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“By mathematical attestation”: Spinoza’s Epiphantic Calculation

Just How Mathematical Was He?

I have been having an interesting conversation over time with Eric Schliesser at Leiden University who holds a minority position on the role mathematics plays in Spinoza’s position on what can be known. He strongly interprets Letter 12 towards a skepticism for just what mathematical calculation (and its attendant applied empirical observation) reveals. As Letter 12 attests, Spinoza regarded mathematics as a “product” of the imagination, come from our imaginary classifications of objects as wholly bound things – cut out from the cloth of Substance, if you will. I wrote on my agreement with this here Spinoza and Mechanical Infinities, but Eric really would like to push the interpretation so far as to restrict all of mathematical endeavours to the realm of the imaginary (the lowest forms of knowledge for Spinoza, the other two being the “rational” and the “intuitional”). This of course comes up against Spinoza’s rather obvious and profound use of mathematics as a model for philosophical investigation and even the higher forms of knowledge as both his method more geometico and his illustrations of higher knowledge both make use of mathematical forms as exemplary.

In large part I really am in agreement with Eric, in the most common Spinoza interpretations the mathematical has somehow risen far above the frame in which Spinoza intended it, but it makes very little sense for me to regard the “products” of imagination as imaginary itself – in the Emendation Spinoza speaks of the production of tools of intellection out of imaginary relations as a positive progression. Clearly mathematical description provides distinct causal understanding of the relations between things, and it is exactly in this vein that the empirical science observations of several centuries does provide a substantive remove from mere superstition, something that Spinoza firmly places himself against. It would seem that Spinoza’s true position lies somewhere in-between, not accepting Galileo’s thought that Nature is written in the language of mathematics, but also grasping that in mathematics (and observation, experiment) there are genuine increases in understanding, freedom, power, joy,  and ultimately for Spinoza, Being.

In this way mathematics is seen as:

1. Both a product of the imagination, and an aid to the imagination [Auxilia imaginationis] (Letter 12).

2. As such they are qualifiable as ens rationis which is what Spinoza calls them in letter 12 [eius modi Entibus rationis], something he is elsewise careful not to be blurred with ens imaginationis (E1App). 

3. It should be admitted that Spinoza does not help things by just prior to distinguishing “number” as a  “things of this reasoning” in letter 12 to Meyer he refers to it as  “nothing more than thinking’s, or better imagining’s, modes” [nihil esse praeter cogitandi, seu potius imaginandi, modos] - he wants to straddle the line here and a little confusedly so.  

4. But reading numbers as ens rationis (distinct from ens imaginationis), these, as Spinoza notes in the all important E2p49 asserting the collapse of volition into the concept of “idea”, are not ens reale (Cor. dem. [III. B (iii)], accept when the latter is understood as an operant, an affirmation and an action. 

5. The real and the rational abstraction that describes it are not to be confused. It is this final distinction that provides the skeptical element towards an ultimate mathematical reduction of Nature.

6. One has to live with the exegetical problem that while Spinoza in the Appendix of Ethics part I makes a strong distinction between things of the imagination, and things of reason, in letter 12 he oscillates, even within sentences, between things of (this) reason and something he undefinedly calls aids (auxilia) of the/to the imagination, never sure how he wants to describe Number.

Are Maths Only Imaginary? What Would that Mean?

By my understanding Eric places increases of power in mathematical description solely within the a “usefulness” category, all the while restricting them to the “imaginary realm”. While I really enjoy this outlier position, the very substantive nature of these increases in usefulness simply remains unexplained. And though this in part may be due to some inconsistency in how Spinoza treats the imagination (and the concept of order itself), I do think that Spinoza saw in mathematics (and scientific testing) genuine increases in the understanding of things, without acceding to the thought that mathematics genuinely reveals the eternal essences of things. For Spinoza we are, even the most scientific among us, like a “worm in blood”, not comprehending entirely the body and indeed the Universe we live in.

