Moving down the same trajectory as in the previous post (see here), our use of quantum set theory becomes important in conjunction with Laruelle, Deleuze, and Badiou.
Since at this plateau we are becoming-mathematicians, note this quick introductory summary of the practicality of set theory which follows (see here):
The purpose of set theory is not practical application in the same way that, for example, Fourier analysis has practical applications. To most mathematicians (i.e. those who are not themselves set theorists), the value of set theory is not in any particular theorem but in the language it gives us. Nowadays even computer scientists describe their basic concept – Turing machines – in the language of set theory. This is useful because when you specify an object set-theoretically there is no question what you are talking about and you can unambiguously answer any questions you might have about it. Without precise definitions it is very difficult to do any serious mathematics.
Furthermore, to apply the “quantumness” aspect to what we have revealed, we find that Ernesto Rodriguez succinctly makes note of the practicality of quantum set theory in his 1984 dissertation abstract:
The work of von Neumann tells us that the logic of Quantum Mechanics is not Boolean. This suggests the formulation of a quantum theory of sets based on quantum logic much as modern set theory is based on Boolean logic. In the first part of this dissertation such a Quantum Set Theory is developed. In the second part, Quantum Set Theory is proposed as a universal language for physics. A Quantum Topology and the beginnings of a Quantum Geometry are developed in this language. Finally, a toy model is studied. It gives indications of possible lines for progress in this program. (see here)
With this understanding of quantum set theory, here we find the answer to what one can formally do with it:
Set theory provides foundations of mathematics in the sense that all the mathematical notions like numbers, functions, relations, structures are defined in the axiomatic set theory called ZFC. Quantum set theory naturally extends ZFC to quantum logic. Hence, we can expect that quantum set theory provides mathematics based on quantum logic. In this talk, I will show a useful application of quantum set theory to quantum mechanics based on the fact that the real numbers constructed in quantum set theory exactly corresponds to the quantum observables. [...] It is shown that all the observational propositions on a quantum system corresponds to some propositions in quantum set theory and the equality relation naturally provides the proposition that two observables have the same value. It has been broadly accepted that we cannot speak of the values of quantum observables without assuming a hidden variable theory. However, quantum set theory enables us to do so without assuming hidden variables but alternatively under the consist use of quantum logic, which is more or less considered as logic of the superposition principle. (see here)
As Terence Blake notes in his paper IS ONTOLOGY MAKING US STUPID?: Diachronic vs Synchronic Ontologies (see here), we have historically had both “diachronic” and “synchronic” approaches available to us. For all the talk of diagrammatic onto-cartographies, we must bear in mind a lesson from the first day of philosophy class: that the map, the chart, the diagram, etc. is not the territory.
By way of our sheaf logic and quantum set theory, it is not only that we only have these two approaches available, but we have also an infinity of approaches that lie somewhere in-between these two ends. Moreover, each of these are to operate on the same even field of play due to the equality or equivalence relation which constitutes them. Everything is on the table, many tables in fact, sometimes indeterminate and unobservable tables, and we may orient ourselves given this radical plurality. Any single ontology, or any given theology (including my Wilderness theology, as we have discovered here) is – as if by definition – a failure.
We may move instead to a roundly post-metaphysical space.
In search of a transfer principle as full as possible, we have come to realize that such an attempt is always necessarily incomplete, as Pavel Florensky noted. In other words, Paradox is embedded not only into our observation of reality, but in the fabric of reality itself. The paradox is what is real. This infinity marks true radicality in how we may interpret, for example, Heisenberg’s uncertainty principle (see here). Quantum set theorist Masahao Ozawa outlines it as follows ”…the uncertainty does not always come from the disturbing influence of the measurement, but from the quantum nature of the particle itself.”
An initial attention to the double-style of Jacques Derrida, and then now to Francois Laruelle, may result from a more basic intuition which grasps this super-positional logic. In Laruelle, there is not only a re-doubling of our interpretation of reality in the text as is the case in Derrida, but there is also a re-doubling of the Real itself. Functioning within the diachronic/synchronic framework we will encounter others who hold sympathies as well as reservations to both prevailing logics, who otherwise do not fit “cozily” in either end. Things are messy and entangled.
There is not merely a relationality between objects (intra-objectivity), but a super-relationality which locates the absolute uniqueness of each neighborhood, of each individual within such neighborhoods, of each locality, of each and every human being as a free absolute or particularity in themselves as a condition of true multiplicity.
Indeed, these diachronic/synchronic ends are the two ends at which we can understand each other most clearly.
But it is important, too, that we attend to the ambiguous cases where logic is inconsistent or incoherent. We must strive to listen even when the logics are neither explicitly diachronic nor synchronic, especially when they are fuzzy. Throughout life we will likely remain somewhere in this intermittent and highly uncertain path, maybe towards one pole over another (i.e. I tend to lean synchronic). Any articulation of a crystal clear diachronic or synchronic logic is only revealed to be momentary at best.
