Wednesday, May 4, 2016

Formal Ontology & The Science of Intelligibility: Irreal Compossibilizations

Here is my final paper for a seminar on the "Ontological Turn." 

Formal Ontology & The Science of Intelligibility:
Irreal Compossibilizations

I, […], being of sound mind and body, hereby solemnly swear on behalf of the Whole Universe: […] that I will interpret every phenomenon as a particular dealing of God with my Soul.
—Oath of the Master of the Temple1

A new mode of thinking is required, a mode that will synthesize and organize all the fruits of thinking into a single kingdom of ethereal possibility. In this paper, the formal contours of theory will be traced from its lowly origins as tools for modeling reality, up through its radical singularization as formal ontology, to the irreal possibilities of thought's self-actualization, and on into the future, to a generic science of intelligibility: formalysis.

Phase I: Abstraction
A theory is often conceptualized as a kind of model. Theory is supposed to somehow represent reality so as to render it manageable. In doing this, it necessarily simplifies the reality it seeks to model. In other words, it reduces or decides something about the Real. A theory of this sort has several parts: a structure, or abstract logical architecture, which differentiates good (justified) from bad (unjustified) moves or deductions; a principle of abstraction or interpretation, which allows for the Real to be reduced to a theoretical object, picking out certain features for use in the theory and ignoring others; and finally, an application to some concrete object or context.

In order to elucidate these three components of theory, it may help to take Karl Marx's economic theory from Capital as an example. The structure of Marx's theory of value largely consists of economic equations and their prescribed interrelations. The primary principle of abstraction in Capital is a kind of economic reductionism, designating how a commodity is to be reduced to its value. Marx abstracts from the use-value of the commodity, resulting in a theoretical object with exchange-value but not use-value. As for application, Marx gathered data from various sources, reduced that data to quantitative terms, and drew theoretical conclusions about the English economy throughout the entire process.

Properly understanding abstraction is very important for theory, perhaps more important than the other two parts; this is especially true for successful modeling. “Abstraction” is the process designated by a principle of abstraction. Abstraction functions as a metaphorical magnifying glass or microscope that specifies or picks out certain features of an object deemed relevant to a theory, necessarily simplifying that object by omitting its irrelevant features. We can define abstraction semi-formally

(Abstraction) A theoretical object Φ is an abstraction from a real object Ψ iff whenever Φ has property π, then Ψ also has π.2

A real object has a set of properties or characteristics. An abstraction from that object, if done correctly, is a theoretical object that has some of that original object's characteristics but no others. A theoretical object is basically just a version of its corresponding real object that has fewer properties and, allows for certain privileged forms of manipulation, both because it has less properties to deal with and because it is theoretical.

Abstraction is the leap of thought that corresponds to representation. Representation is, unfortunately, notoriously difficult to define, especially if the definition is to be philosophically useful. At the very least, a representation is a likeness or reproduction of something else. A representation captures certain features of what it represents, and the more features it captures the better it works as a representation. A painting of oranges, for example, is a representation of oranges (the “of” is doing a lot of work here). The painting represents the oranges just if it captures a context-sensitive subset of properties belonging to the oranges, that is, if the painting and the real oranges have enough of the right things in common.3

Representation also has a more precise theoretical meaning that differs somewhat from the commonsense use of the word. Henceforth, by “representation” I will mean “theoretical representation,” as opposed to, say, visual representation. A model is just a theoretical representation:

(Model) A theory Φ is a model of Ψ iff: i) For every theoretical object φ belonging to Φ, there exists some real object ψ belonging to Ψ such that φ is an abstraction from ψ; and ii) whenever Φ has structural feature λ, Ψ also has λ.

Clause (i) should be clear. Whenever clause (ii) is fulfilled, we say that Φ is a substructure of Ψ. When Φ is a substructure of Ψ and Ψ is also a substructure of Φ, we say that Φ and Ψ are structurally isomorphic.4 A model maybe be better or worse at capturing the structural features of its object, but typically need not be structurally isomorphic to it. While structural isomorphism is an important category for subtractive ontology (explored in the following section), it is impractical overkill when it comes to models.

