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Is Latour an Under-Expressed Spinozist?

Selections from The Prince of Networks

This posting works as something like a hypothetical dialogue, a reading and response to the first twenty pages or so of Graham Harman’s Prince of Networks. Each of the Latourian points below are largely the words and descriptions taken directly from Harman’s forthcoming book on the renowned sociologist of science, presenting him as a metaphysician). Here is an interaction with the metaphysical possibilities of Latour when considered in the context of Spinoza. The aim is to press as coherently as possible the correspondences between these two thinkers, and to find bridges in the analysis of events such that each may inform the other. More specifically could say that this comparison follows from a rough equation between the two which proposes that all of Latour’s actors are well seen as modal expressions of Substance for Spinoza, such that in many respects Spinoza’s philosophy is able to accomodate or even subsume Latourian descriptions. Of course such an overlay is not complete, for no thinker presents the thinking of another, but the similarities are greater than might otherwise be supposed.

Because the Latourian points are specifically drawn from Graham Harman’s description of them, the comparison also serves as a quick introduction to the kinds of characterizations Graham is making in his coming book. Even if you disagree with the Spinozist comparison, orientation to Graham’s coming book is worthwhile orienting oneself towards.

a = Graham’s description of Latour

b = My comment on how Spinoza bears on the same issue.

The Four Axioms

1a. First, the world is made up of actors or actants (which I will also call “objects”).
1b. Spinoza’s modalities.

1.1a All entities are on exactly the same footing. An atom is no more real than Deutsche Bank or the 1976 Winter Olympics, even if one is likely to endure much longer than the others.
1.1b All modal expressions have the same reality as any other, as perfect expressions of Nature, (though some may have greater reality than others, that is be more active and powerful, given number of combinations they can make.)

1.2a This principle ends the classical distinction between natural substance and artificial aggregate proposed most candidly by Leibniz.
1.2b The Leibniz distinction is non-existent in Spinoza, so does not need to be so ended.

2a. Second, there is the principle of irreduction, already cited above. No object is inherently reducible or irreducible to any other.
2b. To know a thing is to understand its causes. In this way no modal expression is “merely” a certain form of description. Which is to say, neither is an apple merely a collection of atoms, or a fruit of the tree, but also because Substance exists and acts through modal expression, an apple is not merely Substance acting and existing, but also the sum total of all modal causation. It can be neither reduced, nor is it irreducible.

2.1a Yet in another sense we can always attempt such explanations, and sometimes they convince others. It is possible to explain anything in terms of anything else, provided we do the work of showing how one can be transformed into the other, through a chain of equivalences that always has a price and always runs the risk of failure.
2.1b I believe that Spinoza would accept this. It is important to understand that all of our causal explanations for things occur in Spinoza within the imaginary horizon of human activity, the social field of actorly agents. It is not at all clear that Spinoza allows human beings to hold completely adequate ideas, so thus all that they can do is build more and more powerful chains of descriptions, marked by their internal coherence to each other.

3a. Third, the means of linking one thing with another is translation.
3b. What links one thing to another is a vectorial degree of power, organized around the power to act, and therefore know: all things are linked to all other things, translation being a question of perspective which is never complete.

4a. Fourth, actants are stronger or weaker not by virtue of an inherent strength or
weakness lying in their private essence. Actants gain in strength only through their alliances
4b. Spinoza agrees. This difference of power is a degree of Being difference along a vector of knowledge (see 1.1b and his General Definition of Affects).

Concepts of Concreteness

5a. [The] four metaphysical axioms all stem from a deeper principle: absolute concreteness. Every actant simply is what it is. This entails that all actants are on the same footing: both large and small, both human and nonhuman. No actant is just fodder for others; each enhances and resists the others in highly specific ways.
5b. Spinoza agrees, the conatus of each thing strives to preserve itself to the optimum of its capacity (this defines its actualization). The human being is on no more “footing” than a peanut. Each are perfect expressions of Substance. In human beings this involves imagining things that give them the greatest power to act, and aligning themselves with things which are imagined to empower them

6a. Though graduate students are usually drilled in the stale dispute between correspondence and coherence theories of truth, Latour locates truth in neither of these models, but in a series of translations between actors.
6b. Spinoza also by-passes the dispute, ultimately because coherence is correspondence, but in human beings it is expressed by degrees of adequacy and power. Spinoza’s theory is not one of Representational knowledge.

7a. Nothing exists but actants, and they are all utterly concrete.
7b. In a deepingly of the flat Latour model, there is a qualification of “exists”. Substance has Being, but its being “exists and acts” through modal actants. So only do actants exist, but the totality of actants are an expression of Substance, individual actants (and their clustered relations) holding degrees of Being powers to act. So the Being of Substance is expressed in the “utterly concrete” expression of the modes.

8a. His philosophy unfolds not amidst the shifting fortunes of a bland human world correlate, but in the company of all possible actants: pine trees, dogs, supersonic jets, living and dead kings, strawberries, grandmothers, propositions, and mathematical theorems. These long lists of actors must continue until their plurality and autonomy is no longer suppressed. We still know nothing about these objects and what they entail.
8b. Spinoza agrees with the plurality of modal expressions, but as we come to understand the causes of any modal expression we ourselves become more active, ultimately aiming to understand the causes of our own modal expression; thus the boundary that keeps us from the world is an non-categorical determination: we are in modal combination with the animate and inanimate.

Not Aristotle’s Notion of Substance

9a. This does not lead Latour to a philosophy of substance. Traditional substance can be defined most easily by contrast with its qualities, accidents, and relations. A substance can easily be distinguished from its qualities, such as warmth or villainy, since these traits may change over time without its becoming a different thing. In fact, one of Aristotle’s best definitions of a substance is that which supports different qualities at different times. In this way, traditional substance suggests something identical beneath all its trivial surface fluctuations. Latour emphatically rejects this rift between a substance and its trivial exterior.
9b. Spinoza changes the Aristotlean approach to substance. First of all, any modal expression of Substance is not trivial, but rather the perfect and determined expression of Substance becoming fully concrete. So the above critique of Substance is understood in terms of Substances, which Spinoza explicitly rejects. There is only one Substance, the only thing that can be conceive through itself being the cause of itself. Modal expressions of Substance in Spinoza are not “trivial surface fluctuations” but the very means by which Substance “exists and acts”.

