Shanna Dobson

Mathematics has a long track record of refining the concepts by which we make sense of the world. For example, mathematics allows one to speak about different senses of "sameness", depending on the larger context. Phenomenology is the name of a philosophical discipline that tries to systematically investigate the first-personal perspective on reality and how it is constituted. Together, mathematics and phenomenology seem to be a good fit to derive statements about our experience that are, at the same time, well-defined, precise, and significant to our inner lives. However, there are difficulties stemming from the fact that phenomenology deals with inherently subjective things. Phenomenological investigations seem to lose their appeal when trying to approach them in the clear-cut way afforded by mathematics. How to overcome this obstacle? We argue that the Arts play a special role in mediating between the precise statements of mathematics and the sometimes fuzzy nature of our experience. Mathematics and art are complementary ways to come to a comprehensive understanding of reality.

We construct a mathematization of Derridian "arche-writing" and Deleuzian "haecceity." We posit an infinity-categorification of exigency (infinity-exigency), a higher-dimensional visual epistemology (infinity-visual epistemology), and infinity-stack Wittgenstein ladder. We reframe haecceities in terms of diamonds, in the sense of Scholze, and mathematize the haecceity-and-arche-writing reflection as a pro-diamond. As an exercise in infinity-visual epistemology, we validate a diamond infinity-stack time signature and infinity-stack harmony.

We construct a mathematization of Derridian "arche-writing" and Deleuzian "haecceity." We posit an infinity-categorification of exigency (infinity-exigency), a higher-dimensional visual epistemology (infinity-visual epistemology), and infinity-stack Wittgenstein ladder. We reframe haecceities in terms of diamonds, in the sense of Scholze, and mathematize the haecceity-and-arche-writing reflection as a pro-diamond. As an exercise in infinity-visual epistemology, we validate a diamond infinity-stack time signature and infinity-stack harmony.

We develop an approach to temporal logic that replaces the traditional objective, agent- and event-independent notion of time with a constructive, event-dependent notion of time. We show how to make this event-dependent time entropic and hence well-defined. We use sheaf-theoretic techniques to render event-dependent time functorial and to construct memories as sequences of observed and constructed events with well-defined limits that maximize the consistency of categorizations assigned to objects appearing in memories. We then develop a condensed formalism that represents memories as pure constructs from single events. We formulate an empirical hypothesis that human episodic memory implements a particular constructive functor.

We recently presented our Efimov K-theory of Diamonds, proposing a pro-diamond, a large stable (∞,1)-category of diamonds (D^{diamond}), and a localization sequence for diamond spectra. Commensurate with the localization sequence, we now detail four potential applications of the Efimov K-theory of D^{diamond}: emergent time as a pro-emergence (v-stack time) in a diamond holographic principle using Scholze’s six operations in the ’etale cohomology of diamonds; a pro-Generative Adversarial Network and v-stack perceptron; D^{diamond}cryptography; and diamond nonlocality in perfectoid quantum physics.

In this paper, we develop a mathematical model of awareness based on the idea of plurality. Instead of positing a singular principle, telos, or essence as noumenon, we model it as plurality accessible through multiple forms of awareness (“n-awareness”). In contrast to many other approaches, our model is committed to pluralist thinking. The noumenon is plural, and reality is neither reducible nor irreducible. Nothing dies out in meaning making. We begin by mathematizing the concept of awareness by appealing to the mathematical formalism of higher category theory. The beauty of higher category theory lies in its universality. Pluralism is categorical. In particular, we model awareness using the theories of derived categories and (infinity, 1)-topoi which will give rise to our meta-language. We then posit a “grammar” (“n-declension”) which could express n-awareness, accompanied by a new temporal ontology (“n-time”). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? Another question which could be re-conceptualized in our model is one of soteriology related to this pluralism: what is a self in this context? A new model of “personal identity over time” is thus introduced.

In this paper, we propose a mathematical model of subjective experience in terms of classes of hierarchical geometries of representations (“n-awareness”). We first outline a general framework by recalling concepts from higher category theory, homotopy theory, and the theory of (infinity,1)-topoi. We then state three conjectures that enrich this framework. We first propose that the (infinity,1)-category of a geometric structure known as perfectoid diamond is an (infinity,1)-topos. In order to construct a topology on the (infinity,1)-category of diamonds we then propose that topological localization, in the sense of Grothendieck-Rezk-Lurie (infinity,1)-topoi, extends to the (infinity,1)-category of diamonds. We provide a small-scale model using triangulated categories. Finally, our meta-model takes the form of Efimov K-theory of the (infinity,1)-category of perfectoid diamonds, which illustrates structural equivalences between the category of diamonds and subjective experience (i.e.its privacy, self-containedness, and self-reflexivity). Based on this, we investigate implications of the model. We posit a grammar (“n-declension”) for a novel language to express n-awareness, accompanied by a new temporal scheme (“n-time”). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? We also examine the notion of “self” within our framework. A new model of personal identity is introduced which resembles a categorical version of the “bundle theory”: selves are not substances in which properties inhere but (weakly) persistent moduli spaces in the K-theory of perfectoid diamonds.