I would be interested in collecting a bunch of examples of mathematical modeling of progress. I think there are probably several of these here, but I don't expect to be able to find all of them myself. I'm also interested to know about any models like this elsewhere.
I was reading the LessWrong 2018 books, and the following posts stuck out to me:
The Year The Singularity Was Cancelled talks about a model which predicted world population quite well, by supplementing a basic population equation with a simple mathematical model of technological progress. To summarize: population carrying capacity is assumed to increase due to technological progress. Technological progress is modeled as proportional to the population: a particular population p leads carrying capacity c to have a derivative pc. (This reflects the idea that carrying capacity multiplies the carrying capacity; if it added to the carrying capacity, we might make the derivative equal p instead.) Population should typically remain close to the carrying capacity; so, we could simply assume that population equals carrying capacity. We then expect hyperbolic growth, IE something like c=1s−t; here, s is the year of the (population) singularity. This model is a decent fit to the data until the year 1960, which is of course the subject of the post.
One of my thoughts after reading this was: wouldn't it make more sense to avoid the assumption that population equals carrying capacity? Population growth can't be greater than exponential. The hyperbolic model doesn't make any sense, and the assumption that population equals carrying capacity appears to be the culprit.
It would make more sense to, instead, use more typical population models (which predict near-exponential growth when population is significantly below carrying capacity, tapering off near carrying capacity). I don't yet know if this has been done in the literature. However, it's commonly said that around the time of the industrial revolution, humankind escaped the Malthusian trap, because progress outpaced birthrates. (I know the demographic transition is a big player here, but let's ignore it for a moment.) If we were modeling this possibility, it makes sense that progress would stop accelerating so much around this point: once progress is increasing the carrying capacity faster than the population can catch up, we no longer expect to see population match carrying capacity.
This would imply that population transitions from hyperbolic growth to exponential growth, some time shortly before the singularity of the hyperbola. Which approximately matches what we observe: a year where the singularity was "cancelled".
However, in the context of AI progress in particular, this model seems naive. Human birthrates cannot keep pace with the resources progress provides. However, AI has no such limitation. Therefore, we might expect progress to look hyperbolic again at some point, when AI starts contributing significantly to progress. (Indeed, one might have expected this from computers alone, without saying the words "AI" -- computers allow "thinking power" to increase, without the population actually increasing.)
Some of the toy mathematical models Paul Christiano discusses in Takeoff Speeds might be used to add AI to the projection.
So, I'm interested in:
Related Question: Any response to Paul Christiano on takeoff speeds?