Fictionalism. Drunken Risibility.

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Applied mathematics is often used as a source of support for platonism. How else but by becoming platonists can we make sense of the success of applied mathematics in science? As an answer to this question, the fictionalist empiricist will note that it’s not the case that applied mathematics always works. In several cases, it doesn’t work as initially intended, and it works only when accompanied by suitable empirical interpretations of the mathematical formalism. For example, when Dirac found negative energy solutions to the equation that now bears his name, he tried to devise physically meaningful interpretations of these solutions. His first inclination was to ignore these negative energy solutions as not being physically significant, and he took the solutions to be just an artifact of the mathematics – as is commonly done in similar cases in classical mechanics. Later, however, he identified a physically meaningful interpretation of these negative energy solutions in terms of “holes” in a sea of electrons. But the resulting interpretation was empirically inadequate, since it entailed that protons and electrons had the same mass. Given this difficulty, Dirac rejected that interpretation and formulated another. He interpreted the negative energy solutions in terms of a new particle that had the same mass as the electron but opposite charge. A couple of years after Dirac’s final interpretation was published Carl Anderson detected something that could be interpreted as the particle that Dirac posited. Asked as to whether Anderson was aware of Dirac’s papers, Anderson replied that he knew of the work, but he was so busy with his instruments that, as far as he was concerned, the discovery of the positron was entirely accidental.

The application of mathematics is ultimately a matter of using the vocabulary of mathematical theories to express relations among physical entities. Given that, for the fictionalist empiricist, the truth of the various theories involved – mathematical, physical, biological, and whatnot – is never asserted, no commitment to the existence of the entities that are posited by such theories is forthcoming. But if the theories in question – and, in particular, the mathematical theories – are not taken to be true, how can they be successfully applied? There is no mystery here. First, even in science, false theories can have true consequences. The situation here is analogous to what happens in fiction. Novels can, and often do, provide insightful, illuminating descriptions of phenomena of various kinds – for example, psychological or historical events – that help us understand the events in question in new, unexpected ways, despite the fact that the novels in question are not true. Second, given that mathematical entities are not subject to spatial-temporal constraints, it’s not surprising that they have no active role in applied contexts. Mathematical theories need only provide a framework that, suitably interpreted, can be used to describe the behavior of various types of phenomena – whether the latter are physical, chemical, biological, or whatnot. Having such a descriptive function is clearly compatible with the (interpreted) mathematical framework not being true, as Dirac’s case illustrates so powerfully. After all, as was just noted, one of the interpretations of the mathematical formalism was empirically inadequate.

On the fictionalist empiricist account, mathematical discourse is clearly taken on a par with scientific discourse. There is no change in the semantics. Mathematical and scientific statements are treated in exactly the same way. Both sorts of statements are truth-apt, and are taken as describing (correctly or not) the objects and relations they are about. The only shift here is on the aim of the research. After all, on the fictionalist empiricist proposal, the goal is not truth, but something weaker: empirical adequacy – or truth only with respect to the observable phenomena. However, once again, this goal matters to both science and (applied) mathematics, and the semantic uniformity between the two fields is still preserved. According to the fictionalist empiricist, mathematical discourse is also taken literally. If a mathematical theory states that “There are differentiable functions such that…”, the theory is not going to be reformulated in any way to avoid reference to these functions. The truth of the theory, however, is never asserted. There’s no need for that, given that only the empirical adequacy of the overall theoretical package is required.

Mathematical Reductionism: As Case Via C. S. Peirce’s Hypothetical Realism.

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During the 20th century, the following epistemology of mathematics was predominant: a sufficient condition for the possibility of the cognition of objects is that these objects can be reduced to set theory. The conditions for the possibility of the cognition of the objects of set theory (the sets), in turn, can be given in various manners; in any event, the objects reduced to sets do not need an additional epistemological discussion – they “are” sets. Hence, such an epistemology relies ultimately on ontology. Frege conceived the axioms as descriptions of how we actually manipulate extensions of concepts in our thinking (and in this sense as inevitable and intuitive “laws of thought”). Hilbert admitted the use of intuition exclusively in metamathematics where the consistency proof is to be done (by which the appropriateness of the axioms would be established); Bourbaki takes the axioms as mere hypotheses. Hence, Bourbaki’s concept of justification is the weakest of the three: “it works as long as we encounter no contradiction”; nevertheless, it is still epistemology, because from this hypothetical-deductive point of view, one insists that at least a proof of relative consistency (i.e., a proof that the hypotheses are consistent with the frequently tested and approved framework of set theory) should be available.

Doing mathematics, one tries to give proofs for propositions, i.e., to deduce the propositions logically from other propositions (premisses). Now, in the reductionist perspective, a proof of a mathematical proposition yields an insight into the truth of the proposition, if the premisses are already established (if one has already an insight into their truth); this can be done by giving in turn proofs for them (in which new premisses will occur which ask again for an insight into their truth), or by agreeing to put them at the beginning (to consider them as axioms or postulates). The philosopher tries to understand how the decision about what propositions to take as axioms is arrived at, because he or she is dissatisfied with the reductionist claim that it is on these axioms that the insight into the truth of the deduced propositions rests. Actually, this epistemology might contain a short-coming since Poincaré (and Wittgenstein) stressed that to have a proof of a proposition is by no means the same as to have an insight into its truth.

