neurologically as a matter of a new circuit connection being activated between the brains different regions. The newly active pathway makes possible further parallel processing – not instantly, not all at once. “Questioning” means, physiologically, dwelling in an incipient state; the pondering brain is considering its circuit options (278-279).

Gregory Bateson

Gregory Bateson

It is not my intention to claim that Spinoza holds a proto-cognitive dissonance theory, though there are some signficant and suggetive correspondences (a social dimension to agreement, determintative conditioning and holistic forces in judgment). Rather, I would like to put it the other way around, to use Sennett’s point about the nature of focal attention to shed some light upon Spinoza’s difficulty in accepting Descartes notion of an ideal crystal clear center of vision. If we simplify, we could say that Descartes was concerned with identifying and constructing means of “clear and distinct” perceptions or thoughts which would define idealvision (mental and otherwise). His engagement with the hyperbolic lens is at least analogically connected to his engagementwithhyperbolic doubt, each designed to focus the mind on a central clarity. What Sennett’s appeal to the craftsman experience of Cognitive Dissonance does is help expose a rift in the very center of focus which Descartes hopes to at least rhetorically stabilize. Focal Attention may be best understood as an irreconcilable line of fragmenting possibility and dys-clarity, and not the consummate moment-after experience of veritability. Modelsof the mind which have most thoroughly drawn upon the visual metaphor for truth mostly have taken the clarity of a perception as the exemplar of correspondence. I see two dogs, and I know that they are twodogs and this clarity is established against a figure-ground constrast. But a Cognitive Dissonance approach seems to suggest is a much finer grain look at what perception is. That is, when our focal attention is turned from this thing to that thing, this aspect to that aspect, it is not clarity which guides our view, but dys-clarity, a fuzziness of the possible and the incomplete. The eye may apprehend the distinction between a figure and the ground but it does not stop there; it continues to trace the significance of relation of elements that both compose the figure, and distinguish it from its context. The processes that give birth to a single distinction carry on in a relational, distinguishing manner. This destabilization of the center, and its resolution through its coherence with a whole, I believe is expressed in several ways in Spinoza thinking.

The depth of field analogy for knowing and Being

The depth of field analogy for knowing and Being

But first a short defintion of the concept

Cognitive Dissonance: “the uncomfortable tension that comes from holding two conflicting thoughts at the same time, or from engaging in behavior that conflicts with one’s beliefs. More precisely, it is the perception of incompatibility between two cognitions.”

Spinoza and the Trace of Consciousness: the grain of wood

Sennett is concerned with the experiences and perceptions which guide the craftsman through his work, the careful notice of differences in materials, possibilities and designs, how a hand passes over wood grain or the mind might connect one part to another part. It is the waythat the mind glides over difficulties and solutions. Taken in its visual state, it is the way that the eye focuses upon this or that, leading itself across the bed of differences. And it is my intuition that Spinoza’s lived practices with craft that gave him a distinct sense of what it means to perceive and distinguish.

What is necessary is to establish just what it is that lies at the center of focus, if it not a crystalline clarity. And there are two selections of the Ethics which I have in mind in response to the Cognitive Dissonance lead. The first is Spinoza’s maxim concerning what it is that we imagine to be the case. It is important to realize that when Spinoza talks about the imagination, he means a confluence of both sensory experiences, and the beliefs we form about them, so much so the latter cannot be separated out from the former. The ideas we hold – or more properly, the ideational states we are in – determine our imaginary, phenomenological experience of the world.

Spinoza writes in part 3 of the Ethics, proposition 12:

The Mind, as far as it can, strives to imagine those things that increase or aid the Body’s power of acting.

This is the key proposition on which Spinoza will found our imaginary relationship to both to the world, but also to others. Through the mind’s force of positive imagining, it brings coherence to our body’s relationships, and thus improves our ability to act more freely (as our own cause). He continues to explain in the demonstration:

Hence, as long as the mind imagines those things that increase or aid our body’s power of acting, the Body is is affected with modes that increase or aid its power of acting, and consequently the Mind’s power of thinking is increased or aided.

In Spinoza’s view, though our relationship to the world may be imaginary (that is, we may not fully understanding the causes and effects involved), if we imagine a relation which improves our power to act, we will experience Joy (defined as an increase in this power, DOA 2), and thus the Mind will tend to continue to imagine in this fashion. Any imaginary improvement, if it results in Joy, is also an improvment in the power of thinking, and thus there is an imaginary, though non-optimal, path to greater power and freedom.

Hopefully the rough connection to Cognitive Dissonance theory will be seen. There is a tendency in perception and belief which determines the mind to think in a more coherent fashion. When there is dissonance – that is, a disjunction between one’sown ideationaland physical states and the states of the world – the imaginary value is to resolve this. In a sense, the imaginationis guided by the resolution of a center of dissonance, bringing the body into concert with its own powers as far as it understands them. (I leave aside the ladder of rational, causal understanding.) 

For Spinoza there is a cohering balast that centers the processes of imaginary experiences of the world. This is reflected in the most characteristic experiences of consciousness, the passing from one thought to other, as if in a chain. When Spinoza presents his General Definition of the Affects, he radically asserts that our chain of thoughts, most generally, are the result of the Mind affirming one state of the Body or another, such that each affirmation leads either to an increase or decrease of the power to act. These changes are the result of affects which express the adequacy of the ideas which compose our mind:

E3: General Defintion of the Affects: An affect, that is called a Passion of the mind is a confused idea, whereby the Mind affirms concerning its Body, or any part thereof, a force for existence (existendi vis) greater or less than before, and by the presence of which the mind is determined to think of this rather than that.

Exp: I say, first, that an Affect or passion of the mind is a confused idea. For we have shown that the mind is only passive, in so far as it has inadequate or confused ideas. (E3P3)
I say, further, whereby the mind affirms concerning its body or any part thereof a force for existence greater [or less] than before. For all the ideas of bodies, which we possess, denote rather the actual disposition of our own body (E2P16C2) than the nature of an external body. But the idea which constitutes the reality of a passion must denote or express the disposition of the body, or of some part thereof, which is possessed by the body, or some part thereof, because its power of action or force for existence is increased or diminished, helped or hindered.
But it must be noted that, when I say a greater or less force for existence than before, I do not mean that the mind compares the present with the past disposition of the body, but that the idea which constitutes the reality of an affect affirms something of the body, which, in fact, involves more or less of reality than before.

