On the Meaning of the Fermi Paradox
Carl L. DeVito
The many extra-solar planets the astronomers have found lends greater poignancy to Enrico Fermi’s famous question: “Where is everyone?”
My paper dealing with this question appeared in a special issue of Futures.
The paper begins with a thought experiment. Suppose that the universe had been scanned continuously, and at many frequencies, for intelligent signals. Suppose the scan had been programed to note any signal detected but then to simply continue the search. We would expect the rate at which societies would be found to depend, in some unknown way, on the number of societies and the rate at which such societies arise. Had this scan begun when the first society arose and continued to the present the number of societies detected would be expressible as a definite integral.
Now, of course, we have not been carrying out such a scan, but if that integral were large we would expect to have detected some society by now. The Fermi paradox may be telling us that that integral is “small”. So let us look at the minimum that integral can be. Problems of this kind arise in physics. Fermat’s principle in optics and Hamilton’s principle in mechanics arise from similar considerations. To solve them one uses the calculus of variations. Applying this to the problem at hand yields an equation with many solutions. Since we know so very little about societies I tried to find the simplest solution obtaining
N(t) = N(0) + ( 2/L) ln ( Lt + 1)
Here zero is the time the first society, or perhaps several societies, arose and N(0) is the number of these. The t is time, and the L is the average time a society continues to signal; the L is the same as the last term in the Drake equation.
The equation shows that N increases steadily but because of the ln (the logarithm base e) very slowly.
I make a point of noting that here our knowledge is meager and what little we know is uncertain. One should not read too much in our mathematical models. This caution certainly applies to the Drake equation where many of the terms are unknown, and it also applies to the equation derived here.
According to our equation the Fermi paradox does not rule out cosmic company. What it might be telling us is that we acquire neighbors so slowly that, given the vastness of space, it is not surprising that we haven’t found them yet. So perhaps, the paradox isn't such a paradox after all.
Carl L. DeVito is a member of the Emeritus faculty of the Department of Mathematics at the University of Arizona.
One of the (many) things we enjoy about Carl is his ability to write and talk about complicated things in engaging and accessible ways.
This is a summary of some of his recent work. The full paper is: Futures. Detectability of Future Earth v106, February 2019. https://www.sciencedirect.com/science/article/pii/S0016328717303907