The singularity as a phase transition (extended abstract)

Béla Nagy, Santa Fe Institute
J.Doyne Farmer, Santa Fe Institute
John Paul Gonzales, Santa Fe Institute

The Finite Time Singularity Scenario

In his seminal article about “The Coming Technological Singularity” Vinge (1993) quotes how Ulam (1958) paraphrased John von Neumann as saying: “One conversation centered on the ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue.”

Of course, we can only guess exactly what kind of singularity the great Hungarian-born American mathematician had in mind, but one such guess is a finite time singularity, i.e. a mathematical singularity that can occur by trying to divide with zero. Graphically, this can be illustrated by a hyperbola approaching a vertical asymptote at the time of the singularity, such as 1/x approaching infinity as x goes to zero.

In this article, we are arguing that when we are talking about the potential outcomes of technological progress, such finite time singularity scenarios are possible, plausible, and should be taken seriously. This is counterintutive. Who in their right mind would seriously consider the possibility of such an explosive acceleration of technological progress that would actually reach infinity in finite time? Clearly, no sane person could or should be expected to entertain such a fantastic scenario. However, this entire analysis is exactly wrong.

The problem stems from a profound misunderstanding of a finite time singularity in this context. For our purposes, being on a hyperbolic trajectory does not mean that the trend will actually hit infinity in finite time. Instead, what it does mean is that the model is not extendable beyond the finite time singularity because there will be a regime change or phase transition where the dynamics will be fundamentally altered, switching into a totally different mode of operation. Note that this is consistent with the prediction wall interpretation of a technological singularity. (Also note the connection with sustainability: a hyperbolic trajectory cannot possibly cross a vertical asymptote to continue beyond the singularity. Hence, once we detect such a trend, we will know that it is not sustainable.)

For example, von Foerster, Mora and Amiot (1960) showed that a hyperbolic growth curve was a very good approximation for human population growth on planet Earth and calculated that the finite time singularity date would be Friday, 13 November, A.D. 2026. However, they did not claim that at that future point in time population will reach infinity. Instead, they predicted a doomsday: the end of humankind. Curiously, this would not be a result of war or famine but a consequence of such a high population density on the surface of the Earth that people would literally be crushed to death! Fortunately, now we can see that other constraints have already kicked in and the latest United Nations projections show that population dynamics is changing quite dramatically as the historical hyperbolic trend is breaking down and (mostly due to falling fertility rates) a fundamental regime change already looks inevitable, long before 2026.

Other excellent examples of finite time singularity dynamics can be found in a variety of fields, e.g. see Meyer and Vallee (1975); Kremer (1993); Johansen and Sornette (2001); Bettencourt, Lobo, Helbing, Kühnert and West (2007). However, this kind of modeling has made virtually no inroads into the minds of those who tend to think about a technological singularity. The recent review article by Sandberg (2010) provides a useful overview of how some of these ideas occurred so far in the technological singularity literature. Vinge (1993) does not follow von Neumann’s cryptic call to explore a mathematical singularity scenario. Curiously, in the appendix of Kurzweil (2005), the author derives mathematical formulas showing possible hyperbolic dynamics, but promptly dismisses such results, saying that “it is hard to imagine infinite knowledge, given apparently finite resources of matter and energy”. Eliezer Yudkowsky expressed similar incredulity that infinite progress is possible in finite time when asked about the finite time singularity scenario after his presentation (Yudkowsky 2007). Of course, as we pointed out earlier, these opinions reveal that both of these authors are hostages to the misguided interpretation. Hence, we can conclude that some of the most influential thinkers in this field do not take finite time singularity scenarios seriously. In our view, this has led to a serious bias for non-finite time singularity scenarios – a distortion that our article is intended to correct.

Our arguments are based on our analysis (Nagy, Farmer, Trancik and Gonzales 2010) of the information technology data in Koh and Magee (2006). Our main finding is that progress in information technology is faster than exponential. This contradicts exponential expectations based on Moore’s Law (Moore 1965, 1975) because if one looks over a sufficiently long span of time, all of the relevant performance metrics appear to improve superexponentially. We analyze six different historical trends of progress for several technologies grouped into the following three functional tasks: information storage, information transportation (bandwidth), and information transformation (speed of computation). Five of the six datasets extend back to the nineteenth century. We perform statistical analyses and show that in all six cases one can reject the exponential hypothesis at statistically significant levels. In contrast, one cannot reject the hypothesis of superexponential growth with decreasing doubling times. Furthermore, one of the simplest functional forms one can fit to such data is the hyperbola leading to a finite time singularity (which is also a better fit to the historical data than Kurzweil’s double exponential more often than not).

The possibility of hyperbolic dynamics is intriguing and raises questions about the sustainability of these trends. Obviously it is physically impossible to reach a singularity, indicating that before that hyperbolic growth must necessarily break down. There may be fundamental physical limits (Levitin and Toffoli 2009) to cause the historical trends to be violated. Other limits might be reached even sooner. For example, according to Jurvetson (2004), “another problem is the escalating cost of a semiconductor fab plant, which is doubling every three years, a phenomenon dubbed Moore’s Second Law”. In conclusion, we are arguing that finite time singularity scenarios are possible, plausible, and should be taken seriously when debating potential forms of a technological singularity.

References

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