There are two significant aspects of this definition I would like to point out. The first is the ateleological view Spinoza takes toward these kind of ideational affirmations of the body. The mind does not arrive at its present affirmation state through a comparison of a present state with a past one, but rather makes of its present existence a repeated and continual “concrescence” (to borrow wrecklessly from Whitehead’s wordsmithing). Any perceptual grasp of the world, insofar as it involves a shift in degrees power and Joy, can be seen as coming from a comprehensive grasp in a sphere of understanding. To put it another way, if we adopt the cognitive dissonance model of perception and belief holding, the running line of potentiated dissonance which guides and centers our focal points of attention becomes repeatedly resolved in the affirmative embrace of a perception/thought/state of the body, made in context with the whole. Clarityarrives not due to the crispness of an axis of perception, but due to the resolution of that line within the panorama either of the visual tableau, or the ideas we hold. Seeing something clearly, thus, is fundamentally a connective and comprehensive apperception.

The ultimate perception is, as Spinoza argues, the perceptual Idea of God, one whose scope and speed of embrace brings clarity to all other affects and imaginations. From the 5th part of the Ethics:

P13: The more an image is joined with other images, the more often it flourishes.

P14 The Mind can bring it about that all the Body’s affection, or images of things, are related to the idea of God.

The second aspect of the General Defintion of the Affects I want to point out is that the chain of thoughts which make our everyday consciousness are not centered upon a Will which controls them, but rather are an expression of the ideas that make up our MInd; thus our ideational states determine the line of imaginary and cognitive processes which include our visual perceptions (clarity of perception cannot be the model of knowing), yet only insofar as these are understood as affirming our physical states. There is no center of vision nor of judgment. Rather there is the confluxof repeated changes in the power to act, something that reveals itself not in binary of Being and Non-Being, but along a gradated spectrum of Being, wherein the power one has is a function of the degree of Being one has.

All this proceeds too fast, for I have not properly connected Spinoza ateleological, affirmational understanding of perceptions and thought-chains to the kinds of curiosity and tensions that arise even the the smallest of conscious distinctions. What a Cognitive Dissonance model of perception and belief provides, I have suggested, is the idea that there is a fissure at the center of the eye’s focus, and that this rift is only closed through the coherent orientation to our experiences at the edge of that rift, in relation to all that lies at the margins. Any philosophical view that in a binary strictly equates focal clarity with Being, and all else with Nothingness or Non-Being, does not fully appreciate the recommendations that a metaphor of visual experience would provide; for at the very center of the eye, if we follow Spinoza’s thinking, lies not the undoubtable truthof one proposition, or the pure assurance of an object seen, but rather the living line of the electric destablized possibility for greater Joy or freedom. Perceptions are a body’s forward lean. In Spinoza’s terms, this line is the shore-point of our realized power to act, and thus occurs along the affects we experience, as they are expressed in both the ideas that make up our mind, and the states our body is in. The very center of focus is our fluxuations in perfection and Joy.

Descartes not Representational Despite His Love of Lens

Now at this point really I would like to take the opportunity to make clear that I have for the sake of contrast been unfair to Descartes, for by and large when he seeks clear thoughts he does not have in mind a clarity which operates independently of other understandings. He, like Spinoza, sees a global and connective sense in truth, one which puts any clear perceptions of the world in the context of the natural dispositions of the Intellect and our soul’s relationship to God. His use of skepticism and doubt is likely at most pedagogical. There has been too much groping at what has become a cadaver of Descartes’ notions of Ideas, without notice of the living relationship such concepts hold in his overall natural science and theological scheme. Nadler, Yolton and Behan (his new piece “Descartes’s Semiotic Realism” forthcoming), all have worked to show that most of our modern conceptions of Descartes’ Representationalism are ill-considered, forwarded by a chain of deformations: first Malebranche, then Reid, and lastly to great effect, Rorty. Much of what we rail against as invidiously “Cartesian” is not really something Descartes would champion. I think the arguments of Nadler et al are very well taken, and expose a tendency of philosophy, for all its sophistication, to organize itself around oppositions simple to grasp. And thus it does us some good to look closer at the forefather of the great Substance divorce between the Mind and the Body.

This is a strange thing to say, considering that much of my contrast between Spinoza’s view of perception and Descartes’ view seems to rely upon representational models of what is known. Spinoza objects to the representational notion of clarity, what he calls “falling into pictures” because he feels that representation simply is inadequate to express what happens when we hold ideas about the world. As I have presented it, Descartes seems too seduced by the visual metaphor of a center of vision becoming clear, a ring of focus, which then can be traced down an ancient heritage of an Ocular philosophy of Presence, where the revealing aletheia of Being stands out from the confusions and negations of Non-Being, playing out the 1s and 2s of dialectical Greek counting. But I would put forward that Descartes is only drawn in this direction against, or at least in tension to, a more comprehensive understanding of perception, one in which the Mind “sees” in a very unrepresentational way, with the “mind’s eye” (a phrase that likely Spinoza takes from Descartes). It is my sense that only Descartes theological commitments to the soul and its freedom of choice expressed through the judgments of the Will which force Descartes away from what he would otherwise be more comfortable with, into an account of vision which emphasize visual clarity along a central axis of focus. It is the need for a localizable edge of judgment, most amenable to an analogy of the otherwise blurred field of view, overdetermined by an essential binary of clear and unclear, which pulls Descartes back into pictures. We see this in the development of his Dioptrics away from the non-representationalistexplanations he begins with.

Descartes’ Blindman

The greatest example of Descartes non-representational concept of mental “seeing” is his analogy of a blindman who sees the world through the use of two sticks, literally feeling the world into accurate appraisal. But first, like Spinoza, Descartes warns us not to fall into pictures. Here he points up the semiotic stimulations of our thought. It is not on the basis of resemblance that we come to know or sense things:

…it is necessary to beware of assuming that in order to sense, the mind needs to perceive certain images transmitted by the objects to the brain, as our philosophers commonly suppose; or, at least, the nature of these images must be conceived quite otherwise than as they do. For, inasmuch as [the philosophers] to not consider anything about these images except that they must resemble the objects they represent, it is impossible for them to show us how they can be formed by these objects, received by the external sense organs, and transmitted by the nerves of the brain…instead we should consider that there are many other things besides pictures which can stimulate our thought, such as for instance, signs and words, which do not in any way resemble the things which they signify (forth discourse, trans. Olscamp)

And then here Descartes draws on the very physical modes of sensing, or seeing through a stick:

It sometimes doubtless happened to you, while walking in the night without a light through places which are a little difficult, that it became necessary to use a stick in order to guide yourself; and you have then been able to notice that you felt, through the medium of the stick, the diverse objects around you, and that you were even able to tell whether they were trees, or stones, or sand…(first discourse)

…just as when the blind man of whom we have spoken above touches some object with his cane, it is certain that the objects do not transmit anything to him except that, by making his cane move in different ways according to their different inherent qualities, they likewise and in some way move the nerves of his hand, and then the places in his brain where the nerves originate. Thus his mind is caused to perceive as many different qualities in these bodies, as there are varieties in the movements that they cause in his brain…(fourth discourse)

Descartes figure 18, Dioptrics

It is Descartes conception of light that the tendency of rays communicate themselves without movement, instanteously across space, just as a blindman’s stick seems to. When rays connect to our eyes, Descartes understands our sensing to be that of connective stimulation. When we see objects, we are seeing like a blindman, with sensations directly transmitted to our nerves. He compares a blindman holding two sticks to the baton centers of vision of each of our eyes, emphasizing that the image itself is not what is directly communicated to the Intellect through the nerves.

