A minor point- but exponential curves never actually asymptote or go vertical, they just keep getting closer to vertical forever. But with a big enough level of growth it might as well be vertical.

True. Making any exponential curve look like a hockey stick is mostly a matter of picking the scale. I could probably create a chart showing how the population of the earth went vertical during the reign of the Caesars. Of course, a logarithmic plot solves that issue.

Exponential functions do not have singularities. There would have to be some additional property of the curve in order to conclude the presence of a singularity.

I agree that "the Singularity" is not a singularity in the mathematical sense of divergence, but it is a singularity in the sense of a singular (i.e. special or unique) event. (If it happens at all, that is.)

How?

Look twenty years out and it appears vertical. But 10 years from now it will look much flatter, and what's then 20 years out (30 years from today) will look vertical.

The grass is always greener 20 years in the future.

As I understand it, "the Singularity" refers to the point in time where AGI becomes better at improving itself than the humans that created it. Then the existing exponential growth gets replaced by another exponential, but with stronger feedback.

It's the difference between a differential equation like x' = x and x' = 10x . In either case you can say "meh, if you normalize on today's value, the increase in 10 years is just a constant multiple", but the transition from one phase to the other will still be noticable, even though things won't "go vertical".

It seems much more likely to me that we're somewhere in the middle of an S-curve, before the limiting factors have kicked in. In terms of processing power, it's already happened to performance per thread, and it doesn't seem unreasonable to assume that there's a plateau to total throughput as well.

Even then, we're undoubtedly going to see some really amazing stuff come out of machine learning in the decades to come, just maybe not the beginning of Ragnarok.

An S-curve still goes "vertical" at some point. That's the idea: that some aspect of reality will be qualitatively weirder when information goes vertical-ish.

My main problem with the predictive power of exponential curves is externalities, usually some tradeoff or some limit to resources.

Try as I might, I probably couldn't put it better than it's done in this talk: http://idlewords.com/talks/web_design_first_100_years.htm