The N Function

Drake’s Equation over Cosmological History

Nick Nielsen
Jan 14, 2020 · 4 min read
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Everyone, I’m sure, is familiar with the Drake equation, which purports to be a formalism that can be employed to estimate the number of communicating technological civilizations in the universe. Many people have tweaked the Drake equation since it was first introduced by Frank Drake to give slightly different results for slightly different circumstances. Many more people have attempted to give values to the variables in order to estimate the number of civilizations in the universe.

In its classic form, the Drake equation looks like this:

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Because we are human beings and usually see things on a human scale, we usually puzzle over N in terms of its value at the present time. This is an instance of what I call the snapshot effect — the temporally narrow nature (and perhaps also the fragmentary nature) of human perception. We see not the world, but a snapshot of the world. Even the grandest of grand views of the world are snapshots: to look into the night sky is to experience a snapshot of cosmology, and to recognize a geological formation is a snapshot of deep time.

Thus when we think of ETI, we think of a snapshot of ETI, i.e., we think of what ETI may be in existence right now with which we might be able to communicate as though with a peer, because peer-to-peer communication is something that occurs over a human scale of time that we can not only conceptualize, but actually internalize. Or, if we are a bit more sophisticated, we think of what ETIs may have existed sufficiently far in the past that their messages might have traveled to us across countless light years, so that we might receive them in real time.

If we step away from this snapshot effect that defines the human intellect in the context of cosmological history, we can see that N is a number that will change over time, and we don’t merely want to know N at an instant, we want to know the value of N over time. We can imagine a flickering nixie display giving the number of communicating civilizations in the universe, and as we pass through the history of the universe, from the big bang to the present day and eventually to thermodynamic equilibrium, the nixie display begins at zero, ramps up to some peak value of civilizations, and then tapers off again to zero.

If we are the only civilization in the universe, then then nixie display shows “0” from the origins of the universe until about 10,000 years ago, then it displays “1” for as long as we exist, and then resets to zero. In this case, the N function is a simple square wave. The N function is the value of N over time, where N is N in the Drake equation and time is the entire history of the universe, divided up into whatever increments are convenient (I would favor a logarithmic history demarcated in years).

If, however, there are many civilizations that appear and disappear over cosmological history, the N function will describe a curve. I will make a spectacularly non-constructive claim and assert that there is an equation that describes the N function for our universe (because, as everyone knows from elementary analytical geometry, a curve can be expressed by an equation, or an equation by a curve). We don’t know what that equation is, and indeed no one could say what that equation is until the last civilization in the universe winks out of existence, but there is some equation that describes the N function of our universe.

If there are multiple universes, then each universe will have an N function, which will be 0 except for those universes consistent with the existence of civilization. If we take the class of all universes that are consistent with the existence of civilization, presumably resembling our own universe to some degree, then each of these universes will have an N function, and the fate of civilization in the multiverse will be given by a function space that is the set of all N functions.

There is a lot that could be done with this simple formalism, but the formalism alone isn’t enough. In order to count civilizations, and to quantify their beginnings and endings, we would need to specify exactly what a civilization is, when it begins, and when it ends. And that is why I spend so much time, and spill so much ink, over the problem of defining civilization. Certainly we can pursue formalisms like the Drake equation or the function space of civilizations in the universe, but the formalisms remain empty until we have done the conceptual work of being able to say exactly what a civilization is in a cosmological context.

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