 So you must not be surprised that objects can be in their true position, even though the picture they imprint upon the eye is inverted; for this is just like our blind man being able to sense the object B, which is two his right, by means of his left hand, and the object D, which is to his left by means of his right hand at one and the same time. And just as the blind man does not judge that a body is double, although he touches it with two hands, so likewise when both our eyes are disposed in this manner which is required in order to carry our attention toward one and the same location, they need only cause us to see a single object there, even though a picture is formed in each of our eyes (sixth discourse).

The eyes using the two batons of central rays of light

[These citations discussed some here: Descartes and The Blind Man’s Cane ]

It would seem that there is within Descartes thought a primary distinction, as Yolton and Behan argue, between signifying and representing; the stimulations of the senses communicate themselves directly through the nerves in a signifying process not based on essential resemblance. The problem is that such a signifying mode of interpretation does not favorably present itself to the requirements of an Individuated and free action of the Will. Where, and before what would the signification process end…the pineal gland? This puts Descartes in tension with himself, as the analogy of visual clarity, embodied by the pursuit of hyperbolic focus in lenses, pulls him back toward representationalist notions. I don’t at all believe that Descartes holds such a representationalist idea of knowledge, but rather suspect that it is only the independence of the freedom of the will which again and again forces its intrusion, under an auspices of directed and establishing clarity. The resting place of hyperbolic doubt, the cogito, assures a clear focus relation on which all relations can be reconstituted, owing to God. 

Conclusion: Spinoza and Craft

What makes this most interesting is that because Spinoza objects to Descartes at the most radical level of the Will itself, denying the rationality of such a theological vestige (Ethics, 2p48s records the critique of boththe will and representation), he remains unencumbered by the need to take from vision a strict Being/Non-Being binary of optical focus and blurring, center and margin. Instead he draws on, if we can be bold enough to assume it, another luminous analogy, that of Plotinus’s Neo-Platonism, the notion that light radiates in a sphere (put forth by Kepler), and that it expresses itself in gradations of ever-weakening power and cohesion, understood as degrees of Non-Being and power. Spinoza positions himself in the Augustinian, Plotinus line of thought which makes of evil a privation, but he does so at the epistemological, yet by virtue of his parallelism, still bodily level, where the degree of the adequacy of our ideas result in real, affectual experiences of the fluctuations of our power and perfection. Instead of a center of vision which affirms a crisp focus of assured clarity, Spinoza’s center of vision is the breaking wave of the affirmation of our own body’s power, its capacity to act, understood within the context of the full scope of tableau of what is “seen”. As our eyes, fingers, ears, mind flits from thing to thing, we are constantly in states of imagined increases of pleasure and power, owed to the coherence of causes and effects. While central clarity may help incise distinctions of importance, these distinctions only grow meaningful and distinct in the full context of the margins.

It is my sense that Spinoza gained something of this metaphysical insight, in addition to the great variety of sources we might name, from his experiences as a craftsman. His patient polishing of propositions not only reflect in form his careful polishing of lenses, but the content of his thought I believe express the sensitive, non-representational experiences of judgment that come from working with materials, designs and tools in a comprehensive fashion. Spinoza’s refusal to admit Descartes Substance divorce of mind and body perhaps came from his bodily experiences of shaping and sensing glass under tension. While Descartes spent much of his time in mathematics and theory, informing and confirming his hypotheses at times through experiments, he lacked hands-on knowledge of what mechanical construction and application required. In a sense, his vision was mechanical, but his hands were not. One cannot help but realize that Spinoza’s cybernetic turnings of the grinding lathe (either with his off-hand or by foot pedal), communicated a complex of sensations and judgments far too subtle and rapid to place the crown of knowing upon a independent and freely functioning Will. Instead, as the lathe was tensioned in a flux of speeds and grits, and his eyes caught the traces of changes, as his hand holding the torquing glass blank felt the moment to moment consequences of his lathe’s turning – in one great curcuit – he necessarily understood the shore of perceptions within a comprehensive and assembled bodily whole of communications. The coherence that a craftsman brings betweeen his own hands, the limits and possibility of tools, the variations and states of material, amid a continuous, creative line of “dissonance”, a hole in the center of the percieved, non-absolute differentiations of grades, their deviations in form, doubtlessly expressed itself in Spinoza’s own embrace of the union between body and mind, and the careful consideration of the moment to moment changes in the body’s capacity to act.

The Ignored Letters of Spinoza

Equine In Thought

I have been working for a few days on a commentary on two of Spinoza’s letters, which I will refer to as his Optical Letters. These two letters are Spinoza’s attempt to answer a question about optics and telescopes put to him by his friend Jarig Jelles. I just want to make a short note here on something that has really struck me as quite astounding about the paths research on Spinoza have taken over the last say, 150 years. During these past few months I have read deeply into the biography of Spinoza, and conducted my own attempt to dig into Spinoza’s connections to lenses and optics. I have found extraordinarily detailed work done on the most obscure elements of Spinoza’s life. Diary entries of people who never met Spinoza are combed for bits of significant facts. Spainish inquistion records reveal hearsay of a meeting he was at and the company he kept. The records of Michael Spinoza’s business dealings before Spinoza was born are carefully noted. Indeed it seems that every trace Spinoza’s echoing life may have made upon the Book of History has been dug up with near archaeological precision, all to a wonderful effect.