9.1a A cat, a tree, or a soul would be substances, but not the nation of Egypt as a whole, or vast pieces of machinery with thousands of parts. But since Latour grants all actants an equal right to existence, regardless of size or complexity, anything in the world must count as an actant, whether natural or artificial, as long as it has some sort of effect on other things.
9.1b As mentioned, Spinoza would grant Substance to only one thing, the Totality of all manifestion, and no modal expression has a metaphysical priority over any other. Indeed, a nation, or a theory, or a the Ethics itself, in that it is an expression of Substance in both Extension and Idea, and thus is a ratio of parts in communication, would all count as a body, having as much right as any other body.

10.a For Latour, an actant is always an event, and events are always completely specific. An actant does not hedge its bets, lying behind current involvements like a substance eluding its surface fluctuations. Instead, an actant is always fully deployed in the world, fully implicated in the sum of its dealings at any given moment.
10.b This is quite in keeping with Spinoza’s ateleological, immanent metaphysics. Each and every moment is an ideational and extensional expression, completely actualized, fully deployed as it is possible to be (under a degree of power/being when considered in isolation, perfectly actualized when considered in terms of the totality.)

11a Unlike a substance, actants do not differ from their accidents, since this would create a hierarchy in which some parts of the world were mere detritus floating on a deeper sea, and Latour’s principle of democracy between actants would be flouted.
11b Modal expressions only differ from Substance in terms of degree. There is no hierarchy though, for Substance exists and acts through modal expressions. The “accidents” are Substance acting. In this way my words, the color of my shirt, my legal standing, all express themselves autonomically as far as their striving can take them.

The Power of Relations

12a Latour’s central thesis is that an actor is its relations. All features of an object belong to it; everything happens only once, at one time, in one place. But this means that Latour rejects another well-known feature of traditional substance: its durability.
12b. For Spinoza Substance is its modal expressions, thus any actor is composed of its relations. The only thing that truly endures is Substance itself. But in the ratio of parts in a communication can be understood to be preserved, what Spinoza calls the ratio of motion and rest which defines a “body”. But because this ratio is mind dependent and ill-defined, it is not clear at all why he cannot be read as an Occasionalist, like Latour. Because each body is only an expression of the totality, ultimately to speak of preservation is a question of perspective.

12.1a We always speak of the same dog existing on different days over many years, but for Latour this would ultimately be no more than a figure of speech. It would entail that we abstract an enduring dog-substance or dog-essence from an entire network of relations or trials of strength in which the dog is involved at each moment of its life. Ultimately the unified “dog” is a sequence of closely related heirs, not an enduring unit encrusted with shifting accidents.
12.1b For Spinoza indeed each thing has an essence which is exhibited in its conatus/striving, and whose existence depends on modal causations external to it, but whether it is the same dog over time, or a series of closely related essences/strivings is ultimately indeterminate in Spinoza (he writes of an aged and mentally stricken poet who seems like the same person, but is not, and how our essence as an adult is different than that when were babies).

13a. Since an actant cannot be split into durable substance and transient accident, it follows that nothing can be reduced to anything else. Each thing simply is what it is, in utter concreteness.
13b. Due to the wide-sense expression of Substance by its modes (not a split), the power of explanation, empowerment through the knowledge of causes, is grounded in immanent expression. This means that modal expressions are not “reduced” to Substance, because modes are the means of Substance’s acting and existing, but they are explained through Substance. This is the ultimate metaphysical grounds which establishes the power of knowing. But in the actualization of modes, human beings necessarily are passive, dependent things, expressing themselves largely through imaginary relations and affective reactions (that is, translation always has its price and effect). The causal connections we make betweeen one modal expression and another though are perspective determined, for all things are connected to all other things.

13.1a We cannot reduce a thing to some privileged inner core by stripping away its inessential features.
13.1b Spinoza is in complete agreement. There is no inessential feature of Substance. Understanding a thing is achieved through understanding its causes in each and miniscule manifestation and effect. As an example, nothing is more useful to man than man, due to the sharing of a nature (the possibilities of connection), but the individual particularities, the causal history of one man and another are signficantly needed to achieve this usefulness.

14a. [For Latour] A theorist is no different from an engineer digging a tunnel through the mountains near Barcelona. One studies the rock, carefully assessing its weak and solid points, the cost of selecting one path over another, the safety concerns of workers, the availability of drill bits needed for specific tunneling methods, and other such factors. The engineer is not a free-floating mastermind of stockpile and calculation, as Heidegger imagines.
14b. Spinoza compares human understanding to being that of a worm in blood. The human being is both a historical being, plagued by imaginary associations and inadequate ideas, but also achieves relative freedom and clarity by understanding that he/she is a expression of Nature, the uniform, parallel expression of thing and idea. In order to theorize something (explain it), the causal history of his/her own ideas, emotions, pictures of the world need to be incorporated, as well as causal expression of the thing to be explained.

14.1a [an] engineer must negotiate with the mountain at each stage of the project, testing to see where the rock resists and where it yields, and is often surprised by the behavior of the rock.
14.1b It is not clear if in Spinoza human beings can hold absolutely adequate ideas at all, but ultimately any attempt to understand something (explain it) is to combine with it, both affectively and ideationally. Theorization is always an experiment in material combination, and not just ideas.

15a Nothing is pure calculation, nothing follows directly from anything else, nothing is a transparent intermediary. Everything is a medium or mediator, demanding its share of reality as we pass through it toward our goal.
15b For Spinoza completely adequate ideas follow cleanly from adequate ideas (as an asymptotic limit), but it is very unclear if human beings can hold a completely adequate idea, so any tracing of explanation occurs with the gradated and real history of one’s necessarily passive position in the world, ever assuming the maximalization of thought and extension, of which one is an expression. Calculation is a material as well as ideational act.

16a A truth is never a simple correspondence between the world and statements that resemble it, since we only link a statement to the world through the most difficult set of displacements.
16b Spinoza agrees. A true idea corresponds to its object, but this correspondence is ever buried in the relations of the human mind to its body and its causal history of the physical world, and his theory is certainly not a representationalist one. Ideas are actions of the mind and body. My ideas about China and Scrabble and Latour are all actually ideas of my body being in various states.

17a Neither is truth a kind of “unveiling,” as in Heidegger’s model, since this still implies that we approach truth asymptotically.
17b Spinoza does not speak of truth as “unveiling” (but he does rarely call his propositions to the Ethics “the eyes of the mind”. Because he refuses a representationalist model of knowledge, running counter to all Idealism, truth is a relation. He does make our relationship to the truth asymptotic, an aymptotology which shows itself concordantly in our power to act (and feel Joy). This asymptotic vector is the actual vector of Being/Power/Activity/Knowing.

