Yes, I was actually thinking of the intelligent explosion type singularity as being the one that resided the most in the map. And the point about the difference between a mathematical singularity and a non-mathematical singularity is a very good one (and the analogy about an event horizon is also interesting. Although even then, there's a strong territory aspect there because once one is inside the event horizon one cannot send a signal out by any means.)
And some are more figurative than others; as I'm sure has been pointed out, Kurzweil's singularity is pretty hard to see in any kind of superexponential growth curve, but perhaps a vertical asymptote can be charitably interpreted as hyperbolic.
Actually, if one starts with a differential equation that has a large rate of growth with respect to the function itself it isn't very hard to force a singularity. But this is a nitpick, such functions don't exist in real life generally, and on the rare occasions when a model has one it is generally an indication that there's a problem with the model, not that there's anything like that in reality.
There's nothing special going on locally at the event horizon. And on the other hand, seed AI FOOM is just about the most singular kind of historical Singularity I can think of. The entire Hubble volume could undergo some kind of phase transition at the speed of light. (Maybe even faster.)
Not actually literally singular, mind you. Because it's 'real', not 'abstract'. But not even literally a 'co-ordinate singularity', figuratively speaking. (Unless Penrose is right about noncomputable physical action being involved in mental activity. [That is a jok...
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