But what amazes me, I say truly amazes me, is that we have in these two letters to Jelles several pages of Spinoza’s thoughts on light, glass, geometry, optics, Cartesian reasoning, perception and the eye, and nearly no one (in English) has thought to ask with any depth: What is Spinoza thinking here, and why does he think it? This resounding silence on these letters is truly extraordinary, given the Armada of Academia’s resources and desires. The disproportionality is extreme. Part of this may be due to the fact that these letters were not completely available to general English readers until Shirley’s translation made them so in the mid nineties (the Elwes translation simply leave off where the optical portions begin!). But I have been assured that even in Dutch there is a dearth of commentary on these letters, and I am not familiar with any other languaged treatment. Despite issues of translation, the reasons for this silence seem to be two-fold, in perhaps the most broad of interpretations. First, Spinoza seems to have been ensconced, crystallized in a certain mystico-rationalist conception that only Philosophy can achieve, strangely divinified such that certain aspects of his thinking simply are unthinkable as relevant. In such a prisming of his life the fact that he ate “Milk-soop” and raisins indeed might have more importance than what he thought about spherical aberration. This is remarkable, for Spinoza has been championed by diverse, incompatable schools of thought throughout the centuries. I think it says something about the aura we receive Spinoza in, no matter where we stand. The other reason seems to be about Academia itself, that issues of science and philosophy are treated so distinctly that not only are questions not asked which might bring these two disciplines together, but also the people posing these questions are either of one background or the other, such that investigative readers of Spinoza who might have looked deeply enough to have found his Optical Letters simply did not possess the knowledge by which to assess them. The letters just were not of interest to either philosophers or historians. And I suspect that there is something to Academic arrogance or supposition – probably because it is a milieu in which one primarily experiences attack from one’s peers, rather than agreement – that is willing to dismiss the relevance of something that is not well understood and does not fit within its program. That such a combination of programs over the years may had this lasting occluding effect is significant. On a more general note, I have been assured by more than one scholar that this research into Spinoza’s lens-grinding is simply the first of its kind. How can this be?

Mirror

If I were to say something more, I suggest that we look into this hole in our perceptions of Spinoza, as they have been recieved and acted out for the last century or more, if only to get a better sense of ourselves, our need to see the people we study through a particular kind of lens (to beat the easy, but meaningful rhetorical drum). This does not mean that Spinoza is among the greatest scientific minds of his time, or even that he may have been able to contribute something to our vast store of scientific knowledge (that narrative tendency of successive victories needs to be amended I suspect). What it means is that if we are to understand Spinoza’s metaphysics (and politics) we must comprehend the full scope of the man: his intellectual concerns and daily devotions, and not just his Propositions. In a Spinozist sense, the fullness of his modal expression needs to be embraced. And if we are to understand Science, the great rise of rational, experimental and mechanical investigation that came out of the 17th century, we are going to have to understand that the truths in the minds of men, were mixed truths drawn from multiple aspects of what now are distinct disciplines. In history there is no dross.

The Optica Promota and Spinoza’s concept of focus

Spinoza: Against Hyperbolic Vision

Optica Promota, figures 1 and 2

Optica Promota, figures 1 and 2, refraction in the densest medium

There are several points of correspondence between Spinoza’s objections to Descartes’ treatment of the magnification of images in the L’Dioptrique in his letters 39 and 40, and the analyses found in James Gregory’s Optica Promota, but perhaps the most significant touching point occurs in Gregory’s conclusion, where he brings up the lone failings of parabolic mirrors and hyperbolic lenses, which Spinoza is sensitive to. Here you have a presentation of the weakness of the hyperbola as an ideal of lenses, and yet the appeal to the human eye as somehow exemplary and natural, which I have pointed out in regards to Kepler’s Paralipomena, [thought about here: A Diversity of Sight: Descartes vs. Spinoza and more broadly here Some Observations on Spinoza’s Sight ]. The entirety of Spinoza’s referential borrowings from Gregory must be fleshed out, (they seem proliferate through these two letters) but for the moment I present this conclusion alone, which marks out the limitations of hyperbolic lenses, apart from their difficulty of manufacture:

But against hyperbolic lenses, it is only objected that nothing will be able to be most clearly seen, except a visible point arising on the axis of the instrument. But this weakness [ infirmitas ] (if it would be worthwhile to call it that) is sufficiently manifested in the eye itself, though not to be impuning Nature, for whom nothing is in vain, but how much all things most appropriately she carries out [ peragit ]. Nevertheless, with conical lenses and mirrors not granted, it shall be rather with spherical portions used in place of spheriods and paraboloids in catoptrics; as with hyperboloids in dioptrics, in which portions of spheres are less appropriate.

With these we go to the stars – His itur ad astra

One can see how Spinoza would have sympathy to the notion that Nature does not err, but the subtle question of the “infirmitas” of human, hyperbolic vision is one that he would pause at. Insofar as human vision is used as a model for mental vision, any weakness “natural” to human beings is not redeemed by that natural state. All states a human being finds herself in are “natural”, but there is a degree of perfection which works as a fulcrum point for change and improvement in Spinoza’s thinking. The question remains whether Spinoza’s appraisal of spherical lenses as ideal is a pragmatic solution to the problem of finding the best lenses possible in the real world, or is a mistaken extension in analogy from his concept of ideal mental vision back down to the question of lens shapes, over-valuing the capacity of spherical lenses to handle otherwise would be paraxial rays.  

Spinoza’s Picture of the Omni-axial Spherical Lens

As Spinoza writes in his letters 39 and 40, following Gregory’s warning about hyperbolics:

Letter 39 – …Perhaps he was silent so as not to give any preference to the circle above other figures which he introduced; for there is not doubt that in this matter the circle surpasses all other figures that can be discovered. If, for example, circle ABCD should have the property that all rays coming from direction A and parallel to axis AB are refracted at its surface in such a way that they thereafter all meet at point B; and also all rays coming from point C and parallel to axis CD are refracted at its surface so that they all meet together at point D, this is something that could be affirmed of no other figure, although the hyperbola and the ellipse have infinite diameters. 

diagram letter 39

And then..

Letter 40 – …Moreover, it is certain that, 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. 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. Now since a definite segment of a circle can bring it about that all the rays coming from one point are (using the language of Mechanics) brought together at another point on its diameter, it will also bring together all the other rays which come from other points of the object, at so many other points. For from any point on an object a line can be drawn passing through the center of a circle…What I here say of the circle cannot be said of the ellipse or the hyperbola, and far less of other more complex figures, since from one single of the object only one line can be drawn passing through both the foci. This what I intended to say in my first letter regarding this matter.

As said, much more than this needs to be assessed regarding these two optical letters, but in this vein we identify a prominent theme of naturalized, hyperbolic vision and its critique, stretching from Kepler, to Descartes, and alternately Gregory to Spinoza.

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.

A Diversity of Sight: Descartes vs. Spinoza

Different Foci of “Clear and Distinct”

This may be premature in my process, but I would like to set down some simple correspondences that have arisen in my reading of Kepler’s Paralipomena to Witelo (1604), correspondences that elicit strong tidal differences between Descartes’ approach to vision – with its attendant metaphysical counterpart, the clear and distinct Idea – and Spinoza’s. That this is found in some of the historically least interesting of Spinoza texts, what has been taken by many to be Spinoza’s blundering into optical theory [ letters 39 and 40 to Jelles, full text ], is suggestive of Spinoza’s critical brilliance. For within Spinoza’s conception of optics and his close-cut rejection of Descartes reasoning seem to be important radical divergences, insofar as vision and light are understood to be more than analogous of metaphysical truths. In these letters Spinoza questions the very vision-philosophy that privileges a central-line of axis, one that fundamentally connects a viewer within to an external and opposing point: a hidden, underpinning assumption of Descartes’ mathematization of the experiential self, Will and the world. Though there is much in Descartes’ philosophy that substantially departs from the Perspectiva epistemology tradition, his enthused embrace of the powers of the hyperbola embody a grounding of the eye, (and therefore the mind), in a centralized perception and knowledge upon which Spinoza places his criticism.  