The Full Deployment of Actors/God

18a Actants are always completely deployed in their relations with the world, and the more they are cut off from these relations, the less real they become.
18b Spinoza’s degree of Being definition of knowing and acting corresponds quite well here. The greater capacity a thing has to act or be acted upon, the greater degree of Being it has. Connections make perfections.

18.1 A Pasteur begins alone in his fight with Liebig over the cause of fermentation, or with Pouchet over spontaneous generation. Yet gradually, Pasteur amasses a formidable army of allies. Since Latour is no Machiavellian, not all of these allies are human. Pasteur’s allies may include mighty politicians who grant him funding, pieces of glassy or metallic equipment, and even bacilli themselves.
18.1b Spinoza would completely agree, both on the level of human and inhuman alliances, but in terms of how alliance functions in the human realm. Though he would insist that alliance is made in the strongest sense by appeal to commonality and a knowledge of causes. A theory is always a material as well as ideational expression, and it shows its power in its ability to combine not only ideas, but also material relations as well. The mind is no more priviledged than the body, and that means any body.

18.2a We become more real by making larger portions of the cosmos vibrate in harmony with our goals, or by taking a detour in our goals to capitalize on the force of nearby actants.
18.2b Spinoza agrees in terms of harmonization of parts, and would judge any interaction which increases our capacity to act a “good” thing.

19a For Latour, the words “winner” and “loser” are not inscribed in the essence of a thing, since there is no essence in the first place.
19b For Spinoza a loser is something that simply was overcome by a stronger force, the ratio of its parts in communication scattered. There indeed are modal essences in Spinoza, but because the existence is not predicated of a modal essence, it existing or not existing is not “inscribed” therein.

20a All actants are equal; all actants can win or lose, though some may have more weaponry at their disposal.
20b For Spinoza this weaponry is largely the knowledge state of a body, its degree of a capacity to act. But contingent circumstances can intervene to upon any partial modal expression of Substance, no matter how knowing/active. That is, the sage might very well have a piano fall on his head. Spinoza has strong Machiavellian influences, arguing that each and everything thing has as much right to the degree of power it can marshall.

Playing with Machiavelli

21a. The impact of Bruno Latour as a thinker is deployed in the bookstores that carry his works, the admirers who recommend them to others, and the careers that are altered by contact with his writings…There is no central point in the network where we encounter the very heart of Latour and his philosophy.
21b Completely so in Spinoza. Spinozist philosophy must be understood as both an extensional expression, materialized in all its manifestions, from books printed, to neurological states in people’s brains, to vibrations of air in conversation.

22a In order to extend itself, an actant must program other actants so that they are unable to betray it, despite the fact that they are bound to do so…. We always misunderstand the strength of the strong. Though people attribute it to the purity of an actant, it is invariably due to a tiered array of weaknesses.” (direct quotation of Latour)
22b I believe that Spinoza would agree because his affinities with Machiavelli, but a primary weapon of such “programming” others, is freeing them from their own illusions so to think more rationally, to understand the causes of the states of the world and their own states, and to ultimately to seek their own advantage. In this way Spinoza is an ethical Machiavellian.

23a Latour scoffs at the notion that the imperialist West succeeded by purifying objective truth from the naïve superstitions that still haunted gullible Indians.
23b For Spinoza it is simply a question of becoming more active and stronger through the understanding of causes and becoming more active. I don’t know what he would say as to why the West ultimately overcame the Indians, historically, for there may have been many intervening contingencies. But he would say that knowing the causes of things (for instance knowing the capacities of gunpowder), played a determinative role. He finds the West in many ways far more superstitious and imaginary than otherwise granted.

24a “It takes something like courage to admit that we will never do better than a politician….[Others] simply have somewhere to hide when they have made their mistakes. They can go back and try again. Only the politician is limited to a single shot and has to shoot in public.” (direct quotation of Latour)
24b Spinoza also finds a “single shot” concept of power manifestation (though the rational understanding of something is a pivotal aspect of the freedom to act). Spinoza’s “single shot” public concept of action is that each and every thought that we have is an affirmation of the power of the body to act, and can provide a change in the degree of reality one has. Conscious thought has no priority over any other mental/bodily action. Our actions are not usually what we think they are.

25a Forces are real, and real tigers are stronger than paper ones, but everything is negotiable.
25b Spinoza provides the same notion of negotiability, but it is between the real increases in Being which result from increases in knowledge, amid seemingly contingent relations to the world. Indeed a pen-stroke can kill, but this depends upon the entire matrix of coherent relations between events, the knowledge of which would only increase one’s own capacity to act.

26a Harmony is a result, not a guiding principle.
26b Harmony is both, a constructed result and a guiding principle. Very often increases in harmony at one level can induce disharmony at another (for instance the reception of Spinoza’s Theological-Political Treatise, which he refused to allow to be translated into Dutch, in much distinction to the ill-fated strategies of his friends the brothers Koerbagh).

27a Even power, the favorite occult quality of radical critics, is a result rather than a substance.
27b Power is not a substance but an expression of Substance.

28a Latour holds that truth itself is a result, not a starting point. “A sentence does not hold together because it is true, but because it holds together we say that it is ‘true.’ What does it hold onto? Many things. Why? Because it has tied its fate to anything at hand that is more solid than itself. As a result, no one can shake it loose without shaking everything else.” We call “true” whatever has attached itself to something more durable, less vulnerable to the resistance of other actants.
28b Spinoza would say that this “solidity” to which true sentences are “tied” to is the very nature of Substance and its expression in parallel Attributes.

29aRecently there has been a tendency to privilege language…. Language was so privileged that its critique became the only worthy task for a generation of Kants and Wittgensteins…. What a fuss! Everything that is said of the signifier is right, but it must be said of every other kind of [actant]. There is nothing special about language that allows it to be distinguished from the rest for any length of time.” (direct quote from Latour)
29b Spinoza does not privilege language either, but makes the power of ideas most distinct from their linguistic expression (or their related images).

30a Since actants are always fully deployed in the universe, with no true reality lying in reserve.
30b God is always fully expressing himself, nothing is reserve.

