Extinction is Guaranteed if We Do Not Colonize Space
The Earth is a dangerous place, and humans make it more so. There are many scenarios, from nuclear war, to designer diseases, to nanotech goo, too environmental catastrophe where we can wipe ourselves out. Further, there are events almost entirely beyond our control, like meteor impacts, which could wipe us out. The Earth is a mass-graveyard: most species which have ever existed are extinct.
The Earth is a single point of failure. If all self-sustainable human breeding populations are on Earth, we are much more likely to go extinct, and far sooner.
Getting of the rock is about human survivability in the longer run. Getting out into the solar system, learning how to create habitats and breeding populations, increases our viability. Spreading to other solar systems, whenever we can, will increase it even further.
On the other hand, if we stay on Earth, especially given how incapable we are of acting in basic racial self-interest (as proved by climate change) our odds of an extinction event, and soon, go way, way up.
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