It seems to be generally accepted that
applying infinite resources to a random process is guaranteed to
produce a particular outcome for which those resources are
necessary. So an infinite number of chimpanzees typing randomly
at an infinite number of typewriters for an infinite length of time
will inevitably produce the complete works of Shakespeare, right?
I don't think so. There is a difference between infinite and
unbounded. I once knew someone who bragged about confusing his
math teacher in high school because whenever she would refer to an
"infinite container" in a math problem, he would insist that such a
container would necessarily contain the entire universe, since it is
infinite. That is utter nonsense. He's confusing infinite
with unbounded, and they are not at all the same thing. An
infinitely tall soup can, for instance, can contain an infinite amount
of soup without ensnaring the entire universe. The boundaries of
the can limit the infinitely tall container so that it takes up an
insignificant fraction of our universe while still being able to hold
an infinite amount of soup.
It is also the case that necessary is not the same as sufficient.
Though it is necessary for something
to type randomly on at least one typewriter to produce the complete
works of Shakespeare, it does not follow that more -- even infinitely
more -- would be sufficient.
Random is just that: random. And even an infinite amount of
randomness is not guaranteed to produce any particular outcome, much
less every conceivable outcome. Claiming that an infinite number
of chimpanzees typing on an infinite number of typewriters for an
infinite length of time will inevitably produce the works of
Shakespeare (or, by inference, every conceivable literary work) is like
saying that an infinitely tall soup can absolutely must contain a dog
(or, by inference, every conceivable thing) simply because it is
If you're not convinced, then consider this: Flip a coin a
hundred times, and count the number of times it comes up heads.
Did it come up heads every single time? Probably not. But
is it possible that it could come up heads every single time?
Though it's not likely to happen, it is, of course, a
possibility. Now flip it a thousand times. Is it still
possible for it to come up heads every time? Yes, it is, though
it is much less likely. Now flip the coin an infinite number of
times. Is it possible that every coin flip throughout eternity
will come up heads? Yes, it is a potential outcome, though the
probability of such an outcome is vanishingly small. But the
point is, it is a statistically valid outcome, even with an infinite number of coin flips.
Similarly, it is a statistically valid, though very unlikely, outcome
for an infinite number of chimpanzees typing randomly at an infinite
number of typewriters for an infinite amount of time to produce
absolutely nothing but pages and pages of the letter "A". Such an outcome,
which, I repeat, is statistically valid, would absolutely preclude
anything like the entire works of Shakespeare from ever being
produced. Therefore, since there exists a statistically valid
outcome that precludes the production of the entire works of
Shakespeare, then production of the entire works of Shakespeare is not
at all guaranteed, even with infinite resources. And if that's
still not enough for you, then consider the fact that an infinity of
pages filled with the letter "A"
is not the only
valid outcome that absolutely precludes the entire works of Shakespeare
from being produced. For example, one valid outcome would be for
no page among all the infinite pages to have more than one instance of
the letter "E", which, as I recall, Shakespeare used quite a lot. There are, indeed, an infinite number of statistically
valid outcomes -- many far more likely than the examples I've cited here -- that would
absolutely preclude the entire works of Shakespeare from being produced.
So if it's a particular result you want, and your process is random,
don't be fooled into thinking that more process -- even infinitely more
process -- will necessarily guarantee the result you want. It is
this fallacious thinking that makes our bosses think that adding more
people to a project that is impossible to complete will make the
project not only possible, but will even shave at least a week or two
off the infinite schedule.
(c) by BSW, 2007. All Rights