things are almost never greater than the sum of their parts Because Reductionism
Isn’t it more like, the value of the sum of the things is greater than the sum of the value of each of the things? That is, f(a+b)>f(a)+f(b) (where perhaps f is a utility function). That seems totally normal and not-at-all at odds with Reductionism.
I think people usually want that sentence to mean something confused. I agree it has fine interpretations, but people by default use it as a semantic stopsign to stop looking for ways the individual parts mechanistically interface with each other to produce the higher utility thing than the individual parts naively summed would (see also https://www.lesswrong.com/posts/8QzZKw9WHRxjR4948/the-futility-of-emergence )
More specifically, for a Jeffrey utility function U defined over a Boolean algebra of propositions, and some propositions a,b, “the sum is greater than its parts” would be expressed as the condition U(a∧b)>U(a)+U(B) (which is, of course, not a theorem). The respective general theorem only states that U(a∧b)=U(a)+U(b∣a), which follows from the definition of conditional utility U(b∣a)=U(a∧b)−U(a).
Oliver specifically wanted me to include the word “naive” because obviously there are sensible things people could mean by this but they phrase things overly strongly and the Lightcone Team’s Autism is Powerful.
Isn’t it more like, the value of the sum of the things is greater than the sum of the value of each of the things? That is, f(a+b)>f(a)+f(b) (where perhaps f is a utility function). That seems totally normal and not-at-all at odds with Reductionism.
I think people usually want that sentence to mean something confused. I agree it has fine interpretations, but people by default use it as a semantic stopsign to stop looking for ways the individual parts mechanistically interface with each other to produce the higher utility thing than the individual parts naively summed would (see also https://www.lesswrong.com/posts/8QzZKw9WHRxjR4948/the-futility-of-emergence )
More specifically, for a Jeffrey utility function U defined over a Boolean algebra of propositions, and some propositions a,b, “the sum is greater than its parts” would be expressed as the condition U(a∧b)>U(a)+U(B) (which is, of course, not a theorem). The respective general theorem only states that U(a∧b)=U(a)+U(b∣a), which follows from the definition of conditional utility U(b∣a)=U(a∧b)−U(a).
Oliver specifically wanted me to include the word “naive” because obviously there are sensible things people could mean by this but they phrase things overly strongly and the Lightcone Team’s Autism is Powerful.
Yes I think your equation looks right.