Yes, I think we should. Why? It'll solve most of the US' problems.
Think about what it was like to live in the 60s-70s during the Mercury, Gemini and Apollo missions. The USA was crazy innovating, frontiers of science and exploration were being broken every
week.
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There were no jobs going overseas. No high tech company was outsourcing work to sweatshops in India or China. Why? Because the nation was innovating so goddamn fast no one else had the capacity to actually figure out what the hell they were doing. Even if they did outsource there was no problem because they'd know there would be something new that the US and only the US could create.
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The education program was booming. You didn't need a special program to tell kids being a scientist or engineer or mathematician was cool. It was printed every day in the newspapers and aired on television that the generation of children and teens who grew up in that era aspired to become scientists and engineers.
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Fields of science. A lot of people point to increasing funding for the NSF because, well, science. However, NASA should be put under consideration as well. NASA, while mainly focused on space, is the frontier of all fields of science and technology. Physics for understanding the core fundamentals of how the universe works, chemists who design a material that can protect a crew from radiation when traveling for several months at a time, engineers solve how to send and set up a base of operations on another planet, biologists to conceptualize what extraterrestrial life might look like and how we'd identify it, geologists to profile planets and look for resources like water, the list goes on and on.
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A cultural benefit. When the photograph "Earthrise Over the Moon" was published, it became the icon for the entire planet and human race. Nonononono, not an icon for a planet with boundaries, war, territories, countries, but for a planet as one. In the next few years as the Apollo project continued, things like the Comprehensive Clean Air Act, Earth Day, the EPA, Doctors Without Borders, Clean Water Act, ban on DDTs, Endangered Species Act, catalytic converters, unleaded gas were introduced, passed and founded. You try to put a price tag on converting our minds to think of Earth as a planet we all share.
And just as a reference from a post I made on my Tumblr

also good videos where a good number of my ideas came from
http://www.youtube.com/watch?v=CbIZU8cQWXc - Talks about the origins of the space race and NASA's influence on the future of the United States, as well as questioning those who claim that the USA cannot afford such a funding increase
http://www.youtube.com/watch?v=BFO2usVjfQc - Focuses mainly on the cultural impact of space exploration