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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.
                       —I RVING J OHN G OOD , “S PECULATIONS C ONCERNING THE F IRST U LTRAINTELLIGENT M ACHINE ,” 1965
     
    To put the concept of Singularity into further perspective, let’s explore the history of the word itself. “Singularity” is an English word meaning a unique event with, well, singular implications. The word was adopted by mathematicians to denote a value that transcends any finite limitation, such as the explosion of magnitude that results when dividing a constant by a number that gets closer and closer to zero. Consider, for example, the simple function
y
= 1/
x
. As the value of
x
approaches zero, the value of the function (
y
) explodes to larger and larger values.

     
    A mathematical singularity:
As
x
approaches zero (from right to left), 1/
x
(or
y
) approaches infinity .
    Such a mathematical function never actually achieves an infinite value, since dividing by zero is mathematically “undefined” (impossible to calculate). But the value of
y
exceeds any possible finite limit (approaches infinity) as the divisor
x
approaches zero.
    The next field to adopt the word was astrophysics. If a massive star undergoes a supernova explosion, its remnant eventually collapses to the point of apparently zero volume and infinite density, and a “singularity” is created at its center. Because light was thought to be unable to escape the star after it reached this infinite density, 16 it was called a black hole. 17 It constitutes a rupture in the fabric of space and time.
    One theory speculates that the universe itself began with such a Singularity. 18 Interestingly, however, the event horizon (surface) of a black hole is of finite size, and gravitational force is only theoretically infinite at the zero-size center of the black hole. At any location that could actually be measured, the forces are finite, although extremely large.
    The first reference to the Singularity as an event capable of rupturing the fabric of human history is John von Neumann’s statement quoted above. In the 1960s, I. J. Good wrote of an “intelligence explosion” resulting from intelligent machines’ designing their next generation without human intervention. Vernor Vinge, a mathematician and computer scientist at San Diego State University, wrote about a rapidly approaching “technological singularity” in an article for
Omni
magazine in 1983 and in a science-fiction novel,
Marooned in Realtime
, in 1986. 19
    My 1989 book,
The Age of Intelligent Machines
, presented a future headed inevitably toward machines greatly exceeding human intelligence in the first half of the twenty-first century. 20 Hans Moravec’s 1988 book
Mind Children
came to a similar conclusion by analyzing the progression of robotics. 21 In 1993 Vinge presented a paper to a NASA-organized symposium that described the Singularity as an impending event resulting primarily from the advent of “entities with greater than human intelligence,” which Vinge saw as the harbinger of a runaway phenomenon. 22 My 1999 book,
The Age of Spiritual Machines: When Computers Exceed Human Intelligence
, described the increasingly intimate connection between our biological intelligence and the artificial intelligence we are creating. 23 Hans Moravec’s book
Robot: Mere Machine to Transcendent Mind
, also published in 1999, described the robots of the 2040s as our “evolutionary heirs,” machines that will “grow from us, learn our skills, and share our goals and values, . . . children of our minds.” 24 Australian scholar Damien Broderick’s 1997 and 2001 books, both titled
The Spike
, analyzed the pervasive impact of the extreme phase of technology acceleration anticipated within several decades. 25 In an extensive series of writings, John Smart has