Re: transitivity -- What if "d(mp′,m′p)" were defined in an interesting (or
pathological) way such that we don't use ≤ but something else ... maybe a
partial order or something even weirder. Then instead of thinking of ε as some
sort of "error", we think of it as a complicated similarity map ... a model in
and of itself? Then maybe there would be some form of transitivity.
Re: the whole thing -- I'm a little worried about the practicalities in all the
symmetric opposites {R_re,Ȓ_c}, {Ȓ_re,R_c}, {Ȓ_c,R_c}, {R_re,Ȓ_re}, {C_c,H_c},
{C_re,H_cr}, {C_re,C_c}, {H_re,H_c}, etc. The reason I'm worried about them is
because they represent the many types of validation and verification beyond the
"data validation" represented by ε-equivalence. Such "behavioral analogies"
(comparing arrows) can be and are scored similarly to the "structural
analogies" considered when comparing the boxes.
I may have missed it in the paper. Where do they talk about the degree to which
the physical form of the abstract objects is arbitrary? I see where they say
there's no need for universality, just sufficiently powerful, accurate,
instantiable, etc. Don't we need such concepts in order to reason out *whether*
there exist a commuting structure for any given abstraction or physical thing?
I.e. just because we can find a commutation with a structural analogy doesn't
imply a behavioral analogy ... and vice versa. And *if* that's the case, then
what does this say about object-behavior (box-arrow) duality? ... if anything?
On 11/7/20 9:34 AM, jon zingale wrote:
> 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.
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