In The Singularity is Near, Ray Kurzweil looked at the natural history of information processing and discovered that the rate of increase in the ability to manipulate information has a logarithmic slope; in other words, the rate of information processing ability increases at an exponential rate. Furthermore, he suggested that within the next 30-40 years we will reach a time at which the pace of technological advance (the current embodiment of information processing) will become so rapid as to make further predictions impossible, a point which he called "The Singularity".
Kevin Kelly, as part of an article critiquing certain aspects of Kurzweil's idea, described the Singularity in fairly accessible terms. (Vinge refers to Verner Vinge, who originated the idea of an Artificial Intelligence Singularity):
In Vinge’s analysis, at some point not too far away, innovations in computer power would enable us to design computers more intelligent than we are, and these smarter computers could design computers yet smarter than themselves, and so on, the loop of computers-making-newer-computers accelerating very quickly towards unimaginable levels of intelligence. This progress in IQ and power, when graphed, generates a rising curve which appears to approach the straight up limit of infinity. In mathematical terms it resembles the singularity of a black hole, because, as Vinge announced, it will be impossible to know anything beyond this threshold. If we make an AI which in turn makes a greater AI, ad infinitum, then their future is unknowable to us, just as our lives have been unfathomable to a slug. So the singularity became a black hole, an impenetrable veil hiding our future from us.
In his response to Kelly's article, Kurzweil wrote a clarifying post and used the Internet to illustrate the concept of a "transformative technology":
Consider the Internet. When the Arpanet went from 10,000 nodes to 20,000 in one year, and then to 40,000 and then 80,000, it was of interest only to a few thousand scientists. When ten years later it went from 10 million nodes to 20 million, and then 40 million and 80 million, the appearance of this curve looks identical (especially when viewed on a log plot), but the consequences were profoundly more transformative. There is a point in the smooth exponential growth of these different aspects of information technology when they transform the world as we know it. [Emphasis mine-SW]
One of Kurzweil's great insights was that almost any information technology, when looked at over time, shows the same pattern of exponential growth, with minimal change until a threshold is hit, followed by explosive transformations.
What does this have to do with Apocalyptic cults and irrational reactions to disruption fueled angst?