I have not encountered any analysis which shows that Descartes’ enthusiasm for the hyperbola derives directly from Kepler’s Paralipomena, yet as I am not fully familiar with the literature perhaps this is a commonplace understanding. Gaukroger, for instance, gives evidence that Descartes read this text around 1620, and allows three possible points of influence, but does not speak to the potential influence of Kepler’s offer of the hyperbola as a natural normative of vision. Yet in the process of reconstructing Spinoza’s conceptions in these two letters of objection to Cartesian optics this connection with Kepler owes be fleshed out. One will see that in the Paralipomena Kepler not only sets out the virtues of the hyperbola’s anaclastic line in terms of its resolution of spherical aberration from focus – the primary and crowning demostration of Descartes’s 8th discourse of La Dioptrique - but he also naturalizes the hyperbola, claiming that the hyperbolic shape of the posterior of the crystalline humor is the very thing that helps establish the soul’s satisfaction with centered-object viewing. In fact, the wide sweep of Descartes’ metaphysics and optics in brief seems signaled by Kepler in his treatment of the hyperbola. And, because I suspect that Spinoza has Kepler’s descriptions in mind when responding to Jelles, making an argument for the importance of a radii of axes of perception, and emphasizing the importance of the angle of incidence in the measure of magnification (see letters linked above), his alternate response to Kepler provides a valuable clue to the stake Spinoza is making against Descartes, both in philosophy and science. In short, Descartes makes a virtue out of the Kepler’s description of the hyperbola and the eye, and Spinoza makes it something of a flaw or limitation. 

Setting Forth Kepler’s Hyperbola

Kepler's illustration exploring the properties of refraction, page 106

First, perhaps it is best to set out what Kepler has to say about the hyperbola so that a comparison can be made. Initially, he embraces the figure in an attempt to assess a law of refraction, which he hopes to find through the virtue of its special properties. He will not attain the law of refraction, but what he does make clear is that the hyperbola alone possesses the capacity to focus parallel rays to a single mathematical point, or vice versa. Thus, he writes of the figure above:

What is required is to say what sort of surface it is upon which these radiations in this position coming forth from α, so that they strike just as do here the lines βλ, μγ, and so on, so that these lines are either tangents to that surface, or lines equidistant to the tangents…These, moreover, are found only in the hyperbola, not in the parabola, which tends toward a straight line parallel to the axis, not one meeting the axis, as X A here. (107-108).

Not to get lost, were are simply establishing that Kepler points out the solitary characteristic of the hyperbola which will make a lens of this shape central to Descartes’ ambitions for providing aberration-free vision through hyperbolic lenses.

The next apparence of the hyperbola in Kepler’s text occurs in chapter 5, where the geometry and anatomy of the human eye is discussed. Here Kepler presents a small, simple drawing of the “crystalline humor” which behind the cornea in the eye is the primary means of focusing light. I enlarge it here for clarity:

Keplers drawing of the crystalline humor

Kepler's drawing of the crystalline humor, page 167

Kepler writes of the shape of the humor:

 On the posterior side…[the] figure is a hyperbolic cononoid, a hyperbola rotated on its axis. For [Johannes] Jessenius thus relates, that it is not spherical as [Felix] Platter said but that it protrudes markedly, and is made oblong stretching up almost into a cone; and that on its anterior face it is of a flattened roundness…(167).

Kepler has emphasized a correction. The humor is not spherical as Platter claimed, but hyperbolic, and made nearly like a cone. It is this particularity of the eye, coupled with his earlier hyperbolic observations that will create a certain naturalization of hyperbolic vision, something that speaks to the very nature of the human soul. The aspherical effect of the crystalline humor on vision is made clear a few pages later where Kepler discusses how the various cones of light coming from each of the points of an object to be seen, interact with and refract upon the cornea and the crystalline humor. What will be shown is that the humor is biased towards the production of clarity oriented towards its central axis, and that light cones that come obliquely to this angle, will appear less clear to the human eye, as they are focused to the borders of the image:

All the lines of the direct 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, 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…(174)

Kepler then takes these facts of refraction and uses them to explain the experience we have of satisfactory vision. What is most notable is that Kepler wants us to understand how the oblique “more confused” images at the borders of an image actually complete and serve the focus of vision around the central axis:

…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). 

The “surroundings of the hemisphere” invite our eye from to this or that. Oblique vision proves satisfying to the soul only to the degree that it inspires the eye to turn its hyperbolically-aided central axis across its field. The concept is that the deprivation of clarity somehow drives the soul to complete its picture. This is an important point when considering the influence of vision as a primary analogy for Cartesian metaphysics.

Kepler Completes the Synthesis of Eye and Lens

Lastly, Kepler will place the facts and inclinations of vision within the context of the powers of hyperbolic focus, and he does this in such a way that it would seem sure that when Descartes looks to solve the problem of spherical aberration he would see in the hyperbola a natural bias towards centralized focus.

Keplers illustration of the spherical aberration of rays

Kepler's illustration of the spherical aberration of rays, page 194

After a protracted investigation of the behavior of light rays in refraction through a crystal globe filled with water, and an explanation of our visual experiences of the images and confusions that result, Kepler will conclude that the hyperbola alone resolves the need for gathering a cone of rays into a single point:

Proposition 24 – Rays converging towards some single point within a denser medium are gathered by the hyperbolic conoidalsurfacebounding the medium to one single point, closer than the former point…These two things [two requirements], however, are accomplished, not by one or another circle but by conic sections…Further, of the conic sections, only the hyperbola or some line very close to it, is the measure of refractions, as was shown in Sect. 5 of chapter 4. Indeed, this very thing was demonstrated there: that the surface making all the rays outside the denser medium parallel is a conoid that does not differ from hyperbolic (198).

Given this remarkable property, Kepler then concludes that “nature’s plan” has endorsed the priorty of the hyperbola. And it seems that there can be little doubt that Descartes worked to synthesize this priorty which his own metaphysical priorty for “clear and distinct” ideas:

Hence it is evident nature’s plan concerning the posterior surface of the crystalline humor in the eye. That is, she wished to gather all the radiation of any visible object entering the opening of the uvea [pupil] into a single point of the retina in order both that the point of the picture might be all the more evident, and that the rest of of the points of the picture might not be confused by extraneous rays whether stray or gathered together. – Chapter 5, proposition 24, corollary (199).