Attempts to disclose the ontology of mathematical objects reveal the following tendency in epistemology of mathematics: Mathematics is seen as suffering from a lack of ontological “determinateness”, namely that this science (contrarily to many others) does not concern material data such that the concept of material truth is not available (especially in the case of the infinite). This tendency is embarrassing since on the other hand mathematical cognition is very often presented as cognition of the “greatest possible certainty” just because it seems not to be bound to material evidence, let alone experimental check.

The technical apparatus developed by the reductionist and set-theoretical approach nowadays serves other purposes, partly for the reason that tacit beliefs about sets were challenged; the explanations of the science which it provides are considered as irrelevant by the practitioners of this science. There is doubt that the above mentioned sufficient condition is also necessary; it is not even accepted throughout as a sufficient one. But what happens if some objects, as in the case of category theory, do not fulfill the condition? It seems that the reductionist approach, so to say, has been undocked from the historical development of the discipline in several respects; an alternative is required.

Anterior to Peirce, epistemology was dominated by the idea of a grasp of objects; since Descartes, intuition was considered throughout as a particular, innate capacity of cognition (even if idealists thought that it concerns the general, and empiricists that it concerns the particular). The task of this particular capacity was the foundation of epistemology; already from Aristotle’s first premisses of syllogism, what was aimed at was to go back to something first. In this traditional approach, it is by the ontology of the objects that one hopes to answer the fundamental question concerning the conditions for the possibility of the cognition of these objects. One hopes that there are simple “basic objects” to which the more complex objects can be reduced and whose cognition is possible by common sense – be this an innate or otherwise distinguished capacity of cognition common to all human beings. Here, epistemology is “wrapped up” in (or rests on) ontology; to do epistemology one has to do ontology first.

Peirce shares Kant’s opinion according to which the object depends on the subject; however, he does not agree that reason is the crucial means of cognition to be criticised. In his paper “Questions concerning certain faculties claimed for man”, he points out the basic assumption of pragmatist philosophy: every cognition is semiotically mediated. He says that there is no immediate cognition (a cognition which “refers immediately to its object”), but that every cognition “has been determined by a previous cognition” of the same object. Correspondingly, Peirce replaces critique of reason by critique of signs. He thinks that Kant’s distinction between the world of things per se (Dinge an sich) and the world of apparition (Erscheinungswelt) is not fruitful; he rather distinguishes the world of the subject and the world of the object, connected by signs; his position consequently is a “hypothetical realism” in which all cognitions are only valid with reservations. This position does not negate (nor assert) that the object per se (with the semiotical mediation stripped off) exists, since such assertions of “pure” existence are seen as necessarily hypothetical (that means, not withstanding philosophical criticism).

By his basic assumption, Peirce was led to reveal a problem concerning the subject matter of epistemology, since this assumption means in particular that there is no intuitive cognition in the classical sense (which is synonymous to “immediate”). Hence, one could no longer consider cognitions as objects; there is no intuitive cognition of an intuitive cognition. Intuition can be no more than a relation. “All the cognitive faculties we know of are relative, and consequently their products are relations”. According to this new point of view, intuition cannot any longer serve to found epistemology, in departure from the former reductionist attitude. A central argument of Peirce against reductionism or, as he puts it,

the reply to the argument that there must be a first is as follows: In retracing our way from our conclusions to premisses, or from determined cognitions to those which determine them, we finally reach, in all cases, a point beyond which the consciousness in the determined cognition is more lively than in the cognition which determines it.

Peirce gives some examples derived from physiological observations about perception, like the fact that the third dimension of space is inferred, and the blind spot of the retina. In this situation, the process of reduction loses its legitimacy since it no longer fulfills the function of cognition justification. At such a place, something happens which I would like to call an “exchange of levels”: the process of reduction is interrupted in that the things exchange the roles performed in the determination of a cognition: what was originally considered as determining is now determined by what was originally considered as asking for determination.

The idea that contents of cognition are necessarily provisional has an effect on the very concept of conditions for the possibility of cognitions. It seems that one can infer from Peirce’s words that what vouches for a cognition is not necessarily the cognition which determines it but the livelyness of our consciousness in the cognition. Here, “to vouch for a cognition” means no longer what it meant before (which was much the same as “to determine a cognition”), but it still means that the cognition is (provisionally) reliable. This conception of the livelyness of our consciousness roughly might be seen as a substitute for the capacity of intuition in Peirce’s epistemology – but only roughly, since it has a different coverage.

Production of the Schizoid, End of Capitalism and Laruelle’s Radical Immanence. Note Quote Didactics.

