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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

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The “ens reale” and the “ens rationis”: Spelling Out Differences

The Pleroma and Creatura: Bateson

Gregory Bateson, a father of modern cybernetic has some very important things to say about the nature of differences, and has been fruitfully appropriated in any number of ways, primarily due to his very powerful defintion of Information as “a difference that makes a difference”. But it should be noted that Bateson’s approach to differences is one that drives a very firm, dualistic line between Mind and Matter, one that follows Carl Jung’s categories of the Pleroma and Creatura:

The significance of all this formalization was made more evident in the 1960s by a reading of Carl Jung’s Seven Sermons to the Dead, of which the Jungian therapist Jane Wheelwright gave me a copy. I was at the time writing a draft of what was to be my Korzybski Memorial Lecture and began to think about the relation between “map” and “territory.” Jung’s book insisted upon the contrast between Pleroma, the crudely physical domain governed only by forces and impacts, and Creatura, the domain governed by distinctions and differences. It became abundantly clear that the two sets of concepts match and that there could be no maps in Pleroma, but only in Creatura. That which gets from territory to map is news of difference, and at that point I recognized that news of difference was a synonym for information. (Angels Fear, Introduction)

For Bateson, the separation is one of processes, and not one of Substance like it is for Descartes, but all the same, it imposes a strict heirarchy which privileges the mental over the physical. A stone simply is restricted to the domain of the Pleroma, while any differential making process, even the simplest of biotic discrimination is given over to the realm of Creatura:

It is, of course, true that our explanations, our textbooks dealing with nonliving matter, are full of information. But this information is all ours; it is part of our life processes. The world of nonliving matter, the Pleroma, which is described by the laws of physics and chemistry, itself contains no description. A stone does not respond to information and does not use injunctions or information or trial and error in its internal organization. To respond in a behavioral sense, the stone would have to use energy contained within itself, as organisms do. It would cease to be a stone. The stone is affected by “forces” and “impacts,” but not by differences. (Mind and Nature, Chapter II)

To most of us this is a perfectly acceptable, perhaps even obvious designation. There seems a powerful instinct that tells us that a stone simply is not in any sense like an amoeba, which is to say, what a stone does (if it does anything at all) is somehow categorically different than what an amoeba does (though both can kill you). The difficulty arises for anyone who wants to theorize in a way that does not privlege the Mind over Matter. This begins perhaps as a desire to not privlege human realities over animal realities, and then ultimately to give over to even the animate some kind of “right”, some play in the game in determining what is “real” and thus “what matters”. When Mind (in some form of Idealism) becomes the heirarchial source point of what matters, somehow this all slips back into a remote solipsism of the merely human world (and then even, the Western world, or the American world, or white upper middle class academic world). If one instanitates a fundamental primacy between the Pleroma and Creatura, wherein the Ceatura determine the status of the Pleroma in heirarchial, a priori fashion, something of the Mind/Bodd, Spirit/Matter dichotomies that have long haunted philosophy are dragged forward (often with explicit political consequences of such binarism).

The Difference that makes a/the Difference

For this reason one must keep in mind the essential metaphysical base from which Bateson is employing his work (Marx makes just such fateful Nature/Culture distinction from the start as well).  If one is going to grant equal footing to the non-human (and non-biotic) actor in the world, this essential binary must be categorically undone. As long as one has divided up the entire world into realms, one realm becomes paramount, and the line merely shifts.

What Bateson has in mind when he speaks of “a difference that makes a difference” is the way that information connects what is “out there” in the world to the “in here” of a cybernetically organized system. To put it most simply, the internal relations within a system form a boundary which is sensitive to only particular kinds of disturbances (a blind person does not turn his head to see someone waving to him from across the street, a tick does not drop from its leaf when a breeze blows). The difference out there in the world that makes a difference in here, is for Bateson the difference that makes a difference, it connects inside to outside.

