Applied category theory (ACT) has many useful tools for creating and layering flexible abstractions, and Topos is seeking ways to apply these tools to important problems. Many in the AI safety community feel a need for new good abstractions to ground their thinking about intelligence, agency, and optimization. This hammer and nail situation suggests an opportunity for collaboration in applying ACT tools to advance human flourishing, and this workshop is intended to start the conversion in earnest.
View a report of the summit here: [PDF]
AI-enabled automation presents the possibility that many features of our world will come under the control of non-human decision-making processes. What do we want their philosophy to be? And how long do we have to make that decision?
Ilya Prigogine classified life as a dissipative structure: we dissipate entropy into our environment as we bring order to our bodies and our lives. We comprise a set of strategies—some in our genome, others learned in life—for capturing negentropy. Whether it be nutritious food or a profound system of thought, we are intrinsically motivated to seek out order and channel it into ourselves. We also convey order to our communities, by bringing in sustenance or new and useful ideas.
This internalized order isn't arbitrary: it's structured so that we can better seek and capture order in the future. Our genome is a record of the biological strategies, and our personality is a record of the social strategies, which together produce the order-seeker we now are. This behavior is not confined to us or even to biological systems, but is also seen in social systems such as organizations, legal systems, education systems, etc. The boundary of agency is blurry, but we clearly cultivate strategies for cultivating strategies.
In this talk, I'll flesh out these ideas and give intuition for why they seem like better abstractions than some others I've seen. For example, I propose shifting the focus away from utility functions and away from individual agents, and toward the autopoietic act itself.
How do systems find their own “right abstractions”? Some first steps towards a compositional theory of active inference (and a discussion of some consequences).
We’ll learn about different frameworks for composing dynamical systems, and conjecture about what this has to do with “thing-y-ness”.
The creation of categorical logic has transformed both category theory and logic, blurring the traditional boundary between syntax and semantics and expanding the reach of logic to new application domains and kinds of semantics. Categorical logic offers a unifying, "plug-and-play" toolkit for understanding old logical systems and creating new ones. In this talk, we illustrate this principle through examples of categorical logic drawn from topics such as algebraic theories, bicategories of relations, graphical linear algebra, and statistical modelling.
“Finding the Right Abstractions" asks for clarification because in principle we don't know "what are these abstractions of?" or "abstractions right for what?" or "how do we know an abstraction is right?" I will use this opportunity to discuss abstractions of problems and of reasoning, assuming problems are what we want to solve. This leads me into Kolmogorov's paper "On the Interpretation of Intuitionistic Logic", which despite its title, is about problems, not logic. I will connect Kolmogorov's problems to objects of the Dialectica construction and show how Dialectica gives you a better approach to Kolmogorov's problems, providing the morphisms that Kolmogorov lacked in 1932. Then I will try to convince you that this coincidence of algebraic structures points to a good way of thinking about multi-agent systems.
Scientists and engineers like to describe processes or systems made of smaller pieces using diagrams: flow charts, Petri nets, electrical circuit diagrams, signal-flow graphs, chemical reaction networks, Feynman diagrams and the like. Many of these diagrams fit into a common framework: the mathematics of symmetric monoidal categories. When we embrace this realization, we start seeing connections between seemingly different subjects. We also get better tools for understanding open systems: systems that interact with their environment. This takes us beyond the old scientific paradigm that emphasizes closed systems.
In everyday life, we abstract the low-level world around us (molecules and photons) into high-level objects and concepts (cars and trees). Can we mathematically characterize these abstractions, in a way which generalizes beyond the specifics of human intuition? This talk will argue that abstraction arises from information relevant "far away" in graphical probabilistic models. We'll see examples from electronic circuits, statistical mechanics, rigid bodies, programming, pure math, and everyday life. We'll talk about the roles of chaos and clustering. Finally, we'll give a precise formulation of "natural" abstractions, and argue that most of the abstractions humans use on a day-to-day basis would likely also be used by alien minds given similar data.