In this discussion there is an interesting, and indeed I think problematic sentence that at least provisionally I would like to retranslate. It is found in the Appendix of the first part of the Ethics, and in it Spinoza appeals to the very mathematical attestation by which we perceive or understand things of the world. He has just finished rebuffing two stages in thinking: addressed are those that feel that astronomically it is the motions of the heavenly bodies themselves that somehow compose [componere] a  harmony, a harmony that God delights in; and then those that from this notion of celestial harmony then find that it is the disposition of the brain alone from which human judgment comes, something which results in a skepticism of knowing in general. One is left with either a physiognomic theory of differences of perceptions (there are as many different kinds of brains as there are palates), or presumably on the other end the “veil of ideas” of proto-idealism.

In contrast to this physiognomic skepticism, Spinoza turns to the very discernment of things [res]:

Res enim si intellexissent, illae omnes teste mathesi, si non allicerent, ad minimum convincerent.

I translate this literally because there is some disagreement in the major English translators on the meaning of Spinoza’s sentence (and I think that both of them are somewhat wrong):

For if things they would have been able to discern, those all by mathematical attestation, if they were not allured, at minimum they would have been convinced.

The two counter translations I provide here. Curley in some rather convoluted restructuring, attempts to emphasize the “all” as an accusative. All these persons would be convinced if they merely discerned things correctly. The things themselves would convince everyone. While Shirley, I think more correctly, places emphasis upon both Spinoza’s mode of argumentation, and mathematical attestation. Here they are:

“For if men had understood them, the things would at least convinced them all, even if they did not attract them all, as the example of mathematics shows.” (Curley)

“For if men understood things, all that I have put forward would be found, if not attractive, at any rate convincing, as Mathematics attests.” (Shirley)

There really is no support for Curley’s inventive transformation of “omnes” into a universal emphasis of agreements, though that may be implied. Really what Spinoza is saying is that indeed contrary to merely the physionomic understanding of judgments (and also a celestial orderliness model), distinct discernments of things has come via the testament of mathematical treatments. While Shirley’s translation grasps the general thought of this, Curley captures the very epiphantic nature of such intellection, it through mathematical treatments that the very nature of “things” appears.

And the little caveat on the nature of how such men will be affected by such discernment is telling. Such fellows will be convinced, though they may be “attracted” to such an interpretation. Here Spinoza seems to be putting his thumb in the eye of those that disagree. There are it would seem libidinal investments in seeing the world other than the way in which it is most arguably so. There is also perhaps a commercial connotative association of liceo, “to buy, to put a price on, to value” which may not be far from Spinoza’s intention.

Teste Mathesi

So what are we to make of this “by mathematical attestation” [teste mathesi]. Clearly, it is by reason of mathematics that philosophers (and scientists) arrived at the notion of a harmony composed of celestial bodies in motion, a sense of harmony that for Spinoza ultimately lead to viewing the brain as the source of all human judgment; so it cannot be by mathematics alone that we come to discern things properly. And Spinoza has in turn used the geometic method in such a way that he seems to feel that he has, via such a mathematical attestation, produced a discernment of things. As Spinoza in Letter 12 strongly calls into question the ultimate knowledge available by mathematical measurement and calculation, there would seem to be only one more meaning remaining. Mathematical attestation is for Spinoza a revelatory one, one in which the coherence itself (what is calls elsewhere a different “standard of truth”) provides the conviction of discerment, but also one in which any mathematical description always remains merely an approximation, a rounding off of the edges. And these are edged through which the epiphany of perception itself shows through. This is in keeping with my general sense that in that all the propositions found in the Ethics are linguistic expressions, none of them actually are adequate ideas. It is rather that the interaction with the Ethics itself, its real, material and ideational body, is to provoke, is to cause, a real material and ideational change in the reader, one which cannot be reduced to the arguments themselves.

In a certain sense, Spinoza’s very intellectual and physical experiences as a craftsman, the precise use of calculation when applied to physical substances in the service of gaining the desired effects is the very thing that would preclude any minimization of mathematics or the testing of experimentation. Craftsmanship is after all where abstracted calculation and experiential rule of thumb come most closely together. And by all testament, Spinoza was a superb and devoted craftsman.

