I’m pretty sure I know the meaning (or at least, so to speak, the structure of the meaning) of the phrase in TWC. Imagine first of all that you have some process that generates energy (from some source, such as the fusion of hydrogen into helium) and that’s just about self-sustaining. Now, suppose you can arrange for some small fraction of the energy that was being generated and radiated out into the world (in, as is the habit of light, straight lines) to be fed back into the system (so it goes around “in a circle” instead). Then the energy content of that system will build up: slowly at first, but at an increasing rate. Eventually it’ll go boom. Initially-small change, positive feedback, large eventual result.
Does this have any connection with the intro to HPMOR? For what it’s worth, I’m guessing not. (Except that maybe that phrase was drifting around in Eliezer’s unconscious and influenced his exact choice of wording.) It’s by no means impossible, but without the extra context in TWC I don’t think there’s any way for “a fraction of a line” to be interpreted to mean anything like “a fraction of something that would have been radiated outward, instead captured to produce positive feedback”, so it would be (so to speak) quite unfair. I expect that whatever the “fraction of a line” before HPMOR chapter 1 is, it can at least be described briefly to people without a mathematics/science/engineering background in such a way that “a fraction of a line” makes sense.
I’m pretty sure I know the meaning (or at least, so to speak, the structure of the meaning) of the phrase in TWC. Imagine first of all that you have some process that generates energy (from some source, such as the fusion of hydrogen into helium) and that’s just about self-sustaining. Now, suppose you can arrange for some small fraction of the energy that was being generated and radiated out into the world (in, as is the habit of light, straight lines) to be fed back into the system (so it goes around “in a circle” instead). Then the energy content of that system will build up: slowly at first, but at an increasing rate. Eventually it’ll go boom. Initially-small change, positive feedback, large eventual result.
Does this have any connection with the intro to HPMOR? For what it’s worth, I’m guessing not. (Except that maybe that phrase was drifting around in Eliezer’s unconscious and influenced his exact choice of wording.) It’s by no means impossible, but without the extra context in TWC I don’t think there’s any way for “a fraction of a line” to be interpreted to mean anything like “a fraction of something that would have been radiated outward, instead captured to produce positive feedback”, so it would be (so to speak) quite unfair. I expect that whatever the “fraction of a line” before HPMOR chapter 1 is, it can at least be described briefly to people without a mathematics/science/engineering background in such a way that “a fraction of a line” makes sense.