Generalized descriptive set theory
My PhD thesis in mathematics was in set theory and model theory (University of Helsinki 2011). My main area is generalized descriptive set theory on which I have been working in collaboration with Tapani Hyttinen (my PhD supervisor), Sy-David Friedman, Miguel Moreno, David Aspero and Yurii Khomskii. The area is based on the original contributions of Jouko Väänänen, Saharon Shelah and Aapo Halko in the early 1990’s. This area belongs to pure mathematical research and is concerned with refining our understanding of uncountable structures, definability and classification complexity. It is an attempt to create a descriptive set theory of the uncountable. My arXiv page should contain an up-to-date list which includes all publications in this area. This is a flourishing area of research with many open questions.
Descriptive set theory and knot theory
I have been interested in classical descriptive set theory as well and my main result in that area is the non-classification of wild knots. A wild knot is a continuous embedding of the unit circle into 3-space (no smoothness assumption). Two knots are equivalent if there exists a homeomorphism of the entire space taking one knot to the other. The result says that this equivalence relation on wild knots is strictly harder to classify than the isomorphism of countable structures (e.g. countable groups). More precisely it is shown that a turbulent equivalence relation can be Borel-reduced to the knot-equivalence. An open question remains whether the homeomorphism of open subsets of the 3-space can be classified by countable structures. I conjectured that the methods that I developed in the aforementioned knot-paper could work to solve this. I haven’t been able to do it so far.
* Project of ‘mathematizing’ enactivism (this is my main project!)
As a cognitive scientist I am interested how brain works. As a logician I am interested in meaning, content and semantics. As a cognitive scientist I find in the enactivists’ theory of formation and emergence of contents promising. As a mathematician I want to have a mathematical theory of how that’s happening. The aim is to take enactivist concepts such as affordances and active perception, the enactivist way of thinking such as the primarity of sensorimotor coupling over representation, and translate them into mathematical language. The only publication (pages 8-18 here) I have on this topic so far is already 4 years old. There is a growing number of ideas, notebook scribbles and drafts which might be enough to turn into a paper soon. The long term goal is to create a unifying mathematical and logical framework for enactive approach to cognition which will help in the analysis and design of complicated machine learning systems. Mathematical tools include causality (as developed by Judea Pearl and others), dependence logic (following Jouko Väänänen’s conceptualization), dynamical systems (especially Markov processes), information theory and game theory, but first and foremost we start from very simple analysis of toy-model examples.
Philosophy of mathematics
I have several side-projects. With Markus Pantsar we are working on a philosophical argument about the nature of mathematical and abstract reasoning. Our insight is that the concepts of epistemological emergence and reducibility can be applied to mathematical concepts. Sometimes a mathematical concept can be completely described and exhausted by existing concepts in which case it is reducible. For example the concept of prime number is reducible to the concepts of arithmetic such as division, multiplication and so on. Sometimes, however, a mathematical concept seems to be irreducible. For example the real number line is such a concept. Even though it can be “defined” in set theory, for example as the collection of Dedekind cuts, the intuitions and metaphors necessary to grasp the concept are not exhausted by the reduction to set theory.
I have developed a Hebbian neural network model to account for various assymmetries in the Stroop effect. I am developing several other ideas involving Hebbian learning. In relation to the “enactivism project” above, I am working on a philosophical account of representations in deep learning and other machine learning systems. I recently co-authored a paper on probabilistic team semantics (again, see my arXiv-page).