Just like an abstraction tends to have fewer properties than its real object, a model tends to display fewer structural features than what it models. Though the given definitions of abstraction and model might tend towards unrealistic idealization, what it paid for in artificiality is more than made up for in clarity and rigor. As a result, we can formally and quantitatively compare the adequacies of models to one another, or the degrees of abstraction of different theoretical objects. Furthermore, when there is a change in the real object of a relatively successful model, that model should allow for its own change in parallel.

Theoretical models are powerful analytical tools that can increase knowledge of both particular and general objects. To put the process in clear but schematic terms: To create a model, abstract real objects into theoretical ones and then plug them into a structure. The structure will then allow for both theoretical manipulation of the objects (like running a simulation) and valid sentential or propositional deduction.

Phase II: Subtraction
Instead of representing (i.e. modeling) its object, theory might not represent anything at all. It seems intuitively correct that if a theory is non-representational, it must be presentational, and vice verse; there does not seem to be a third option between presentation and representation.5 Theory, then, might simply present (itself).

Presentational theories have no need of abstraction, and are therefore made up of less parts than are their representational cousins. Abstraction is the ultimate ground of representation because abstractive thinking sets itself up against its object, which it encounters as something to be reproduced in theory. Presentational theory, then, is identified simply with structure. Presentational theory would be altogether quite boring if it were only structure, if it were simply the unapplied part of a model. In that case, presentation would just be an unremarkable subset of representation. But presentation is actually fundamentally dissimilar to representation, due entirely and radically to subtraction, abstraction's austere doppelganger.6

Before defining “subtraction,” it will be useful to clear up a possible confusion regarding the meaning of “predicates,” and how those predicates differ from the formal features of a theory's structure. The subtraction from predicates does not indicate a mystical withdrawal into an undifferentiated trivial structure in which nothing can be deduced. On the one hand, that structure has features indicates the presence of predicates. On the other hand, these features must be kept conceptually distinct from situational predicates because the former features enable the possibility of modeling and therefore of predicative representation in the first place. Predicates are on this formalist account internal to the model, and do not really apply to structures (especially insofar as they are formal). Although this line of thought seems obscure, its implications will make much more sense once subtractive formal ontology is more fully defined; for now, bear with me.

(Subtraction) An iterative process Δ is subtractive to an object Φ iff for stages δ1 and δ2 of Δ such that δ1 precedes δ2, Φ has fewer predicates at δ2 than it has at δ1.

Subtraction, in practice an iterative process, comes in degrees. The two extremes are maximal subtraction, where the object is left with no predicates whatsoever, and minimal subtraction, where it sheds only a single predicate. The process of subtraction can be likened to singularization. With subtraction/singularization, things may be either internal or external to the process. From the outside, subtraction might look like abstraction, possibly until total subtraction is achieved.7

Ontology is, first of all, a proper subset of theory. Different philosophical traditions assign totally different and contradictory meanings and roles to ontology. For my purposes here, however, I will provide a rather strict definition:

(Ontology) A theory Φ is an ontology iff i.) Φ is (only) a structure; and ii.) Φ is fully subtracted.

This definition is useful for several reasons. First, it provides a clear and rigorous way to distinguish ontology as the science of being from metaphysics as a general philosophical approach to reality. Second, ontology as the most general science should treat being in its most general features. Since the most general feature of being is isness, i.e. being qua being, this requirement immediately discounts a number of “ontologists” who under this definition are really just doing the old representationalist metaphysics under a groovy new aegis.8 Ontology's strict generality implies that this or that difference, whatever those differences are, is simply not under its purview.

Obviously, full subtraction is incompatible with modeling. Partial subtraction, however, is formally equivalent to abstraction. The result of any non-maximal subtraction is a theoretical object whose predicates constitute a subset of its previous (whether real or theoretical) predicates. Now it is obvious that one could get to the same result going the other direction, using abstraction to specify the intended subset.

However, as activity, the two have varying implications: subtraction proceeds from infinity (the infinity of the complex, real object) to the infinitesimal; abstraction always begins from a finite number, a finite specification of predicates, and as it becomes “concretized” (that is, as it becomes more abstract), more predicates are added. Subtraction is thus an extraordinarily powerful mechanism, though as noted, incompleted subtraction is indistinguishable from abstraction, at least from the outside.