31a Latour dismisses any distinction between literal and metaphorical meanings of words
31b Spinoza would make a strong distinction between the imaginary relationship (affects) caused by words, and the ideational linkings, which allow us to see the causes of things. I do argue that there is room for the power of metaphor in Spinoza, since it is through metaphor and imagination that human beings are bound to each other in increasingly powerful relations, but there is a deep distinction.

32a Like the works of Whitehead, Nietzsche, or Leibniz, Irreductions views objects as individual perspectives striving to impose their viewpoints on the rest.
32b Spinoza is in 100% agreement, we want others to love what we love.

33a “I don’t know how things stand. I know neither who I am nor what I want, but others say they know on my behalf, others, who define me, link me up, make me speak, interpret what I say, and enroll me. Whether I am a storm, a rat, a rock, a lake, a lion, a child, a worker, a gene, a slave, the unconscious, or a virus, they whisper to me, they suggest, they impose an interpretation of what I am and what I could be.” (direct quote of Latour)
33b I am ultimately an expression of Substance as it expresses itself concretely, and thus what I am is also expressed in all other things. As I go through this seemingly contingent world, passively exposed to things I have little control of, as a mode of Substance my power to act comes from my ability to combine with any other mode of Substance.

Conclusion

If there is a difference between the two it is that Spinoza grants greater ontological changes in being and power to the understanding of the causes of things, whereas Latour would like causal explanations to be understood much more haphazardly, productions of chance and not needing explanation themselves. One might ask, Does not with the relative suppression of the need to explain the power of understanding the causes of things come the suppression of an ethics of communication, the way in which our ability to form networks with others and objects primarily occurs through attributions of interpretive charity? One might also ask, if indeed actants are to be understood to only as real as the power they exhibit in their networks, has Latour privided enough traction for the strategies for self-determination and liberation, the kinds of which grant freedom through the understandings of our own causes?

[all selections on Latour are quotations from an unpublished PDF of Graham Harman’s Prince of Neworks to be published this Spring by re.press]

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Spinoza’s Substance Stripped Bare

Duchamps The Bride Stripped Bare By Her Bachelors Even (The Large Glass) (1915-23)

(above Duchamp’s “The Bride Stripped Bare by Her Bachelors Even,” 1913 -23)

Just a Blob of nothing, an intellectual Sleight of Hand…?

Levi, over at Larval Subjects has a well-worded summation of the possible difficulties and assumptions contained in Spinoza’s Proposition 5 (Ethics, part I: ) “Proposition Five: Questions of Individuation”, in particular how they reflect upon just what Individation is. He seems to feel that if one accepts this proposition (and its referenced assumptions) one is by the force of logic to accept a great deal of what follows in Spinoza’s philosophy. So he sees this as something of a keystone. If one can effectively challenge it, the entire edifice of Spinoza thinking is threatened to collapse. I can’t say that I agree with this because I read the rationalistic cohension of Spinoza’s Ethics a little differently than most, but he does raise interesting points.

I commented extensively on the posting (much in greater detail than I expected), so it seemed best to re-present the issues here, with a bit more quoted material. I think it worthwhile to dig into this proposition as Levi has given us the lead to do, but in the end I am not sure as to the final spear point of his objection.

First off, let’s give the proposition, and then I’ll post the context of my comments:

In rerum natura non possunt dari duae aut plures substantiae euisdem naturae sive attributi.

In the nature of things they are not able to be granted two or multiple substances of the same nature or attribute.

I provide the Latin and literal translation so one can see the lexical doubling that Spinoza performing, as well as the “of things” individuation which shows the proposition to be an explanation of things we already perceive as distinction, but Curley translates a bit less literally and much more fluidly,

In nature there cannot be two or more substances of the same nature or attribute.

The reason for this that Spinoza puts forth is that it is the attribute itself that tells us exactly what a thing is, its essence. It is the attribute which grounds all our other attributve properties. If there were multiple substances which had the same attribute (the same conceptual manner of distinction), there remains no specific additional qualification which distinguishes them from each other. I will reference and cite Della Rocca’s treatment below, for his presentation is a good clean exposition. And it is his argument I will follow. What Spinoza has in mind here is Descartes’ somewhat unspecified assertion that there actually are two kinds of Substance, the Aristotlelian kind of individual things which are dependent upon other things for their existence, and then the soon-to-be Spinozist kind, the kind that is self-caused. The move that Spinoza is making here is turning against the notion that it is Attributes themselves that distinguish things as individual kinds, but rather it is modal expressions alone. Descartes’ two kinds of Substance simply can’t be rationally supported. Unfortunately for the Christian, this leaves of of creation to be literally part of God. There is no gap between God and the world. Once we remove the unjustified kind of Substance inherited from Aristotle, we are just left with an ultimate and immanent ground.

Mammoth Hot Springs

Anyways, that is where Spinoza is going. But what Levi objects to, after a thorough engagment with the problems with the argument is that there seems to be a kind of non-sensicalness of Substance itself, the way that if we say that an object in the world (and he uses his friend Melanie), is stripped of all her qualities, we really are left with nothing at all. What would remain under Spinoza’s description, is somehow blob-like and indistinct. Spinoza has provided us with a concept that seems to do nothing. Here is a quote from Levi’s post, and my consideration that follows:

Levi: ” Suppose I strip my friend Melanie of all her affections or qualities. In striving to think Melanie as a substance, I ignore all of her physical properties, her quirks of thought, her personal history, her mannerisms, her love of okra, etc., so as to think this hypothetical “Melanie-substance” in and through herself. What am I left with at the end of this exercise? Absolutely nothing!. In other words, a substance subtracted from all of its affections turns out to be nothing but a formless void.”

Kvond:…I’m not sure that I follow exactly your objection here. The complaint that you make as to the blobness of Substance is actually very close to the one that Descartes made against Medieval Aristotelian “Prime Matter”, a completely non-quality “stuff” which is suppose to inertly just be there as a support for inhering form and qualities. As Della Rocca tells it, it was this seeming superfluousness of Prime Matter that got Descartes to just do away with it. Instead, a Substance simply had a form, was defined by its form, which in Descartes was its Principal Attribute.

