The usual line of attack for answering this question is the Drake equation [but see the next post for a bit on its origins --D.H Oct 2018], which breaks the question of "How many intelligent civilizations are there in our galaxy?" down into a series of factors that can then be estimated and combined into an overall estimate.
Let's take a simpler approach here.
The probability of detecting extraterrestrial intelligence given our efforts so far is the product of:
- The probability it exists
- The probability that what we've done so far would detect it, given that it exists
(For any math geeks out there, this is just the definition of conditional probability)
Various takes on the Fermi paradox (why haven't we seen anyone, if we're pretty sure they're out there?) address these two factors
- Maybe intelligent life is just a very rare accident. As far as we can tell, Earth itself has lacked intelligent life for almost all of its history (one could argue it still does, so feel free to substitute "detectable" for "intelligent").
- Maybe intelligent life is hard to detect for most of the time it's around (See this post for an argument to that effect and this one for a bit on the distinction between "intelligent" and "detectable"). A particularly interesting take on this is the "dark forest" hypothesis, that intelligent civilizations soon figure out that being detectable is dangerous and deliberately go dark, hoping never to be seen again. I mean to take this one on in a bit, but not here.
- One significant factor when it comes to detecting signs of anything, intelligent or otherwise: as far as we know detectability drops with the square of distance, that is, twice as far away means four times harder to detect. Stars are far away. Other galaxies are really far away.
- Maybe intelligent life is apt to destroy itself soon after it develops, so it's not going to be detectable for very long and chances are we won't have been looking when they were there . This is a popular theme in the literature. I've talked about it here and here.
- Maybe the timing is just wrong. Planetary time scales are very long. Maybe we're one of the earlier ones and life won't develop on nearby planets for another million or billion years (basically low probability of detection again, but also an invitation to be more rigorous about the role of timing)
At first blush, the logic of the Fermi paradox seems airtight: Aliens are out there. We'd see them if they were out there. We haven't seen them. QED. But we're not doing a mathematical proof here. We're dealing in probabilities (also math, but a different kind). We're not trying to explain a mathematically impossible result. We're trying to determine how likely it is that our observations are compatible with life being out there.
I was going to go into a longish excursion into Bayesian inference here, but ended up realizing I'm not very adept at it (note to self: get better at Bayesian inference). So in the spirit of keeping it at least somewhat simple, let's look at a little badly-formatted table with, granted, a bunch of symbols that might not be familiar:
| We see life (S) | We don't see life (¬S) | ||
| Life exists (L) | P(L ∧ S) | P(L ∧ ¬S) | P(L) |
| No life (¬L) | P(¬L ∧ S) | P(¬L ∧ ¬S) | P(¬L) |
| P(S) | P(¬S) | 100% |
P is for probability. P(L) is the probability that there's intelligent life out there we could hope to detect as such, at all. P(S) is the probability that we see evidence strong enough that the scientific community (whatever we mean by that, exactly) agrees that intelligent life is out there. The ¬ symbol means "not" and the ∧ symbol means "and". The rows sum to the right, so
- P(L ∧ S) + P(L ∧ ¬S) = P(L) (the probability life exists is the probability that life exists and we see it plus the probability it exists and we don't see it)
- P(S + ¬S) = 100% (either we see life or we don't see it)
Likewise the columns sum downward. Also "and" means multiply (as long as the two probabilities are independent; they are here, since we allow for false positives), so P(L ∧ S) = P(L)×P(S). This all puts restrictions on what numbers you can fill in. Basically you can pick any three and those determine the rest.
Suppose you think it's likely that life exists, and you think that it's likely that we'll see it if it's there. That means you think P(L) is close to 100% and P(L ∧ S) is a little smaller but also close to 100% (see conditional probability for more details) . You get to pick one more. It actually turns out not to matter that much, since we've already decided that life is both likely and likely to be detected. One choice would be P(¬L ∧ S), the chance of a "false positive", that is, the chance that there's no life out there but we think we see it anyway. Again, in this scenario we're assuming false positives should be unlikely overall, but choosing exactly how unlikely locks in the rest of the numbers.
It's probably worth calling out one point that kept coming up while I was putting this post together: The chances of finding signs of life depend on how much we've looked and how we've done it. A lot of SETI has centered around radio waves, and in particular radio waves in a fairly narrow range of frequencies. There are perfectly defensible reasons for this approach, but that doesn't mean that any actual ETs out there are broadcasting on those frequencies. In any case we're only looking at a small portion of the sky at any given moment, our current radio dishes can only see a dozen or two light years out and there's a lot of radio noise from our own technological society to filter out.
