One of the pleasures of life on the modern web is that if you have a question about, say, the history of the Fermi paradox, there's a good chance you can find something on it. In this case, it didn't take long (once I thought to look) to turn up E. M. Jones's "Where is Everybody?" an Account of Fermi's Question.
The article includes letters from Emil Konopinski, Edward Teller and Herbert York, who were all at lunch with Enrico Fermi at Los Alamos National Laboratory some time in the early 1950s when Fermi asked his question. Fermi was wondering specifically about the possibility that somewhere in the galaxy some civilization had developed a viable form of interstellar travel and had gone on to explore the whole galaxy, and therefore our little blue dot out on one of the spiral arms.
Fermi and Teller threw a bunch of arguments at each other, arriving at a variety of probabilities. Fermi eventually concluded that probably interstellar travel just wasn't worth the effort or perhaps no civilization had survived long enough to get to that stage (I'd throw in the possibility that they came by millions of years ago, decided nothing special was going on and left -- or won't come by for a few million years yet).
Along the way Fermi, very much in the spirit of "How many piano tuners are there in Chicago?" broke the problem down into a series of sub-problems such as "the probability of earthlike planets, the probability of life given an earthlike planet" and so forth. Very much something Fermi would have done, (indeed, this sort of exercise goes by the name "Fermi estimation") and very similar to what we now call the Drake equation.
In other words, Fermi and company anticipated much of the subsequent discussion on the subject over lunch more than fifty years ago and then went on to other topics (and presumably coffee). There's been quite a bit of new data on the subject, particularly the recent discovery that there are in fact lots of planets outside our solar system, but the theoretical framework hasn't changed much at all.
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