Most uses of the word “insight” mean something similar to “seeing into the nature of things,” but it’s not clear that the particular use you have here meshes well with at least one other common use of the word. Eliezer captured it well:
an “insight” is a chunk of knowledge which, if you possess it, decreases the cost of solving a whole range of governed problems.
As a simple example, let’s say you were trying to prove the statement “there are infinitely many primes.” To progress on this problem at all, you’ll probably need to realize:
Insight 1 - The statement “there are infinitely many primes” can be re-expressed as “it is not the case that there are finitely many primes.”
Insight 2 - A statement of the form “not P” can sometimes be proven by assuming “P” and showing that this assumption leads to contradiction.
After assuming there are finitely many primes (i.e. there exists an n such that P = {p1, p2, …, pn} is the set of all primes), insight again comes into play when one realizes:
Insight 3 - Every integer > 1 can be expressed as a product of primes, so we can find a prime not in P (i.e. a contradiction) by finding an integer that is not divisible by any prime in P.
In this latter case, the insight consisted in using the fundamental theorem of arithmetic to transform the previous goal of “deriving a contradiction” to a more specific goal of “finding an integer that is not divisible by any prime in P.”
I realize that the context of problem solving is somewhat removed from the context of assessing the probability of hypotheses, but perhaps we should clarify what particular usage of the word “insight” is meant if we’re going to be analyzing it in detail.
Most uses of the word “insight” mean something similar to “seeing into the nature of things,” but it’s not clear that the particular use you have here meshes well with at least one other common use of the word. Eliezer captured it well:
As a simple example, let’s say you were trying to prove the statement “there are infinitely many primes.” To progress on this problem at all, you’ll probably need to realize:
Insight 1 - The statement “there are infinitely many primes” can be re-expressed as “it is not the case that there are finitely many primes.”
Insight 2 - A statement of the form “not P” can sometimes be proven by assuming “P” and showing that this assumption leads to contradiction.
After assuming there are finitely many primes (i.e. there exists an n such that P = {p1, p2, …, pn} is the set of all primes), insight again comes into play when one realizes:
Insight 3 - Every integer > 1 can be expressed as a product of primes, so we can find a prime not in P (i.e. a contradiction) by finding an integer that is not divisible by any prime in P.
In this latter case, the insight consisted in using the fundamental theorem of arithmetic to transform the previous goal of “deriving a contradiction” to a more specific goal of “finding an integer that is not divisible by any prime in P.”
I realize that the context of problem solving is somewhat removed from the context of assessing the probability of hypotheses, but perhaps we should clarify what particular usage of the word “insight” is meant if we’re going to be analyzing it in detail.