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Spinoza’s “Spring Pole” Lathe: Experience to Metaphysics and Back

Spinoza’s Practiced Knowing

I mentioned in my recent post on the likely design of Spinoza’s grinding lathe that the dynamics of the Hevelius’ spring pole lathe may be tied to Spinoza’s ideas of Substance and the modes, such that one would be able to see how the epistemo-kinetic experiences Spinoza had during his many hours, days and years of lens grinding on such a lathe may have bore influence upon his metaphysical conceptions.

Here I want to take up this intuition, and perform the appropriate visualizations that would allow us, if for a moment, to picture what Spinoza’s body went through in communications with his device. In this way we might place ourselves, materially and affectively, in a relationship to his ideas, such that reading them alone in text would not allow (even if this goes against what one could argue is Spinoza’s rationalist program of understanding). That is, in short, one hopes to understand through body and affect what ideas Spinoza thought at the most abstract of levels, through their causal origins; for we can follow what Spinoza wrote, “experience can determine our mind to think…of certain essences of things” (Ep. 10), and assume that a similiarity of experiences may determine us to think of a similarity of essences. If we attain the experiences Spinoza underwent which determined him to think of certain essences (in his terms), this I believe can provide clarification to the same thoughts reached through his geometric pedogogy alone. 

The Horizontal and the Vertical: An Initial Philosophical Platform

Further below is an illustration of the tension dynamic of a spring pole lathe, as taken from Hevelius (second diagram). But first in order to follow my thinking here I want to insert a most revealing point made by Gatens and Lloyd in their critique of Hegel’s critique of Spinoza. As they point out, Hegel, while in great praise of Spinoza, felt that he did not embrace the full reality of the “negation”, Hegel’s personal contribution to the progress of philosophy and mankind. Spinoza’s description from Hegel’s point of view simply collapses into an acosmism, an undifferentitated whole of Substance, leaving no specific reality for either rocks, lakes or most importantly Man. In examining Hegel’s objection, Gatens and Lloyd introduce the vectoral notion of verticality and horizontality. It is suggested that Hegel’s problem is that he is only thinking of the “vertical” relationship of the modes to Substance, their individual, expressive relationship to the Totalizing whole. I quote at length here from the two authors because it is a very good paragraph, as they make an extremely important point:

Hegel’s critique of Spinoza thus focuses on the relation between individual mode and Substance. His complaint is that Spinoza cannot coherently articulate that relation without collapsing the infinite mode back into Substance; Substance remains undetermined, undifferentiated, while the individual mode is merely negative. But this is to miss the other dimension of Spinoza’s treatment of the finite modes -their mutual interaction, in which the determining force of Substance is mediated through the whole interconnected network of modes. Hegel’s critique of Spinoza is oriented, as it were, to the vertical relation between Substance and individual mode, rather than to the horizontal relation in which finite modes act on and are acted on by one another. Here, along the horizontal axis of finite modes, the claim that determination involves negation can be seen not as a repudiation of finite individuals but as an insight into their interdependence (71)

– Moira Gatens and Genevieve Lloyd, Collective Imaginings, chapter 3 “Re-imagining Responsibility”

Gatens and Lloyd are making clear the a key emphasis Spinoza places on modal expressions, as real, when included in our own constructive and immanent projects of freedom. I would graph out the differentiation that they find in Spinoza in this way:

I would parenthetically add to Gatens and Lloyd’s point that the reason why Hegel is thinking solely about the verticality of the modes is that the binary of individual/God, (or individual/world) is one of the primary schemas of the intersubjective investigations and inculcations of European Christianity. The salvation of the soul, through its relationship to an all-encompassing God, (via various institutional mediations), was the essential dogmatic concern, and this lengthy heritage necessarily brings the philosophical focus back to this binary difficulty: how does the individual soul return to God. Hegel, because he sees man at the apex of history, primarily must read the problematic in terms of a vertical dynamic, the individual vs. the whole. But Spinoza, because he is not burdened by the primacy of man or his reflective powers, allows another dimension of analysis which Hegel cannot see: that of the horizontal. The institutional mediations of the State, Church, Family and their imaginary relationswhich simply interpose themselves as aids to an essential verticality, are by Spinoza exploded out to the ultimate horizontal limit: the infinite expression of the modes. And the imaginary status of their mediating ideas become proliferate and constitutive vectors of power, degrees of freedom, across the field of Being (if one could put it that way without too much obfuscation).

