If I’m calculating the trajectory of the sun, and two other stars that are ten light-years away happen to be identical with each other, then this shouldn’t make my calculations any less valid.
Two issues here.
First, in this case, we’re asking about one specific query: the trajectory of the sun. In general, abstraction is about validity of some class of queries—typically most-or-all queries which can be formulated within the high-level model. So validity of the star-abstraction would depend not just on the validity of the calculation of the sun’s trajectory, but also the validity of the other stars’ trajectories, and possibly other things depending on what else is in the model.
(Of course, we can choose just one query as our class, but then the abstraction won’t be very useful; we might as well just directly compute the query’s answer instead.)
For a star-abstraction, we expect the abstraction to be valid for any queries on kinematics of the stars, as long as the stars are far apart. Queries we don’t expect to be valid include e.g. kinematics in situations where stars get close together and are torn apart by tidal forces.
Second, it’s generally ok if the low-level structures happen to be identical, just as two independent die rolls will sometimes come out the same. The abstraction breaks down when the low-level structures are systematically correlated, given whatever information we have about them. What does “break down” mean here? Well, I have some low-level data about one star, and I want to calculate what that tells me about another star. Using the abstraction, I can do that in three steps:
Use my low-level data on star 1 to compute what I can about the high-level variables of star 1 (low-level 1 → high-level 1)
Use the high-level model to compute predictions about high-level variables of star 2 from those of star 1 (high-level 1 → high-level 2)
Update my low-level information on star 2 to account for the new high-level information (high-level 2 → low-level 2)
… and that should be equivalent to directly computing what the low-level of star 1 tells me about the low-level of star 2 (low-level 1 → low-level 2). (It’s a commutative diagram.) If the low-level structures are not independent given the high-level summaries, then this fails.
Concretely: if I’m an astronomer studying solar flares and I want to account for the influence of other stars on the solar flare trajectory, then my predictions should not depend on the flare activity of the other stars. If my predictions did depend on the flare activity of the other stars, then that would be a breakdown of the usual star-abstraction. If the flare activity of two stars just-so-happens to match, that would be unusual but not a breakdown of the abstraction.
Please let me know if that didn’t fully address the question; I expect other people are wondering about similar points.
Ah, looking at earlier posts in your sequence, I realize that you are defining abstraction in such a way that the properties of the high-level abstraction imply information about the low-level details - something like “a class XYZ star (high-level classification) has an average mass of 3.5 suns (low-level detail)”.
That explains my confusion since I forgot that you were using this definition, and was thinking of “abstraction” as it was defined in my computer science classes, where an abstraction was something that explicitly discarded all the information about the low-level details.
Two issues here.
First, in this case, we’re asking about one specific query: the trajectory of the sun. In general, abstraction is about validity of some class of queries—typically most-or-all queries which can be formulated within the high-level model. So validity of the star-abstraction would depend not just on the validity of the calculation of the sun’s trajectory, but also the validity of the other stars’ trajectories, and possibly other things depending on what else is in the model.
(Of course, we can choose just one query as our class, but then the abstraction won’t be very useful; we might as well just directly compute the query’s answer instead.)
For a star-abstraction, we expect the abstraction to be valid for any queries on kinematics of the stars, as long as the stars are far apart. Queries we don’t expect to be valid include e.g. kinematics in situations where stars get close together and are torn apart by tidal forces.
Second, it’s generally ok if the low-level structures happen to be identical, just as two independent die rolls will sometimes come out the same. The abstraction breaks down when the low-level structures are systematically correlated, given whatever information we have about them. What does “break down” mean here? Well, I have some low-level data about one star, and I want to calculate what that tells me about another star. Using the abstraction, I can do that in three steps:
Use my low-level data on star 1 to compute what I can about the high-level variables of star 1 (low-level 1 → high-level 1)
Use the high-level model to compute predictions about high-level variables of star 2 from those of star 1 (high-level 1 → high-level 2)
Update my low-level information on star 2 to account for the new high-level information (high-level 2 → low-level 2)
… and that should be equivalent to directly computing what the low-level of star 1 tells me about the low-level of star 2 (low-level 1 → low-level 2). (It’s a commutative diagram.) If the low-level structures are not independent given the high-level summaries, then this fails.
Concretely: if I’m an astronomer studying solar flares and I want to account for the influence of other stars on the solar flare trajectory, then my predictions should not depend on the flare activity of the other stars. If my predictions did depend on the flare activity of the other stars, then that would be a breakdown of the usual star-abstraction. If the flare activity of two stars just-so-happens to match, that would be unusual but not a breakdown of the abstraction.
Please let me know if that didn’t fully address the question; I expect other people are wondering about similar points.
Ah, looking at earlier posts in your sequence, I realize that you are defining abstraction in such a way that the properties of the high-level abstraction imply information about the low-level details - something like “a class XYZ star (high-level classification) has an average mass of 3.5 suns (low-level detail)”.
That explains my confusion since I forgot that you were using this definition, and was thinking of “abstraction” as it was defined in my computer science classes, where an abstraction was something that explicitly discarded all the information about the low-level details.