At the intersection of category theory, cybernetics, and dialectical reasoning lies a profound framework for understanding computation and control. This paper examines how categorical structures—particularly adjoint functors and fixed points—illuminate the nature of feedback and control in both mathematical and philosophical contexts. Through an analysis of Lawvere’s fixed point theorem, Bayesian Open Games, and modern approaches to categorical cybernetics, we develop a unified perspective that bridges computation, control, and dialectical reasoning. We demonstrate the practical implications of this theoretical framework through (...) a compiler pipeline that targets modern GPU architectures. (shrink)

A quick introduction to category theory and topos theory, axiomatically. These notes are adapted from Prof. Pettigrew’s notes.

Johan Georg Granström’s Treatise on Intuitionistic Type Theory represents a landmark contribution to our understanding of the philosophical foundations of Per Martin-Löf’s intuitionistic type theory (ITT). The work is particularly noteworthy for its careful exposition of how ITT emerges as a sophisticated codification of Brouwerian intuitionism while simultaneously advancing a distinctly Kantian program in the philosophy of mathematics. This philosophical grounding, drawing heavily on both Kantian and Husserlian phenomenology, offers valuable insights into the nature of mathematical knowledge and computation. The (...) significance of this work cannot be overstated. At a time when type theory is gaining increasing prominence in both mathematics and computer science, Granström provides a philosophical foundation that helps explain why type-theoretic approaches have proven so successful. By situating ITT within the broader philosophical tradition, particularly in relation to Kant’s views on mathematical knowledge, the book illuminates both the historical development of constructive mathematics and its contemporary relevance. The work comes at a crucial moment in the development of mathematical foundations. The increasing importance of computer-assisted proof and formal verification has led to renewed interest in constructive approaches to mathematics. Granström’s analysis helps explain why type-theoretic foundations are particularly well-suited to meet these contemporary challenges. His careful exposition of the philosophical foundations of ITT provides essential context for understanding both its historical development and its future potential. Moreover, the book represents a significant contribution to our understanding of the relationship between philosophy and mathematics. By showing how deep philosophical insights about the nature of mathematical knowledge inform the development of formal systems, Granström demonstrates the continuing relevance of philosophical reflection for mathematical practice. This is particularly valuable at a time when the relationship between philosophy and mathematics is often questioned. (shrink)

Andr ́e Weil viewed mathematics as deeply intertwined with metaphysics. In his essay ”From Metaphysics to Mathematics,” he illustrates how mathematical ideas often arise from vague, metaphysical analogies and reflections that guide researchers toward new theories. For instance, Weil discusses how analogies between different areas, such as number theory and algebraic functions, have led to significant breakthroughs. These metaphysical underpinnings provide a fertile ground for mathematical creativity, eventually transforming into rigorous mathematical structures. -/- Alexander Grothendieck’s work, particularly in ”R ́ecoltes (...) et Semailles,” resonates with Weil’s ideas by emphasizing the organic and generative aspects of mathematical creation. Grothendieck sees mathematical work as akin to cultivating a garden, where ideas grow and develop in a nurturing environment. He describes the process of mathematical discovery as a deeply personal and creative journey, involving both solitary reflection and collaborative effort. -/- Grothendieck’s concept of creating mathematical ”houses” aligns with his broader philosophical view that mathematics provides structures within which new ideas can flourish. His work in category theory and algebraic geometry exemplifies this approach, where he developed vast, interconnected frameworks that have become foundational in modern mathematics. Grothendieck’s analogy of constructing houses for others to live in highlights the mathematician’s role in creating abstract frameworks that others can inhabit and explore. This metaphor emphasizes the constructive nature of mathematics, where researchers build theoretical structures that form the basis for further exploration and application by the mathematical community. (shrink)

Due to the wide scope of (in particular linear) homotopy type theory (using quantum natural language processing), a metatheory can be applied not just to theorizing the metatheory of scientific progress, but ordinary language or any public language defined by sociality/social agents as the precondition for the realizability of (general) intelligence via an inferential network from which judgement can be made. How this metatheory of science generalizes to public language is through the recent advances of quantum natural language processing, but (...) the traditional metalogical encodings (of Tarski) is relatively comparable through universes such as the type of types or type of types of types found in the inherent inferentialism of UF/HoTT via infinity-groupoids. The "computational trinity" of proofs=programs=algebra in HoTT also means reason is defined functionally as "what it does" by what it computes (proofs are programs). Following Reza Negarestani's recent book, through the self's self-realizability, achieving self-consciousness and consciousness beyond selfhood, "geistig manifestation" is achieved in the form of general intelligence, but only through an inferential network of social agents intrinsically encoded through computation. (shrink)