About me

A picture of me

I am a mathematician. My work revolves around understanding meaning-making systems: how to recognize them, how to design them, and how to make sense of them within our own interpretive frame. I ponder questions like what does a market want? or how do we know the reasoning trace of an LLM is faithfully representing its behaviour? I am a theory builder so I approach these questions from the foundations. My tools of choice are category theory and logic. I worked on a meta-theory of systems and cybernetic systems (especially games and learners), and more recently on logical foundations for quantitative methods.

Whereabouts. I am in the process of submitting my PhD thesis at the University of Strathclyde, where I was part of the MSP group. Meanwhile, since 2024, I am part of UK ARIA's 'Safeguarded AI' programme as an independent researcher. I am PI on my own project 'Doubly Categorical Systems Logic: A Theory of Specification Languages' (£120,000+) and collaborator on a second project 'Quantitative Predicate Logic as a Foundation for Verified ML' (£500,000+) where I lead a collaboration to develop quantitative logic.

News. I will be serving as PC Chair for ACT 2027 together with Tobias Fritz.

Active lines of research

This is a more of less up-to-date list of the topics I'm working on. For a list of talks, preprints and papers, check out this page. For dormant research topics, see here.

I'm open to collaborations in any of the topics listed below, you can get in touch.

research topicDouble Operadic Theory of Systems

Double Operadic Theory of Systems (DOTS), neé (Doubly) Categorical Systems Theory ((D)CST), is a formal mathematical theory of systems. A theory of systems is the algebra of a double operad: the operations of the operad describe how to compose systems, while the algebra substantiates these operations into actual functors between categories of systems. This structure combines both the compositional and structural aspects of systems theory, with the operad part describing compositionality and the categorical part describing the structure of systems.

DOTS being formal means we can work at a level of generality at which we can meanigfully talk about the subjects of systems theory—systems, processes, interfaces, behaviours, etc.—without ever having to commit to a specific way to model them (coalgebras, machines, decorated cospans, behaviour types, etc.).

I'm working on DOTS within the ARIA project 'Safeguarded AI' in close collaboration with Topos Research UK and the folks there (David Jaz Myers, Sophie Libkind, Jason Brown, José Siqueira, etc.)

Reading list

research topicQuantitative Logic

Most attributes of systems aren't qualitative absolutes, but quantitative statements. How far things are from each other, how much error there is in an approximation, what is the probability of a property being satisfied, and so on. Thus adapting the structural methods of logic, category theory, and categorical systems theory to a quantitative setting is crucial for making them practically useful.

Meanwhile, many ideas and phenomena in statistics, machine learning, and functional analysis have a logical flavour. Can we turn this hunch into a tool to understand such settings in a new light?

Together with Bob Atkey and Ekaterina Komendantskaya I am developing quantitative logic in the sense of F. William Lawvere, i.e. as an 'enriched' logic. We started from a propositional theory, which we dubbed quantitative linear logic (QLL), and we are now moving on to predicate logic.

We put a strong emphasis on softness, thus relaxing the hard properties of lattice connectives (\(\lor , \land \)) to accomodate more useful quantitative connectives (like \(+\) and \(\times \)).

Thomas Flinkow also helped us investigating the role quantitative logic can have in property-driven learning, with very encouraging results.

Reading list

research topicStructural primitives for cybernetics

Cybernetic systems self-adapt through the observations they make of the 'environment' which interacts with them. In games, this dynamics brings players to play equilibria. In machine learning, it makes models learn from a dataset and RL agent adapt to an environment. In control theory, it keeps systems in a viable state.

Category theory can put the mathematical treatment of these systems on strong and flexible foundations. We can then use string diagrams to describe systems compositionally, categorical logic to impose guarantees on their behaviour, and functional programming to produce efficient and effable programs to analyse them.

My main research focus is develop structural primitives for cybernetics, working in the groove of categorical systems theory. A concrete goal is characterizing good regulators abstractly, and develop a modern, model-free internal model principle.

Reading list

  1. The series of posts on open cybernetics on this blog
  2. Towards Foundations of Categorical Cybernetics by me, Bruno Gavranovic, Jules Hedges, Eigil Fjeldgren Rischel
  3. From categorical systems theory to categorical cybernetics, by me
  4. Mathematical Foundations for a Compositional Account of the Bayesian Brain, by Toby St Clere Smithe
  5. A Bayesian Interpretation of the Internal Model Principle, by Manuel Baltieri, Martin Biehl, Nathaniel Virgo, and me

Shiny links I collected

Favourite quotes

Ce qui fait la qualité de l'inventivité et de l'imagination du chercheur, c'est la qualité de son attention, à l'écoute de la voix des choses. Car les choses de l'Univers ne se lassent jamais de parler d'elles-mêmes et de se révéler, à celui qui se soucie d'entendre.

A. Grothendieck

We have as an elemental part of our nature the ability to observe imperfection and from that imperfection deduce the laws of Nature–deduce perfection. This ability is our capacity to 'abstract' and it may well be the case that a different capacity to abstract, even in the face of a universal order of things, would lead to a different perception of that order.

G. Boole
Pungent commentary on CT folklore

There's no doubt that they do exist but you can't poke and prod them except by thinking about them. It's quite astonishing and I still don't understand it, despite having been a mathematician all my life. How can things be there without actually being there?

J. H. Conway, when asked 'Do mathematical objects exist?'

The core question is not how you do math but how does the unconscious do it. How it is that it's demonstrably better at it than you are? You work on a problem and then you put it away for a while. But it doesn't go away. It reappears at lunch. Or while you're taking a shower. It says: Take a look at this. What do you think? Then you wonder why the shower is cold. Or the soup. Is this doing math? I'm afraid it is. How is it doing it? We don't know. How does the unconscious do math?

'Stella Maris', C. McCarthy

After seeing a problem like this, your response should be 'please, may have a better problem?'

C. McBride

The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise

E. Dijkstra