Arthur Petron

I’ve been thinking a lot about comath recently as it’s clear the convex hull allows a category theorist to actually, finally, touch all of math. The convex hull, and the density of the theorems in the convex hull, and the densification by language models, is a REALLY good thing. LLMs will prove a million useless theorems on the way towards what you want. If you can navigate with taste, then the denser the convex hull becomes, the faster you can move through ideation Recently I’ve been creating a hct rewrite engine. It has been a process of 1) generalizing your structures to gain proper categorical semantics 2) elucidating the categorical semantics needed to abstractly present the data you work with, compactly 3) leaving certain choices of abstract representation open to choice, those choices living in some higher/structured category 4) having the llms choose the exact object of that category yielding a presentation guided towards the formalization of an HPC algorithm And then repeating this process over and over, to guide both your generalizations, and the choices you need to bake into the structures for the sake of high performance algorithms. Making your system both more general, while remaining in the realm of projecting down to HPC I have been calling this “carnot ideation” the carnot cycle is a cycle made of isothermal and adiabatice curves. Carnot is a curve in PV space. I have been seeing mathematics as low volume high pressure, and discovery as high volume low pressure. So I’m inclined to think of P= formalization and HPC requirements and V= generalization and discovering more expressive categorical semantics. The adiabatic is a curve along generalization and tighter formalization for the sake of HPC implimentation without loss of expressivity, generality or potential for HPC