The Fermi Paradox and Singularities

Robert Pisani
Department of Statistics
University of California
Berkeley, California
robert pisani <r.pisani@mac.com>


There are many billions of galaxies that contain
even more billions of star systems that can support
life. The universe has existed for 13+ billion years,
and the time that humans needed to grow from single
celled creatures to what they are today is just a few
million years, an instant in the life of the universe.
If life is as common as is now thought, we would not
be the first civilization to have arisen in the
universe. Any civilization that has advanced to our
stage must produce radio waves, microwaves, etc. But
we don't find any. Enrico Fermi said, "Where is
everybody?"

It was thought at one point in the 1950's that
nuclear weapons had the potential to end human life on
earth. Knowledgeable insiders assessed the chance of
reaching the 21st Century as "about 50%". Whether or
not such thinking was justified, new technologies like
DNA manipulation and nanotechnology and molecular
manufacturing clearly do have such potential. These
technologies and others could cause extinction of the
human race, and in fact all life on our planet, either
through deliberate application or technical
malfeasance. Following is a framework for beginning
to think rigorously about the Fermi Paradox.

Kurzweil's Law says that technology grows at a
double exponential rate. Denote the level of
technology at time t by yt.

(0) yt = exp(exp(t)).

Given a sequence of times t0 < t1 < t2 < . . .
define
(1) Epoch(n) = the period from tn to tn+1 and
(2) pn = the chance of total annihilation during
Epoch(n)

Assuming that survival chances in different Epochs
are independent, given that a civilization has
survived to the end of Epoch(k-1) and thus the
beginning of Epoch(k), if n „ k the chance that it
will survive until the end of Epoch(n) is:

(3) s(n) = ½ {(1-pi), i = k, n}

And the chance it will survive for an infinite number
of epochs is

(4) lim n ý ƒ s(n) = lim n ý ƒ ½ {(1-pi), i = k, n},
which is greater than 0 if and only if

(5) _ ln (1-pi) converges.

For small p, ln(1-p) ~ - p, so (loosely) for small p,
(5) converges if and only _ pi converges. If (5)
diverges, the probability of ultimate extinction is
1.0.

Let yn = the number of different weapons available at
time n which will cause extinction if used. Certainly
defensive measures against such weapons will also be
available.

If the probability of any given weapon being
applied successfully against all of its defenses in
Epoch(n) is q > 0, then the chance that any given
weapon is not used successfully in Epoch(n) is 1-q,
and the chance that at least one is used successfully,
and the civilization perishes, is pn = 1 - (1-q)x,
where x = yn.

Since ln(1-q) < 0 and yn ý ƒ ,

ln(1 - pn) =ln (1-q)x = x ln(1-q) = yn ln(1-q) ý -
ƒ, and (5) diverges

Suppose that defensive measures yield Epoch(n)
probabilities qn which decline to 0 as n grows larger.
Still, if (5) is to converge, we must have
yn ln(1- qn) ~ - yn qn ý 0.

and to avoid extinction, we must have necessarily (but
not sufficiently)

qn < 1/ yn = 1/exp(exp(n)) for n> n0, some n0

Even more restrictively, we must have
yn qn < 1/n, and thus

qn < 1/nyn = 1/(n exp(exp(n)) for all n larger than
some value.

And so on.

Clearly the qn must decline to 0 very rapidly,
and (0) clearly places an extreme burden on any
program of defense that seeks to avoid the
annihilation of the civilization. Should other
considerations show that such rapid decline is
structurally not possible, (4) will then necessarily
equal 0, explaining the Fermi Paradox.

=========

Discuss...