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These are eclectics of the production, eclectics of the repetition, eclectics of the difference, where the fecundity of the novelty would either spring forth, or be weeded out. There is ‘schizoproduction’ prevalent in the world. This axiomatic schizoproduction is not a speech act, but discursive, in the sense that it constrains how meaning is distilled from relations, without the need for signifying, linguistic acts. Schizoproduction performs the relation. The bare minimum of schizoproduction is the gesture of transcending thought: namely, what François Laruelle calls a ‘decision’. Decision is differential, but it does not have to signify. It is the capacity to produce distinction and separation, in the most minimal, axiomatic form. Schizoproduction is capitalism turned into immanent capitalism, through a gesture of thought – sufficient thought. It is where capitalism has become a philosophy of life, in that it has a firm belief within a sufficient thought, whatever it comes in contact with. It is an expression of the real, the radical immanence as a transcending arrangement. It is a collective articulation bound up with intricate relations and management of carnal, affective, and discursive matter. The present form of capitalism is based on relationships, collaborations, and processuality, and in this is altogether different from the industrial period of modernism in the sense of subjectivity, production, governance, biopolitics and so on. In both cases, the life of a subject is valuable, since it is a substratum of potentiality and capacity, creativity and innovation; and in both cases, a subject is produced with physical, mental, cognitive and affective capacities compatible with each arrangement. Artistic practice is aligned with a shift from modern liberalism to the neoliberal dynamic position of the free agent.

Such attributes have thus become so obvious that the concepts of ‘competence’, ‘trust’ or ‘interest’ are taken as given facts, instead of perceiving them as functions within an arrangement. It is not that neoliberal management has leveraged the world from its joints, but that it is rather capitalism as philosophy, which has produced this world, where neoliberalism is just a part of the philosophy. Therefore, the thought of the end of capitalism will always be speculative, since we may regard the world without capitalism in the same way as we may regard the world-not-for-humans, which may be a speculative one, also. From its inception, capitalism paved a one-way path to annihilation, predicated as it was on unmitigated growth, the extraction of finite resources, the exaltation of individualism over communal ties, and the maximization of profit at the expense of the environment and society. The capitalist world was, as Thurston Clarke described so bleakly, ”dominated by the concerns of trade and Realpolitik rather than by human rights and spreading democracy”; it was a ”civilization influenced by the impersonal, bottom-line values of the corporations.” Capitalist industrial civilization was built on burning the organic remains of ancient organisms, but at the cost of destroying the stable climatic conditions which supported its very construction. The thirst for fossil fuels by our globalized, high-energy economy spurred increased technological development to extract the more difficult-to-reach reserves, but this frantic grasp for what was left only served to hasten the malignant transformation of Earth into an alien world. The ruling class tried to hold things together for as long as they could by printing money, propping up markets, militarizing domestic law enforcement, and orchestrating thinly veiled resource wars in the name of fighting terrorism, but the crisis of capitalism was intertwined with the ecological crisis and could never be solved by those whose jobs and social standing depended on protecting the status quo. All the corporate PR, greenwashing, political promises, cultural myths, and anthropocentrism could not hide the harsh Malthusian reality of ecological overshoot. As crime sky-rocketed and social unrest boiled over into rioting and looting, the elite retreated behind walled fortresses secured by armed guards, but the great unwinding of industrial civilization was already well underway. This evil genie was never going back in the bottle. And thats speculative too, or not really is a nuance to be fought hard on.

The immanence of capitalism is a transcending immanence: a system, which produces a world as an arrangement, through a capitalist form of thought—the philosophy of capitalism—which is a philosophy of sufficient reason in which economy is the determination in the last instance, and not the real. We need to specifically regard that this world is not real. The world is a process, a “geopolitical fiction”. Aside from this reason, there is an unthinkable world that is not for humans. It is not the world in itself, noumena, nor is it nature, bios, but rather it is the world indifferent to and foreclosed from human thought, a foreclosed and radical immanence – the real – which is not open nor will ever be opening itself for human thought. It will forever remain void and unilaterally indifferent. The radical immanence of the real is not an exception – analogous to the miracle in theology – but rather, it is an advent of the unprecedented unknown, where the lonely hour of last instance never comes. This radical immanence does not confer with ‘the new’ or with ‘the same’ and does not transcend through thought. It is matter in absolute movement, into which philosophy or oikonomia incorporates conditions, concepts, and operations. Now, a shift in thought is possible where the determination in the last instance would no longer be economy but rather a radical immanence of the real, as philosopher François Laruelle has argued. What is given, what is radically immanent in and as philosophy, is the mode of transcendental knowledge in which it operates. To know this mode of knowledge, to know it without entering into its circle, is to practice a science of the transcendental, the “transcendental science” of non-philosophy. This science is of the transcendental, but according to Laruelle, it must also itself be transcendental – it must be a global theory of the given-ness of the real. A non- philosophical transcendental is required if philosophy as a whole, including its transcendental structure, is to be received and known as it is. François Laruelle radicalises the Marxist term of determined-in-the-last-instance reworked by Louis Althusser, for whom the last instance as a dominating force was the economy. For Laruelle, the determination-in-the-last-instance is the Real and that “everything philosophy claims to master is in-the-last-instance thinkable from the One-Real”. For Althusser, referring to Engels, the economy is the ‘determination in the last instance’ in the long run, but only concerning the other determinations by the superstructures such as traditions. Following this, the “lonely hour of the ‘last instance’ never comes”.