In a modern sense, we might want to say that for Spinoza the Universe is not a linear mathematical thing, but that the coherences of cognitions and communications between things is at best brought out by linear mathematical treatments (those only known of the day), treatments that in the end must also then be compared with man’s own finitude as a creature. As a craftsman perhaps he not only understood the way in which calculation and figure could be used to control and shape material, but also understood the often unexpected, unique and eruptive form of material itself, the way in which the glass, bubbled and fogged as it is, defies the curve of optical imprint of the lens grinding form. For Spinoza there are always non-linear magnitudes within magnitudes, beyond any one boundary-making, linear abstraction. But this does not prevent mathematics itself to produce reductive epiphanies unto the relationship between things.

Some follow-up thoughts: Spinoza “Following the Traces of the Intellect”: Powers of Imagining

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.

}∅{ The Full Set

The Full-ness of a Body

This is not what I intend to write on the Subject of Infinity, but it is a projective chain of thoughts. It is a tracing of a yet un-consolidated line of interpretation.

Fido the Yak presents a symbol/concept which – and I’m not sure of its origin or complete meaning - is quite intriguing in the light of my recent readings on Spinoza, the Infinite and mathematics. What I would like to call the Full Set. This is how Fido describes his imagination of it

}∅{ is like an in-cept, yet it emerges discursively as a response to the arche. How does it originate? Autopoietically? Or do we acknowledge that it is a con-cept with the arche? (Like its withness with the empty set.) I talk about }∅{ extemporaneously because the extemporaneous describes it. If you can be in a state of consciousness that includes nothing while excluding nothing then }∅{ can describe such a state of consciousness, or a goal of thinking, a guideline, better. Maybe you’d want it to represent an empty concept, I don’t know. I say now the }∅{ expresses the acknowledgment that the explanation never exceeds its explained, which is a way of saying it never accomplishes what it sets out to do, and that “foundation” is a metaphor—you may see why I call it “the breach.”

We are running in similar directions, but I prefer in my own thinking to not think of it as a breach, so much as a whole plenitude, close to Deleuze’s Full Body, the Body without Organs. (Here my thoughts proceed from his in-spiration, in perhaps an appropriation.) It is closely related to Spinoza’s notion of the Infinite as something that cannot be broken. So, in my hands, it would describe the infinite proximity between any two limits.

It emerges from Spinoza’s diagram of bound infinities, which proclaims that within any bounds there are an infinity of magnitudes which are themselves divisible:

It is important to see that for Spinoza these are magnitudes(and not simply points on an imaginary line, which are at best ab-stractions). As magnitudes, they are FULL. Now, if we are to play with symbols, the requisite symbol for Spinoza’s point about bound infinities would be something like }∞{, which is to say, between any imposed limits, there are an infinity of magnitudes buried. Also importantly, and somewhat divergent from Badiou’s concept of Count-as-one, the symbol should not be {∞}, because for Spinoza any (abstract) internal bounds already, already refers to, or references to some degree a determination that lies outside of it, as in his Letter 12 diagram

The sub-section between AB and CD {}, already includes comprehension of the circle circumstance itself, }{. Or, the internal count-as-one (set), really is composed, determined by the bordering edges of something beyond it, as a mode of comprehension, consciousness itself.

Playing With Symbols

So, while Spinoza in Letter 12 seems to be presenting something of a }∞{ determination, what would be the intuition of the full set }∅{, which is our subject here mean? It is not that between any two abstract limits there is some sort of nothing, or emptiness (for Spinoza denies the ontological consistency of the void). It is rather that the act of distinction and limitation itself drives itself toward the impossibility of separation: as we enter into the infinity of magnitudes (let us start with Badiou’s count-as-one teeming/erupting with multiplicities) {∞}, we are pre-positedly forced both outward…}∞{…but also inward to the full set itself…}∅{…the way in which what lies between simply cannot be divided at all and remains unbroken. It is not the sheer multiplicity that lies between any borders (inside, or outside), but rather the implicition that there is no “gap”, the very fullness of Being, from which mathematics, figure drawing and set-making composes only an abstraction, an imaginary class. It is for this reason that for Spinoza rational thought leads ultimately to an Intuition of immanent wholes and a speed of thought. 

