I first saw mention of The Singularity on the Facebook page of Sophrosyne Stenvaag, one of Second Life’s foremost Extropians. I thus tend to associate Singularitarianism with Extropianism, whereas they are two separate (albeit related) doctrines.

Extropianism would seem to be a form of Transhumanism, which foresees radical use of technology to improve human mental and physical capacities, including the possibility of immortality. But what is meant by “the Singularity”?

Mathematical Singularity

The term “singularity” expresses several different meanings in mathematics and science. The meaning referred to by the Singularitarians is that of a mathematical singularity as seen in hyperbolic growth.

A mathematical singularity is a point at which the value of a given mathematical function is undefined, such as for example the value of y = 1/x when x = 0.

A quantity growing towards a singularity is said to undergo Hyperbolic growth. For example, the function y = 1/x undergoes hyperbolic growth as it approaches the singularity at x=0. When x approaches zero, the value of y soars towards infinity, approaching a vertical asymptote.

The function y = 1/x is a rather poor example of hyperbolic growth, since a positive x must decrease in order for y to approache the asymptote. A better example of hyperbolic growth would be y = 1/(1-x) , because in this case y goes to infinity as x increases towards the value of 1.

Hyperbolic Growth versus Exponential Growth

Some theories of the Singularity refer to exponential growth, rather than asymptotic growth. There is a fundamental difference between these two. In exponential growth y grows faster and faster as x increases, but y becomes infinite only when x does. There is therefore no singularity. Certain Singularitarians thus specify that The Singularity will be a point of fastest growth, rather than a true vertical asymptote.

Growth is called exponential (or geometric) when the growth rate depends upon the current value of the function. Classic examples of exponential growth are population increase, where the growth rate is proportional to the size of the existing population, and interest accumulation, where the amount of newly added interest is proportional to the total current value of the loan together with the accumulated past interest.

Note that polynomial functions, such as y = x^2 , are not exponential functions, even though they contain exponents. Exponential growth will always outstrip polynomial growth in the long run.

One commonly-used formula for exponential growth is the following:

y = y(0) * e^kt

In this formula the new value of y is obtained by multiplying the initial value y(0) by the number “e” raised to an exponent which increases regularly with elapsed time. (Here ”t” for “time” has replaced “x” as the variable along the horizontal axis.) It is this steady increase of the exponent that causes exponential growth to outstrip any kind of polynomial growth. To see this, imagine a case in which y, instead of being always equal to x^2 , is at one moment equal to x^2, at the next equal to x^3, at the next equal to x^4, and so on. (The exponent is in fact attached to the “e”, rather than to the x, but the result is the same.) It is this progressive increase of the exponent that gives exponential growth its overwhelming force.

I.J. Good and Artificial Superintelligence

A singularity is thus a moment in time when a growth curve becomes so steep as to approach being vertical.

This idea was first applied to technical evolution by the mathematician I.J. Good, writing about artificial intelligence in 1965. Good wrote:

Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an ‘intelligence explosion,’ and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make, provided that the machine is docile enough to tell us how to keep it under control.

Good used the word “explosion” to denote the moment when the rate of technical change will spike upwards, due to the creation of the first ultraintelligent machine. Good’s vision assumes that true “artificial intelligence” is in fact possible. Today, more than 50 years later, it is still easy to argue that computers are in fact very dumb, because they can only follow the instructions that we humans program into them. Machine intelligence can extend human intelligence, particularly by collecting and processing vast amounts of information, but it seems unlikely that machines will ever surpass humans in real intelligence.

Good’s idea of an artificial superintelligence was taken up by writers of science fiction. William Gibson’s cyberpunk novel Neuromancer is about the emergence of a superintelligence within a planetary computer network or “cyberspace” (a term which Gibson invented). This superintelligence finds a way to overcome the barriers that humans deliberately set up to prevent it from emerging.

The setting of Gibson’s “cyberpunk” novel is a sordid, crime-ridden society (like our own when seen from below), which the newly emerged superintelligence leaves completely unchanged. The superintelligence seeks only its own well-being, rather than that of humans. These latter are to be counted lucky if the superintelligence is satisfied with just leaving them alone!

Vernon Vinge and Intelligence Amplification

In 1993 the mathematician Vernor Vinge updated Good’s ideas in a short paper, which can be found online here. At the beginning of this paper, Vinge concisely states his thesis as follows:

The acceleration of technological progress has been the central feature of this century. I argue in this paper that we are on the edge of change comparable to the rise of human life on Earth. The precise cause of this change is the imminent creation by technology of entities with greater than human intelligence.

Vernor Vinge was the first to call such a change event “the Singularity”. He wrote:

I think it’s fair to call this event a singularity (”the Singularity” for the purposes of this paper). It is a point where our old models must be discarded and a new reality rules.

