The sky seems so vast, and space beyond it. You leave the city, head out to the country, and discover the blanket of stars. You see the Hubble Deep Field, an image of a seemingly empty spot of sky, revealed to be full of galaxies, clustered like jewels. There’s a Hubble eXtreme Deep Field, too, which reveals even more.

So how could we be alone, with all this space? It seems impossible. It seems impossibly lonely, too.

As humanity has come to understand the universe, we’ve come to see, again and again, that we are not special. In 1543, Copernicus proved that Earth is not the center of the universe, and so it began. Again and again, our understanding of the world was disturbed (sometimes completely demolished), and humanity was shuffled toward the edge. The Earth is not the center of the solar system; the solar system is not the center of the galaxy; the Milky Way is just one of hundreds of billions of galaxies — not at the center of the universe, and maybe not even the only universe. To think, then, that we are the only thinking life in the universe is obvious hubris.

There is the wisdom of hindsight and centuries of discovery, and then there is the evidence — or the resounding lack thereof. Not just for intelligent life, but any other life at all. Evidence could come in the form of fossilized microbes on Mars, methane-swimming bacteria in the depths of one of Jupiter’s moons, a galactic starship gliding into view. But nothing yet. Just decades of thought experiments, tempering hope with what we know and our best guess at extrapolation.

This story feels like it should be apocryphal, and maybe some of the details are. In 1950, four physicists took a lunch break from their work at Los Alamos. They joked about the recent spate of UFO sightings, a New Yorker cartoon suggesting a connection to a trend of trash-can lid disappearances. The speculation led to more serious conversation:

Do you think travel beyond the speed of light is possible?

I think the odds of us observing faster-than-light travel, by some object, by 1960, would be one-in-a-million.

I would put the odds at one in 10.

And so on. Lunch was ordered, the conversation moved on. But Enrico Fermi, the Nobel Prize–winning physicist, was lost in some calculations he was scribbling on a napkin. After a while, he looked up looked up and said, “Where is everybody?”

That question has come to be known as the Fermi paradox. The universe is very old, but the Earth is not. Nor are the solar system and our sun. There are planets and solar systems far older than ours. And life on Earth found its foothold very quickly, just as soon as conditions allowed it. There has been enough time for life on other planets to be millions of years ahead of where we are.

Fermi calculated that a single civilization could colonize the galaxy in 10 million years, and subsequent calculations have put it between 1 million and 13 million years — a daunting length of time, but the galaxy is a thousand times older than that.

There is hardly a human population that hasn’t, when it flourished, also spread. Animals and plants, too — vast mats of fungal colonies, even bacterial spores on the wind. Sometimes this is a simple arithmetic of population, resources, and space; sometimes it’s human acquisition and conquest. We are expanders and explorers. We look across the ocean and across the gulf of space and imagine ourselves on the other side. One reason to think we should see evidence of alien civilizations is that we assume they share this bug.

So: Where is everybody?

A sidebar: It’s the galaxy we’re talking about, yes, and not the universe, because as vast as the universe is, answers could only come from nearby — a signal, a visit, a microbe unlike any we’ve known. If the universe is infinite, as physicists say it might be, then there is other life. There is every permutation of possibility, infinite versions both unfathomably strange and identical to you (which, I suppose, is also unfathomable). But whether the universe is infinite or not, the observable universe has a boundary, and even that is terribly vast. If we want proof, it will likely have to come from our own galaxy. (That’s all that was ever traversed on Star Trek, anyway.) So we keep our search — and our hopes — close to home.

With all the ways we could spend energy searching — sampling missions to Mars and the moons of Jupiter, tilting radio telescopes toward every promising exoplanet in the sky — it’s worth asking, “Is it worth it? What are the odds?”

Fermi’s paradox is a paradox because even if the odds are incredibly low, even if the number of spacefaring, exploring, colonizing alien civilizations is one, they should likely be everywhere by now. Of course, there are plenty of hypothetical solutions to the Fermi paradox, reasons why they’re not. Physicist Stephen Webb’s book If the Universe Is Teeming with Aliens … WHERE IS EVERYBODY? catalogs 50 of them. They are grouped into three categories: Oh, actually, they’re here; they exist, but they haven’t contacted us; and they do not exist at all. The table of contents is a dizzying read; chapters run the gamut from from “they are here and meddling in human affairs” to “they are observing us secretly” to “they stay at home and surf the net.” But an entire section of the book is devoted to reasons extraterrestrial civilizations might not exist. And the list is long. As you read it, these once-infinite odds of contact start to feel quite small.