Spinoza: “the eye is not so exactly constructed..” (letter 40)

To pull back for a moment. If this analysis is correct, that a decisive rift in Cartesian philosophy could be seen in Spinoza’s letter to a friend discussing what at most appears to be a trivial oversight on Descartes’ part is striking. Many scholars seem to have struggled over the meaning of Spinoza’s words in these letters (39 and 40) sensing that there is an elementary blunder in Spinoza’s thinking, but, as seen in their lack of a careful examination of it, there is an inability to locate just what this blunder would be. Instead, because Spinoza has been read as solely a metaphysician, his foray into optical matters in these two letters was largely dismissed as Spinoza simply wading in too deep a water. With very few exceptions, most only assumed that Spinoza was unfamiliar with the issues at stake, and his in-concordant use of terms appeared to prove this; yet a change in scholarship is occurring. Perhaps our examination of these two letters can add to this shift of perspective.

What does Spinoza mean in his letter 40 talk about the inexactness of the human eye?:

Moreover, it is certain that, 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. 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 (letter 40, full text)

We have seen from Kepler that the likely reference is to the distortions of “more confused images” at the borders of vision, in part due to the aspherical, single axis nature of the eye’s lenses. This stands in contrast to Spinoza’s diagram, which he takes to be an ideal spherical refraction: 

Spinozas diagram of the virtue of spherical refraction, Letter 39

Spinoza's diagram of the virtue of spherical refraction, Letter 39

Spinoza is extolling the comprehensive capacities of spherical refraction, and his embrace of this concept marks out a distinct divergence from Descartes’ naturalized emdorsement of a center-focused vision.  Descartes will develop a theory of clear and distinct thinking which narrows the field of mental vision. This vision is imagined to be in concert with what Nature had planned in having given the eye its own hyperobolic lens; conversely, Spinoza will take from Descartes the notion of “clear and distinct”, but the concept of vision in which it is to be deployed is dramatically different. Spinoza emphasises the clarity of a connective, hemispherical scope; Descartes is aimed at the close focus on the surest of things.

Looking With the “Mind’s Eye”

This difference perhaps can be made more distinct by considering the use of the phrase “the Mind’s eye” by both thinkers. For instance, in his Regulae, after his very influential Rule 8 [the full text of which I post and briefly engage here ], Descartes tells us in Rule 9 how we must compare improvements in thinking by attending to how we naturally see things. Human vision provides the exemplar of how of mental vision is, and the issue is one of central focus:

Rule 9: We must concentrate our mind’s eye totally upon the most insignificant and easiest matters, and dwell upon them for long enough to acquire the habit of intuiting the truth distinctly and clearly.

…We can best learn how mental intuition is to be employed by comparing it with ordinary vision. If one tries to look at many objects at one glance, one sees none of them distinctly. Likewise, if one is inclined to attend to many things at the same time in a single act of thought, one does so with a confused mind. Yet craftsmen who engage in delicate operations, and are used to fixing their eyes on a single point, acquire through practice the ability to make perfect distinctions between things, however minute and delicate. The same is true of those who never let their thinking be distracted by many different objects at the same time, but always devote their whole attention to the simplest and easiest of matters: they become perspicacious.

- Descartes, The Regulae, Rule 9

It is not without significance that Descartes in his metaphor of close focus appeals to artisans and craftsmen to praise the powers of concentrated vision. These are not a class of persons that he would embrace in society- his attitude toward de Beaune and Ferrier is well known – but their analogous use in a hierarchy of powers is in keeping with his concept of knowledge being like an ascent of mechanized complexity, from primitive mental tools to those most intricate (Rule 8, Regulae). The same artisan trope is found again in his La Dioptrique, where the limits of the eye are by craftsmen strained and improved by the exercise of ocular muscles. I quote at length below because it is an important passage. For one, it is found at the end of the 7th Discourse which is the text that Spinoza and Jelles are commenting on in their letters, and is at the cusp of his presentation of the 8th Discourse praise of the hyperbola. Secondly, here the virtue of a dispersive vision (which Kepler describes as limited by the hyperbolic shape of the lens) is denied: seeing more is not seeing better, as is testified by experience; and thirdly, Descartes treats the possible alteration of the limits of crystalline humor and pupil’s capacities as being achievable, not in the direction that Spinoza would like (tpward a more-than-human breadth of clear vision), but in terms of an exceedingly close narrowing, achieved by trained specialists:

There is only one other condition which is desirable on the part of the exterior organs, which is that they cause us to perceive as many objects as possible at the same time. And it is to be noted that this condition is not in any way requisite for the improvement for seeing better, but only for the convenience of seeing more; and it should be noted that it is impossible to see more than one object distinctly at the same time, so that this convenience, of seeing many others confusedly, at the same time, is principally useful only in order to ascertain toward what direction we must subsequently turn our eyes in order to look at the one among them which we will wish to consider better. And for this, Nature has so provided that it is impossible for art to add anything to it.

I have still to warn you that the faults of the eye, which consist in our inability to change sufficiently the shape of the crystalline humor or size of the pupil, can bit by bit diminish or be corrected through practice: for since this crystalline humor and the membrane which contains this pupil are true muscles, their functions become easier and greater as we exercise them, just like those of other muscles of our body. And it is in this way that hunters and sailors train themselves to look at very distant objects, and engravers or other artisans who do very subtle work to look at very close ones. 

- Descartes, Seventh Discourse

In the highlighted passage Descartes repeats Kepler’s summation of the benefits of a hemisphere of vision, that it simply leads to a field of vision which helps serve our central axis: Kepler: “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 “. Yet Descartes has changed the emphasis some and placed the notion of “willing” at this central axis (a literal conflation). The widening of a view only provides the occasion for free choice, which will express itself in the turning of the single axis of the eye. (It is thus clear that Descartes and Spinoza mean different things by “the faults of the eye” and “the eye is not so exactly constructed”.)

Spinoza: “so that we must not fall into pictures”

What is compellingly consistent is that Spinoza’s own use of Descartes’ “Mind’s eye” phrasing will directly address the Cartesian issue of the freedom of the will, not to mention the pictorial conception of clarity and distinctness. The phrase is most pointedly found in his Ethics, just where the distinction between willing and understanding is by Spinoza denied (E2p48 and 49):

E2p48 – In the mind there is no absolute, or free, will, but the Mind is determined to will this or that by a cause which is also determined by another, and this again by another, as so to infinity.