Interlude: Model Theory
It is instructive to consider these propositions through the lens of (mathematical) model theory. It is worth quoting at length from the preface of Maria Manzano's Model Theory, a classic textbook of the discipline:

The classical way of describing the task of model theory is to say that in it one studies the relations between formal languages, on the one hand, and the realities about which those languages speak, on the other hand. That is correct, provided that one bears in mind that both the languages and the realities mentioned above are not languages and realities in the usual sense of the words, but rather in another sense which is distinct and sui generis.9

This “reality” is in fact the “absolute mathematical universe.”10 Furthermore, Manzano comments on the correct relation of model theory to reality: “Model theory is not a semantical theory which relates natural languages to the physical and social reality, but rather a mathematical theory which relates some mathematical structures to other mathematical strucures.”11 A model in this sense, then, is not a representation of some mathematical structure, but that very structure which is represented by a formal language. As Zachary Fraser notes in the introduction to Badiou's The Concept of Model:

What a careful study of the formalizations presupposed by any consideration of 'models' reveals is the incorporation of both formal syntaxes and semantic structures into a single mathematical situation, a situation exhausted in the deployment of notational differences.12

To summarize quite a bit, Badiou's argument in The Concept of Model boils down to the conclusion that representation and what it represents are internal to mathematics, in other words that the proper understanding of model theory takes it to be anti-representational. This conception of “model” (theory) as anti-representational and sui generis is precisely in line with what I have been calling ontology.13

Phase III: The Irreal
The object of ontology is just ontology itself. It is a fully subtracted, and therefore fully singular, discipline. The study of the object of ontology is the study of ontology itself. Ontology, because it is fully subtracted and theoretical, must be differentiated from other forms of (real, non-theoretical) presentation. Subtracted presentation is the same as strict self-representation. Ontology represents itself and can represent only itself.14 Self-representation, in turn, implies something about this object, namely that it is “minimal.” We can define this formally as follows:

(Minimal Object) A theoretical object Φ is a minimal object iff i.) Φ is fully subtracted; and ii.) Any theory Ψ that encounters Φ is expressively complete with respect to Φ.15

By “expressively complete,” I mean that all that can be said of Φ can be said in the theory in which Φ exists, without structural modification. This is just to say that a minimal object is created entirely ex nihilo by the theory in which it plays a part.16 If that object in encountered by a theory, it must have been created by that theory, and hence that theory is expressively complete with regard to it.17 It follows from this definition that only ontology, and only ontologies, can have minimal objects.18

Expressive completeness should seem similar to our earlier definition of structural isomorphism.19 Hence, we can say that ontology is structurally isomorphic with regard to its object, that is to being, presentation, or in other words to itself (with only a minimal gap or remainder internal to it).

Because a minimal object is created wholly in the act of its theorization, and just is that theorization, it constitutes a decision. This decision determines what objects will be treated/created by the deciding ontology. This decision is what allows ontology to think. It forecloses certain possibilities and opens up others. A formal logic that can derive every sentence is no logic at all, and similarly for a logic that can derive no sentences; both are trivial. Hence certain principles must be adopted (i.e. axioms or axiom-like entities) to enact an ontological decision and allow thought to gain some traction against itself.

On the other hand, this exclusion constitutes a problem for the generality and indeed the genericity of ontology by deciding for a single system and disallowing the ontological adequacy of another system—each system is fully singular on its own terms, but which terms should one adopt? Decision is relative, and perhaps an absolute is wanted.

Alain Badiou is a case in point. His preferred axiomatization of set theory, namely Zermelo-Fraenkl plus the Axiom of Choice, determines wholly the laws of being that he will support, i.e. the ontological structure he will treat as singular. Badiou himself is fully cognizant of the role of decision in ontology.20 External to that axiomatization, his decision seems arbitrary, and may not even appear fully singular.21 We can state the problem as an incongruency on Badiou's part as well: On the one hand, Badiou wants to uphold the Parmenidean thesis of the identity of thinking and being;22 on the other hand, he also holds fast to Zermelo-Fraenkl set theory as the correct ontology. But correct by what measure? Surely other modes of thinking are possible, and therefore other modes of being, and therefore also ontologies?

