The universe is calculated to be be roughly 13.8 billion years old with the observable universe approximately 93 billion light-years in diameter. The Earth itself is 4.5 billion years old with basic life estimated to have emerged at 3.4 billion years ago. Humans have existed for ~200,000 years, and have only recently begun transmitting radio signals into interstellar space within the last century. The Fermi Paradox is the apparent contradiction between the size and age of the universe and the seeming absence of any other intelligent life other than what resides on our own planet. In simplest words: "If the universe is so old and so vast, where is everybody?". I've spent some time looking into this question and the following are some notes. There have been lot of solutions to this paradox suggested (including the Simulation Argument), though I will only focus on one here.
Why would we think the universe is teeming with life? According to the mediocrity principle, our sun, planet, and the circumstances that have brought forth life is unremarkable. It is the notion that "if an item is drawn at random from one of several sets or categories, it's more likely to come from the most numerous category than from any one of the less numerous categories". The idea is to assume mediocrity, rather than starting with an assumption that an observation is exceptional. Therefore we would expect life to pop up many times elsewhere, long before here on Earth. Yet, we've never found a single bit of evidence for it.
Looking at the Drake equation, we only begin to realize just how many barriers exist in the universe that will prevent even simplest forms of life from developing. The Drake Equation is an argument based on probability that estimates the number of intelligent civilizations with the technical ability to communicate (N). It looks at the rate of star formation in the galaxy (R*), the fraction of those stars that contain planets (fp), the number of planets orbiting that star with an environment that is capable of developing organic life (ne), the fraction of those suitable planets that actually develop life (fl), the fraction of habitable planets where intelligent life can actually appear (fi), the fraction of those civilizations of intelligent life that can communicate (fc), and the length of time that those civilizations send signals (L). The equation looks like this:
N = R* · fp · ne · fl · fi · fc · L
The Drake equation originally estimated that between 1,000 and 100,000,000 planets with civilizations existed within our galaxy, which further begs the question "where is everybody?". This leads us to what's known as the rare earth hypothesis where it's suggested that complex life is extremely unusual (but not impossible) and adds additional constraints to the Drake question.
In order for abiogenesis and complex life to even emerge, there are a host of fortuitous circumstances that must occur. For example, some very unlikely circumstances that have occurred to make Earth a potentially habitable world includes having a moon to shield the planet and create tidal pools, volcanic activity and plate tectonics, being in a region of the galaxy with less radiation than the center, the axial tilt of our planet which may have stimulated evolution, ocean to land ration, just to name a few.
The probability of intelligent life emerging and becoming a technologically advanced civilization makes the likelihood even more improbably. Some requirements are the dinosaurs being wiped out as the dominant species of the planet, the control of fire by early humans, and fossil fuels for the industrial revolution, but there are dozens.
With these constraints, we could come up with an equation that's much longer than the Drake equation and it would make the probability of a technologically advanced civilization even more improbable. That being said, even at an extremely low probability, we would would still expect life to occur somewhere - and here we are. We would have to exist in a universe capable of supporting intelligent life, no matter how incredibly rare, otherwise no one would be here to ask the question.
The Rare Earth hypothesis is the Fermi paradox solution I personally subscribe to. When I think about what exact circumstances were needed for intelligent life to emerge here on Earth, the probability of it actually occurring makes it exceptionally rare that it's likely that most galaxies don't spawn any civilizations.
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