I actually had some similar alarm bells go off for conflation of concepts in the op, especially because the post specifically gestures at one concept and doesn’t give explanations of the different examples where this might come up.
However, on second thought I think I do like the concept this builds. To phrase it in your formal terms, I think it’s very useful to notice all the systems in which the Taylor series for f has b>0, ESPECIALLY when it’s comparably easy to control f via b∗x rather than just a.
In this light, you can view momentum, exponential growth, heavy-tails, etc., as all cases where a main component of controlling or predicting future x is by paying attention to the b∗x term, and I claim this is an important revelation to have at a variety of levels.
Perhaps more relevant to your actual crux, I also get shudders when people overload physics terms with other meanings, but before they were physics terms they were concepts for intuitive things. Given that we view the world through physical metaphors, I think it’s quite important for us to use the best-fitting words for concepts. Then we can remind people of the different variants when people run into conflationary trouble. If we start off by naming things with poor associations we hold ourselves back more. If you have alternative name to “momentum” for this that you also think have good connotations though, I’d love to hear them.
The second thing first: ”...but before they were physics terms they were concepts for intuitive things” is actually not true in this case: momentum did not mean anything, before being coined in physics. Than, it become used in a metaphorical way, but mostly congruently with the original physics concepts, as something like “mass”x”velocity”. It seems to me easy to imagine vivid pictures based of this metaphor, like advancing army conquering mile after mile of enemy territory having a momentum, or a scholar going through page after page of a difficult text. However, this concept is not tied to the b∗x term (which is one of my cruxes).
To me, the original metaphorical meaning of momentum makes a lot of sense: you have a lot of systems where you have something like mass (closely connected to inertia: you need great force to get something massive to move) and something like velocity—direction and speed where the system is heading. I would expect most people have this on some level.
Now, to the first thing second: I agree that it may be useful to notice all the systems in which the Taylor series for f has b>0, ESPECIALLY when it’s comparably easy to control f via b∗x rather than just a. However, some of the examples in the original post do not match this pattern: some could be just systems where, for example, you insert heavy-tailed distribution on the input, and you get heavy-tailed distribution on the output, or systems where the a term is what you should control, or systems where you should actually understand more about f(x) than the fact that is is has positive first derivative at some point.
What should be a good name for b∗x>0 I don’t know, some random prosaic ideas are snowballing, compound, faenus (from latin interest on money, gains, profit, advantage), compound interest. But likely there are is some more poetic name, similarly to Moloch.
I actually had some similar alarm bells go off for conflation of concepts in the op, especially because the post specifically gestures at one concept and doesn’t give explanations of the different examples where this might come up.
However, on second thought I think I do like the concept this builds. To phrase it in your formal terms, I think it’s very useful to notice all the systems in which the Taylor series for f has b>0, ESPECIALLY when it’s comparably easy to control f via b∗x rather than just a.
In this light, you can view momentum, exponential growth, heavy-tails, etc., as all cases where a main component of controlling or predicting future x is by paying attention to the b∗x term, and I claim this is an important revelation to have at a variety of levels.
Perhaps more relevant to your actual crux, I also get shudders when people overload physics terms with other meanings, but before they were physics terms they were concepts for intuitive things. Given that we view the world through physical metaphors, I think it’s quite important for us to use the best-fitting words for concepts. Then we can remind people of the different variants when people run into conflationary trouble. If we start off by naming things with poor associations we hold ourselves back more. If you have alternative name to “momentum” for this that you also think have good connotations though, I’d love to hear them.
The second thing first: ”...but before they were physics terms they were concepts for intuitive things” is actually not true in this case: momentum did not mean anything, before being coined in physics. Than, it become used in a metaphorical way, but mostly congruently with the original physics concepts, as something like “mass”x”velocity”. It seems to me easy to imagine vivid pictures based of this metaphor, like advancing army conquering mile after mile of enemy territory having a momentum, or a scholar going through page after page of a difficult text. However, this concept is not tied to the b∗x term (which is one of my cruxes).
To me, the original metaphorical meaning of momentum makes a lot of sense: you have a lot of systems where you have something like mass (closely connected to inertia: you need great force to get something massive to move) and something like velocity—direction and speed where the system is heading. I would expect most people have this on some level.
Now, to the first thing second: I agree that it may be useful to notice all the systems in which the Taylor series for f has b>0, ESPECIALLY when it’s comparably easy to control f via b∗x rather than just a. However, some of the examples in the original post do not match this pattern: some could be just systems where, for example, you insert heavy-tailed distribution on the input, and you get heavy-tailed distribution on the output, or systems where the a term is what you should control, or systems where you should actually understand more about f(x) than the fact that is is has positive first derivative at some point.
What should be a good name for b∗x>0 I don’t know, some random prosaic ideas are snowballing, compound, faenus (from latin interest on money, gains, profit, advantage), compound interest. But likely there are is some more poetic name, similarly to Moloch.