My heart longs to work more directly to help animals. I’m doing things that are meta-meta-meta-meta removed from actions that feel like they’re actually helping anyone, all the while the globe is on moral fire. At times like this, I need to remind myself of Ahmdal’s Law and Inventor’s Paradox. And to reaffirm that I need to try to have faith in my own judgment of what is likely to produce good consequences, because no one else will be able to know all the relevant details about me, and also because otherwise I can’t hope to learn.[1]
Ahmdal’s Law tells you the obvious thing that for a complex process, the maximum percentage speedup you can achieve by optimising one of its sub-processes is hard limited by the fraction of time that sub-process gets used. It’s loosely analogous to the idea that if you work on a specific direct cause, the maximum impact you can have is limited by the scale of that cause, whereas if you work on a cause that feeds into the direct cause, you have a larger theoretical limit (although that far from guarantees larger impact in practice). I don’t know a better term for this, so I’m using “Ahmdal’s Law”.
The Inventor’s Paradox is the curious observation that when you’re trying to solve a problem, it’s often easier to try to solve a more general problem that includes the original as a consequence.[2]
Consider the problem of adding up all the numbers from 1 to 99. You could attack this by going through 99 steps of addition like so: 1+2+3+...97+98+99.
Or you could take a step back and look for more general problem-solving techniques. Ask yourself, how do you solve all 1-iterative addition problems? You could rearrange it as:
(1+99)+(2+98)+...+(48+52)+(49+51)+50=100×49+50
For a sequence of N numbers, you can add the first number (n0=1) to the last number (nN=99) and multiply the result by how many times you can pair them up like that (n0+nn)+(n1+nn−1)+(n2+nn−2)+... plus if there’s a single number left (50) you just add that at the end. As you’d expect, this general solution can solve the original problem, but also so much more, and it’s easier to find (for some people) and compute than having to do the 99 steps of addition manually.
“The more ambitious plan may have more chances of success […] provided it is not based on a mere pretension but on some vision of the things beyond those immediately present.” ‒ Pólya
Self-confidence feels icky and narcissistic when I just want to be kind and gentle. But I know how important it is to cultivate self-confidence at almost any cost. And I don’t want to selfishly do otherwise just because it’s a better look. Woe is me, rite.
The reasons why and when Inventor’s Paradox works is something that deserves deep investigation, in case you could benefit from the prompt. Lmk if you find anything interesting.
My heart longs to work more directly to help animals. I’m doing things that are meta-meta-meta-meta removed from actions that feel like they’re actually helping anyone, all the while the globe is on moral fire. At times like this, I need to remind myself of Ahmdal’s Law and Inventor’s Paradox. And to reaffirm that I need to try to have faith in my own judgment of what is likely to produce good consequences, because no one else will be able to know all the relevant details about me, and also because otherwise I can’t hope to learn.[1]
Ahmdal’s Law tells you the obvious thing that for a complex process, the maximum percentage speedup you can achieve by optimising one of its sub-processes is hard limited by the fraction of time that sub-process gets used. It’s loosely analogous to the idea that if you work on a specific direct cause, the maximum impact you can have is limited by the scale of that cause, whereas if you work on a cause that feeds into the direct cause, you have a larger theoretical limit (although that far from guarantees larger impact in practice). I don’t know a better term for this, so I’m using “Ahmdal’s Law”.
The Inventor’s Paradox is the curious observation that when you’re trying to solve a problem, it’s often easier to try to solve a more general problem that includes the original as a consequence.[2]
Consider the problem of adding up all the numbers from 1 to 99. You could attack this by going through 99 steps of addition like so: 1+2+3+...97+98+99.
Or you could take a step back and look for more general problem-solving techniques. Ask yourself, how do you solve all 1-iterative addition problems? You could rearrange it as:
(1+99)+(2+98)+...+(48+52)+(49+51)+50=100×49+50
For a sequence of N numbers, you can add the first number (n0=1) to the last number (nN=99) and multiply the result by how many times you can pair them up like that (n0+nn)+(n1+nn−1)+(n2+nn−2)+... plus if there’s a single number left (50) you just add that at the end. As you’d expect, this general solution can solve the original problem, but also so much more, and it’s easier to find (for some people) and compute than having to do the 99 steps of addition manually.
Self-confidence feels icky and narcissistic when I just want to be kind and gentle. But I know how important it is to cultivate self-confidence at almost any cost. And I don’t want to selfishly do otherwise just because it’s a better look. Woe is me, rite.
The reasons why and when Inventor’s Paradox works is something that deserves deep investigation, in case you could benefit from the prompt. Lmk if you find anything interesting.