I just listened to XYZ talking about "How Universal Is the Idea of Numbers?", and bashing the concept as an accidental historical artifact. He suggested that totally different computational primitives could exist, that are sometimes more useful than numbers.
Modern CPUs provide efficient "circuit-parallel" primitives for addition, subtraction, multiplication, division, square-root, and inverse-square-root. I believe those are all contained in TC0. The primitives provided by modern CPUs also include address logic (multiplexer-circuits), which I believe are contained in AC0. The arithmetic primitives are actually also useful for address logic related tasks (and related data structures), at least to a certain extent.
Assuming that those totally different computational primitives conjectured by XYZ really exist, how plausible is it that they are actually contained in TC0 (or NC1), such that it would have been possible for modern CPUs to provide them as "useful primitives"? (Maybe the multiplexer-circuits give some hint that there really could be such primitives, if we allow that the number of input bits can be totally different from the number of output bits.)