humans to the cosmos." He noted: "The waiting period between the mindsteps is getting shorter. One can't help noticing the acceleration." Hawkins' empirical 'mindstep equation' quantified this, and gave dates for (to him) future mindsteps. The date of the next mindstep (5; the series begins at 0) he cited as 2021, with two further, successively closer mindsteps in 2045 and 2051, until the limit of the series in 2053. His speculations ventured beyond the technological: Vinge's exponentially accelerating change The mathematician Vernor Vinge popularized his ideas about exponentially accelerating technological change in the science fiction novel Marooned in Realtime (1986), set in a world of rapidly accelerating progress leading to the emergence of more and more sophisticated technologies separated by shorter and shorter time intervals, until a point beyond human comprehension is reached. His subsequent Hugo award-winning novel A Fire Upon the Deep (1992) starts with an imaginative description of the evolution of a superintelligence passing through exponentially accelerating developmental stages ending in a transcendent, almost omnipotent power unfathomable by mere humans. His already mentioned influential 1993 paper on the technological singularity compactly summarizes the basic ideas. Kurzweil's The Law of Accelerating Returns In his 1999 book The Age of Spiritual Machines, Ray Kurzweil proposed "The Law of Accelerating Returns", according to which the rate of change in a wide variety of evolutionary systems (including but not limited to the growth of technologies) tends to increase exponentially. He gave further focus to this issue in a 2001 essay entitled "The Law of Accelerating Returns". In it, Kurzweil, after Moravec, argued for extending Moore's Law to describe exponential growth of diverse forms of technological progress. Whenever a technology approaches some kind of a barrier, according to Kurzweil, a new technology will be invented to allow us to cross that barrier. He cites numerous past examples of this to substantiate his assertions. He predicts that such paradigm shifts have and will continue to become increasingly common, leading to "technological change so rapid and profound it represents a rupture in the fabric of human history". He believes the Law of Accelerating Returns implies that a technological singularity will occur before the end of the 21st century, around 2045. The essay begins: The Law of Accelerating Returns has in many ways altered public perception of Moore's law. It is a common (but mistaken) belief that Moore's law makes predictions regarding all forms of technology,