[inserted from Della Rocca’s Spinoza  a selection which lays out Descartes’ thinking on Substance and attribute in terms of prime matter]:

But why must all the properties of a substance be subsumed under a fundamental feature? Why can’t there be a feature of substance that does not presuppose the principle attribute of the substance, but is nonetheless a feature of that substance? Thus, for example, why can’t an extended substance also having some thinking features, features that cannot be understood through extension? Descartes does not, as far as I know, explicitly address this question, but its clear what his answer would be: there would be no good account what makes this free-floating thinking feature a feature of this extended substance. What would bind this thinking feature to this extended substance? For Descartes, the conceptual connection provided by an attribute furnishes the link to make a particular property of a given substance. Without the link afforded by an attribute, we cannot see a property as belonging to a substance. In other words, Descartes insists that there be this over-arching feature because otherwise there would be no explanation of why a given feature is a feature of a particular substance.

Because the principle attribute helps us to understand all the properties of a substance, it tells us what kind of thing the substance is, what its essence is. And for this reason, purely formal features of a substance do not count as attributes in this sense. Each substance has features, let us say, of existing and being powerful to some degree. But exitence and power are not principal attributes for Descartes. This is because these features do not tell us what kind of thing a substance is and do not tell us what kinds of more particular properties it has.

In this way we can see that on Descartes ontology of substance and attribute, substances are explanatory engines. Each substance has a nature that can be articulated or explained in terms of its principal attribute, and this principal attribute in turn articulates or explains all the properties of the substance. Thus for Descartes each substance is fully conceivable. Everything about a substance must be capable of being understood and what it is understood in terms of is its principal attribute.

This is, of course, a rationalist dimension of Descarte’s ontology, and we can appreciate this dimension by contrasting Descartes’s view with a broadly Aristotelian account of substance. On the Aristotelian account (or at least on the Aristotelian account as it is developed by medieval philosophers such as Aquinas), a corporeal substance consists of prime matter and a substantial form. The substantial form, is in some ways, like a Cartesian principal attribute: it tells us the nature of a substance and the kinds of properties it can have. But the form is not the only constituent of substance. The substantial form must somehow inhere in the subject and this subject is prime matter, a featureless, bare subject for a substantial form. The prime matter is a thing is some sense, but, precisely because it is featureless, it cannot be articulated or explained. Literally, prime matter is no “kind” of thing, and precisely for this reason Descartes rejects the notion as unintelligible (see CM I 91, 92/AT XI 33, 35). Marleen Rozemond sums up the view here nicely:

“Since Descartes eliminates prime matter from the hylomorphic conception of corporeal substance, the result in Aristotelian terms is that a substance just consists in a substantial form. In Descartes own terms, the result is that substance just consists in a principal attribute” (Spinoza, 2008; 38)

Prime Matter, Begone!

[continuing my response] But as Prime Matter was done away with because it lacked explanatory value, we have to ask the same of Spinoza’s overriding Substance itself. If we strip Melanie of all her qualities are we left with Prime Matter, or with Substance, and what would be the difference?

There are a few ways to proceed. As you know, Substance is what it is because it is the only thing that is its own cause, by virtue of nothing lying “outside” of it (I don’t know if you accept this, but it is fundamental to answering the question). As such, it is the only thing which has existence in its very nature (it does not depend on something other than itself to exist), it must, logically and ontologically exist. So, in a certain sense, the question being asked has something of a non-sequitor in it. Because Substance “exists and acts” through its modal determinations, asking what Melanie is (if merely Substance) without her modal determinations, in a way does not follow. In Spinoza’s universe, Melanie must have certain modal properties, given the state of the rest of the universe, which has determined her to be a certain way.

Now there is a kind of aporia we run into here, for in Spinoza’s framework it is not entirely clear why Melanie when she is five years old and has a cool-aid stain on her mouth, and Melanie when she is 33 and has a broken arm, is the very same thing (has the same essence). It is perfectly conceivable that from moment to moment or stage to stage, there are different essences expressing themselves. It seems that only Spinoza’s definition of a body as a specific ratio of motion in communication between parts that restricts this possibility. And because this “ratio” is unspecified and really unidentifiable, this is a rather tenuous barrier. So there is a very real sense in which Spinoza’s depiction can be read as a kind of Occasionalism.

But generally, when thinking about Melanie, sub specie aeternitatis, what she is in or out of existence, this is a modal “essence”, a certain beingness which depends upon a provisional modal interaction with other modal essences, each bringing each other into being in a kind of co-dependent fashion, what Gatens and Lloyd term “horizontally”.

Is this very close to the blob of Prime Matter? It doesn’t seem so. Because Substance itself is an expressional thing, a thing which by its very nature determines itself to exist, if you do the thought experiment and ask what any one modal expression is without its current state of modal expressiveness, one is left with the explanatory ground of Substance, its very capacity to press forward in existence and acts.

Indistinguishable Melanie

Now is this a bit of a slight of hand? Has Spinoza just made up a buried capacity of a hypothetical under- or over- thing? Perhaps one can say that. But what he has in mind (and one cannot undervalue this), is that things must have an explanatory context for what they are. If you are going to say something like:

“Sure, you tell me that Gravity is some mysterious force which causes this apple to fall with such and such a rate at such and such at time, but what then is this apple-event if stripped of all its qualities, its rate and timing?…It is just a blob of a force called Gravity”

If you take away what is being explained, and then ask what good is the explanation, one might really be dissatisfied with the answer. So in answer to what Melanie is in or out of existence requires that we define what she is in existence. And for Spinoza this answer is a conatus, a striving. She is pure striving (expressed in human beings as either appetite or desire). That is her existential essence. It is the diagnosis of this striving that gives weight to Spinoza’s view of Substance as explanatory. What is Melanie’s striving, her conatus, stripped of all the particular “strivings for”? It is the existential strivings of Substance itself. But there is no blobness to it, for the strivings of Substance must be particularized, that is expressed in determined modal forms. Substance does not collapse on itself, or meld into one great sea of potentiality. It is always particularized in concrete, existential manifestation.

You [Levi] bring this up when you conclude:

[Levi writing]:”However, again, we run into the same problem: Is an attribute such as extension thinkable independent of all spatial determinations (modes)? Again, the thought of space without any spatial things turns out to be the thought of nothing or the absence of all determination. The conclusion then would be that the idea of an affectionless substance- such as Spinoza evokes in 1p5 -is an incoherent idea that functions as a sleight of hand, rather than a genuine concept.”

