I'm not sure why you picked hasidic. It sounds like you are confuing hasidic with haredi which is roughly speaking the general ultra-orthodox population. The hasidim are a specific strain of orthodox Judaism which arose in the last 1700s. But many haredim are not hasidim. In any case, the Shabbat goy has nothing to do with either the larger category or the smaller category. Note also that neither category describes all of Orthodox Judaism.
You are incidentally correct that orthodox Judaism does not believ that gentiles have to obey the Torah. However, I'm not sure that's not due to a universal moral rule. The belief is generally defended by claiming that there is something metaphysically different about Jews or physically different. For example, there are hasidic sources which argue that keeping the specific commandments for Jews have direct reverberations on the world which make things in general better. That doesn't happen for non-Jews. The distinction between a universal moral law and a non-universal moral law isn't so clear. For example, if my universal moral law contains the rule "people who are HIV+ should not have unprotected sex" arguably this is just a non-universal law for HIV+ people rather than a universal. What quantifiers and qualifiers are acceptable in such rules for them to be universal is not clear to me.
(Incidentally, if we are talking about hasidim in particular, classical hasidic thought has some things which unambiguously separate Jews from non-Jews in really nasty ways. For example, some hasidim believe that Jews have a holier soul than non-Jews. Others declare non-Jews to be intrinsically less moral. For example, the Tanya, one of the formative texts of the Lubavitch hasidic movement, says that non-Jews can't engage in actual altrusim but will only be altruistic for selfish reasons whereas Jews can engage in genuine altruism).
Wikipedia says, "Haredi Judaism comprises a diversity of spiritual and cultural orientations, generally divided into Hasidic and Lithuanian-Yeshiva streams from Eastern Europe, and Oriental Sephardic Haredim." I wasn't aware. Changed to just say "some orthodox".
Exodus 22:25, Leviticus 25:36, and Deuteronomy 23:20-21 forbid Jews from charging interest on loans to "your brother" (other Jews). (This is to me the most convincing argument against Judaism and Christianity, because it's too simple to argue around. That proscription is just wrong, in exactly the way you would expect laws written by uneducated tribal people to be wrong.)
Roman Catholics believe they must follow the Old Testament laws, except for the ones they don't have to follow; but during much of the middle ages in Western Europe, this was one of the ones they had to follow. They interpreted "your brother" as meaning "brother Christians". So Jews could lend to Christians with interest (and, presumably, Christians could lend to Jews). This was convenient for everyone. The Jews were necessary to work around an irrational moral prohibition of the Christians.
Of course, the Jews had to take on the guilt of violating the moral code, even though it was for the benefit of the Christians. (This was also convenient; it meant that after some Jews had loaned you an especially large amount of money, you could kill or expel them instead of paying them back, as the Spanish monarchy did in 1492).
Later on, some orthodox Jews hired goyim to turn lightswitches and other electric devices on and off for them on the Sabbath. They're called Shabbos goy, the Sabbath goy (thanks, Alicorn!).
JCVI is considering moving from an on-site hardware grid, to cloud computing. There are lots of reasons to do this. One is so that Amazon can be our Shabbos goy.
We develop lots of bioinformatics software that we're supposed to, and would like to, give out to anyone who wants it. But if you don't have 800 computers at home, connected using the Sun Grid Engine with a VICS interface and using a Sybase database, with exactly the same versions of C++ and Perl and every C++ and Perl library that we do, you're going to have a hard time running the software.
We can't put up a web service and let anybody send their jobs to our computers, because then some professor is going to say to their freshman class of 200 students, "Today, class, your assignment is to assemble a genome using JCVI's free genome assembly web service."
If we could charge users just a little bit of money, just a fraction of the cost of running their programs, we could probably do this. Then people wouldn't be so cavalier about running a program repeatedly that takes 500 CPU hours each time you run it.
But we can't, because we're an academic institution. So that would be evil.
So we need a Shabbos goy. That's Amazon. We can release our software and tell users, "All you have to do to run this is to get an account on the Amazon cloud and run it there. Of course, they'll charge you for it. They're evil."
(The Amazon cloud is evil, BTW. They charged me for 21G of RAM and then only gave me 12, and charged me for 24 1GHz processors and gave me about 1/4 of that. I spent over $100 and was never able to run my program; and they told me to stuff it when I complained. But that's another story.)
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