Heveliuss Selenographia (1647) Spring Pole, Foot Pedal Lathe, illustrated

Spinoza’s Lathe

I want to return to the assumed spring pole, foot pedal lathe (pictured above), to get a much more concrete, affective sense of the vertical and the horizontal in Spinoza, and how the practices of Spinoza’a lens grinding may have helped construct his metaphysical conceptions. One must recall that Spinoza spent hours upon hours at such work lens work. To grind, polish and re-polish a lens could take several days, much of it in non-stop and highly repetitive, one might even say meditative, action (Auzout in 1664 records a time of 15 days for a single objective lens). So let our attention be called to the internal dynamics depicted above, found in focus of the dotted yellow frame. Here one should picture Spinoza seated or standing at length, his foot rhythmically pushing down a vertical tension from the spring pole at the ceiling. The vertical rising and falling motion forms a kinetic warp which not only works to orient the body spatially to the height of the room, connecting consciousness from the floor boards to the ceiling, but also creates and punctuated temporality, a timed ratio to the work. Transverse to this warp is the weft of horizontal action. The oscillations of the grinding form are pulled across the body by the push and pull connectionsof the foot the ceiling pole, and distinctly lateral to the focus of concentration. Again and again for hours these up-down, left-right actions literally weave a room of fluctuating conscious attentions, in which the craftsman necessarily embraces the lived experiences of the space, aware that whatever precise spotlight of focus he may have, it is merely a part of much larger, wider degrees of perception. Against Hegel’s fear of an individual’s collapse into the undifferentiated, the agentized craftsman becomes the draw-string of every quarter of the room, a focal point of at least two vectors action, from which and to which he is a differentiated, yet interconnected and expressive part. The melding of the craftsman and his tool is more than a metaphor. It is an experiential and metaphysical certainty.

Within this dotted frame of loomed space, at its center is a rotating circle. It whips at varying speeds in response to both the intensities of the leg, and the limits of the spring pole above, in a concentric motion. It is no secret that Spinoza had great love for the circle as a diagramed exemplar of the relationship between the modes and Substance (Ethics 2p8s, pictured below), but also as an ideal of vision and the actions of optical focus (Letter 39 to Jelles, March 3rd 1667). Here, for hours on end Spinoza would stare determinitively as a rotating circular form which remained both fixed (stable in its ratio), and changing, expressing in both the consummation of the vectors of the room’s actions. One cannot help but think that such concentrated attention upon the spinning form would leave at least a conceptual imprint upon the philosophical craftsman, especially as he considered the modal expressions to be causal interactions immanent to the whole, just as internal rectangles can be considered immanent to the properties of a circle (his diagram below). It is most suggestive to see that the rotating circular form becomes a bed of friction and idea, producing realized changes in the material of glass held in Spinoza’s sensing hand.

from Ethics 2p8s
The Turning Lap

To carry our instructive analogizing further, one must look closer at the actions immediate to Spinoza’s attention as he worked his glass into the required shape.

In the scalloped metal form likely an abrasive would be applied to aid in the grinding, the light blue arrow above represents the hand’s actions upon the circular rotation. The horizontal and vertical tensions are vortexed into an oscillating circularity. There a recipe of frictions and intelligenced experiences interact to bring about an ideal result. What Antonio Negri calls the “concrete…unique terrain of reality, [the] fruit of the paradoxical determination [a metaphysical dilation of unity and multiplicity]” (The Savage Anomaly, 127), the modal “surface of the sea”, occurs here, in the turning scalloped dish, a product of the cybernetic expressions of a room, a mechanism, and man, in which the craftsman’s hand performs a living shore of perceptual action.

The Sublime Tool

If this notional leap from mechanism to conceptual metaphysics seems too great, too fantastic, I believe that this is because we do not have a strong enough sense of the bodily, affective, imaginary foundational means of immanent abstract thought, something that Spinoza’s own metaphysics works to make more clear. Further I believe we must adjust ourselves from thinking of Spinoza merely in terms of propositions and proofs, though the rhetorical form of his work certainly at first or even second glance invites us to think of him in this way. Richard Sennett for instance in his recent pragmatic and near-poetic book, The Craftsman (2008), perhaps gives us a bridge for thinking about craft and abstraction as part of one constitutive process. He invites us to understand how human progress and freedom comes by thinking through one’s tools, how tools help frame our questions and solutions. In fact this is very much how Spinoza has conceived of abstract thinking itself, as he followed Descartes’ analogy found in the 8th rule of the Regulae : just as how a blacksmith’s tools had to be originally made by simpler tools themselves, so too simple tools of the intellect are needed to make other, more complex tools of the intellect (On the Emendation of the Intellect ). In a certain sense, one needs something that hammers in order to make a hammer:

