There is a theorem that shows that it is impossible for an algorithm to jointly satisfy the statistical fairness criteria of Calibration and Equalised Odds non-trivially. But what about the recently advocated alternative to Calibration, Base Rate Tracking? Here, we show that Base Rate Tracking is strictly weaker than Calibration, and then take up the question of whether it is possible to jointly satisfy Base Rate Tracking and Equalised Odds in non-trivial scenarios. We show that it is not, thereby establishing (...) an even more general impossibility theorem. (shrink)

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. (shrink)

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. (shrink)

If episodic memory is constructive, experienced time is also a construct. We develop an event-based formalism that replaces the traditional objective, agent-independent notion of time with a constructive, agent-dependent notion of time. We show how to make this agent-dependent time entropic and hence well-defined. We use sheaf-theoretic techniques to render agent-dependent time functorial and to construct episodic 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 episodic memories as pure constructs from single events. We formulate an empirical hypothesis that human episodic memory implements a particular time-symmetric constructive functor, and discuss possible experimental tests. (shrink)

Motivated by Scholze and Fargues' geometrization of the local Langlands correspondence using perfectoid diamonds and Clausen and Scholze's work on the K-theory of adic spaces using condensed mathematics, we introduce the Efimov K-theory of diamonds. We propose a pro-diamond, a large stable (infinity,1)-category of diamonds D^{diamond}, diamond spectra and chromatic tower, and a localization sequence for diamond spectra.

We investigate the idea of sleep as the protostate, and posit the idea of dark consciousness where dark is a 2-fold hybrid. We model dark consciousness as a 2-topos in p-adic time, and outline perfectoid and diamond-like versions. We then introduce and illustrate implications of Dark Imaginarium, which is a higher order Curiosity Artificial Intelligence, an Infinity-Curiosity Type, that thinks in infinity categories.

We propose a quantum curiosity algorithm as a means to implement quantum thinking into AI, and we illustrate 5 new quantum curiosity types. We then introduce 6 new hybrid quantum curiosity types combining animal and plant curiosity elements with biomimicry beyond human sensing. We then introduce 4 specialized quantum curiosity types, which incorporate quantum thinking into coding frameworks to radically transform problem-solving and discovery in science, medicine, and systems analysis. We conclude with a forecasting of the future of quantum thinking (...) in AI and illustrate an example of how to apply the new curiosity types: General Collaborative Networks. (shrink)

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.

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. (shrink)

We present a colorful and novel discussion of mathematical techniques of visualizing a fourth spatial dimension. We first discuss notions of dimensionality including the homotopy dimension for objects in an (infinity,1)-topos. We try to visualize the fourth spatial dimension using color, and illustrate this with four-dimensional ice-cream. We apply categorical negative thinking to what we have called (infinity,1)-visual epistemology. The aim is that visualizations of higher spatial dimensions can occur functorially. We illustrate with five images five conjectural methods for how (...) to visualize the fourth spatial dimension. (shrink)

We present colorful illustrations of particular properties of functorial diamonds, in the sense of Scholze; namely profinite reflections as categorical colors.We discuss sight as site using representable functors in the condensed formalism. We illuminate diamonds using our novel constructions of categorical Ozma and Cinderella, the site of Oz, and condensed Through the Looking-Glass.

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 propose Qurio, which is our new model of pedagogy incorporating the principles of quantum mechanics with a curiosity AI called Curio AI equipped with a meta-curiosity algorithm. Curio has a curiosity profile that is in a quantum superposition of every possible curiosity type. We describe the ethos and tenets of Qurio, which we claim can create an environment supporting neuroplasticity that cultivates curiosity powered by tools that exhibit their own curiosity. We give examples of how to incorporate non-locality, complementarity, (...) and quantum lateral thinking into epistemology. We then futurecast the epistemology of curiosity and quantum pedagogy by way of curiosity’s event horizon. (shrink)

Commensurate with our work on dark consciousness, we present a small extension of our previous formalism to model the mixed-frequency states as time crystals. We then posit shared intelligence as an Everettian phone, which is a looking-glass p-adic periodicity in time.

The cognitive sciences tell us that the self is a construct. The visual arts illustrate this fact. Mathematics give it full expression, abstracting the self to a Grothendieck site. This self is a haecceity, an ephemeral this-ness and now-ness. We make up our minds and our histories. That our acts are public, that they communicate effectively, becomes a dialetheic paradox, a deep paradox for our times.

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. (shrink)

A comprehensive review of diamonds, in the sense of Scholze, is presented. The diamond formulations of the Fargues-Fontaine curve and Bun_G are stated. Principal results centered on the diamond formalism in the global Langlands correspondence and the geometrization of the local Langlands correspondence are given. We conclude with a discussion of future geometrizations, and conjecture a diamond reformulation of quantum computational complexity towards a diamond ER = EPR.

Time and space are conflated in time-warps when asleep we dream. Our wakeful cognitive ability to keep them separate indicates different ways of envisaging self-hood. Awareness that dream-time and life-time are separate is itself a propensity of human minds that has evolved by natural selection with adaptive developments in cerebral neuronal circuitry that underpin human behavioural complexity. The contrast is highlighted between how memory of our spatio-temporal experiences appears to be treated by our brain one way when we are wide-awake (...) and thus well aware of our unfolding personal life-history, and in a different way when, fast asleep, our dreaming dissolves any such awareness and our oneiric self-hood seems different from our real-life self-hood. The difference is considered from two angles. Regarding simultaneous memory-storage for an experience recalled from alternative perspectives, a theoretical possibility is proposed of a pro-system comparable to Cantor Ternary Set topology. Regarding the brain, the difference is considered in relation to the temporo-spatial theory of consciousness and the compatible concept of active inference forestalling surprisal by minimising variational free energy according to the free energy principle. (shrink)