Ultimately, the “non-dissociative” demand for such rigorous clarity of thought, when all else remains unclear, is often inappropriate when approaching dissociation. The way in which we think about dissociation matters significantly in the way we approach the situation. This phenomenon is well studied in medical papers such as Trauma-related dissociation: conceptual clarity lost and found (2004) by van der Hart O, Nijenhuis E, Steele K, and Brown D (see here), whose objective is to get across the notion that “Imprecise conceptualizations of dissociation hinder understanding of trauma-related dissociation.”
What we need is a certain versatility which variably escapes rigorous notions of identification, abstraction, conceptualization, and categorization.
With this logic of superposition, we encounter unstable states which are “in-between” stable ones, and therefore to some degree or another incomprehensible no matter our classification schema. This is a happening which brings us all the way back to the poesy of Novalis. With a non-Laruellean non-philosophy (see here), we achieve the possibility of a high degree of artistic, poetic, and otherwise creative expression and understanding in these moments of Crisis.
With this quantum set theory also comes not merely what Laruelle calls dualysis, but what we may brand as the hallmark of the non-Laruellean: radical paralysis. To those who see Laruelle as a “dead-end”, they are in one sense correct in doing so — yet to turn away because there’s “no more theory” beyond this point seems absurd. It means putting down the texts and integrating them into “felt and bodily” practices in the new World. Just as there is revealed an identification with the “foreigner” as found in post-colonial discourses such as that which can be found in work of Homi Bhabha or R. Radhakrishnan, of the one cast outside of the boundaries, there is also in non-Laruellean non-philosophy a certain identification with those who remain within the scene of the familiar, who are also outside.
Consider Peter Sloterdjk’s book You Must Change Your Life. From his chapter “Only Cripples Will Survive: Unthan’s Lesson”, pages 40-60:
Living in the Nonetheless imposes an ostentatious zest for life on those who are determined to succeed. The fact that things may be different on the inside is no one’s business. The land of smiles is inhabited by cripple artistes. [...] The vaudeville people know more about ‘real life’ because they are those who have been thrown to the margins, the fallen and the battered. These ‘jostled humans’ are perhaps the only ones who still exist authentically. [...] Thanks to them, the circus becomes an invisible church. In a world of fellow travellers complicit in the collective self-deception, the circus performers are the only ones who are not swindlers – someone walking on a tightrope cannot pretend for a moment. [...] It is not walking upright that makes humans human; it is rather the incipient awareness of the inner gradient that causes humans to do so.
So to shift registers again, as Dr. Ozawa wrote in his 2007 paper (see here), quantum set theory builds upon the Many-valued Boolean logics of Takeuti, locating in through this transfer principle the realization that “[d]espite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model.” We are to weave through our concern for liberty a thread of equivalence and its equality.
In this traumatic site, it may be said that everything is still on a table. It is mere a question of “At which table we will offer ourselves up to be eaten?” It is not that either Deleuze or Badiou or any of these thinkers are “wrong” per se; however, it is just that becoming the luxury meal of high-theory and politics is not even remotely available in many of the circumstances with which we face in our everyday immanence.
We may seek to stay in tune with the Spirit, oscillating between a concern for radical immanence and alchemical spirituality as the Crisis commands of us. This movement maps a ever-so-difficult position of tension to maintain, to keep the Vision-beyond-Vision of a skilled archer, or perhaps the balance of an acrobat as Sloterdijk elegantly wrote. It is the position of being-in-tension whose name is stability. It is one whose pragmatics are modeled upon a sheaf logic of approximations.
Mutualization and Communization, therefore, may be named the twin pillars which are to be set in place in order to keep the other from going too far astray in the wrong direction.
Sin – missing the mark?
To transfer in between these two is to see as closely as possible and to orient oneself properly with respect to the vision-in-One. When acting from the place of the One, to be inside this sphere in particular, there is a whole circumference of possible actions which can be made with our sheaf logic.
Can one inhabit, like the purple bacteria of Seth Lloyd’s Quantum Life at 34:00-37:00, a “decoherence-free subspace” with others?
An archer would fire from a distance, and another would shout Sin! in the event the “target” was missed. This analogy doesn’t seem to capture the monstrosity of reductive logics. It is not simply an act of missing the mark which occurs when the Spirit ceases its movement, or a mere failing to hit the “target”; rather, it is akin to turning around and firing arrows around in a random, decoherent manner, in an act of absolutely sublime or otherwise surrealist form of Violence.
By contrast, there is Unitive strength found within a “concatenated quantum code” of collective solidarity, and this is the intended structure of the “church in the Wilderness”.
Moving forth from quantum set theory, we may arrive at a possible interpretation of physics itself. The employment of set theory in particular is of great use to us because the understanding of infinite sets allows us to conceptualize elementary topology. Our aim is to think about otherwise neglected elements of spatiality in a way that best attends to the place of Crisis so as to transform the Wilderness.