Does ontological decision decide about the Real?23 If it is really singularly subtracted, ontology obviously does not decide directly or explicitly about the Real, since it would then only be able to decide about itself. But nonetheless something about the Real might seem to be implied through the adoption of certain principles of intelligibility.24 Decision delimits one's horizons, and what one can't be thought for a singular structure like ontology, simply isn't—and this could be disastrous. Exclusion from thought, and hence from being, would certainly seem to decide about the Real, not through the internal decision of the axioms of the ontology, but through the brute fact that there are other axiomatizations.

There is thus a sort of incongruence within ontology itself: it proclaims its absolute unbinding from anything external to it (from the Real), but nonetheless ends up an apparently arbitrary and, despite its best protestations, decisional structure. But what could singular thought do if it were to avoid this parochial restriction? A properly irreal project with regard to ontologies in general, a thinking external to them and yet internal to them all, could seal the gulf between the singular adequacy of the minimal object and the particularity of the leap of thought that creates it.

Phase IV: Formalysis
Let us call the speculative para-ontological enterprise that aims to pursue this possibility “formalysis.”25 Formalysis has two goals: first, it aims to singularize ways of thinking, rendering them fully subtracted and generically intelligible ontologies; second, it seeks to compossibilize these different ontologies and think them together.26 These two goals taken together constitute formalysis as not merely the science of being, but the science of intelligibility in general.

First, a note about formal methods. The formal methods and techniques now extant may not be strong or nuanced enough to singularize certain structures found within continental philosophy. Derrida's logic of deconstruction, for example, is probably not formalizable using current techniques. Ever more complex formal systems of reasoning must be developed that can account for this sort of informal complexity. Within analytic philosophy the situation is rather different, since analytics tend to adopt a commonsense classical logic when judging between arguments and systems, no matter what the logics internal to those arguments and systems happen to be. The analytic mode of reasoning is in almost all cases the reasoning of classical metalogical and mathematical proofs. Continental philosophy, in contrast, mixes together object languages and metalanguages, neither of which need be classical. Despite the difficulty of this project, I refuse to leave the formal logical structure of argument to analytic quibbling.27 What this project needs are methods to enact a generalization of the “Grand Style” of mathematical philosophy, a designation coined by Badiou to describe his own use of mathematics, as opposed to the “Little Style” of the analytic philosophers, in particular the logical empiricists.28 It is in this sense that I claim Formalysis as a kind of generalization of Badiousian formal ontology to include the thinking of any intelligibility (any ontology) whatsoever.

In order to singularize a mode of thought, formalysis must identify the principles of intelligibility at work within that mode.29 Speaking loosely, intelligibility is simply what can be thought insofar as it can be thought, irrespective of interpretation, representation, and abstraction—hence its importance for ontology. Generic intelligibility extends this definition into the irreal, making thought irrespective of any conceivable reality or non-reality.

Ontologies are differentiated above all by their different principles of intelligibility. A principle of intelligibility is necessarily a restriction, a decision that allows thought to encounter something other than itself (i.e. to externalize itself) and therefore to produce particular thoughts.30 While formalysis must think through and with these particular principles, it must in the last instance refuse them all, or rather relativize their scope.

If formalysis were to have a principle of intelligibility of its own, this contradict the whole aim and method of formalysis, which is to avoid deciding between ontologies. We are therefore faced with a dilemma: How does formalysis think a compossibilization that is beyond the ontologies it compossibilizes, without merely positing a higher principle of intelligibility and therefore a higher ontology?

Formalysis cannot have a principle of intelligibility, but for this it substitutes a “prescription of intelligibility.” This prescription is a kind of conditional logic which applies not on the “level” of formalysis itself (whatever that would be) but to the local process of subtraction that governs the transformation of theory into ontology. We can understand this prescription as the injunction to subtract.31 There is no global principle of intelligibility because there is no intelligible globe; there is only an infinite multiplicity of possible subtractions.

These considerations are captured beautifully by the Oath of the Master of the Temple, a vow taken by advanced occultists in the Golden Dawn tradition.32 To advance between a certain magickal grade, one must swear to interpret every single thing that happens as an intensely personal, intimate, and infinitely meaningful communication with God. To put it in philosophical terms: There is no such thing as nonsense.