In 1960, radio astronomer Frank Drake began listening for signals of alien technology—first with Project Ozma and later with SETI, which continues today. Drake was hopeful — the search for life always is — but he needed to frame the usefulness of his quest. How many transmitting civilizations might be out there at any moment? What were the odds he’d catch a signal?

Drake wanted to calculate — estimate, really — the number of technologically active civilizations in the galaxy. He devised what’s called the Drake equation, which looks like this:

N = R* ∙ fp ∙ ne ∙ fl ∙ fi ∙ fc ∙ L

You take the rate at which stars form in the galaxy and multiply that by the fraction of stars that have planets, the number of planets per star that are suitable for life (or “biofelicitous”), the fraction of habitable planets on which life appears, the fraction of alien life that evolves to intelligence, and the fraction of intelligent life that develops technology that puts proof of existence into space. When you multiply by fractions, your result gets smaller.

There is one more term in Drake’s equation that constrains the final result: L, or the average lifespan for a technological civilization. Do they last the millennia it takes to colonize the stars? Or do they blow themselves up in a century? I won’t dwell on our own sustainability so far as a model; suffice it to say, we seem to be very good at skating on that edge.

The fact is, almost all of these factors are guesses. Scientists estimate the rate of star formation as between five and 10 per year; in the years since Drake began his work, we’ve discovered that exoplanets turn out to be bountiful. But beyond that, we still don’t know.

In 1961, Drake took an extraordinarily optimistic stab at things — 100 percent of biofelicitous planets developing life, 100 percent of those going on to develop intelligent life — and ended up with between 1,000 and 1 billion civilizations in the galaxy at any given time. In his book Cosmos, Carl Sagan went with his hopes for civilization’s ability to persevere and estimated values that ended up with N, the number of civilizations out there in the stars, at several million. (Fermi’s paradox rears its head.)

The Drake equation isn’t inherently optimistic, though. The optimism comes in with the estimations for each factor. And perhaps with the brevity of the equation itself.

This is what the pessimists do: They show you how much we do not know. They dissect the idea of biofelicity and show you how much more it means than the right amount of sunlight and some water. On Earth, we are in the right part of the galaxy, not too close to the central black hole or forceful clusters of stars or lethal gamma ray bursts; Jupiter may deflect comets and other bombardment from the inner solar system, absorbing interlopers into its own great gas body; our large moon stabilizes Earth’s axis, and thus its seasons; Earth’s tectonic activity drives evolution, gives us mountains and oceans, cloaks our planet in a protective magnetic field, and maintains the balance of carbon in our air. If that seems like a lot, there’s even more.

The pessimists enumerate the steps between life and intelligent life, too: the mysterious leap from primitive prokaryotic cells to eukaryotes with their nucleii, which took a billion and a half years after life’s emergence to happen; the five times life on Earth has been almost eradicated, by celestial bombardment or other cataclysm, but not quite.

Listen to the pessimists and you’ll start to fear that the odds are impossibly small. They leave you practically aghast that you exist, it’s just so improbable. Improbable and absurd and amazing. The sky seems empty and cold. We, at least, are here, but it’s hard to be hopeful of anything else.

It’s impossible to know what the odds are. They’re interesting to think about — the Drake equation forces us to grapple with questions about stellar and planetary formation, geology, chemistry, and evolution, as well as the origin and future of life on Earth. It’s ostensibly about looking outward, but it’s a mirror for ourselves, too.

We try to estimate the odds, as a thought experiment or as a guideline for how we search for extraterrestrial life, where we put whatever resources can be found. But the odds aren’t the endgame.

Let’s say we found one signal. Or one little probe floated into the solar system. The Drake equation or any pessimistic list of improbabilities would cease to matter. Do we really care how many other lives there are out there? Or do we just care whether or not there are?