Suspicion on Consciousness as an Immanent Derivative

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The category of the subject (like that of the object) has no place in an immanent world. There can be no transcendent, subjective essence. What, then, is the ontological status of a body and its attendant instance of consciousness? In what would it exist? Sanford Kwinter (conjuncted here) here offers:

It would exist precisely in the ever-shifting pattern of mixtures or composites: both internal ones – the body as a site marked and traversed by forces that converge upon it in continuous variation; and external ones – the capacity of any individuated substance to combine and recombine with other bodies or elements (ensembles), both influencing their actions and undergoing influence by them. The ‘subject’ … is but a synthetic unit falling at the midpoint or interface of two more fundamental systems of articulation: the first composed of the fluctuating microscopic relations and mixtures of which the subject is made up, the second of the macro-blocs of relations or ensembles into which it enters. The image produced at the interface of these two systems – that which replaces, yet is too often mistaken for, subjective essence – may in turn have its own individuality characterized with a certain rigor. For each mixture at this level introduces into the bloc a certain number of defining capacities that determine both what the ‘subject’ is capable of bringing to pass outside of itself and what it is capable of receiving (undergoing) in terms of effects.

This description is sufficient to explain the immanent nature of the subjective bloc as something entirely embedded in and conditioned by its surroundings. What it does not offer – and what is not offered in any detail in the entirety of the work – is an in-depth account of what, exactly, these “defining capacities” are. To be sure, it would be unfair to demand a complete description of these capacities. Kwinter himself has elsewhere referred to the states of the nervous system as “magically complex”. Regardless of the specificity with which these capacities can presently be defined, we must nonetheless agree that it is at this interface, as he calls it, at this location where so many systems are densely overlaid, that consciousness is produced. We may be convinced that this consciousness, this apparent internal space of thought, is derived entirely from immanent conditions and can only be granted the ontological status of an effect, but this effect still manages to produce certain difficulties when attempting to define modes of behavior appropriate to an immanent world.

There is a palpable suspicion of the role of consciousness throughout Kwinter’s work, at least insofar as it is equated with some kind of internal, subjective space. (In one text he optimistically awaits the day when this space will “be left utterly in shreds.”) The basis of this suspicion is multiple and obvious. Among the capacities of consciousness is the ability to attribute to itself the (false) image of a stable and transcendent essence. The workings of consciousness are precisely what allow the subjective bloc to orient itself in a sequence of time, separating itself from an absolute experience of the moment. It is within consciousness that limiting and arbitrary moral categories seem to most stubbornly lodge themselves. (To be sure this is the location of all critical thought.) And, above all, consciousness may serve as the repository for conditioned behaviors which believe themselves to be free of external determination. Consciousness, in short, contains within itself an enormous number of limiting factors which would retard the production of novelty. Insofar as it appears to possess the capacity for self-determination, this capacity would seem most productively applied by turning on itself – that is, precisely by making the choice not to make conscious decisions and instead to permit oneself to be seized by extra-subjective forces.

Of Magnitudes, Metrization and Materiality of Abstracto-Concrete Objects.

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The possibility of introducing magnitudes in a certain domain of concrete material objects is by no means immediate, granted or elementary. First of all, it is necessary to find a property of such objects that permits to compare them, so that a quasi-serial ordering be introduced in their set, that is a total linear ordering not excluding that more than one object may occupy the same position in the series. Such an ordering must then undergo a metrization, which depends on finding a fundamental measuring procedure permitting the determination of a standard sample to which the unit of measure can be bound. This also depends on the existence of an operation of physical composition, which behaves additively with respect to the quantity which we intend to measure. Only if all these conditions are satisfied will it be possible to introduce a magnitude in a proper sense, that is a function which assigns to each object of the material domain a real number. This real number represents the measure of the object with respect to the intended magnitude. This condition, by introducing an homomorphism between the domain of the material objects and that of the positive real numbers, transforms the language of analysis (that is of the concrete theory of real numbers) into a language capable of speaking faithfully and truly about those physical objects to which it is said that such a magnitude belongs.