Scholium – We must investigate, I say, whether there is any other affirmation or negation in the Mind except that which the idea involves, insofar as it is an idea – on this see the following Proposition [49] and also D3 – so that our thought does not fall into pictures. For by ideas I understand, not the images that are formed at the back of the eye (and, if you like, in the middle of the brain), but concepts of Thought [NS: or the objective Being of a thing insofar as it consists only in Thought]; – trans. Curley

Descartes’ notion of looking with the “Mind’s Eye” requires learning what distinctness is in terms of our experience of human vision, a lesson that requires that we focus closer and closer upon obvious things, training our eye to become more and more exact, a lesson which in turn gives us to understand the centrality of the focus of a single axis, and the use of a breadthof vision as merely the field for a freedom of choices; yet in Spinoza the human eye itself is seen as “inexact” in its singular axis of focus. And looking with the Mind’s eye is for Spinoza not so much a process of learning to see clearer and clearer pictures, (or even holding one clear idea or another in mental vision), but rather learning to look in a way quite unlike the way of the human eye, within a matrix of conceptual understandings; and this matrix is one which decenters the central axis of vision (and one could say the “self”), strives to achieve something akin to an infinity of axes of vision, (nothing more than a breadth understanding of the order of Adequate Ideas and thus the causes of the phenomena we witness, and that we affectively experience). While Descartes would say, following closely our analogous experiences of human vision, that seeing more is not seeing better, Spinoza would say that one is only seeing better if one is seeing more: hence his thought moves very, very quickly to the intuition of the Adequate Idea of God.

A Difference in Method

One grasps this if one compares Descartes’ notion of method with Spinoza’s own early Emendation notion of method, which is in response to it. The below passage is important because it follows several points which bear 8th Rule influence (a focus upon human powers, viewing knowledge building like blacksmithing, for instance). While Descartes is interested in focusing the mind on simple truths which may serve ultimately to connect one to a transcendent God, Spinoza’s method is not one of focus upon this or that truth, but upon the standard of truth itself as it immediately directs one’s attention to a maximalizationof thought: a most perfect Being. It is the distinction between one kind of perception and all others, which throws the vision wide:

That is, the most perfect method will be one which shows how the mind should be directed according to the standard of a given idea of the most perfect Being…From this one can readily understand how the mind, as it understands more things, at the same time acquires other tools which facilitate its further understanding. For, as my gathered from what has been said, there must first of all exist in us a true idea as an innate tool, and together with the understanding of this idea there would likewise be an understanding of the difference between this perception and all other perceptions. Herein consists on part of our method. And since it is self-evident that the more the mind understands Nature, the better it understands itself, it clearly follows that this part of our method will become that much more perfect as the mind understands more things, and will become then most perfect when the mind attends to, or reflects upon, the knowledge of the most perfect Being. (trans. Shirley 39)

The web of truths that “Mind’s eye” vision focuses on is a breadth of vision, governed by a comprehension of determined causes. And in a sense, this begins with God, God as a totalizing reality of Being from which we are not separate. A field of vision, which for Descartes provides an array of choices which an axis-eye then willfully judges and picks its way through, for Spinoza is an incandescent weave of causes and effects, any adequate understanding of which leads to all others. It is a spherical conception of a refraction along an infinity of axes, in which the Will plays no part.

Descartes’s Hyperbolic Doubt and Hyperbolic Lens

In considering Kepler’s introduction of hyperbolic lenses and his Nature’s single-axis plan for the eye, and then Descartes synthesis of the two, there is the happy result of support found for a contested interpretation of Descartes offered by Graham Burnett, in his book Descartes and the Hyperbolic Quest . Professor Burnett offers that Descartes’ obsessive, mechanized pursuit of the grinding of a hyperbolic lens, and his project of legitimatizing his Natural Philosophy through “hyperbolic doubt” are something more than a mere conflation of uses of the word “hyperbolic”. Burnett tells us, citing Gaukroger’sbiography, that the two may correspond to a single conception of mind, (quoting at length):

What configuration of mind of mind allows natural light to coalesce into a clear and distinct idea? The answer…is hyperbolic doubt. If once we saw, as in a glass darkly, and if at some (beatific) point we will see face to face, for the time being the best we can seem to do is to see through the right kind of glass that one that does not distort or obscure: and this just might be, at least least initially, the focusing glass of hyperbolic doubt. To play out the suggestion then: Descartes greatest scientific success lay from his perspective, in his systematic investigation of optics and the perfection of human vision those investigations promised; his optics presented an instanteous light that could be focused into clear and distinct images by means of the imposition of ahyperbolic form. 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 the human mind received, via natural light of reason, an instanteous, clear and distinct illumination, but only by means of the interposition of another hyperbolic focusing device – the hyperbolic doubt.

I do not wish to overemphasize the signification of the parallelism, tantalizing as it is. Following 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 (126-127).

Indeed, if we follow professor Burnett’s conclusion, and allow it to have a substance greater than mere lexical coincidence, we find that when armed witha knowledge of Kepler’s antecedent approach to the hyperbola such a reading begins to cohere. Descartes’ embrace of Kepler’s hyperbola of the human eye shows that “extreme doubt” and the focus of the Will that it accomplishes is for Descartes truly both a mechanism for focusing the mind upon simple truths and a naturalized legitimization of a will-centered, single axis organized perception. What professor Burnett intuited through his study of Descartes’ life-long pursuit of an automated, hyperbolic lens-grinding machine, is given traction when the genealogy of Descartes’ conception of the importance of the hyperbola is traced back to Kepler, its orgin. And this is exposed in two almost-ignored letters written by Spinoza on a subject he long had been considered deficient in, as the force of Spinoza’s attack upon both Descartes’ metaphysics andhis optics is to be considered as being of one cloth. Spinoza has Kepler in mind because Descartes had Kepler in mind. This is suggestive of the power of Spinoza’s critique, and the level at which he carried it forth. It touched not only the abstraction of Descartes’ metaphysics, but also the optical-theory origins of Descartes preoccupation, that human vision was somehow naturally hyperbolic and therefore offering a guide toward the perfection of the mind. Because our inheritance of the optical trope of human vision is so rooted in our conceptions of the world, and our acceptance of Descartes’ approach to thought and mechanism is so pervasive, Spinoza’s optical critque proves promising of a radical importance at the very least.

Lasting Questions

None of this goes any distance toward proving whether Spinoza’s critique of Descartes in letters 39 and 40 was correct in the terms he meant it by, speaking of how light and lenses behave. Clearly Spinoza was well informed about the nature of Descartes’ claim as to the importance of the hyperbola. He had read and followed Johannes Hudde’s Specilla circularia, which dismissed the importance of spherical aberration in a mathematically exact way, minimizing Descartes’ impractical solution; and he was likely familiar with ChristiaanHuygens’ own complaints about Descartes’ failures in treating telescope magnification accurately. Additionally, it seems quite likely that Spinoza was familiar with the Ur-source of Descartes’ own embrace of the hyperbola, Kepler’s Paralipomena to Witelo, since he addresses specific terms of its explanation, and the argument he presents in brief cuts to the quick of the virtues of the hyperbola presented there: the idea that the human eye’s hyperbola somehow expresses Nature’s plan which in Cartesian hands would naturalize a priorty of a single-axis, will-driven priority of focus and choice. By arguing for the “inexactness” of the eye, Spinoza is undermining a primary vision/knowledge metaphor which helps form part (but most certainly not all !) of Descartes’ metaphysics of clarity.

