This is the question in the course where the temptation to overclaim is strongest, in both directions. One camp treats the size of the universe as settling it: billions of galaxies, so of course there is life. The other treats the silence as settling it: nobody has called, so we are alone. Neither inference works, and the machinery of lessons five, six and thirteen is what shows why.
The Drake equation, honestly
Frank Drake wrote his equation in 1961 for the agenda of a small meeting at Green Bank. It expresses the number of communicating civilisations in the galaxy as a product of factors: the rate of star formation, the fraction with planets, the number of habitable planets per system, the fraction where life arises, the fraction where life becomes intelligent, the fraction that develop detectable communication, and the length of time they stay detectable.
Drake was explicit that it was not a calculation. It was a way of organising a discussion by naming the unknowns, so that a meeting of ten people could work through them in order.
That is exactly how it should be used, and almost never is. The first three factors are now measured reasonably well, and the news is good: planets are common, and estimates from the Kepler mission put roughly one in five Sun-like stars as hosting an Earth-sized planet in the habitable zone, though the figure depends heavily on how the zone is defined.
The last four are unknown. Not imprecisely known, unknown, in the sense that credible published estimates differ by ten or twenty orders of magnitude. When a product contains a factor uncertain by twenty orders of magnitude, the product is uncertain by twenty orders of magnitude, and reporting a single number from it is not a result.
Example. Someone says the universe contains 10^22 stars, so even if life is a one in a billion chance, there must be trillions of civilisations. What is wrong with the reasoning?
The "one in a billion" was invented. It sounds conservative, and conservatism is a rhetorical property rather than an epistemic one: nobody knows whether abiogenesis is a one in a billion event, a near certainty on any wet rock, or a one in fluke that happened once. Assuming a number in the range that makes the argument work is assuming the conclusion. Notice that the same trick runs in reverse, since a pessimist can pick a tiny probability and derive that we are alone with equal apparent rigour. The honest statement is that the number of stars is known and multiplies an unknown, and that a large known times an unknown is unknown.
Now you. We have one data point: life arose on Earth, apparently quite early in its history. Why is this weak evidence that abiogenesis is easy?
Answer
Because of the selection effect from lesson six. We could only be on a planet where life arose, so the observation was guaranteed regardless of how improbable abiogenesis is, and by lesson five the likelihood ratio is 1. The early timing is a subtler case and was argued by Brandon Carter to point the other way: if intelligence requires a long series of improbable steps, then the ones that happened early would be those that could happen fast, and we would expect to find ourselves on a planet where the sequence just fit into the star's lifetime. Recent work using better estimates of the remaining habitable lifetime of Earth has been read both ways. The safe conclusion is that a single self-selected data point constrains very little, and that anyone claiming it settles the question has not taken the selection effect seriously.
Fermi's question
The story is that Enrico Fermi, at Los Alamos in 1950 over lunch, listened to a conversation about flying saucers and interstellar travel and asked: where is everybody?
The question is sharper than the equation. Take the galaxy at roughly 100,000 light years across and 10 billion years old. A civilisation expanding at even one thousandth the speed of light, which requires no new physics, crosses it in 100 million years. That is one percent of the galaxy's age. So if technological civilisations arise with any regularity at all, the galaxy should have been settled many times over, and it visibly has not been.
The force of this is that it does not depend on radio. It is an argument from the absence of physical presence, over a timescale where the expansion would have been finished long before the Earth formed.
Example. Someone replies that advanced civilisations would have no interest in expanding, so Fermi's argument fails. Assess it.
The reply is legitimate and it has to be much stronger than it sounds to work. It is not enough that most civilisations lose interest in expansion, because the argument only needs one that does not, over ten billion years, and expansion is self-perpetuating once started. So the premise required is that essentially every civilisation, everywhere, permanently declines to expand or to send self-replicating probes, with no exceptions across the whole history of the galaxy. Universal claims about the motives of unknown beings are exactly the sort of thing lesson six warns about, since the reasoning is anthropocentric in both directions: projecting our expansionism onto them, and then projecting our imagined wisdom onto them. The reply survives as a possibility and cannot be leaned on as a solution.
Now you. Why is the Fermi question a better argument than the observation that SETI has found nothing?
Answer
Because of what each observation could have shown. SETI has searched a small fraction of the sky, over a narrow range of frequencies, for a few decades, for signals of a kind we would recognise and that would have to be aimed or very powerful. A null result from that search is close to what you would expect even in a galaxy full of chatty neighbours, so the likelihood ratio is near 1 and lesson five says it moves almost nothing. The Fermi observation is different in kind: physical colonisation over ten billion years would leave consequences that cannot be missed by searching in the wrong band, namely a galaxy that is visibly occupied, including this solar system. An observation that could have come out dramatically otherwise is doing evidential work, and one that was nearly guaranteed is not.
The Great Filter
Robin Hanson's 1996 framing is the most useful way to hold the problem. Somewhere between dead matter and a galaxy-spanning civilisation there is at least one step that is extraordinarily improbable. The question is where.
If the filter is behind us, in abiogenesis, or in the transition to complex cells, or in the emergence of intelligence, then we have passed it and the future may be open. The single origin of eukaryotic cells, apparently from one endosymbiotic event that the evolution course describes, is a candidate: it happened once in four billion years, which is what a hard step looks like.
If the filter is ahead of us, then technological civilisations reliably destroy themselves or are destroyed, and we should expect to as well. This is why the Fermi question is not idle: it bears on how worried to be about our own prospects.
Nick Bostrom drew the uncomfortable corollary. Finding independent life on Mars would be bad news, since it would move probability away from abiogenesis being the hard step and toward the filter being somewhere later, possibly ahead. The more life we find, and the more easily it arises, the worse the news.
The rare Earth response, from Peter Ward and Donald Brownlee in 2000, argues the filter is behind us and is made of many small steps: a large moon stabilising the axial tilt, plate tectonics recycling carbon, a gas giant in the right orbit, a quiet stellar neighbourhood. Each is plausible, none is established, and the hypothesis has the shape lesson nine warned about, since it can absorb any new discovery by adding another requirement.
Example. A 2018 paper by Sandberg, Drexler and Ord argued that we should not be surprised by the silence. What did they do differently?
They propagated the uncertainty instead of multiplying point estimates. If you take the scientific uncertainty in each Drake factor seriously and represent it as a distribution, then multiply the distributions rather than best guesses, the result is not a number but a very wide spread, and a substantial portion of it lies below one civilisation per galaxy. So the appearance of a paradox came from the arithmetic: multiplying central estimates of quantities with enormous log-scale uncertainty produces a confident answer that the underlying uncertainty does not support. The result does not show we are alone, it shows that being alone was never surprising and the Fermi paradox may be an artefact of doing the sum badly.
Now you. Does the silence tell us anything at all?
Answer
Yes, and less than is usually claimed. What it rules out with reasonable confidence is a galaxy densely populated by expansionist civilisations that colonise physically, since the timescales above mean we would see the results rather than merely fail to hear signals. What it does not rule out is a galaxy with a handful of civilisations, or with civilisations that do not expand, or that are unrecognisable, or that are quiet deliberately. And the radio searches have covered a tiny fraction of the relevant volume, frequency and time, so "we have listened and heard nothing" is much weaker than it sounds. The disciplined statement is that the observation shifts credence away from one specific and rather anthropocentric model of what a civilisation does, which lesson six should make us suspicious of having assumed in the first place, and leaves most of the space untouched.
That completes the three questions. What remains is to put the pieces together and hold a position, which is what the last lesson asks you to do.