A mistake in this critique is it assumes an exponential: a constant proportional rate of growth. It is true that, in some sense, an exponential always seems to

A mistake in this critique is it assumes an exponential: a constant proportional rate of growth. It is true that, in some sense, an exponential always seems to be accelerating while infinity always remains equally far away. But this is a bit of a straw man. Mathematical models of the technological singularity [1], along with the history of human economic growth [2], are super-exponential: the rate of growth is itself increasing over time, or at least has taken multiple discrete leaps [3] at the transitions to agriculture and industry, respectively. A true singularity/infinity can of course never be achieved for physical reasons (limited stuff within the cubically-expanding lightcone, plus inherent limits to technology itself), but the growth curve can look hyperbolic and traverse many orders of magnitude before those physical limits are encountered. [1] https://www.nber.org/system/files/working_papers/w23928/w23928.pdf https://www.nber.org/system/files/working_papers/w23928/w239... [2] https://docs.google.com/document/d/1wcEPEb2mnZ9mtGlkv8lEtScUw1k_dI0akbuu1ltb0gM/edit?tab=t.0 https://docs.google.com/document/d/1wcEPEb2mnZ9mtGlkv8lEtScU... [3] https://mason.gmu.edu/~rhanson/longgrow.pdf https://mason.gmu.edu/~rhanson/longgrow.pdf