However, when two previously separate and mutually unintelligible principles of intelligibility are first brought together, they may result in an apparent disfiguration of one another with regard to the single structure in which they are placed. The practical task of formalysis in those cases is to think with and through the disrupting process, and finally to come out the other side with the principle of intelligibility once more intact. Again: There is no such thing as nonsense. The word “nonsense,” like “real,” is strictly relative to structure, while formalysis is absolute.

In a sense, formalysis is to generic thinking what formal ontology is to being.33 Formal ontology deals with the being qua being, i.e. with being insofar as it is, everywhere in the same way (or equivalently, nowhere at all). In contrast, formalysis is conditional, exploring the possibilities opened or closed by various structures, but never deciding about those structures with regard to the Real. It does not do to treat theoretical structures insofar as they are, i.e. insofar as they have being; this would tell us absolutely nothing internal to the structure.

The brute fact that there is a multiplicity of ontologies exhorts us towards formalysis. This facticity constitutes formalysis in its external or compossibilizing mode. It is to be distinguished from formalysis in its internal or singularizing mode. The difficult task of formalysis is to think the two together. To treat of structures only insofar as they have being (i.e. in the external mode) erases singularity and results finally in the tautology that “everything exists.” On the other hand, to treat of structures only on their own terms results in decision. Singularization and compossibilization are two sides of the same coind: generic intelligibility.

Compossibilization is the thinking-together of ontologies. As more ontologies are compossibilized, pure new modes of thought are produced, which in turn leads to more ways to subtract and to compossibilize.34 This is a good result both in itself and for non-ontological thought, since it makes available both new modes of thinking and perhaps even new possibilities for (formal) propositional composition. Furthermore, since formalysis is indifferent to the Real, the compossibilization it carries out is perfectly universal, formal, and general.

The dedication to generic intelligibility, the prescription of intelligibility, can be put in ethical terms: There is no greater sin than to deny intelligibility to a conceptual presentation, whether from a position internal (indicating not unintelligibility but a refusal to think) or external (indicating a parochial and aristocratic view) to that system. Instead of quibbling over who can understand the least (as certain ordinary language analytic philosophers seem happy to do), we should seek to indicate where systems differ from one another, and thereby to elaborate a theory of “conceptual work” that would allow us to determine whether a shift in vocabulary results in a corresponding change in what can be thought and in the methods for thinking it. Sometimes changes in vocabulary give the impression that nothing new is being said. Sometimes new words are adopted that do not correspond to the minutest change in what is intelligible. Formalysis therefore only critiques insofar as a system substitutes language for thinking.

To critique formalysis itself, one would have to avoid decision and hence the critique would also be formanalytical. On the other hand, it is possible to avoid this problem if one attacks the decision to adopt the prescription of intelligibility instead. Denying legitimacy to this line of attack would import the prescription into formalysis, singularize it, and transform it into a principle. Deciding for this prescription is different from a decision about the Real because it is reflexive, not with regard to its object but to its act. “Decision” about the prescription of intelligibility is deciding not to decide. It is thought deciding itself in the most universal and generic way possible. Of course ontology, insofar as it is decisional, refuses to recognize the irreality of this decision.35 In the occult tradition it is understood that will produces desire and desire produces belief; a good occultist, just like a good formanalyst, tinkers directly with will (decision) and explores what it can do.

The objects treated by formalysis are individual formal ontologies, constituted via a radical processual subtraction and brought into speculative non-relation via structural compossibilization.36 Formalysis is in some sense the (proposed) realization of Deleuze and Guattari's definition of philosophy as conceptual creation; but rather than merely creating concepts, it creates structures, modes of thought, conceptual pathways, or in other words, full-fledged formal ontologies.37 And it creates inventive modes of thinking-together.

It seems to me a fundamentally reasonable position to hold that structures (or logics of some abstract and diverse type) govern everything, and that they may be thought as singularities, not just self-creating, self-justifying or self-representing, but also utterly unique.38 Surely there is are “logics” not just of philosophy, but of music, of film, of psychedelic experience. The key to the gate is to find God in all things: “Every number is infinite; there is no difference.”39 Each singular principle is the hidden God of the atom, umanifest; the infinitely diffuse prescription is the azure Goddess of the sky. And when the two are united the Real is dissolved.

