essential inequality vs functional inequivalence

i percieve two paradigms of computational/mathematical thinking, which ramificate into a bunch of fields. i believe the core of the distinction is captured by comparing two mathematical perspectives: essential inequality such as what distinguishes two vertices in a graph, and functional inequivalence such as what distinguishes two general mathematical functions by comparing their sets of outputs for every input.

functional inequivalence (hereby FI) seems to me like the default: if two objects cannot be distinguished by how they relate to other things, then they might as well be considered the same. for example, f(x) = 2 * x and g(x) = x + x, or in some sense vertices a and b in the graph V={a,b,c}, E={a→a,a→c,b→b,b→c}

essential inequality (hereby EI) is the perspective where elements get to have a special pointer or uniqueness essence that make them essentially different from all others, even when things are completely equivalent if you swap them around. for example, two vertices in any graph are usually thought of in that way; other examples include memory locations in a turing machine's tape, pairs in a LISP (with their eq? comparability), and others.

i've notably seen people use EI as an argument for worrying about consciousness with regards to teleportation, uploading, or other supposedly continuity-breaking events: instead of the vertices or memory locations being roughly the same ones, or at least being causally "nearby" the ones they were being computed on just before, they're suddenly being computed on a wholly different set of locations/vertices.

as for myself, i usually am on the FI side, especially when it comes to fundamental cosmos stuff. for example, i tend to deduplicate for the purpose of ethics. for anthropics, however, my prior allows for caring not just about first instance, but about number of instances of a given experience or moral patient, which could be in contradiction with FI. in addition, because most computational models seem to have some notion of essential location which could be a basis for EI, i find myself mildly updated towards EI — though i still mostly fall on the FI side, just less strongly so.

more importantly, regardless of whether the cosmos fundamentally "has" FI or EI, i don't believe that consciousness/qualia/soulstuff have their continuity particularly broken when they're moved to being computed elsewhere. i believe that, in order to coherently believe they're being particularly broken, you need to believe in some immaterial magical soul-type thing, and i have a good reason to not assume that, on top of occam disfavoring it. thus, i believe there are only functional processes, either being computed or not.

(FI vs EI might have also ramifications for SIA vs SSA, but it's not at the moment clear to me which ones if any)

one argument in favor of FI is that it could be implemented on top of EI, or that some "implementation details" of the cosmos (such as persistent data structures) de-reify FI by automatically deduplicating compute.

see also: psi rewriting, in which i offer an FI alternative to wolfram's hypergraph rewriting.

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