One thing which confused me momentarily—I looked at your differential equations and mentally substituted I(t) with f(R(t)), to get something just in terms of R, for convenience. Then I was temporarily confused by your graphs in terms of I, because I was getting very different graphs (graphs in R) working things out in my head.
This pointed me at the question: should we be graphing in terms of I or R?
A large part of the analysis is to pinpoint where we get sublinear growth vs superlinear, and subexponential vs superexponential. This quantifies different meanings of explosive growth (ie the “explosion” in “intelligence explosion”). But perhaps we should be looking at the growth of R, instead of I.
It seems like, in this model, R represents capabilities—if you know a lot of concrete things, you can do a lot.I represents the pace at which capabilities increase. An explosion in capabilities could be alarming despite a rather modest graph of intelligence increase.
Put simply, you’re graphing the derivative of capabilities. What happens when we graph capabilities?
Considering the cases you look at:
f(x)=x: This is the one case where the two graphs are just the same anyway.R grows exponentially, just like I.
f(x)=log(x): capabilities see very nearly linear growth (since the derivative is very nearly constant).
f(x)=x1/3: capabilities grow like x3/2.
f(x)=√x: capabilities grow like x2.
f(x)=x2/3: capabilities grow like x3.
f(x)=xa: capabilities grow polynomially for a<1, exponentially at a=1, and hyperbolically at a>1.
This gives a very different picture: some sort of superlinear growth seems almost inevitable. We get an explosion unless returns are extremely diminishing. On the other hand, the crossover from subexponential to superexponential happens at exactly the same point.
Of course, “cababilities” is a rather ambiguous notion. What does it really entail? Perhaps the salient feature of the world ends up being the log of capabilities.
One thing which confused me momentarily—I looked at your differential equations and mentally substituted I(t) with f(R(t)), to get something just in terms of R, for convenience. Then I was temporarily confused by your graphs in terms of I, because I was getting very different graphs (graphs in R) working things out in my head.
This pointed me at the question: should we be graphing in terms of I or R?
A large part of the analysis is to pinpoint where we get sublinear growth vs superlinear, and subexponential vs superexponential. This quantifies different meanings of explosive growth (ie the “explosion” in “intelligence explosion”). But perhaps we should be looking at the growth of R, instead of I.
It seems like, in this model, R represents capabilities—if you know a lot of concrete things, you can do a lot.I represents the pace at which capabilities increase. An explosion in capabilities could be alarming despite a rather modest graph of intelligence increase.
Put simply, you’re graphing the derivative of capabilities. What happens when we graph capabilities?
Considering the cases you look at:
f(x)=x: This is the one case where the two graphs are just the same anyway.R grows exponentially, just like I.
f(x)=log(x): capabilities see very nearly linear growth (since the derivative is very nearly constant).
f(x)=x1/3: capabilities grow like x3/2.
f(x)=√x: capabilities grow like x2.
f(x)=x2/3: capabilities grow like x3.
f(x)=xa: capabilities grow polynomially for a<1, exponentially at a=1, and hyperbolically at a>1.
This gives a very different picture: some sort of superlinear growth seems almost inevitable. We get an explosion unless returns are extremely diminishing. On the other hand, the crossover from subexponential to superexponential happens at exactly the same point.
Of course, “cababilities” is a rather ambiguous notion. What does it really entail? Perhaps the salient feature of the world ends up being the log of capabilities.