Does the success of applying mathematics in the study of the physical world mean that this world has a mathematical structure in an ontological sense, or does it simply mean that we find in mathematics nothing but a convenient practical tool for putting order in our representations of the world? Neither of the answers to this question is right, and this is because the question itself is not correctly raised. Indeed it tacitly presupposes that the endeavour of our scientific investigations consists in facing the reality of “things” as it is, so to speak, in itself. But we know that any science is uniquely concerned with a limited “cut” operated in reality by adopting a particular point of view, that is concretely manifested by adopting a restricted number of predicates in the discourse on reality. Several skilful operational manipulations are needed in order to bring about a homomorphism with the structure of the positive real numbers. It is therefore clear that the objects that are studied by an empirical theory are by no means the rough things of everyday experience, but bundles of “attributes” (that is of properties, relations and functions), introduced through suitable operational procedures having often the explicit and declared goal of determining a concrete structure as isomorphic, or at least homomorphic, to the structure of real numbers or to some other mathematical structure. But now, if the objects of an empirical theory are entities of this kind, we are fully entitled to maintain that they are actually endowed with a mathematical structure: this is simply that structure which we have introduced through our operational procedures. However, this structure is objective and real and, with respect to it, the mathematized discourse is far from having a purely conventional and pragmatic function, with the goal of keeping our ideas in order: it is a faithful description of this structure. Of course, we could never pretend that such a discourse determines the structure of reality in a full and exhaustive way, and this for two distinct reasons: In the first place, reality (both in the sense of the totality of existing things, and of the ”whole” of any single thing), is much richer than the particular “slide” that it is possible to cut out by means of our operational manipulations. In the second place, we must be aware that a scientific object, defined as a structured set of attributes, is an abstract object, is a conceptual construction that is perfectly defined just because it is totally determined by a finite list of predicates. But concrete objects are by no means so: they are endowed with a great deal of attributes of an indefinite variety, so that they can at best exemplify with an acceptable approximation certain abstract objects that are totally encoding a given set of attributes through their corresponding predicates. The reason why such an exemplification can only be partial is that the different attributes that are simultaneously present in a concrete object are, in a way, mutually limiting themselves, so that this object does never fully exemplify anyone of them. This explains the correct sense of such common and obvious remarks as: “a rigid body, a perfect gas, an adiabatic transformation, a perfect elastic recoil, etc, do not exist in reality (or in Nature)”. Sometimes this remark is intended to vehiculate the thesis that these are nothing but intellectual fictions devoid of any correspondence with reality, but instrumentally used by scientists in order to organize their ideas. This interpretation is totally wrong, and is simply due to a confusion between encoding and exemplifying: no concrete thing encodes any finite and explicit number of characteristics that, on the contrary, can be appropriately encoded in a concept. Things can exemplify several concepts, while concepts (or abstract objects) do not exemplify the attributes they encode. Going back to the distinction between sense on the one hand, and reference or denotation on the other hand, we could also say that abstract objects belong to the level of sense, while their exemplifications belong to the level of reference, and constitute what is denoted by them. It is obvious that in the case of empirical sciences we try to construct conceptual structures (abstract objects) having empirical denotations (exemplified by concrete objects). If one has well understood this elementary but important distinction, one is in the position of correctly seeing how mathematics can concern physical objects. These objects are abstract objects, are structured sets of predicates, and there is absolutely nothing surprising in the fact that they could receive a mathematical structure (for example, a structure isomorphic to that of the positive real numbers, or to that of a given group, or of an abstract mathematical space, etc.). If it happens that these abstract objects are exemplified by concrete objects within a certain degree of approximation, we are entitled to say that the corresponding mathematical structure also holds true (with the same degree of approximation) for this domain of concrete objects. Now, in the case of physics, the abstract objects are constructed by isolating certain ontological attributes of things by means of concrete operations, so that they actually refer to things, and are exemplified by the concrete objects singled out by means of such operations up to a given degree of approximation or accuracy. In conclusion, one can maintain that mathematics constitutes at the same time the most exact language for speaking of the objects of the domain under consideration, and faithfully mirrors the concrete structure (in an ontological sense) of this domain of objects. Of course, it is very reasonable to recognize that other aspects of these things (or other attributes of them) might not be treatable by means of the particular mathematical language adopted, and this may imply either that these attributes could perhaps be handled through a different available mathematical language, or even that no mathematical language found as yet could be used for handling them.

Concepts – Intensional and Extensional.

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Let us start in this fashion: objects to which concepts apply (or not). The first step in arriving at a theory for this situation is, to assume that the objects in question are completely arbitrary (as urelements in set theory). This assumption is evidently wrong in empirical experience as also in mathematics itself, e.g., in function theory. So to admit this assumption forces us to build our own theory of sets to take care of the case of complex objects later on.

Concepts are normally given to us by linguistic expressions, disregarding by abstraction the origin of languages or signals or what have you. Now we can develop a theory of concepts as follows. We idealize our language by fixing a vocabulary together with logical operators and formulate expressions for classes, functions, and relations in the way of the λ-calculus. Here we have actually a theory of concepts, understood intensionally. Note that the extensional point of view is by no means lost, since we read for e.g., λx,yR(x,y) as the relation R over a domain of urelements; but either R is in the vocabulary or given by a composed expression in our logical language; equality does not refer to equal extensions but to logical equivalence and reduction processes. By the way, there is no hindrance to apply λ-expressions again to λ-expressions so that hierarchies of concepts can be included.

Another approach to the question of obtaining a theory of concepts is the algebraic one. Here introducing variables for extensions over a domain of urelements, and calling them classes helps develop the axiomatic class calculus. Adding (two-place) relations again with axioms, and we can obtain the relation calculus. One could go a step further to polyadic algebra. These theories do not have a prominent role nowadays, if one compares them with the λ-calculus or set theory. This is probably due to the circumstance that it seems difficult, not to say actually against the proper idea behind these theories, to allow iteration in the sense of classes of classes, etc.

