Yes. Those same questions were asked a few million years ago in Africa then again a few hundred years ago in Spain, England and Portugal. And they were asked innumerable times within that intrval. The answer is that some stayed behind. Some did not.
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These are great questions. As a space nut let me try some answers. The final purpose is..... To explore, to find answers, to increase our knowledge, to get out of the madding crowd, a hundred reasons from ridiculous to sublime. And after exploring the universe what do we accomplish? We may adapt, evolve into different shapes, and increase our tolerance for strangeness. And we may begin to finally understand ourselves.
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Are we talking about sending people into the cosmos to live?
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More that just live. They will form communities, governments, societies. They will adapt, overcome challenges, love, have babies, fight and die. They will repeat all the triumphs and tragedies we have, but they will be elsewhere. In time it will be more than mere colonization, it will be diaspora begetting diaspora.
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Are we going to mine the surface of Mars for materials not available or in dwindling supply here on Earth?
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Short answer, yes..But. There needs to be something really short, or totally non-existant to justify the cost of Mars Earth transit at the present cost of space freight. The main reason to go to mars is to learn about planetary origins, some really strange land forms, and later possibly terraforming. But there is another reason. To make space travel cheap enough that the average citizen can afford to go to space. Then come the colonies, the societies, etc.
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Are we going ot find something that justifies the huge expense and the manhours put into it or is it going to be another series of moon missions (i.e. we managed to do some really neat stuff. Now we're bored with it).
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First let me debunk the huge expense myth. The Total NASA budget is .7% of the annual US budget. The space exploration piece of that starts small and increases over time to about .5% of the annual budget. This a very small investment over a long time to improve mans chances of surviving another impact the likes of which killed off the dinosaurs. Next the Apollo missions to the moon were more political than scientific because we were in a race with the Soviets. We won, we hung up the trophy, took a picture of an American flag, and stopped going to the moon. It is not that the scientists got bored, they were overcome by political reality. In all the missions only one scientist went to the moon. Harrison Schmitt on Apollo 17.