[FRIAM] A/R theory

[jon zingale]

This work does seem to be relevant, up to 𝜀-equivalence, to many of the
fibers in recent threads :) As the authors point out, the question of
deciding which diagrams 𝜀-commute is the business of experimental science à
la EricC's commentary on the history of chemistry. Also, the ideas expressed
in this paper appear to point in a similar direction to the
(model-theoretic) ideas I was attempting to land in the *downward-causation*
discussion from last week. Lastly, the thesis is related to questions of how
extensional (or purely-functional) computation arises from the intentional
(maximally-stateful) variations of a substrate. So, thanks.

𝜀-equivalence itself is interesting because it comes with a *competence
constraint* that prevents it from being a transitive relation, that in
general a =𝜀 b ^ b =𝜀 c ⊬ a =𝜀 c is crucial to the theory. In other
words, while there may be a wide range of arm shapes that can be used as
bludgeons, one can evolve themselves out of the sweet spot. Dually, the
𝜀-equivalence condition provides a route to modeling *exaptation*, via
modal possibility. As p's belonging to the Physical domain vary, images in
the abstract theory vary into or out of 𝜀-equivalence with values belonging
to other problem domains. In particular, if we imagine that the R-map in the
paper is *actually* a structural functor as it seems to imply, we can
imagine another functor R' which specifies yet another problem space.
Natural transformations then, up to 𝜀-equivalence, provide a model of
exaptation. Because of the experimental nature of 𝜀-equivalence, I suspect
we would slowly discover an underlying Heyting algebra which would extend to
a topos via studying relations on sieves of 𝜀-equivalent structures. This
approach would formalize *how far from competent* a structure is wrt
*proving* a particular computation.



--
Sent from: http://friam.471366.n2.nabble.com/