The matter stands on the same footing as the making of material tools, which might be argued about in a similar way. For, in order to work iron, a hammer is needed, and the hammer cannot be forthcoming unless it has been made; but, in order to make it, there was need of another hammer and other tools, and so on to infinity. We might thus vainly endeavor to prove that men have no power of working iron. But as men at first made use of the instruments supplied by nature to accomplish very easy pieces of workmanship, laboriously and imperfectly, and then, when these were finished, wrought other things more difficult with less labor and greater perfection; and so gradually mounted from the simplest operations to the making of tools, and from the making of tools to the making of more complex tools, and fresh feats of workmanship, till they arrived at making, with small expenditure of labor, the vast number of complicated mechanisms which they now possess. So, in like manner, the intellect, by its native strength, makes for itself intellectual instruments, whereby it acquires strength for performing other intellectual operations, and from these operations gets again fresh instruments, or the power of pushing its investigations further, and thus gradually proceeds till it reaches the summit of wisdom.

But is our reading of Spinoza’s metaphysics anything more than simply a coincidence of horizontal and vertical vectors in the lens-grinding lathe, and Gatens and Lloyd’s horizontal and vertical spatialization of his metaphysics in defense against Hegel? Is this conflation merely accidentally bolstered by the analogy of ideas to be taken read tools in Spinoza’s very early work? I think there is much more to this than that, and that aside from the notion of the horizontal and the vertical there is a multiplicity of core principles that seem to stem from Spinoza’s unique, and one must say classed, artisan experiences. In particular we must understand that Spinoza, unlike Descartes, was through and through a practiced craftsman, an artisan by trade and value, who repetitious and refining practices which he took rather seriously must have influenced his guiding conceptions of Mind, Body, Idea and Power. In fact, it seems that it is a tooled notion of idea and body that I believe informs his vital definition of the power of the body, a defintion which will reconceptualize any of our instrumental approaches to material augment of the human body or mind:

Whatever so disposes the human Body that it can be affected in a great many ways, or renders it capable of affecting external Bodies in a great many ways, is useful to man; the more it renders the Body capable of being affected in a great many ways, or of affecting other Bodies, the more useful it is; on the other hand, what renders the Body less capable of these things is harmful.

– E4p38

Think on how this expressive yet instrumental numerical notion of power can be found within the most elementary experiences of tool use, as Richard Sennett tells us about the wonders of the flat-edged screwdriver:

…in its sheer variety this all-purpose tool admits all manner of unfathomed possibilities: it, too, can expand our skills if only our imagination rises to the occasion. Without hesitation, the flat-edged screwdriver can be described as sublime – the word sublime standing, as it does in philosophy and the arts, for the potentially strange. In craftwork, that sentiment focuses especially on objects very simple in form that seemingly can do anything (195)

The Craftsman, Chapter Six “Arousing Tools”

I believe that Spinoza’s lifelong craft experiences with the lens-grinding lathe (among so many other simple tools) had a lasting effect on his conceptions of Mind and Body, and their necessary unification. The grinding lathe, with its intimate, indeed cybernetic, interweave of body, mind and material construction, its concentric use of the spinning semi-sphere, must have struck Spinoza as sublime in the sense that Sennett tells us. There is the evidence of Spinoza’s resistance to the sophisticated, semi-automated designs of his brilliant and wealthy neighbors the brothers Huygens ( EP 15/32 ) which tells us that Spinoza was quite hesitant to leave behind the interface of the machine with the understanding and felt hand. But it is more than this. It seems that the grinding lathe leaves its conceptual, kinetic trace all the way up through to the most abstract, and most radical of conceptions. In fact there is the very real sense in which we may read Spinoza’s Ethics (as it exists both in idea and extension) as a tool which can affect and be affected in the greatest number of ways. 

