Let us define the “Clarke Horizon” as the point at which technology can be considered to operate in an essentially magical fashion. By way of example and explanation, I’ll use the sentence in which the term was coined: “It isn’t really proper to say that Dune takes place in a post-Singularity civilization, as the idea of the Singularity hadn’t been proposed when the novel was written. However, it’s fair to say that the setting is well past the Clarke Horizon.”

Not only is this potentially an interesting metaphor for the far future, but I think it gives us a potentially useful new way to conceptualize the Singularity itself.

Let be the time between the present, , and the Clarke Horizon. Clearly , though is not necessarily linear, or even monotone (though, of course, ). Now, consider a time . We can now say that the Singularity occurs if for some .

As a side note, our normal conception of “progress” probably boils down to the statement that , while my own suspicion is that , .