But seem to have inverted the reasoning. It is precisely because one cannot conceive of space without its spatial determinations that Substance must be an expressive grounds of spatial things, in the Attribute of Extension. It is precisely that there are spatial things, and that they can only be understood fully by understanding their cause, that Substance is what it is. It seems that you have reversed the Explanans and the Explanandum, and argued that the Explanans is meaningless without the Explanandum, but it the requirement of the Explanans due to the existence of the Explanandum [the nature of things], that grants it its coherence. It is the very fact of its explanatory nature that Substance logically must express itself in the concrete things that it is explaining, that gives the argument its force.

Michael Della Rocca, Chair of Philosophy at Yale

Michael Della Rocca, Professor of Philosophy at Yale

To end I would like to reprint a lengthy selection from Della Rocca’s book that deals particularly with 1p5 so as to give immediate context to my points, but also to provide a place of comparison for much of the same ground covered by Levi’s also worthwhile summation. At the very least it will give those unfamiliar with Spinoza’s argument one more clear presentation of the issues at hand in the notions of Substance, Attribute and mode, and their possible objections. Its interesting, but when I first got Della Rocca’s book I was a bit disappointed and distracted from it. It possessed none of the verve of his first book, Representation and the Mind-Body Problem in Spinoza (1996): But as I have turned to it in reference, it really has grown on me. In its quietude one can feel the delicate care of Della Rocca’s mind as he weighs the meanings and implications of Spinoza’s assertions, and is invited to consider them as he does.

Thus let’s take 1p5 first: “In Nature there cannot be two or more substances of the same nature or attribute.” To prove this proposition, Spinoza considers what is required in order to individuate two substances, i.e. what is required in order to explain their non-identity. For Spinoza, the distinctness between two distinct things must be explained by some difference between them, some difference in their properties. In the case of the individuation of substances, this amounts to the claim that they must be individuated via a difference either in their attributes or in their modes. Thus Spinoza says in 1p4d:

“Two or more distinct things are distinguished from one another, either by a difference in the attributes of the substances or by a difference in their affections.”

In 1p5d, he makes clear that such a difference in properties is needed for two things to be “conceived to be” – i.e. explained to be – “distinguished from one another.”

In insisting on some difference in properties between two things, Spinoza endorses the Principle of the Identity of Indiscernibles. This is a principle – more often associated with Leibniz that with Spinoza – that if a and b are indiscernible, i.e. if a and b have all the same properties, then a is identical to b. One can see that this principle turns on the notion of explaining non-identity and, as such, one can see its roots in the PSR [Principle of Sufficient Reason]. Non-identities, by the PSR, require explanation, and the way to explain non-identity is to appeal to some difference in properties.

Thus two substances could be individuated either by a difference in their attrributes or in their modes. Spinoza dismisses right away any differentiation of substances in terms of their attributes because he says we are considering whether two substances can share an attribute. Thus a case in which substances might have different attributes might seem irrelevant to the case at hand. However, as we will see in a moment, this dismissal may be too hasty. Spinoza then considers whether they can be distinguished by their modes. Spinoza eliminates this possibility as well, offering the following argument.

Since a substance is prior to its modes (by 1p1), we are entitled, and indeed obligated, to put the modes to the side when we take up the matter of individuating substances. Thus, with the modes to one side and with the attributes already eliminated as individuators, it turns out that there are no legitimate grounds for individuating substances with the same attribute, for explaining why they are distinct. Thus, since substances with the same attribute cannot legitimately be individuated, there cannot be any sharing of attributes.

Obviously this argument turns crucially on the claim that we should put the modes to one side. But what justifies this claim? Spinoza appeals here to the notion of priority introduced in 1p1. What exactly does this priority amount to? For Spinoza, as well as Descartes, it is a conceptual priority. One can have the idea of a substance without having ideas of its modes.

Thus, we can see why Descartes would have a problem individuatin, say, two extended substances. All Descartes could appeal to in order to individuate the substances is the modes, but given Descartes’ own explanatory notion of substance, according to which all of a substance’s modes are explained through its attributes, such an appeal is illegitimate.

Of course Descartes might at this point simply give up the claim that the non-identity of substance is explicable. Fair enough. After all, Descartes does not explicitly assert the Principle of the Identity of Indiscernibles. But Descartes’s rejection of prime matter is in the spirit of such a principle. For Descartes, there is no way to articulate what prime matter is precisely because it has no qualities. In the same way, there is no way to articulate what the non-identity of a and b consists in because no qualities are available to do the job of individuation. Thus, even on his own terms, Descartes should feel the force of this Spinozistic argument that rules out a multiplicity of substances sharing an attribute.

But even if substances that share an attribute are not individuated by their modes, perhaps such substances are individuated by attributes they do not share. Spinoza does allow, after all, that a substance can have more than one attribute. So why can’t we have the following scenario: substance 1 has attributes X and Y and substance 2 has attributes Y and Z. On this scenario, while the two substances share an attribute (i.e. Y) they differ with regard to other attributes and can thus be individuated after all. So perhaps then, contrary to 1p5, there can be some sharing of attributes by different substances. This objection was first raised by Leibniz, one of the most acute readers of Spinoza.

This objection is harder to answer than the charge that substances that share an attribute can be individuated by their modes, but Spinoza clearly has the resources to handle this objection too. To see why, let’s assume that Leibniz’s scenario is possible. If so, then attribute Y would not enable us to pick out or conceive of one substance in particular. The thought “the substance with attribute Y” would not be a thought of one substance in particular, and thus attribute Y would not by itself enable us conceive of any particular substance. For Spinoza, such a result would contradict the clause in the definition of attribute according to which each attribute constitutes the essence of substance. As Spinoza says in 1p10s, a claim that he clearly sees as following form the definition of attribute, “each [attribute of a substance] expresses the reality or being of substance.” So for Spinoza, if a substance has more than one attribute, each attribute by itself must enable us to conceive of the substance, and this can by the case only if each attribute that a substance has is unique to that substance. Thus Leibniz’s scenario is ruled out (46-48)

 

Spinoza’s Circle and the Interior

Some Ruminations on Spinoza’s “Simplest” Thought

A single figure keeps returning to me as I contemplate Spinoza’s optical concepts and metaphysics, in view of the prevailing Cartesian science of the age, that elementary figure that Spinoza provides to help explain how he conceives of the reality of the modes, even the modes of ideas for non-existent things, in relationship to the ultimate reality of Substance, God or Nature. As Spinoza explains, the figure stands for something that has not parallel, not example, because it is a fundamental relationship which is unique and totalizing.