But out of a completely unintended difference in the way that Bateson has framed his defintion of Information, I would like to use his notion of difference differently. Because I am not interested in giving priority of mind over matter, I am less concerned with the way that mental systems exercise dominance over physical structures (picking out what matters so as to eventually predict and control it), I am not going to follow the breadcrumbs of difference from outside to inside. This is far too Idealist for me. Rather, I want to see if we can talk about differences in such a way that the things a stone is doing, and the things that an amoeba are doing, are in someway signficantly related (and such that the actions of each are given footing).

Bateston states his defintion of Information in at least two ways in separate works.

1. A difference that makes a difference.

2. The difference that makes the difference.

It might sound trivial, but in the spirit of acknowledging even the smallest of differentiations, of this variation between the definite and indefinite article, I would like to spin out a profound distinction which maps onto a fundamental ontological distinction of Medieval Scholasticism. Much of Scholasticism spent its time trying to iron out the remarkable, but underdeveloped semiotic point that Augustine made, that signs transcend the Culture/Nature dichotomy. There are natural signs, and their are signs of convention. And (natural and cultural) signs are defined as:

“a sign is something which, offering itself to the senses, conveys something other to the intellect,” (Signum … est res praeter speciem quam ingerit sensibus, aliud aliquid ex se faciens in cogitationem venire) (Augustine De doctr. chr. II 1, 1963, 33)

Attempting to work out the full consequences of an ontology of the semiotic which transcended the Nature/Culture barrior, Scholastic philosophy realized that there must not only be material signs “out there”, but also mental signs “in here,” and much ado was made on how to connect the two (until in modern times gradually questions of signification became a questions of representation…many like to put this at the foot of Descartes, or even the Locke, but it is not altogether clear that this is the case).

A product of this debate was the two classifications Ens Reale and Ens Rationis. A real thing, and a rational thing. These are treated in various ways, often as the difference between “physical being” and “logical being”, but I want to speak much more broadly, without precision. An ens reale is a thing in the world, and an ens rationalis is a thing in the mind. Is here that I want to propose a loose though hopefully enlightening homology.

1. A difference that makes merely a difference  is an ens reale.

2. The difference that makes the difference is an ens rationis.

Leaving behind Bateson’s use of information as the thing that connects inside to outside, as an ontologist I want to speak of differences in their variety of states. Following Plato’s initial definition of being as the capacity for anything to affect or be affected, as found in the Sophist, the general sense of the reality of differences is that anything that makes a difference in general, “a difference” has being, and is ens reale. But any difference that is strictly internal  to a closed horizon relation of parts, is an ens rationis, that is which is to say, it is a difference that makes the difference, recursively. In this way, and event out there in the world, perhaps lightning strike, is an ens reale difference insofar as it is not taken with in an overarching internal circuit of relations, and its effect upon the human organism, that actual internal differences which are within the horizon of person, are each ens rationis. It is important to keep track though, that every ens rationis is an ens reale. The question is: Is every ens reale also an ens rationis. I think they are.

Spinoza’s Bodies as Certain or Fixed Ratios

As I mentioned previously, Spinoza’s defintion of Body is far more rich that it is often taken to be. More than simply a billiard ball image of circulating motions (which is how it appears at first glance), his panpsychic metaphysics grants some degree of mind (Idea) to any extensional expression, such that even the simplest of bodies in composite have a foothold in the mental. Here is the definition in bodily terms:

Definition: When a number of bodies of the same or different magnitude form close contact with one another through the pressure of other bodies upon them, or if they are moving at the same or different rates of speed so as to preserve an unvarying relation of movement among themselves, these bodies are said to be united with one another and all together to form one body or individual thing, which is distinguished from other things through this union of bodies. E2p13a2d

It is quite interesting that Spinoza finds what separates out one body or individual from another is a certain or fixed ratio, certa ratione. It seems safe to say that not only living things preserve for Spinoza through a certa ratione, but also taken to be inanimate things. We have here the potential for categorical description that crosses through the Pleroma/Creatura divide that Bateson privleges. The ultimate question is: Do abiotic wholes which do preserve through a certa ratione, also achieve within that horizon of “individual” an order of differences that allows us to say that they are each ens rationis.

