We can say “a monotonic map, Φ∈mono(QP) is a phenomenon of P as observed by Q”, then, emergence is simply the impreservation of joins.
Given preorders (P,≤P) and (Q,≤Q), we say a map in mono(QP) “preserves” joins (which, recall, are least upper bounds) iff ∀ab∈P,Φa∨QΦb=Φ(a∨Pb) where by “x=y” we mean x≤y∧y≤x.
Suppose Φ is a measurement taken from a particle. We would like for our measurement system to be robust against emergence, which is literally operationalized by measuring one particle, measuring another, then doing some operation on the two results and getting the exact same thing as you would have gotten if you smashed the particles together somehow before taking the (now, single) measurement. But we don’t always get what we want.
Indeed, for arbitrary preorders and monotone arrows, you can prove Φa∨QΦb≤QΦ(a∨Pb), which we interpret as saying “smashing things together before measuring gives you more information than measuring two things then somehow combining them”.
In the sequences community, emergence is a post-it note that says “you’re confused or uncertain, come back here to finish working later” (Eliezer, 2008 or whatever). In the applied category theory community, emergence is also a failure of understanding but the antidote, namely reductions to composition, is prescribed.
This is all in chapter 1 of seven sketches on compositionality by fong and spivak, citing a thesis by someone called adam.
preorders as the barest vocabulary for emergence
We can say “a monotonic map, Φ∈mono(QP) is a phenomenon of P as observed by Q”, then, emergence is simply the impreservation of joins.
Given preorders (P,≤P) and (Q,≤Q), we say a map in mono(QP) “preserves” joins (which, recall, are least upper bounds) iff ∀ab∈P,Φa∨QΦb=Φ(a∨Pb) where by “x=y” we mean x≤y∧y≤x.
Suppose Φ is a measurement taken from a particle. We would like for our measurement system to be robust against emergence, which is literally operationalized by measuring one particle, measuring another, then doing some operation on the two results and getting the exact same thing as you would have gotten if you smashed the particles together somehow before taking the (now, single) measurement. But we don’t always get what we want.
Indeed, for arbitrary preorders and monotone arrows, you can prove Φa∨QΦb≤QΦ(a∨Pb), which we interpret as saying “smashing things together before measuring gives you more information than measuring two things then somehow combining them”.
In the sequences community, emergence is a post-it note that says “you’re confused or uncertain, come back here to finish working later” (Eliezer, 2008 or whatever). In the applied category theory community, emergence is also a failure of understanding but the antidote, namely reductions to composition, is prescribed.
This is all in chapter 1 of seven sketches on compositionality by fong and spivak, citing a thesis by someone called adam.