For the mathematical purposes and for the use of logics, the appropriate way is to restrict a theory of concepts to a theory of their extensions. This has a good reason, since in an abstract theory we are interested in being as neutral as possible with respect to a description or factual theory given beforehand. There is a philosophical principle behind this, namely that logical (and in this case set theoretical) assumptions should be as far as possible distinguishable from any factual or descriptive assumption.

Topological Drifts in Deleuze. Note Quote.

Brion Gysin: How do you get in… get into these paintings?

William Burroughs: Usually I get in by a port of entry, as I call it. It is often a face through whose eyes the picture opens into a landscape and I go literally right through that eye into that landscape. Sometimes it is rather like an archway… a number of little details or a special spot of colours makes the port of entry and then the entire picture will suddenly become a three-dimensional frieze in plaster or jade or some other precious material.

The word fornix means “an archway” or “vault” (in Rome, prostitutes could be solicited there). More directly, fornicatio means “done in the archway”; thus a euphemism for prostitution.

Diagrammatic praxis proposes a contractual (push, pull) approach in which the movement between abstract machine, biogram (embodied, inflected diagram), formal diagram (drawing of, drawing off) and artaffect (realized thing) is topologically immanent. It imagines the practice of writing, of this writing, interleaved with the mapping processes with which it folds and unfolds – forming, deforming and reforming both processes. The relations of non-relations that power the diagram, the thought intensities that resonate between fragments, between content ad expression, the seeable and the sayable, the discursive and the non-discursive, mark entry points; portals of entry through which all points of the diagram pass – push, pull, fold, unfold – without the designation of arrival and departure, without the input/output connotations of a black boxed confection. Ports, as focal points of passage, attract lines of resistance or lines of flight through which the diagram may become both an effectuating concrete assemblage (thing) and remain outside the stratified zone of the audiovisual. It’s as if the port itself is a bifurcating point, a figural inflected archway. The port, as a bifurcation point of resistance (contra black box), modulates and changes the unstable, turbulent interplay between pure Matter and pure Function of the abstract machine. These ports are marked out, localized, situated, by the continuous movement of power-relations:

These power-relations … simultaneously local, unstable and diffuse, do not emanate from a central point or unique locus of sovereignty, but at each moment move from one point to another in a field of forces, marking inflections, resistances, twists and turns when one changes direction or retraces one’s steps… (Gilles Deleuze, Sean Hand-Foucault)

An inflection point, marked out by the diagram, is not a symmetrical form but the difference between concavity and convexity, a pure temporality, a “true atom of form, the true object of geography.” (Bernard Cache)

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Figure: Left: A bifurcating event presented figurally as an archway, a port of entry through order and chaos. Right: Event/entry with inflexion points, points of suspension, of pure temporality, that gives a form “of an absolute exteriority that is not even the exteriority of any given interiority, but which arise from that most interior place that can be perceived or even conceived […] that of which the perceiving itself is radically temporal or transitory”. The passing through of passage.

Cache’s absolute exteriority is equivalent to Deleuze’s description of the Outside “more distant than any exterior […] ‘twisted’, folded and doubled by an Inside that is deeper than any interior, and alone creates the possibility of the derived relation between the interior and the exterior”. This folded and doubled interior is diagrammed by Deleuze in the folds chapter of Foucault.

Thinking does not depend on a beautiful interiority that reunites the visible ad articulable elements, but is carried under the intrusion of an outside that eats into the interval and forces or dismembers the internal […] when there are only environments and whatever lies betwen them, when words and things are opened up by the environment without ever coinciding, there is a liberation of forces which come from the outside and exist only in a mixed up state of agitation, modification and mutation. In truth they are dice throws, for thinking involves throwing the dice. If the outside, farther away than any external world, is also closer than any internal world, is this not a sign that thought affects itself, by revealing the outside to be its own unthought element?

“It cannot discover the unthought […] without immediately bringing the unthought nearer to itself – or even, perhaps, without pushing it farther away, and in any case without causing man’s own being to undergo a change by the very fact, since it is deployed in the distance between them” (Gilles Deleuze, Sean Hand-Foucault)

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Figure: Left: a simulation of Deleuze’s central marking in his diagram of the Foucaultian diagram. This is the line of the Outside as Fold. Right: To best express the relations of diagrammatic praxis between content and expression (theory and practice) the Fold figure needs to be drawn as a double Fold (“twice twice” as Massumi might say) – a folded möbius strip. Here the superinflections between inside/outside and content/expression provide transversal vectors.

A topology or topological becoming-shapeshift retains its connectivity, its interconnectedness to preserve its autonomy as a singularity. All the points of all its matter reshape as difference in itself. A topology does not resemble itself. The möbius strip and the infamous torus-to-coffe cup are examples of 2d and 3d topologies. technically a topological surface is totalized, it can not comprise fragments cut or glued to produce a whole. Its change is continuous. It is not cut-copy-pasted. But the cut and its interval are requisite to an emergent new.

For Deleuze, the essence of meaning, the essence of essence, is best expressed in two infinitives; ‘to cut ” and “to die” […] Definite tenses keeping company in time. In the slash between their future and their past: “to cut” as always timeless and alone (Massumi).