It is widely suggested (see for example Wikipedia) that Vinge was making an analogy with the idea of a gravitational singularity, which is a location in space where the quantities which are normally used to measure the gravitational field become infinite (such as within a black hole). However, Vinge makes no reference to gravitation in his 1993 paper, and instead refers to John von Neumann’s use of the term “singularity” in the 1950s.

Vinge builds on Good’s prediction that computers will become “superhumanly intelligent.” More interesting is that Vinge adds a number of other possible ways to create “entities with greater than human intelligence.” He lists these as follows:

• Large computer networks (and their associated users) may “wake up” as a superhumanly intelligent entity.
• Computer/human interfaces may become so intimate that users may reasonably be considered superhumanly intelligent.
• Biological science may provide means to improve natural human intellect.

Vinge calls these other possible paths to the Singularity “Intelligence Amplification” (IA).

While the development of computers that have true intelligence remains unlikely, the amplification of human intelligence by use of computers is already a part of our daily reality. The first two of the above bullet-points concern the emergence of superintelligent humans through the use of computers, new computer interfaces, and the Internet. The third bullet point is more controversial, and concerns using biological science to augment human intelligence, as the transhumanists advocate.

Vinge himself admits that his paper is frightening. He evokes a future world where “IA for individual humans creates a rather sinister elite.” This hierarchical system of intelligences would only be livable if all intelligences adopt a “Meta-Golden Rule” first proposed by I.J. Good as follows: “Treat your inferiors as you would be treated by your superiors.” Vinge reports that most of his friends react skeptically to the idea of such a Rule, since “the game-theoretical payoff is so hard to articulate.”

Kurzweil’s Law of Accelerating Returns

After Vinge, Ray Kurzweil further refined the theory of the Singularity. Kurzweil is an inventor and entrepreneur who developed in the 1970s the first all-font optical character recognition system.

Kurzweil’s new contribution was to reformulate the theory of the Singularity in terms of the exponential growth of technology as a whole. He presented his arguments in 2001 in a paper called The Law of Accelerating Returns.

Kurzweil states that the rate of technological change is itself changing. The introductory paragraph to his 2001 paper expresses this as follows:

An analysis of the history of technology shows that technological change is exponential, contrary to the common-sense “intuitive linear” view. So we won’t experience 100 years of progress in the 21st century — it will be more like 20,000 years of progress (at today’s rate).

Kurzweil argues that technology always grows exponentially because of positive feedback loops whereby the existing technology permits more rapid development of new technologies. He writes:

Exponential growth is a feature of any evolutionary process, of which technology is a primary example. One can examine the data in different ways, on different time scales, and for a wide variety of technologies ranging from electronic to biological, and the acceleration of progress and growth applies.

Thus he concludes that technology on the whole always exhibits exponential growth. However, the subtlety of Kurzweil’s analysis is that he recognizes that there are limits to the growth of any given technology. He cites as an example Moore’s Law:

Gordon Moore, one of the inventors of integrated circuits, and then Chairman of Intel, noted in the mid 1970s that we could squeeze twice as many transistors on an integrated circuit every 24 months. Given that the electrons have less distance to travel, the circuits also run twice as fast, providing an overall quadrupling of computational power.

After sixty years of devoted service, Moore’s Law will die a dignified death no later than the year 2019. By that time, transistor features will be just a few atoms in width, and the strategy of ever finer photolithography will have run its course.

Moore’s Law thus describes the rapid growth phase of what will in the long run be seen to be an S curve, which is the typical curve for growth which reaches some fixed limit or ceiling.

The overall progression of any specific technology is an S curve, ending when that technology attains its limits. But evolutionary systems escape such limits by switching to another approach, carrying out a paradigm shift:

A specific paradigm (a method or approach to solving a problem, e.g., shrinking transistors on an integrated circuit as an approach to making more powerful computers) provides exponential growth until the method exhausts its potential. When this happens, a paradigm shift (i.e., a fundamental change in the approach) occurs, which enables exponential growth to continue.

Thus Kurzweil arrives at the exponential equivalent of a mathematical singularity: “As exponential growth continues to accelerate into the first half of the twenty-first century, it will appear to explode into infinity….” The Singularity is technological change that becomes so rapid that it represents a rupture with everything that came before.

Is Kurzweil Right?

In his 2001 paper, Kurzweil also elaborates on the type of future the Singularity will bring, continuing along the same lines as Vinge. The introduction states for example:

The implications include the merger of biological and nonbiological intelligence, immortal software-based humans, and ultra-high levels of intelligence that expand outward in the universe at the speed of light.

This vision of the future has been strongly criticized, notably by Bill Joy and Bruce Sterling.

But what interests me personally is the accuracy of Kurzweil’s prediction that technology as a whole will continue to grow exponentially. Some critics have argued that the overall rate of technological innovation seems in fact to be slowing down. I hope to investigate this in a future blog post.