Ethics part 2, prop 8 schol. — The nature of a circle is such that if any number of straight lines intersect within it, the rectangles formed by their segments will be equal to one another; thus, infinite equal rectangles are contained in a circle. Yet none of these rectangles can be said to exist, except in so far as the circle exists; nor can the idea of any of these rectangles be said to exist, except in so far as they are comprehended in the idea of the circle. Let us grant that, from this infinite number of rectangles, two only exist. The ideas of these two not only exist, in so far as they are contained in the idea of the circle, but also as they involve the existence of those rectangles; wherefore they are distinguished from the remaining ideas of the remaining rectangles. (Elwes trans)

This has always been a convenient image for the explanation how the modes dependently are an expression of the totality which is all of what there is. One practically sees how the modes, while causally interacting with each other to produce current states of being, immanently rely upon a larger comprehesivity. And if one wants to even leave off from the narrowness of Spinoza’s explicit thought, the chords D and E could be read as vectors which produce an intersecting point (unlabeled), the modal expression of two lines of force, a point of focus, this seems at least arguable when considering much of the rest of Spinoza’s view.

But two other realizations come to me as well. The first is that the rectilinearitythat Spinoza presents within the borders of the circle very well may for Spinoza have represented the linear analyses of motion provided by Descartes. Part of the challenge presented to Spinoza, as he tried to heal a perceived breech between Body and Soul, Thought and Extension, was how to bridge mathematical descriptions with affective realities. In a certain sense the rectilinearity with the circle of Substance for Spinoza represents the ideational crispness of mathematical description understood as a partial, yet sufficient expression of substance. The mathematics is to some degree subsumed. Just as Huygens and others struggled to assume the advantages of the Perspectiva  tradition in optics, yet still fully explain the properties of light’s propagation (which Huygens would show to be that of a wave or a pulse), Spinoza too sought to preserve the mechanized sureties of Descartes mathematizations, yet with a view that could account for the affective propagations living bodies (as early as Kepler’s Paralipomena light is described as an “affectus”).

Secondly though is the realization that the circle does not only represent Substance as a whole, but also any conatus expression of Body (defined by Spinoza as a preservation of a ratio of motion and rest over time). The circle above can also be read as the recursive coherence of a body, and the internal causal relationships of which it is composed, immanent to its essence. 

In other words, a body, in the ordinary as well as in the Cartesian sense, preserves its physical integrity in just the same way the whole universe preserves it: they differ only with regard to the degree of probability that each can ‘survive’ externally caused modifications. Only the unique individual of level-ω can be modified in infinite ways without giving up its identity.

The Physics of Spinoza’s Ethics, David Lachterman

What level-ω is in Lachterman’s nicely conceived reduction is the last border of an entire body. Level-2 is the most elementary kind of body for Spinoza, a hypothetical composition of what one supposes is at least two of the most simple bodies “corpora simplicissima”. The human body, and all such bodies we regularly recognize as such are what Lachterman regards as level-k bodies (2 < k < ω). Following these descriptions, the circle above from the Ethics could be read not only as the schema of the level-ω Body, but also of any typical level-k Body. It is only the degree of power, freedom and being that preserves such a circle (ratio) that distinguishes it from the ultimate whole. Spinoza world can be seen as one of circles within such circles. The modal causal interactions are immanent expressions not only the totality of Substance or Nature, but also are expressions of the ratio of motion and rest of which a said body (level-k) is composed. The existential modality of ideas and extensions which is interior to the circle (level-k) can be read as contingent to the essence of that Body taken as a whole, as it is determined to exist.

While it is fair and best to understand the diagram exactly in the way that Spinoza meant it, I think it is important to realize that in a figure such as this, representing something for which Spinoza says there is no parallel, the image is likely overdetermined by several other Spinoza concerns and conceptions. The circle and its interior floats through Spinoza’s thinking process (additionally). The diagram can be read across at least these two layers: the affective subsumption of Cartesian mathematicization of Nature in terms of propogating, interacting bodies; and, the immanence of constituent modes of bodies under gradations of Being and power. And I would end that the suggestive recursivity of this body (level-k) opens itself perhaps to Autopoietic descriptions, at least for those bodies considered to be alive, and Autopoiesis itself to Spinozist re-description.

A Conflation of Spinoza Diagrams

How Spinoza Thought of the Eye, the Lens and The Modes

Perhaps this is an irresponsible and trite comparison, but sometimes the mind indeed works visually, even in authors as exacting and deductive as Spinoza attempts to be. It is striking that Spinoza uses two very similar diagrams to illustrate on the hand, the powers of spherical lenses to most ideally focus rays across an infinity of axes, (the manifestation of which is subject to the properties of real lenses):

Text not available

Letter 39 to Jelles, March 3rd 1667
Benedicti de Spinoza opera quotquot reperta sunt quotquot reperta sunt By Benedictus de Spinoza, Baruch Spinoza, Johannes van Vloten, Jan Pieter Nicolaas Land

Depicted above are the hypothetical intersection of rays, in two sets taken to be parallel, as they arrive at the surface of a spherical lens. Such rays are taken to be then focused at the back of the circumfrance, as the would be at the back of the eye, or as part of the refractions of a lens.

 

In this diagram, Spinoza illustrates how each contingently expressive mode – what is usually taken to have come into existence and then will pass away – are implied by, that is caused by as immanent to, the Idea of an infinity of points that make up a circle. In this way, the rectangles that are immanent to a circle’s circumfence are by analogy seen to be dependent upon that circle. The rectangles come and go, the circle remains eternal. As explained in Ethics IIp8s:

The nature of a circle is such that if any number of straight lines intersect within it, the rectangles formed by their segments will be equal to one another; thus, infinite equal rectangles are contained in a circle. Yet none of these rectangles can be said to exist, except in so far as the circle exists; nor can the idea of any of these rectangles be said to exist, except in so far as they are comprehended in the idea of the circle.

There is the simple coincidence of using a circle to diagram both physical effects, and metaphysical effects (which for Spinoza are of course commensurate). But if one allows a conflation, one that may have occurred within Spinoza’s thinking, in the first we have the effects what occur within the eye, as it interacts with events outside of it, and in the second, we have the effects (modes) as expressed immanent to the circle that contains them.

