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the argument between Beit Hillel (BH) & Beit Shammai (BS) was whether we follow the qualified majority based on the daat of man, or the most correct answer. R Eliezer(RE) was right and the rab...
Answer
#1: Initial revision
the argument between Beit Hillel (BH) & Beit Shammai (BS) was whether we follow the qualified majority based on the daat of man, or the most correct answer. R Eliezer(RE) was right and the rabbis led by R Yehoshua(RY) knew it. RE brought proofs from the heavens that he was saying the truth. RY replies that he knows that RE is right but we are not deciding like him as according to the halachic minds of man down here it makes more sense not like RE. See the introduction to the Kzot for more on this concept. So in the disagreements between BH & BS - RE holds that BS are correct as they are sharper. But the majority holds that they go like BH. Since there is no resolution to this argument, they ostracized RE so there would not be 2 torahs in Israel. See mesiliat yesharim.