Sources Cited

Badiou, Alain. Being and Event. Trans. Oliver Feltham. London; New York: Continuum, 2005. Print.

–-. The Concept of Model: An Introduction to the Materialist Epistemology of Mathematics. Ed. and Trans. Zachary Luke Fraser and Tzuchien Tho. Melbourne:, 2007. Print.

–-. Theoretical Writings. Ed. and Trans. Ray Brassier and Alberto Toscano. London; New York: Continuum, 2004. Print.

Brassier, Ray. “Axiomatic Heresy: The Non-Philosophy of François Laruelle.” Radical Philosophy 121.01 (2003): 24-35. Web. 2 May 2016.

Crowley, Aleister. Liber AL vel Legis: The Book of the Law. San Fransisco, CA: Weiser Books, 1976. Print.

Deleuze, Gilles and Félix Guattari. What is Philosophy? Trans. Hugh Tomlinson and Graham Burchell. New York: Columbia University Press, 1994. Print.

Fraser, Zachary. “The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of Forcing and the Heyting Calculus.” Cosmos and History: The Journal of Nature and Social Philosophy 2.1-2 (2006): 94-133. Web. 3 May 2016.

Laruelle, François. Principles of Non-Philosophy. Trans. Nicola Rubczak and Anthony Paul Smith. London; New York: Bloomsbury Academic, 2013. Print.

Manzano, María. Model Theory. Trans. Ruy de Queiroz. Oxford: Clarendon Press, 1998. Print.

Morton, Timothy. Hyperobjects: Philosophy and Ecology After the End of the World. Minneapolis: University of Minnesota Press, 2013. Print.


1 AKA Oath of the Magister Templi. Source:

2 It would be useful here to have a working theory of the difference between real and theoretical objects.

3 Usually, different senses are weighted and different properties given different values; vision in particular is a privileged sense when it comes to representation.

4 One could also define these in terms of “more or less,” and allow for deviations on either the real or theoretical side. Defining things that way might be more intuitive, but it really doesn't make a difference in this instance.

5 I am open to the possibility of a third option. Depending on its nature, an alternative might or might not affect the current argument.

6 See Badiou's “On Subtraction,” which can be found in his Theoretical Writings.

7 This properly experiential aspect is one of the most complicated parts of the theory of singularization. It can by no means be solved here.

8 A prime example is Timothy Morton's Hyperobjects, or really any of the Object-Oriented Ontologists.

9 Maria Manzano, Model Theory ix.

10 Ibid.

11 Manzano, Model Theory x. The “semantical theory” referred to is almost certainly the use of model theory by analytic philosophers, wherein a (mathematical) model is a set-theoretical structure that “represents” external reality, typically in order to study the use of language, whether for philosophical problems or otherwise. The use of model theory within analytic philosophy is not, however, limited to this vulgar representationalism. Among logicians, model theory is used (and this is first and foremost why it was produced) to study the logical features of a formal language. The offending use is consequently to identify the models present within model theory as representations.

12 Zachary Fraser, in Alain Badiou, The Concept of Model xxxvii.

13 This line of thought surely has numerous implications for the relation between syntax and semantics, or between proof-theory and model-theory. However, here I have neither the space nor the expertise to explore either avenue with the detail they warrant.

14 This is by no means meant in anything even remotely resembling the phenomenological way.

15 A brief comment about the word “encounters”: Here, it is assumed that the non-ontological cannot properly encounter a subtracted object, since that object is determined by the same subtraction and at the same time that its would-be encounterer makes itself subtractive. Note that this definition also accounts for the “totally subtracted” talk regarding ontology in section III.

16 The obvious example is the set within Badiou's ontology. What a set is emerges solely from the axioms of set theory and is not dependent in any way on a definition or intuition of what it is.

17 Of course incompleteness, where it applies to ontology, tells us that not every object is presentable by the theory; this is an interesting take on the “minimal gap” between ontology and being in higher-order ontologies.

18 This obviously implies that any sufficiently strict formalism is already an ontology.

19 See Phase I.

20 Badiou, Being and Event 6.

21 See Zachary Fraser's “The Law of the Subject” for an interesting alternative to Badiou's theory of the subject hashed out in intuitionistic logic. This is precisely the kind of alternative thinking I am advocating.

