**Positive Geometry** refers to a mathematical framework that explores the underlying geometric structures of certain physical and abstract systems. It posits that the geometry of these systems is constrained by "positivity" conditions, meaning that only certain configurations are allowed, often those that yield meaningful or physically relevant outcomes. This concept is particularly significant in fields such as theoretical physics, cognitive science, and the study of consciousness. See [[Apply SMT to analyze Positive Geometry]]. ### Positive Geometry and Perception: In the realm of cognitive science and the philosophy of perception, Positive Geometry suggests that the structures underlying our perceptions are not arbitrary but are shaped by specific geometric principles that optimize functionality rather than reflect an objective reality. Building on ideas from cognitive scientist [[Donald Hoffman]], this approach posits that what we perceive as reality is a construct—a simplified interface shaped by evolutionary pressures. This "positive" aspect of geometry refers to the idea that these structures are optimized for survival and utility, rather than providing an accurate depiction of an underlying objective world. [[Donald Hoffman]] also talks about [[POSet]] a general case of [[Lattice]], which are effectively intervals of possibilities, which can always be computed in finite time using [[Abstract Interpretation]]. ### Positive Geometry in Theoretical Physics: In theoretical physics, Positive Geometry is exemplified by concepts like the **[[Amplituhedron]]**. The Amplituhedron is a geometric object that dramatically simplifies calculations in quantum field theory, particularly in the computation of scattering amplitudes. Unlike traditional methods that rely on the complexities of spacetime, the Amplituhedron operates in a purely geometric space (within the space of [[Polytope]]), where positivity conditions on the geometry itself lead to meaningful physical predictions. This structure reveals how certain constraints, rooted in geometry, can lead to a deeper understanding of fundamental interactions without the need for an underlying spacetime framework. ### Connections and Parallels: 1. **Optimization and Structure:** - Both in perception and in physical theories like the Amplituhedron, Positive Geometry reflects an optimization process. In perception, it optimizes for survival and cognitive efficiency; in physics, it optimizes the way we calculate interactions, simplifying complex processes into elegant geometric forms. 2. **Evolutionary Fitness and Positivity:** - Just as Hoffman's Positive Geometry suggests that our perceptions are shaped by evolutionary fitness, the positivity conditions in structures like the Amplituhedron are not arbitrary but are crucial for ensuring that the geometric object corresponds to physically meaningful amplitudes. In both cases, positivity serves as a guiding principle that filters out configurations that do not contribute to the system's effectiveness—whether in survival or in predicting physical phenomena. 3. **Geometric Representation:** - In the context of the Amplituhedron, physical processes are represented within a high-dimensional geometric object where only positive combinations of vectors are allowed. Similarly, in Hoffman's theory, our perceptions can be thought of as existing within a geometric space constrained by positive structures that optimize our interaction with the world. ### Summary: Positive Geometry, whether applied to cognitive science or theoretical physics, is a concept that emphasizes the role of geometric structures constrained by positivity in shaping our understanding of reality. In the case of perception, this geometry shapes the interface through which we interact with the world, optimizing for survival rather than accuracy. In the case of the Amplituhedron, Positive Geometry simplifies the complexities of quantum interactions into elegant, manageable forms. In both fields, Positive Geometry reveals the power of abstract mathematical structures to inform and optimize our understanding of the world, whether through the lens of human perception or the fundamental interactions of particles. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Positive Geometry") sort title, authors, modified ```