Or, if put another way, any conception of emptiness, or lack, or nothing {∅} (whether it be mere psychological wanting, or mathematical 0), diverges upon the fullness of being, showing itself to be a figment.

Spinoza on the Infinite, the Unbound: Part I


Key to understanding Spinoza’s approach to the Infinite is appreciating that for him, primarily and speaking generally, The Infinite is Unbroken. And following this, modifications of the Infinite (how Spinoza defines the modes) do not break that unbroken state. For Spinoza, any treatment of Substance must follow from this understanding.

What makes this compelling, and ultimately germane to any assessment of the status of rational knowledge as it is found in logically related descriptions, and at the seeming apex of such, mathematical descriptions, is that insofar as numerical designation indicate a limitation, a bound, a break in The Infinite, this is an imaginary product and is not adequate knowledge. Perhaps, penultimately: mathematical, scientific knowledge stands at a skeptical remove from the true nature of Nature. Which is not to say that mathematical relations, and the various fields of mathematical description, do not play a significant role the human being coming into some form of absolute knowledge of God/Substance/Nature. At most, the internal coherence and powerful indications of mathematical forms act as Augustine’s finger, pointing to Substance’s moon.

One can see this in Meyer’s preface to Spinoza’s early career more geometrico treatment of Descartes’ philosophy, The Principles of Cartesian Philosophy. Meyer explicitly sets out Spinoza’s distance from Descartes’ matho-scientific treatment of Nature:

This [work] must not be regarded as expressing our Author’s own view. All such things, he holds, and many others even more sublime and subtle, can not only be conceived by use clearly and distinctly but can also be explained quite satisfactorily, provided that the human intellect can be guided to the search for truth and the knowledge of things along a path different from that which was opened up and leveled by Descartes. And so he holds that the foundations of the sciences laid by Descartes and the superstructure that he built thereon do not suffice to elucidate, and resolve all the most difficult problems that arise in metaphysics. Other foundations are required if we seek to raise our intellect to that pinnacle of knowledge.

I had to get this basic beginning out so that I can move on. Hopefully to follow soon: Spinoza’s hyper-proximity to Badiou (and Badiou’s intention misreading), Cantor’s attempt to in-concretize Spinoza’s Infinite (as transfinite), and, the Place of Mathematics within Spinoza’s theory of the Intellect and Knowledge, the weight of Letter 12  (or something of these sorts).

Analog and Digital Intellect: Threshold Intensity, or Either/Or


Analogical Co-munications

I came across (now twice, but this time investigated) this wonderful collection of Deleuze-inspired writing and exhaustive explications, Pirates and Revolutionaries. Some of the very best stuff on the internet for instance on Spinoza’s concept of infinity. This article though on the difference between Analogical and Digital thinking is immensely clear and open-ended, for any of those who have not considered deeply the two modes of intellect. Below is one small snippet in a wide-ranging summation and positioning:

We will first address the research on animal communication that Gregory Bateson discusses in his Steps to an Ecology of Mind, a text that Deleuze cites when distinguishing analog and digital language. According to Bateson, the ‘messages’ that animals convey refer not to objects but to their social relations; for example, the cat’s mewing does not mean milk, but ‘dependence.’ A more compelling illustration is his story of a wolf-pack leader catching an inferior male who broke the code of hierarchies, and achieved coitus with a female, which involves being locked-in with her. Bateson explained previously how an adult wolf weans young puppies by crushing them down with its jaw. Then, in the case of the leader finding his subordinate infringing upon his mating prerogatives, instead of attacking, the leader simply crushed the male down as though weaning him. This communicates their social relationship by analogy: ‘just as a father is to a puppy, I am to you.’ In general, most animals normally convey their interrelations by means of such an analogical language, which consists of paralinguistic and kinesthetic expressions (body language) that communicate magnitudes of social relations (such as being more or less dominant) by means of analogous changes of magnitudes in bodily expression. Deleuze himself defines analogical language as one of relations, which consist of “expressive movements, paralinguistic signs, breaths and screams, and so on.

