We have plenty of robots doing our work already.
They have taken lots and lots of jobs away from lower-middle-class workers.
That's part of what's fueling this anger.
A lower-middle-class, public-school-education-in-a-bad-neighborhood cashier who just lost their job and had to go on welfare or starve because a robot made by a giant corporation and now owned by his/her ex-employer took that job - that person is very willing to listen to what Trump or Nigel Farage has to say, and the usual rhetoric by nativists about how once all the immigrants go away, things will be golden again. That is not typically a historical reality very long, as this kind of up-the-locals attitude has played out throughout history when international leaders neglect society sufficiently. The violent, unfriendly nativist locals often do take over, but whether they are economic failures (Robespierre and much of the early French Revolution), economic successes (ironically enough, Nazi Germany, at least at first), or somewhere in the middle (Mao Zedong and early Communist China) - it generally ends in tears, war, or autocracy in the end. The hate in the society eats it alive.
Nazi Germany still stands as the scariest example, because they actually made their pro-German, slave-races economy *work* enough somehow to become such a major militarist power that it took most of the rest of the free world to stop them and the countries it subverted.
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You're picturing something like Zanarkand, man, with some kind of "robots do all the work, universal income system" model, where everyone can party all day because the robots take care of everything...the universal income system isn't there, nor is it unquestionably a good idea without debate, or possibly even a good idea at all. I mean, yes, FFX is fiction, but shit, look at that plotline. Scripted, or not, that's plausible.
The robots really can't take care of everything yet, nor should they probably - I mean, look at something like Mother Brain from Phantasy Star II as well, a plot point that's really relevant here if you played it. Power corrupts, and so does the influence of humankind's rampant emotionality and tendencies towards anger and hate. Power can corrupt a centralized robocracy as fully as a human society; we are making the things, after all, and teaching them how to think. Remember our discussion about Microsoft's Tay AI recently?
I don't think anyone wants her running our traffic system or crop waterers, at least until the "0 to neo-Nazi in 24 hours under the influence of too much humanity" problem is solved.
Things like that need debate. So far the roboticization of labor has *not* been economically a good thing for all but a handful of megacorporations that can afford the megaprices of a private army of labor robots.
“I would unite with anybody to do right and with nobody to do wrong."
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Frederick Douglass-