Picturing Work and the Work of Picturing

Once we have a vivid sense of the kinds of material engagements Spinoza had concerned himself with, his bodily practices of concentrated creation and refinement, we get a better sense of how Spinoza conceived of his own Rationalist, propositioned philosophical aims. From there we can place ourselves with the lived historical space of the man who lived at the cusp of our modernity, and feel something of the material and pragmatic focus of his articulations of freedom. Below is Spinoza’s rented room in Rijnsburg where he lived roughly from age 29 to 31 having fled the upheavals of Amsterdam, perhaps with concern for the return of his tuberculosis from remission. It is today’s Spinozahuis museum. As mentioned before the wood turners lathe depicted there is NOT the kind of lathe Spinoza would have used, but if you look to the upper center left of the photo you can see a hypothesized spring pole, the vertical vector of his practice. It is not known if Spinoza’s later rooms in the village of Voorburg, where it is thought that he did his most concentrated grinding work, were of this size, but the combination of the Rijnsburg room and Hevelius’s illustration gives us I believe some determinative sense of the internal dynamics of Spinoza’s lived experiences as a craftsman and thus as a thinker; they directs us to the material and conceptual causes that may have privileged Spinoza treatment of the Mind and Body over his predecessor, Descartes. There is much that divided these two thinkers from each other, but perhaps even more than joined them. Each was concerned with lens-grinding, optics and the improvement of the telescope, but only one of them was a practiced maker of lenses and instruments. Only one of them touched the glass.

The Rijnsburg Wood-turners Lathe, Spinozahuis

Hevelius's Spring Pole Lathe, from the Selenographia (1647)


The Rijnsburg Wood-turners Lathe, another angle

The Rijnsburg Wood-turners Lathe, another angle


Spinoza’s Eyes of the Mind and the Grinding Form

Three References to Sight

Still, we sense and experience ourselves to be eternal. For the mind no less senses those things that through thinking it grasps, than those it has through memory. For the mind’s eyes, by which it sees and observes things, are demonstrations themselves…

Ethics, 5P23s

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…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, at the middle of the brain), but concepts of Thought [or the objective Being of a thing, insofar as it exists only in Thought]Ethics, 2p48s.

…after I have come to know the nature of vision and realise that it has the property of making us see one and the same thing as smaller at a distance than if we were able to see it near at hand, we infer that the sun is bigger than it appears.

The Emendation of the Intellect

To follow up on yesterday’s thinking, these are the three substantial references to mental vision in Spinoza which one must come to terms with if we are to understand the effect the grinding of lenses and the making of optical instruments had upon his thinking. Spinoza speaks very little about light, in terms of an explanatory metaphor, and when it takes up the idea of vision, it is in very selective ways.

Considering the first two quotations above, we see his firm resistance to any “picturing” of truths, something he relegates solely to the experiences of the imagination. Thinking, or having a truth thought, is not to be understood in any sense of having a picture of the world (and he distances himself either from Descartes, or popular conceptions of Descartes in this passage). But when considering the second of these, see still understand that the Mind, expressed in the succession of ideas of which it is composed, actually senses. We sense, and the Mind senses (both sentio ). But if we follow the pairs, “we” sense and experience, although the mind “senses and observes”, giving clue to the shift Spinoza is thinking of. In our collective sensing, we passively experience the world, but in our intellection, we observe the world. (Yet part of the process of intellection, as Spinoza conceives of it, is the realization that we are not distinct from it.)

I am still caught by the notion that the demonstrations or proofs of propositions are literally the eyes of the mind (remember the apodictic connecton between logical demonstration, and a craftsman’s demonstration: what “showing” means). I recall a study that I read on Plotnius, and the author was talking about how or why we read philosophers of the past. He suggested that as one submits themselves to their architecture of arguments a subtle, perceptual shift happens. The world, our very perceptual experience of it, alters. This is how Spinoza conceives of his propositions and proofs operating. They are not so much seen as indefatigable statements that secure a fact, (although most certainly he conceives of them as correct and true), they are meant much more as a interaction, a pedagogy, which alters one’s experiences of the world, promoting intuitions which will grant greater freedom and greater capacity to act. They are cybernetic combinations. I have used that world before, but intentionally so, because Spinoza defines a body as a communication of parts. For Spinoza, I believe, the parts of the Ethics, combine in a useful way with the parts of our Body-Mind, to produce an interaction, a communication, which empowers the person, in assemblage. In a sense, the conatus that gave rise to the Ethics, the conatus that preserves the Ethics in print and discussion, combines with the conatus that preserves the reader, to produce One, collective, meditative, thinking and acting body. The material dimension of this must be understood.