Add the individuating “to shift” to the infinitives that reside in the timeless zone of indetermination of future-past. Given the paradigm of the topological-becoming, how might we address writing in the age of copy-paste and hypertext? The seamless and the stitched? As potential is it diagram? A linguistic multiplicity whose virtual immanence is the metalanguage potentiality between the phonemes that gives rise to all language?

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An overview diagram of diagrammatic praxis based on Deleuze’s diagram of the Foucaultian model shown below. The main modification is to the representation of the Fold. In the top figure, the Fold or zone of subjectification becomes a double-folded möbius strip.

Four folds of subjectification:

1. material part of ourselves which is to be surrounded and folded

2. the fold of the relation between forces always according to a particular rule that the relation between forces is bent back in order to become a relation to oneself (rule ; natural, divine, rational, aesthetic, etc)

3. fold of knowledge constitutes the relation of truth to our being and our being to truth which will serve as the formal condition for any kind of knowledge

4. the fold of the outside itself is the ultimate fold: an ‘interiority of expectation’ from which the subject, in different ways, hopes for immortality, eternity, salvation, freedom or death or detachment.

Carnap’s Topological Properties and Choice of Metric. Note Quote.

Husserl’s system is ontologically, a traditional double hierarchy. There are regions or spheres of being, and perfectly traditional ones, except that (due to Kant’s “Copernican revolution”) the traditional order is reversed: after the new Urregion of pure consciousness come the region of nature, the psychological region, and finally a region (or perhaps many regions) of Geist. Each such region is based upon a single highest genus of concrete objects (“individua”), corresponding to the traditional highest genera of substances: in pure consciousness, for example, Erlebnisse; in nature, “things” (Dinge). But each region also contains a hierarchy of abstract genera – genera of singular abstracta and of what Husserl calls “categorial” or “syntactic” objects (classes and relations). This structure of “logical modifications,” found analogously in each region, is the concern of logic. In addition, however, to the “formal essence” which each object has by virtue of its position in the logical hierarchy, there are also truths of “material” (sachliche) essence, which apply to objects as members of some species or genus – ultimately, some region of being. Thus the special sciences, which are individuated (as in Aristotle) by the regions they study, are each broadly divided into two parts: a science of essence and a science of “matters of fact.” Finally, there are what might be called matters of metaphysical essence: necessary truths about objects which apply in virtue of their dependence on objects in prior regions, and ultimately within the Urregion of pure consciousness.

This ontological structure translates directly into an epistemological one, because all being in the posterior regions rests on positing Erlebnisse in the realm of pure consciousness, and in particular on originary (immediate) rational theoretical positings, i.e. “intuitions.” The various sciences are therefore based on various types of intuition. Sciences of matters of fact, on the one hand, correspond to the kinds of ordinary intuition, analogous to perception. Sciences of essence, on the other hand, and formal logic, correspond to (formal or material) “essential insight” (Wesensschau). Husserl equates formal- and material-essential insight, respectively, as sources of knowledge, to Kant’s analytic and synthetic a priori, whereas ordinary perceptual intuition, the source of knowledge about matters of fact, corresponds to the Kantian synthetic a posteriori. Phenomenology, finally, as the science of essence in the region of pure consciousness, has knowledge of the way beings in one region are dependent on those in another.

In Carnap’s doctoral thesis, Der Raum, he applies the above Husserlian apparatus to the problem of determining our sources of knowledge about space. Is our knowledge of space analytic, synthetic a priori, or empirical? Carnap answers, in effect: it depends on what you mean by “space.” His answer foreshadows much of his future thought, but is also based directly on Husserl’s remark about this question in Ideen I: that, whereas Euclidean manifold is a formal category (logical modification), our knowledge of geometry as it applies to physical objects is a knowledge of material essence within the region of nature. Der Raum is largely an expansion and explication of that one remark. Our knowledge of “formal space,” Carnap says, is analytic, i.e. derives from “formal ontology in Husserl’s sense,” but our knowledge of the “intuitive space” in which sensible objects are necessarily found is synthetic a priori, i.e. material-essential. There is one important innovation: Carnap claims that essential (a priori) knowledge of intuitive space extends only to its topological properties, whereas the full structure of physical space requires also a choice of metric. This latter choice is informed by the actual behavior of objects (e.g. measuring rods), and knowledge of physical space is thus in part a posteriori – as Carnap also says, a knowledge of “matters of fact.” But such considerations never force the choice of one metric or another: our knowledge of physical space also depends on “free positing”. This last point, which has no equivalent in Husserl, is important. Still more telling is that Carnap compares the choice involved here to a choice of language, although at this stage he sees this as a mere analogy. On the whole, however, the treatment of Der Raum is more or less orthodoxly Husserlian.

Diffeomorphism Invariance: General Relativity Spacetime Points Cannot Possess Haecceity.

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Eliminative or radical ontic structural realism (ROSR) offers a radical cure—appropriate given its name—to what it perceives to be the ailing of traditional, object-based realist interpretations of fundamental theories in physics: rid their ontologies entirely of objects. The world does not, according to this view, consist of fundamental objects, which may or may not be individuals with a well-defined intrinsic identity, but instead of physical structures that are purely relational in the sense of networks of ‘free-standing’ physical relations without relata.

