What do you do when you want to follow the stringencies of two opinions, and they contradict?! Bottom line principle: Follow Beit Hillel! What if you like Beit Shammai's position better? Sure, go ahead. What you should *not* do is follow only the leniencies of both Beit Hillel and Beit Shammai. *Nor* only the stringencies of both. Example: Is the new year for the trees on the first or the fifteenth of Shevat? There are practical differences... and Rabbi Akiva kept the stringencies of both dates. Also: The intertextuality of this Gemara - going back to the principle that the ruling is always in accord with Beit Hillel, as told by a bat kol. But that position only works as long as you accept the notion that a bat kol is authoritative - which R. Eliezer does not. Our daf presumes knowledge or the background... And is thereby engaging in a nuanced discussion about the multiplicity of approaches within halakhah.