It is hard to know exactly what Spinoza has in mind: when he describes this perpetuation of communicated motion, for instance, is it a different sense of body than that brought about by external causes in the earlier part, When a number of bodies of the same or different magnitude form close contact with one another through the pressure of other bodies upon them, or it is simply the internal specification of those external forces? What we can do is use the definition as tellingly as possible. What I suggest is that differences that are internal to an object or body as Spinoza sees it, are differences that are indicative of a mutuality of effects. A change in this part of the body effects a change in another part of the body, and then another, and so forth, such that the whole is still maintained. And there need not be the cybernetic closure that Bateson enjoys with Creatura. The entire world would seem vectored with communicated balances between bodies that however briefly or enduringly remain in ratio with each other. These mutuality of communications I hold is the threshold for an ens reale to be an ens rationalis. The cybernetic closures which map a territory are certainly different kinds of internal organizations of horizons, but rocks, breeze patterns, neuron rhythms, photon pathways, planetary equalibriums, dust corners, electron loops, all possess an internal coherence of differences which is preserved, and in which a single difference (I would say) semiotically indicates consequences of internal coherence. Stones “think”.

Stone Cognitum

There is a perspective of stones, one that is not reducible to the way in which differences in stones make differences upon us. In this sense, as Graham Harman says in Latourian fashion, stones translate other stones when they encounter each other. (I do not see how such a claim can be separated out from panpsychism.) The internal relations that make up a stone (semiotic, of each an ens rationalis), are also each an ens reale (a difference that makes a difference) which can make a difference that makes the difference to us (or some other internal set of relations), is itself also a difference as ens realis.

There are several interesting ways to procede from this, but the one that I would like to take up follows through from my last post on Spinoza, and that is that any ens realis (a difference that makes a difference), is not only already a difference that makes the difference in the internal expression of Substance as a modal whole, and thus an ens rationalis. But it is already caught up in any number, perhaps an infinite number, of ens rationalis horizoned closures. In this way, differences which are semiotic to an internal whole of differences, are also because real, differences that are internal to a plethora of bodies that cross cut that body. That “fixed ratio” is tugged at from any number of other “fixed ratio” directions, as parts of its coherence respond not only to an external horizon of differences, but also to their participant share in a cross-sectioning fixed ratio, communication whole. Any ens rationalis is Semiotically Conjoined to a variety of mentalizations.

For this reason, it is not just that the totality of coherent differences that make up a body are occluded from us, selected out by our cybernetic, ratioed closure, but also that the semiotic investment of those differences is occluded from that body itself, the coherence of its inside/outside closure. And the same is said of our own body (bodies, really).

There is another aspect which should be grasped so that we don’t fall too deeply into any Subject/Object binary. And this is something I will develop later. Because ultimately an entia rationales closure is itself a perspective, when one or many entia rationales closures come into supportive relationships to each other they can be read as forming new bodies. This is to say, when we come to know something else and intimately relate to it in a bodily, the boundary between us and it at least is semiotically problemized (if we seek to keep them completely distinct). Thus, it is not merely the case that the “kernel” of relations of an object we engage is kept from us, like a forever retreating shadow, but also the case that as we engage an object (an aspect of our environment), we at a very real, semiotic level (that is, at the level of entia rationales), become it.

Thus, as the carpenter uses his hammer, or the lens grinder his grinding lathe, there is a communication of motions which exceed the boundary of bodies, forming one of two (to some degree). The world is felt, mutually, through the performance union of both bodies. It is for this reason that Tommaso Campanella tells us: To know is to be, cognoscere est esse. This is not a metaphorical transformation of the subject into the object, but rather a real, substantial in-form-ation, binding the two bodies both epistemologically and ontologically, through the ordering of their mutual coherences. If the object of the hammer remains somewhat blind to the carpenter (some of its variety of aspects still hidden), these aspects must be accorded their place within the causal, and hence semiotic, internal relations of the body (body + body). Ultimately, these differences can only be the differences of Conjoined, and thus often silent, Semotic inherence at the bottom of any entia rationales closure, the way that an ens rationalis is necessarily polyvalent to a variety of cognitioning, and therefore persisting, bodies.