Advocates of ROSR have taken at least three distinct issues in fundamental physics to support their case. The quantum statistical features of an ensemble of elementary quantum particles of the same kind as well as the features of entangled elementary quantum (field) systems as illustrated in the violation of Bell-type inequalities challenge the standard understanding of the identity and individuality of fundamental physical objects: considered on their own, an elementary quantum particle part of the above mentioned ensemble or an entangled elementary quantum system (that is, an elementary quantum system standing in a quantum entanglement relation) cannot be said to satisfy genuine and empirically meaningful identity conditions. Thirdly, it has been argued that one of the consequences of the diffeomorphism invariance and background independence found in general relativity (GTR) is that spacetime points should not be considered as traditional objects possessing some haecceity, i.e. some identity on their own.

The trouble with ROSR is that its main assertion appears squarely incoherent: insofar as relations can be exemplified, they can only be exemplified by some relata. Given this conceptual dependence of relations upon relata, any contention that relations can exist floating freely from some objects that stand in those relations seems incoherent. If we accept an ontological commitment e.g. to universals, we may well be able to affirm that relations exist independently of relata – as abstracta in a Platonic heaven. The trouble is that ROSR is supposed to be a form of scientific realism, and as such committed to asserting that at least certain elements of the relevant theories of fundamental physics faithfully capture elements of physical reality. Thus, a defender of ROSR must claim that, fundamentally, relations-sans-relata are exemplified in the physical world, and that contravenes both the intuitive and the usual technical conceptualization of relations.

The usual extensional understanding of n-ary relations just equates them with subsets of the n-fold Cartesian product of the set of elementary objects assumed to figure in the relevant ontology over which the relation is defined. This extensional, ultimately set-theoretic, conceptualization of relations pervades philosophy and operates in the background of fundamental physical theories as they are usually formulated, as well as their philosophical appraisal in the structuralist literature. The charge then is that the fundamental physical structures that are represented in the fundamental physical theories are just not of the ‘object-free’ type suggested by ROSR.

While ROSR should not be held to the conceptual standards dictated by the metaphysical prejudices it denies, giving up the set-theoretical framework and the ineliminable reference to objects and relata attending its characterizations of relations and structure requires an alternative conceptualization of these notions so central to the position. This alternative conceptualization remains necessary even in the light of ‘metaphysics first’ complaints, which insist that ROSR’s problem must be confronted, first and foremost, at the metaphysical level, and that the question of how to represent structure in our language and in our theories only arises in the wake of a coherent metaphysical solution. But the radical may do as much metaphysics as she likes, articulate her theory and her realist commitments she must, and in order to do that, a coherent conceptualization of what it is to have free-floating relations exemplified in the physical world is necessary.

ROSR thus confronts a dilemma: either soften to a more moderate structural realist position or else develop the requisite alternative conceptualizations of relations and of structures and apply them to fundamental physical theories. A number of structural realists have grabbed the first leg and proposed less radical and non-eliminative versions of ontic structural realism (OSR). These moderate cousins of ROSR aim to take seriously the difficulties of the traditional metaphysics of objects for understanding fundamental physics while avoiding these major objections against ROSR by keeping some thin notion of object. The picture typically offered is that of a balance between relations and their relata, coupled to an insistence that these relata do not possess their identity intrinsically, but only by virtue of occupying a relational position in a structural complex. Because it strikes this ontological balance, we term this moderate version of OSR ‘balanced ontic structural realism’ (BOSR).

But holding their ground may reward the ROSRer with certain advantages over its moderate competitors. First, were the complete elimination of relata to succeed, then structural realism would not confront any of the known headaches concerning the identity of these objects or, relatedly, the status of the Principle of the Identity of Indiscernibles. To be sure, this embarrassment can arguably be avoided by other moves; but eliminating objects altogether simply obliterates any concerns whether two objects are one and the same. Secondly, and speculatively, alternative formulations of our fundamental physical theories may shed light on a path toward a quantum theory of gravity.

For these presumed advantages to come to bear, however, the possibility of a precise formulation of the notion of ‘free-standing’ (or ‘object-free’) structure, in the sense of a network of relations without relata (without objects) must thus be achieved.  Jonathan Bain has argued that category theory provides the appropriate mathematical framework for ROSR, allowing for an ‘object-free’ notion of relation, and hence of structure. This argument can only succeed, however, if the category-theoretical formulation of (some of the) fundamental physical theories has some physical salience that the set-theoretical formulation lacks, or proves to be preferable qua formulation of a physical theory in some other way.

F. A. Muller has argued that neither set theory nor category theory provide the tools necessary to clarify the “Central Claim” of structural realism that the world, or parts of the world, have or are some structure. The main reason for this arises from the failure of reference in the contexts of both set theory and category theory, at least if some minimal realist constraints are imposed on how reference can function. Consequently, Muller argues that an appropriately realist stucturalist is better served by fixing the concept of structure by axiomatization rather than by (set-theoretical or category-theoretical) definition.