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The Fermi Paradox Robert Pisani, Ph.D

The Fermi Paradox: Three Models

Robert Pisani

Introduction

   There are many billions of galaxies that contain billions of star systems that can support
life. The universe has existed for 13+ billion years. 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?"

   At one point after the first atomic bombs were detonated, it was considered that nuclear
weapons had the imminent potential to end human life on earth. Various analysts assessed
the chance of humankind reaching the 21st Century as “about 50%” because of this
potential. Whether or not such thinking was justified, new technologies like DNA
manipulation, nanotechnology and molecular manufacturing clearly do have such potential.

   A number of universal laws apply to technologies and other societal structures that must
be common to all advanced civilizations, and probabilistic arguments based upon these
laws may lead to a small or zero probability of civilizations growing beyond a certain stage.

           Humankind is faced with numerous and interrelated existential threats: extreme
climate change, nuclear holocaust, biowarfare, natural plague, nanotechnological weaponry,
etc., which can be triggered either by natural causes, deliberate malignant activity, or
technical malfeasance. One of the chief threats arising from human activities is the
extermination of human life, and perhaps all life, by advanced technological instruments.

   The following notes suggest a framework for beginning to think rigorously about the Fermi
Paradox. Note that, even if life is rare, even if we are the only instance of life in the entire
universe, under the posited assumptions the below probability arguments nonetheless apply
to us.

I – Empowering the Masses

1 - At any time s in history, let f(s) = the maximum number of people that can be killed with
one application of the most powerful weapon in existence,
regardless of the defenses available at that time. f(s) is an increasing function of s and is
bounded only by the human population of the world.

2 - At any times s and t, with t > s, let c(s,t) = the cost at time t of
the most powerful weapon in existence at time s, measuring the cost by the
percentage of the population that cannot afford to acquire such a weapon.
Thus c(s,s) ~ 100%, and c(s,t) decreases monotonically as t --> ∞ to
nearly 0%. That is, at some time later than s almost every human can acquire the most
powerful weapon available at time s.

3 - Let p(t) = the percentage of the population at time t willing to destroy
the world. Note that p(t) is never declines to 0 and is probably bounded below.

4 -Since f(s) increases with s without bound, at some point s*, f(s*) > the number of people in
the world.

5 - And at some point t*> s*, c(s*,t*) < p(t*).

   The inevitable conclusion of this clearly simplistic argument is
that, at some point t*, the means to destroy the entire population of the
world will be available to anyone who wishes to do so.

II – Kurzweil’s Law

   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)

   Following are two analyses using this framework.

III – Kurzweil’s Law, Model I

   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) = ∏ {(1-pi), i = k, n}

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

(4) lim n  ∞ s(n) = lim n  ∞ ∏ {(1-pi), i = k, n}, which is greater than 0 if and only if

(5) Σ ln (1-pi) converges.

For small p, ln(1-p) ~ - p, so (loosely) for small p, (5) converges if and only Σ pi converges. If
(5) diverges, the probability of ultimate extinction is 1.0. Note that if pi is bounded away from
0, (5) diverges.

   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  ∞ ,

ln(1 - pn) =ln (1-q)x = x ln(1-q) = yn ln(1-q)  - ∞, 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  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.


III – Kurzweil’s Law, Model II

   The following assumption is plausible.

(1)         If Epoch(j) runs from sj to tj and Epoch(k) runs from sk to tk and if y(tj)/y(sj)   = y(tk)/y
(sk) = a, then pj = pk.

   If we assume (1), the question arises: what sequence of times t0 < t1 < t2 < . . . is defined
by

(2)        y(tn+1) = a * y(tn) for all n

   Convergence of this sequence of times to a finite value would of course be highly
unsettling.

(2) implies that y(tn) = y(t0) *an, so that

exp(exp(tn)) = y(t0) * an

exp(tn) = ln(y(t0)) + n*ln(a)

and

tn = ln{ln(y(t0)) + n*ln(a)} = O(ln(n))

   So, at least if (1) is assumed to be the case, tn does not converge to a finite value.
However, if p = the common probability of annihilation between ti and ti+1 the chance that the
civilization will survive for n Epochs is:

(3)        (1-p)n

   And n Epochs last until real time tn+1, which is roughly ln(n+1). Thus, (3) approaches 0
very quickly in real time.

   Finally, a careful structural examination may show that (1) is optimistic. And if Kurzweil’s
Law should dramatically underestimate the growth of technology, the extreme growth of y(t)
could cause the ti to converge to a finite value. This last seems certain to be the case after
the Singularity and perhaps even in some neighborhood this side of the Singularity,

IV A Non-Technical Explanation of III with a=2

   Kurzweil’s Law applies to any technological civilization and to every aspect of technology:
the number of computer chips per square inch, the speed of computer chips, computational
efficiency, the number of scientific advances, and the development of weapons. Kurzweil’s
Law says that technology grows at an ever increasing rate, and not only does it grow more
quickly all the time, the acceleration in its growth also increases with time.

   Whereas by some measures the world’s technology doubled between the year 1800 and
the year 1950, and it has doubled again between 1950 and today, per Kurzweil’s Law at
some point technology will double every five years, and then every year, and then every
month, and sooner or later every ten seconds and eventually so quickly that it’s doubling
cannot be measured with human oriented time stamps.

   We already have technologies with the capability of ending human life on earth. With each
doubling of technology, new such technologies are created and there is some chance that
one of them will activate.

   So imagine that, instead of conducting the immense experiment of the human race
advancing with all of its political, religious, environmental, social and scientific complications,
you replace that grand experiment with a simple probability experiment: each time
technology doubles, you roll two dice and if they come up snake eyes humanity perishes.
And if you prefer not to assume such a great chance of this happening (with two dice it is 1 in
36, a little less than 3%), then replace the two dice with eight dice (with eight dice the chance
all will come up “1” is a little less than 1 in a million), or a number of dice whose chance of all
coming up “1” corresponds to your estimate of the true probability of total annihilation in a
period of technology doubling.

   Where one can consider that we survived one roll between 1950 and 2000 with chance
perhaps 50% of getting “multiple snake eyes”, another roll is now taking place in the time
between 2000 and, say, 2010. And if we survive that roll, another roll will take place between
2010 and an even closer date in the future, perhaps 2015. If you roll the dice repeatedly, no
matter how small the chance of getting multiple snake eyes on any single roll, sooner or later
that event must occur.

   If the chance of total annihilation on each roll is in any doubling period is always the same,
then no matter how small that chance is, total annihilation will ultimately occur with 100%
probability*. Only in the case that the successive probabilities of annihilation become
infinitesimally small, and quickly, will the civilization survive.

   Because of Kurzweil’s Law, the dice will be rolled more and more frequently as time
progresses. A time will come when they will be rolled once a year, and then every month,
and sooner or later every ten seconds and eventually so quickly that the doubling of
technology, and the rolling of the dice, cannot be measured with human oriented time
stamps.

•        If the chance is so small that the expected time until a multiple snake eyes event is
longer then the expected life the earth, this is not true, but such low probabilities are not
relevant as even far larger chances, such as a chance of 1 in a million, are probably too low
an estimate for this chance of a catastrophic event.
•
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