Because Adequate Ideas are understood by Spinoza to be Ideas uncaused by something external to them, I don’t think it is too big of a leap to understand that when Spinoza is diagramming the effects of light with the eye (and for a lens, post-angle of incidence), he is thinkingof the second diagram. It is perhaps for this reason that Spinoza is not obsessed with the crystality of vision that occupied Descartes in his quest for the hyperbolic lens. The sharpness of an image is but a part played in an assemblage of knowledge. However clearly one’s eye, or lenses work, this simply is not clear thinking. Of course Descartes understood this as well, but there is something to how Descartes and Spinoza each responded to spherical aberration which reveals a difference of emphasis in the very project of mental and physical liberation. I believe in this co-incidence of diagrams, a profound conflation is being accomplished in Spinoza’s process of thinking.

I see hear as well an interesting graphic subsumption of the scattering of rays that occur with spherical aberration, as in being focused they tend about a “mechanical point” [Johannes Hudde]. Much as rays are never entirely focused to a mathematical point (even with real, hyperbolic lenses), so too we never possess wholly adequate ideas. The focus rays as seen in the first diagram (again, if we allow an analogical thought), appear to enact indices found in the second diagram. Is Spinoza at some level conceiving of rays of focus as being parallel to the adequacy of ideas? And is Spinoza’s theoretical acceptance of spherical aberration [a la Hudde] a product of his acceptance of the fundamentally inadequate nature of ideas we hold? Is his mechanical project of lens focusing analogous to a mechanical – that is, pragmatic, rational and crafted – construction of human freedom? These are large and obscured questions.

This certainly does not make up an argument either for Spinoza’s position, or for an interpretation of Spinoza’s position. It is really more an intuition into the kinds of thought processes Spinoza may have been engaged in, in part elicited by the diagrams he used to make things clear. Meant is a direction for future analysis.

Spinoza, On The Immortality of the Soul

Controversially, the question of the immortality of the soul/mind arises in Spinoza’s writings, and with it the definition of personal identity. At one point he takes up thoughts about a poet who has, in an Altzheimer’s way, has lost contact with his person. In what sense is the poet still himself? To answer this, Spinoza argues for the existent essence of non-existent modes, a position which Deleuze sums as such:

“A mode’s essence is not a logical possibility, nor a mathematical structure, nor a metaphysical entity, but a physical reality, a res physica. Spinoza means that the essence, qua essence, as an existence. A modal essence has an existence distinct from that of the corresponding mode.” Expression in Philosophy (192)

Despite Deleuze’s assurance that this reality is not mathematical, Spinoza does take recourse to mathematical analogy to make clear his meaning, for instance (cited below), the existence of essence of an infinity of equal rectangles within the essence of a circle (Theorem 35, Euclid) which exist even if only one or even none exist modally.

 

So the essence of a mind is said to exist within the mind of God, eternally, despite its own limited duration. What this does is give the human mind a kind of eternity, an existence outside of the brief flicker of expression, but what this also does is place that eternal existence in relation to all other essences, of all other things, animate and inanimate, which are also produced by God/Nature. The human mind is eternal in essence as all other things are eternal in essence. But further, (as is shown in the note to EIV39 below), identity itself, our preservation of ourselves as ourselves in duration, is also not guaranteed, and is in fact likely an illusion of perspective. Just as his Spanish poet has died to himself, despite the continuity of his body, unable to recognize even his own writings, we too would only be an infinite series of eternal essences – slight modifications of a rectangle within its circle – defined only by our momentary consonance of parts – both ideational and extended. It is not so much that Spinoza has awarded undue eternity to the human mind, but rather has radically (categorically) undermined the basis upon which the human mind privileges itself to be unique among things in this world, given eternal life, but a life fused with all other things, capable as alien to its own “past” as akin to another thing. I list below relevant passages and definitions to this thinking:

EV29- The human mind cannot be absolutely destroyed with the body, but there remains of it something which is eternal.

(Proof) There is necessarily in God a concept or idea which expresses the essence of the human body, which, therefore is necessarily something appertaining to the essence of the human mind. But we have not assigned to the human mind any duration, definable by time, except insofar as it expresses the actual existence of the body, which is explained through duration, and may be defined by time – that is we do not assign to it duration, except while the body endures. Yet there is something, notwithstanding, which is conceived by a certain eternal necessity through the very essence of God; this something, which appertains to the essence of the mind, will necessarily be eternal.

EIV39note – …But here it should be noted that I understand the Body to die when its parts are so disposed that they acquire a different ratio of motion and rest to one another. For I dare not deny that – even though the circulation of the blood is maintained, as well as the other [signs] on account of which the Body is thought to be alive – the human Body can nevertheless be changed into another nature completely different from its own. For no reason forces me to think that the Body does not die unless it is changed into a corpse. And, indeed, experience seems to urge the opposite conclusion. Sometimes a man undergoes such changes that I should hardly believe that he was the same man. For example, I have heard tell of a Spanish poet who was struck by an illness; though he recovered, he remained so oblivious to his past life that he did not believe the tales and tragedies he had written were his own. He might have been taken for a grown-up infant had he also forgotten his native tongue.

EIp8 – By eternity, I mean existence itself, insofar as it is conceived necessarily to follow solely from the definition of that which is eternal. (Explanation) – Existence of this kind is conceived as an eternal truth, like the essence of a thing, and, therefore, cannot be explained by means of continuance or time, though continuance may be conceived without a beginning or end.

EIp24- The essence of things produced by God does not involve existence. (Corollary)… God must be the sole cause, inasmuch as to him alone existence appertain.

EIp25 – God is the efficient cause not only of the existence of things, but also of their essence.

EIIp8 – The ideas of particular things, or of modes, that do not exist, must be comprehended in the infinite idea of God, in the same way as the formal essences of particular things or modes are contained in the attributes of God. Note – If anyone desires an example to throw more light on this question, I shall, I fear, not be able to give him any, which adequately explains the thing of which I here speak, inasmuch as it is unique; however, I will endeavour to illustrate it as far as possible. The nature of a circle is such that if any number of straight lines intersect within it, the rectangles formed by their segments will be equal to one another; thus, infinite equal rectangles are contained in a circle. Yet none of these rectangles can be said to exist, except in so far as the circle exists; nor can the idea of any of these rectangles be said to exist, except in so far as they are comprehended in the idea of the circle. Let us grant that, from this infinite number of rectangles, two only exist. The ideas of these two not only exist, in so far as they are contained in the idea of the circle, but also as they involve the existence of those rectangles; wherefore they are distinguished from the remaining ideas of the remaining rectangles.