There are therefore a few questions that remain. For one, Kepler does not merely serve as a source for a negative critique of Descartes, insofar as he has followed Kepler. For instance Kepler’s conception of light in many ways diverges from Descartes’, and could be said to have concepts which presage Christiaan Huygens’ own wave theory, which would eclipse Descartes’ optics. It remains to be seen if Spinoza’s optical understanding stems directly from Kepler in a positive sense, that is, if Spinoza’s holds optical concepts which were superior to Descartes’ theories due to Kepler’s influence. Spinoza has used Kepler to undercut Descartes’ metaphysics, but where does Spinoza stand in terms of contemporary optics? For this to be answered, Spinoza’s praise of the versatility of an infinity of axes has to be set up against the contemporary science of telescope construction. For though Descartes’ hyperbolic lenses were nearly impossible to make at that time, in theory at least they would have offered an advantage. Spinoza’s objection is that Descartes is incomplete in his analysis of magnification, and that the capacity of a lens to handle a variety of axes is important in compound telescope magnification. Such a possible importance remains unaddressed, though all existing telescopes obviously achieved their magnification without hyperbolic lenses (and notably, Christiaan Huygens had privately solved the issue of spherical aberration using only spherical lenses in the summer of 1665, when Spinoza and he were closest).

The other question that remains is to determine the large scale consequences of Spinoza’s rejection of a naturalized, single-axis concept of hyperbolic vision, upon his own preoccupation with lens-grinding and instrument making. The grinding of a lens, after all, is exactly the kind of “craftsman” or “artisan” practice that Descartes lauded in his 9th Rule, one that lead to an acuity of vision. The purpose of a lens is most often to achieve a magnification which concentrates the vision at a local point. And this is the mode of narrow focus to which Spinoza seems to making his objection. But if we allow the analogy between craftwork and mental tools found in the works of both Descartes and Spinoza, the careful refinement of a proposition, such as those found in Spinoza’s Ethics, would be read as a kind of development of perspicuity. Permitting the Ethics to stand as our model for complex, intricate knowledge, by analogy any grinding of glass into a polished shape must be seen as part of an interlocking of all other actions, ideas and material states; for just as there is extensive cross-reference in Spinoza’s Ethics, Spinoza’s own daily preoccupation with lens-grinding and instrument building must be seen as cross-referenced to an infinity of other causes and actions, all leading to an theorized increase in freedom. As Spinoza clarified a piece of glass and made it capable of magnfication, was it that he was concerned not just withthe the lens, but how this magnification fit with other lenses, in devices, with phenomena to be discerned, and the Ideas we hold as we use them? It would seem at that this is so, but this question has to be answered more fully.

 

 

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.

Spinoza: Letter 40 and Letter 39

Below the pertinent, optically related passages from letter 40, for reference sake. Spinoza in this letter attempts to answer a follow-up question posed to him by his friend Jarid Jelles, regarding an answer he had already supplied in Letter 39. Both letters from Jelles are lost. The ultimate subject I take to be the seventh discourse of Descartes La Dioptrique (that passage here). The translation is Shirley’s:

Spinoza’s Letter 40

…I now proceed to answer your other letter dated 9 March, in which you ask for a further explanation of what I wrote in my previous letter concerning the figure of a circle. This you will easily be able to understand if you will please note that all the rays that are supposed to fall in parallel on the anterior of the glass of the telescope are not really parallel because they all come from one and the same point. But they are considered to be so because the object is so far from us that the aperature of the telescope, Read more of this post

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

[For a fuller treatment of the topic read "Deciphering Spinoza's Optical Letters"]

Telescopes and Turning a Flea into a Elephant

To offer context to the question that Jelles poses in a letter we have lost, regarding the size of objects on the retina, I post here the likely text that Jelles has in mind, and to which Spinoza is responding. Spinoza writes in answer:

I have looked at and read over what you noted regarding the Dioptica of Descartes. On the question as to why the images at the back of the eye become larger or smaller, he takes account of no other cause than the crossing of the rays proceeding from the different points of the object, according as they begin to cross one another nearer to or further from to eye, and so he does not consider the size of the angle which the rays make when they cross one another at the surface of the eye. Although this last cause would be principle (sit praecipua ) to be noted in telescopes, nonetheless, he seems deliberately to have passed over it in silence, because, I imagine, he knew of no other means of gathering rays proceeding in parallel from different points onto as many other points, and therefore he could not determine this angle mathematically.

Perhaps he was silent so as not to give any preference to the circle above other figures which he introduced; for there is not doubt that in this matter the circle surpasses all other figures that can be discovered (letter 39)

It is the 7th discourse that Spinoza and Jelles are discussing. Here is a portion of the relevant passage:

As to the size of images, it is to be noted that this depends solely on three things, namely, on the distance between the object and the place where the rays that it sends from its different points towards the back of the eye intersect; next on the distance between this same place and the base of the eye; and finally, on the refraction of these rays (trans. Olscamp).

At this point in the explanation Descartes seems to have touched on the factor of the “the size of the angle which the rays make when they cross one another at the surface of the eye” for this would seem implicit in a discussion of the refraction of rays. But Spinoza seems to have focused on what follows, which leaves off any concern for this factor:

Using this diagram, Descartes continues:

“Thus it is evident that the image RST would be greater than it is, if the object VXY were nearer to the place K, where the rays VKR and YKT intersect, or rather to the surface BCD, which is properly speaking the place where they begin to intersect, you will see below; or, if we were able to arrange it so that the body of the eye were longer, in such a way that there were more distance than there is from its surface BCD, which causes the rays to intersect, to the back of the eye RST; or finally, if the refraction did not curve them so much inward toward the middle point S, but rather, if it were possible, outward. And whatever we conceive besides these three things, there is nothing which can make this image larger.”

Here, Descartes has claimed to total all possible means of enlarging an image. He indeed has talk about the surface of the lens (BCD), but perhaps in keeping to Spinoza point, has not talked about the “size of the angle” that the rays make at the surface of a lens. (An issue Spinoza would like to make regarding the powers and functions of a telescope, it would seem.)

Descartes continues, detailing the kinds of improvements of magnfication that are possible:

“Even the last of these [the refraction curving outward from point S] is scarcely to be considered at all, because by means of it we can augment the image no more than a little bit, Read more of this post

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