Questions (last update: january 2021)
Here are some questions I’m currently interested in. These questions may be superficial, meaningless or may even have been already solved.

How are logic/computation related to space/time? Linear Logic seems to lead to an understanding of Logic in general through the primitive intuition of space and time. Mathematical proofs which were seen as sequential mental acts evolving through time because proofnet which are purely structural objects. The lost of time seems to be recovered with polarisation and focalisation, exhibiting a mechanism of dialogue between positive and negative connectives. What does this polarity mean in the context of proofnets (how are the geometry of interaction and game semantics for linear logic related)? What does the “time” mean in the practical reasoning.

How are Logic and Computation related to Cognition and Nature? Are links between Quantum and Logic superficial or is it truly possible to relate them? Logic and computation seem to be part of the nature but does the nature itself computes or make use of logic for physical constraints? The Transcendental Syntax exhibits a mechanism of nondeterministic interaction in logic, similarly to atomic/chemical reactions or biological interactions. Do logic, computation, biology and physics obey to the same laws/mechanisms? Can it be related to a computational theory of mind?

What is the root/origin of historical prejudices and misconceptions of Logic? What does the history of Logic teach us about the way we think and what/why we believe? Similarly to Physics and Biology, should Logic be considered as a science observing “coherent interaction” as a natural phenomenon? It may be interesting to investigate the biological, psychological and sociological aspects of Logic. Why was Logic thought and organised the way it was? How can we learn from the past, in the context of Logic?

What makes something computationally difficult or easy? Implicit Computational Complexity allows definitions of computational complexity classes by syntactical restrictions on formal languages. It would be interesting to explore the “deep structure” of these restrictions. It seems that algebraic tools may give an understanding we do not have with the algorithmic and classical point of view.

Is there a common ground of syntactical formats? Some concepts were formalised in a certain way, mainly due to historical or practical reasons but are equivalent to other ones. For instance, the inductive description of regular languages, finite state automata, regular grammars, regular expressions and readonly Turing machines all define regular languages. Can they be abstracted to only keep their interactional potential or use of information/data? Although equivalent, these formats are far from being identical. How do they differ? How do concepts (specifications) become syntax (usable/sharable entities)? Is a “meta”study of the “administrative” aspects of mathematical practice and formats possible?

What is nondeterminism and what is its role in the human/machine duality? A sort of nondeterminism seems to exist within classical logic especially in the classical sequent calculus. Although there is a kind of connexion between the two branches of the law of the excludedmiddle (prove ~A then focus on A by retraction), to prove a statement of the shape (A or ~A) amount to try to prove A, then ~A, then return to A. We cannot predict the outcome (if there is one). We can also remark that classical logic (in its pure form) lacks computational content but is the most natural way of reasoning for humans (even though it was rejected by intuitionism). Can it be related to the separation between humans and machines?

How do mathematical concepts emerge from thoughts? How do intuitive concepts such as sets, numbers and functions emerge? Numbers for instance, have several definitions (Church, Parigot, Peano, Von Neumann…). Is there any kind of “dialogue” involving our mind (subject/subject) and our environment (subject/object) making this emergence possible? Can we classify concepts emerging through these interactions? For instance, classify into categories such as “plurality”, “unity”, “existence”, “universality” etc (Kant’s categories). How is it related to our conception of space and time? (do numbers emerge from our conception of time and sets from our conception of space?).