The Singularity Is Near: When Humans Transcend Biology (91 page)

The solution to this is:

and world knowledge accumulates at a double exponential rate.

Now let’s consider some real-world data. In
chapter 3
, I estimated the computational capacity of the human brain, based on the requirements for functional simulation of all brain regions, to be approximately 10
¹⁶
cps. Simulating the salient nonlinearities in every neuron and interneuronal connection would require a higher level of computing: 10
¹¹
neurons times an average 10
³
connections per neuron (with the calculations taking place primarily in the connections) times 10
²
transactions per second times 10
³
calculations per transaction—a total of about 10
¹⁹
cps. The analysis below assumes the level for functional simulation (10
¹⁶
cps).

If we factor in the exponentially growing economy, particularly with regard to the resources available for computation (already about one trillion dollars per year), we can see that nonbiological intelligence will be billions of times more powerful than biological intelligence before the middle of the century.

We can derive the double exponential growth in another way. I noted above that the rate of adding knowledge (
dW/dt
) was at least proportional to the
knowledge at each point in time. This is clearly conservative given that many innovations (increments to knowledge) have a multiplicative rather than additive impact on the ongoing rate.

However, if we have an exponential growth rate of the form:

where
C
> 1, this has the solution:

which has a slow logarithmic growth while
t
1/ln
C
but then explodes close to the singularity at
t
= 1/ln
C
.

Even the modest
dW/dt
=
W
²
results in a singularity.

Indeed any formula with a power law growth rate of the form:

where
a
> 1, leads to a solution with a singularity:

at the time
T
. The higher the value of
a
, the closer the singularity.

My view is that it is hard to imagine infinite knowledge, given apparently finite resources of matter and energy, and the trends to date match a double exponential process. The additional term (to
W
) appears to be of the form
W
× log(
W
). This term describes a network effect. If we have a network such as the Internet, its effect or value can reasonably be shown to be proportional to
n
× log(
n
) where
n
is the number of nodes. Each node (each user) benefits, so this accounts for the
n
multiplier. The value to each user (to each node) = log(
n
). Bob Metcalfe (inventor of Ethernet) has postulated the value of a network of
n
nodes =
c
×
n
²
, but this is overstated. If the Internet doubles in size, its value to me does increase but it does not double. It can be shown that a reasonable estimate is that a network’s value to each user is proportional to the log of the size of the network. Thus, its overall value is proportional to
n
× log(
n
).

If the growth rate instead includes a logarithmic network effect, we get an equation for the rate of change that is given by:

The solution to this is a double exponential, which we have seen before in the data:

(15)
W
= exp(
e
^t
)

Notes

Prologue:
The Power of Ideas

1
. My mother is a talented artist specializing in watercolor paintings. My father was a noted musician, conductor of the Bell Symphony, founder and former chairman of the Queensborough College Music Department.

2
. The Tom Swift Jr. series, which was launched in 1954 by Grosset and Dunlap and written by a series of authors under the pseudonym Victor Appleton, continued until 1971. The teenage Tom Swift, along with his pal Bud Barclay, raced around the universe exploring strange places, conquering bad guys, and using exotic gadgets such as house-sized spacecraft, a space station, a flying lab, a cycloplane, an electric hydrolung, a diving seacopter, and a repellatron (which repelled things; underwater, for example, it would repel water, thus forming a bubble in which the boys could live).

The first nine books in the series are
Tom Swift and His Flying Lab
(1954),
Tom Swift and His Jetmarine
(1954),
Tom Swift and His Rocket Ship
(1954),
Tom Swift and His Giant Robot
(1954),
Tom Swift and His Atomic Earth Blaster
(1954),
Tom Swift and His Outpost in Space
(1955),
Tom Swift and His Diving Seacopter
(1956),
Tom Swift in the Caves of Nuclear Fire
(1956), and
Tom Swift on the Phantom Satellite
(1956).

3
. The program was called Select. Students filled out a three-hundred-item questionnaire. The computer software, which contained a database of about two million pieces of information on three thousand colleges, selected six to fifteen schools that matched the student’s interests, background, and academic standing. We processed about ten thousand students on our own and then sold the program to the publishing company Harcourt, Brace, and World.

4
.
The Age of Intelligent Machines
, published in 1990 by MIT Press, was named Best Computer Science Book by the Association of American Publishers. The book explores the development of artificial intelligence and predicts a range of philosophic, social, and economic impacts of intelligent machines. The narrative is complemented by twenty-three articles on AI from thinkers such as Sherry Turkle, Douglas Hofstadter, Marvin Minsky, Seymour Papert, and George Gilder. For the entire text of the book, see
http://www.KurzweilAI.net/aim
.

5
. Key measures of capability (such as price-performance, bandwidth, and capacity) increase by multiples (that is, the measures are multiplied by a factor for each increment of time) rather than being added to linearly.

6
. Douglas R. Hofstadter,
Gödel, Escher, Bach: An Eternal Golden Braid
(New York: Basic Books, 1979).

Chapter One: The Six Epochs

1
. According to the Transtopia site (
http://transtopia.org/faq.html#1.11
), “Singularitarian” was “originally defined by Mark Plus (’91) to mean ‘one who believes the concept of a Singularity.’ ” Another definition of this term is “ ‘Singularity activist’ or ‘friend of the Singularity’; that is, one who acts so as to bring about a Singularity [Mark Plus, 1991;
Singularitarian Principles
, Eliezer Yudkowsky, 2000].” There is not universal agreement on this definition, and many Transhumanists are still Singularitarians in the original sense—that is, “believers in the Singularity concept” rather than “activists” or “friends.”