I could model this as a further conditional probability, but it's probably best just to keep in mind that P(S) is the probability of having detected life after everything we've done so far, and so includes the possibility that we haven't really done much so far.
To make all this concrete, let's take an optimistic scenario: Suppose you think there's a 90% chance that life is out there and a 95% chance we'll see it if it's out there. If there's no chance of a false positive, then there's an 85.5% chance that we'll see signs of life and so a 14.5% chance we won't (as is presently the case, at least as far as the scientific community is concerned). If you think there's a 50% chance of a false positive, then there's a 90.5% chance we'll see signs of life, including the 5% chance it's not out there but we see it anyway. That means a 9.5% chance of not seeing it, whether or not it's actually there.
This doesn't seem particularly paradoxical to me. We think life is likely. We think we're likely to spot it. So far we haven't. By the assumptions above, there's about a 10% chance of that outcome. You generally need 99.99994% certainty to publish a physics paper, that is, a 0.00006% chance of being wrong. A 9.5% chance isn't even close to that
Only if you're extremely optimistic and you think that it's overwhelmingly likely that detectable intelligent life is out there, and that we've done everything possible to detect it do we see a paradox in the sense that our present situation seems very unlikely. But when I say "overwhelmingly likely" I mean really overwhelmingly likely. For example, even if you think both are 99% likely, then there's still about a 1-2% chance of not seeing evidence of life, depending on how likely you think false positives are. If, on the other hand, you think it's unlikely that we could detect intelligent life even if it is out there, there's nothing like a paradox at all.
My personal guess is that we tend to overestimate the second of the two bullet points at the beginning. There are good reasons to think that life on other planets is hard to detect, and our efforts so far have been limited. In this view, the probability that detectably intelligent life is out there right now is fairly low, even if the chance of intelligent life being out there somewhere in the galaxy is very high and the chance of it being out there somewhere in the observable universe is near certain.
As I've argued before, there aren't a huge number of habitable planets close enough that we could hope to detect intelligent life on them, and there's a good chance that we're looking at the wrong time in the history of those planets -- either intelligent life hasn't developed yet or it has but for one reason or another it's gone dark.
Finding out that there are potentially habitable worlds in our own solar system is exciting, but probably doesn't change the picture that much. There could well be a technological civilizations in the oceans of Enceladus, but proving that based on what molecules we see puffing out of vents on the surface many kilometers above said ocean seems like a longshot.
With that in mind, let's put some concrete numbers behind a less optimistic scenario. If there's a 10% chance of detectable intelligent life (as opposed to intelligent life we don't currently know how to detect), and there's a 5% chance we'd have detected it based on what we've done so far and a 1% chance of a false positive (that is, of the scientific community agreeing that life is out there when in fact it's not), then it's 98.6% likely we wouldn't have seen clear signs of life by now. That seems fine.
While I'm conjecturing intermittently here, my own wild guess is that it's quite likely that some kind of detectable life is out there, something that, while we couldn't unequivocally say it was intelligent, would make enough of an impact on its home world that we could hope to say "that particular set of signatures is almost certainly due to something we would call life". I'd also guess that it's pretty likely that in the next, say, 20 or 50 or 100 years we would have searched enough places with enough instrumentation to be pretty confident of finding something if it's there. And it's reasonably likely that we'd get a false positive in the form of something that people would be convinced it was a sign of life when there in fact wasn't -- maybe we'd figure out our mistake in another 20 or 50 or 100 years.
Let's say life of some sort is 90% likely, there's a 95% chance of finding it in the next 100 years if it's there and a 50% chance of mistakenly finding life when it's not there, that is, a 50% chance that at some point over those 100 years we mistakenly convince ourselves we've found life and later turn out to be wrong. Who knows? False positives are based on the idea that there's no detectable life out there, which is another question mark. But let's go with it.
I actually just ran those numbers a few paragraphs ago and came up with a 9.5% chance of not finding anything, even with those fairly favorable odds.
All in all, I'd say we're quite a ways from any sort of paradoxical result.
One final thought occurs to me: The phrase "Fermi paradox" has been in the lexicon for quite a while, long enough to have taken on a meaning of its own. Fermi himself, being one of the great physicists, was quite comfortable with uncertainty and approximation, so much so that the kind of "How many piano tuners are there in Chicago?" questions given to interview candidates are meant to be solved by "Fermi estimation".
I should go back and get Fermi's own take on the "Fermi paradox". My guess was he wasn't too bothered by it and probably put it down to some combination of "we haven't really looked" and "maybe they're not out there".
[As noted above, I did in fact come across something --D.H Oct 2018]
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