The "Fermi Paradox" was an assertion by famous physicist Enrico Fermi that if intelligent life exists elsewhere in the Universe it would have visited Earth already. Since it hasn't it must not exist.
It is, obviously, silly. It makes assumptions without evidence as to the ease of travel between star systems, the large prevalence of life elsewhere if it exists at all, the widespread desire of other lifeforms to visit Earth, and at its heart it misunderstands (indeed, sort of makes a mockery of) positivism as a scientific principle.
Boing Boing links to a post by Charlie Stross on his blog Charlie's Diary that takes Fermi's paradox apart imagining it as the philosophy of a tapeworm wondering if there are other tapeworms. It is, in Stross's words, "the Fermi paradox, mired in shit." He takes apart its errors point by point.
What we know of the world is well reasoned conjecture based on observation. The observations on which the conjecture that life exists elsewhere in the Universe are the scale of the Universe, the number of galaxies and stars it contains, the likelihood that Earth-like planets exist, and what we know of the circumstances where life could arise. Life elsewhere, somewhere, seems likely.
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