22 Badiou quotes Parmenides as saying “It is the same to think and to be” (Theoretical Writings 177).

23 See Laruelle for the inspiration of this question.

24 More on the term “principle of intelligibility” will come in the following section.

25 I end this paper with a few musing on what this enterprise might look like, what it might have to do. Necessarily, this part of the exposition will be less rigorous and rather more utopian than the other sections. This is because formalysis is properly not yet in existence as a discipline. It is largely a future project and a hope for what thought can do.

26 Badiou himself claims that the role of philosophy is “compossibilization,” but I must admit I find it hard to view some of his analyses, say, of art, as anything but philosophical reductions of, say, a poet to an intellectually flattened and fanciful thinker of the event. For this sort of overly-philosophizing reading, see Badiou's Being and Event, 191 (for Mallarmé) and 255 (for Hölderlin).

27 These days, there are many non-classical logics being developed and studies, from paraconsistent logics (which reject that the law of non-contradiction is true only), to intuitionistic logics and mathematics (which reject the law of excluded middle), to free logics (which reject the ontological commitments inherent in classical first-order logic). However, the complexity of continental reasoning remains under-formalized.

28 See Badiou's “The Grand Style and the Little Style” in Theoretical Writings. The (analytic) philosophers of the Little Style view mathematics and logic as just special cases or objects for their commonsense philosophical analyses, rather than being truly conditioned by them.

29 Intelligibility is a characteristic of subtracted structures, a speculative unity of form and content (in very much a Hegelian sense) that need be adequate only to itself. The category corresponding to intelligibility in the Science of Logic is “absolute form,” the absolute indifference to the question of form and content. Content, as that which is not form, can therefore be identified, perhaps counter-intuitively, with the whole complex of abstraction, representation qua model and that which is represented rolled into one. Hegel's deduction of absolute form is, when conceived in this way, actually itself a subtraction and not a concretization, as most readings of the Hegelian dialectic would have it. If form is not properly subtracted from content, if the non-relation inherent in intelligibility is not carried through, then speculation can never think itself on its own terms and would therefore refuse its own intelligibility. Hence it would not truly be immanent speculation.

30 Because this formalization is subtractive rather than abstractive/representational, one can imagine taking well-nigh anything as an inspiration for formalization, whether it be a black metal album, a horror film, a mystical experience, or whatever else.

31 The “qua” (as in “thinking qua thinking” or “appearance qua essence”) has a privileged status with regard to formalysis' prescription of intelligibility. In particular, the form “X qua X” is fascinating, in part because of its subtractive character. The “qua,” is in this reflexive guise the operator of singularity. In its other guise (“X qua Y”) it is abstractive. Perhaps a typology of this operator could be developed, leading to alternatives beyond the abstraction/subtraction distinction.

32 See the epigraph. For more on the profound and sometimes terrible effects of that Oath, see the story of Charles Stansfeld Jones.

33 Generic thinking is just undecided being.

34 This could be thought obscenely abstract, but nonetheless it can be related quite easily to Badiou's claim that mathematics/ontology itself is a situation that is periodically subjected to events and truth-processes, and hence radical reconfigurations. But Badiou's vision of ontology is always of a one—there is one proper ontology for a given universe, it seems, and of course Badiou has a monopoly on it. Formalysis differs in this as well as its project of actively singularizing theory, upon which it is parasitic (compare this to Laruelle's treatment of philosophy as found in Principles of Non-Philosophy).

35 This is similar to the way philosophy treats Laruelle's non-philosophy.

36 The precise implications of this “speculative non-relation” cannot be fully explored here. Suffice it to say that subtraction is a form of de-relationalization. The specifically structural aspects of this non-relation must unfortunately wait for another time.

37 See Deleuze & Guattari's What is Philosophy, 22. Of course this is no critique of Deleuze and Guattari. Rather, it indicates a shift in perspective from the conceptual to the formal.

38 It is highly possible that those two modes of singularity always go together, since something self-creating would presumably be unique, not created after a mold of any kind.

39 Liber AL vel Legis: The Book of the Law, I.4.

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