If one is to draw from the lens-grinding analogy, a lens is ostensibly made by grinding a glass-blank in one’s free hand, against a metal (copper or iron) form or lap, which holds the curvature that is desired. The off-hand either drives the spinning of the lathe so that it spins something like a potter’s wheel. (Or this can be accomplished with a foot petal, leaving the off-hand free.) As one drives the metal dish in concentric revolutions, it clearly manifests a changing, yet fixed state, turning but stable. Against this metal form, upon which abrasives of various grits are poured, the glass is pressed, until it gathers the shape desired. The fineness of the grits and processes of polish are made in stages.

This is something how I imagine Spinoza’s Ethics was to work. The first armature of arguments for Substance, and then continuations on to knowledge and power, are meant to grind down the rough edges. This is where the greatest resistance comes I believe in many readers. The propositions seem to lack application, and may appear to indulge in terminology that grates. Yet, the work progresses to grades of increasing finesse, until the human emotional condition is addressed, and then the glass is given a subtle polish, in hopes of increasing its speed and capacity.

The propositions and the proofs exist to be ground upon. One tests one’s mind against them, pushing against their rigid character, and their tight cross-reference. It is not so much that one proposition is right, and convincing, but rather that their very atomic, veined fabric is meant to be engaged in, as one presses against its spinning form. The hardness of the material and its ideational webbing wears away the rough edges of the mind, and suddenly you are seeing something, however mutely, or blurred. This is how I imagine that Spinoza thought of his text, and why he used the geometrical method for his master work. He was not thinking to convince, but rather to engage, to cause the mind to “sense and observe” in a different manner.

How Platonic this may sound, the meeting of a Form to rough material, but the dynamic, material character of the form itself denies any transcendence. The interaction is seen as an aid, but also as a combination. Here the third quotation from the top of the page comes into view. By acquiring knowledge about the nature of vision, we come to see differently. We come to see the sun as further away, such that we can act more freely in regards to it. We sense and observe in a continuity. The propositions and proofs that we have pressed ourselves against have polished us, not because there is perfect vision, but because there is better vision, a vision that is understood as material.

If we compare this “mind’s eyes” notion to Descartes’ “mind’s eye”, from which it likely drew influence, we detect a radical difference:

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

The Regulae

Like Spinoza, Descartes wants to promote an intuition, the capacity to see clearly with great immediacy. Yet Descartes imagines a mental vision that is closely focused on what is before one. Whether this be a small, causal relation, or an indubitable fact, Descartes wishes to build surety upon surety. Spinoza though wants mental vision to be comprehensive and vast. Any close comprehension moves very quickly to a totality of perceptions, from which all else gains clarity. The over-arching Coherence is that from which individual distinction and merit comes from. And it is for this reason that his optical conceptions were based on the notion that vision requires a wide-view clarity of focus, refining both the center and the borders. And it was for this reason that he composed his Ethics vastly, not so that he could cover every base, but so engagement with it would produce a symmetrical, whirling, grinding curvature of the mind, one that could focus all events, as best as possible, yet ever with a view to the expresssion of the body, as a body. And part of this final understanding is that not only are we formed by the things we are ground against, in the frictions of our living, but we are ever in communication with them. And I suspect Spinoza wants to say, in telling us that demonstrations (of every kind) are the eyes of the mind, that because of this, our organs of “sight” extend far beyond us, into the infinity of affects and motions that communicate themselves to us, each along a tensioned, discerning line of power.

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.



Descartes’ 8th Rule: A Spinoza Touchstone

Rules For the Direction of the Mind, gives clue to Spinoza’s concept of the instrumentation of knowledge

I post here Descartes’ 8th Rule from his Regulae, as a marker to important influences upon both Spinoza’s philosophy in general – it contains the seeds of his argument for Three Knowledges, for instance – and his approach to technology, as it possesses the analogy that knowledge production is like the development of blacksmithing. This is not to mention the inclusion of possible paths to the discovery of the importance of the anaclastic, significant for the subject of Spinoza’s objection to Descartes’s claim for the importance of the hyperbolic lens. I highlight the relevant places of correspondence:

Full Text of the 8th Rule with Annotations and Highlights

Rule 8: If in the series of things to be examined we come across something which our intellect is unable to intuit sufficiently well, we must stop at that point, and refrain from the superfluous task of examining the remaining items.

The three preceding Rules prescribe and explain the order to be followed; the present Rule shows when order is absolutely necessary, and when it is merely useful. It is necessary that we examine whatever constitutes an integral step in the series through which we must pass when we proceed from relative terms to something absolute or vice versa, Read more of this post