For reasons that aren't even clear to me, I'm about 1/3rd into Barry Jay's Reflective Programs in Tree Calculus and about 1/10th into Frank Waaldijk's Natural Topology, which I've been reading side-by-side. Smashing these two into each other is pretty fascinating, even as I'm not sure what to make of the mixture yet.

Having sat with the notion for about six months now, I think Jay's critique of the Church-Turing thesis has legs. I don't see clearly yet exactly where and how the limits of computation manifest in his own system(s), which of course they must. But I think he's correct that this thesis as it's colloquially presented (and taught to students, including me!) is misleading at best and false in a certain important sense. Apparently he is regularly called a crackpot for forwarding this critique even though it's straightforwardly demonstrated.

Waaldijk's book is more of a constructive mathematics exploration. In this it is closely related to computer science, but it's focused on traditionally mathematical notions like topological space. The latter is usually quite complicated, but Waaldijk shows that the core concept of compact space can be represented with finitely-branching trees, making these spaces amenable to computation. Since we imagine physics taking place in spaces that are topological (among other things) there's potentially an interesting bidirectional flow of ideas between computer science and physics.

Jay calls his central notion "natural trees". Waaldijk calls his central notion "natural spaces". In both cases I think the intended sense is "with minimal artifice".

#ComputerScience #computation #math #mathematics #topology

all those top. spaces, while i'm looking for a bottom. space
#topology #math