“Deleuze’s Analog and Digital Communication; Isomorphism; and Aesthetic Analogy”

Analogical/Digital Oscillation

What is interesting for my processes is that here in the treatment of the analog and digital I am finding the confluence of two divergent studies. Last month I found myself troubled by Hoffmeyer’s notion of the life-defining Digital and Analog concretizations of an individual, touched on in my review of Morten Tonnesson’s essay on Bio-morality Bioethics, Defining the Moral Subject and Spinoza. I very much wanted to write a piece on the kind of distortion Hoffmeyer was performing when reducing the individual into an almost entirely digital (DNA) state, a capacity he felt that was only something that living things could achieve. I had a strong intuition of what I wanted to say about what was problematical in this, but time and circumstance dragged me away.

My objection to Hoffmeyer stemmed from my Spinozist position of the parallel postulate that the order of things and of ideas is the same, and that, at least from a Spinozist position, it was nonsensical to say that an individual existed in primarily a digital state. If Spinoza is correct, one can never have a primarily digital state of an individual, as the material, bodily dimension follows it explicitly. At the time of my original intuition I simply roughly equated Spinoza’s “idea” with digitality. But in the long loop I’ve run into discussions with Eric Schliesser who is organizing a paper to be presented on Spinoza’s skepticism towards mathematical capacities to describe Nature (at first a counter-intuitional position given the mathematic-like forms of Spinoza’s reasoning, and his dependent use on mathematical examples). Our talks gave me to look closer at Spinoza’s letter 12 to Meyer (which Corry Shores does an incredible job of summarizing in cross-reference fashion, treatment I would like to return to). There, famously, Spinoza puts numbers and mathematics to be the products of the Imagination, the lowest forms of knowledge in his coming trinity of knowledges, found in the Ethics). There is no space/time here to go into these investigations, though it is good to mention that they touch on Badiou’s deep misreading of Spinoza and Badiou’s Ontology of Mathematics. It is enough to say that Spinoza denies the Substance itself cannot be discretely divided, and that even the discrete operations of which mathematics specialize fail at capturing the infinity of the taken-to-be finite modes. The order and connection between ideas (and things) is not a numerically ordinal connection. Mathematical discretions are imaginary constructs by Spinoza’s reasoning, as must be the digital reductions/abstractions that much of conceptual philosophy concerns itself with.

In this sense any digital abstraction of analog expressions/relations itself must be materialized. This makes Hoffmeyer’s digital/analog oscillations that are supposed to define life in further jeopardy, at least from a Spinozist perspective, for digital discretion does not even correspond to the notion of “idea” ordering. Rather, Spinoza’s take on infinities under which a maximum and minimum are known, turns digital processes into extreme analogical ones.

This leads me to minimize the entire latter portion of Corry Shores appreciation of Deleuze’s digital/analog analysis of modern painting, on Spinozist grounds. Even the most binary reductions are not “safe distance” processes, but rather are products of the imaginary under specific thresholds. They are felt in topographies, as any viewer can attest. The digital is always felt. The calculation is ever an impression on the material of the body seen through the discretion of its organized thresholds. One can see that there is a certain “faculative disorder” in the (digital) peak tracing of diagrammic representations, but, following Spinoza, these can only be analogical, which is to say continual, conjoinings. If Spinoza’s treatment of the infinite which disjoins the imaginarily discrete (mathematical) infinity from the real, expressive causal infinity, tells us anything, it is that diagrammic dis-organization and re-organization are imaginary processes which ever seek a continuity in the body itself, the body an infinite expression of magnitudes which press nestled upon each other. But unlike Deleuze’s pursuit of the chaotic elements (and this may only be an aesthetic difference), looking with the Intellect, as Spinoza would, is seeing-through these connections, not as bound, but as continually out-flowing and unitary. In this sense the ordering of numbers is a pale, imaginary imitation of the density of continuity in all things, a mechanism for our continual re-orientation.


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