Our Sages taught: The mitzvah of Chanukkah is a [single] candle for [each] man and his household [ביתו can also translate as wife], and those who are scrupulous: A candle for each and every one, and the scrupulous among the scrupulous: Beit Shammai says: The first day he lights eight, from here going [forward], he goes on reducing, and Beit Hillel says: The first day he lights one, from here going [forward], he goes on adding. Ulla said They argued about it - two Amoraim from the West [i.e., Eretz Yisrael]: Rabbi Yosi ben Avin and Rabbi Yosi ben Zevida. One said: the reasoning of Beit Shammai is: to correspond to the days which are entering,and the reasoning of Beit Hillel is: to correspond to the days which are going out; [the other] one said: the reasoning of Beit Shammai is: to correspond to the bulls [offered on Sukkot] and the reasoning of Beit Hillel is that [we] raise up in holiness and [we] do not lower. Rabbah bar Bar Chanah said that Rabbi Yochanan said:
(יז) כָּל מַחֲלֹקֶת שֶׁהִיא לְשֵׁם שָׁמַיִם, סוֹפָהּ לְהִתְקַיֵּם. וְשֶׁאֵינָהּ לְשֵׁם שָׁמַיִם, אֵין סוֹפָהּ לְהִתְקַיֵּם. אֵיזוֹ הִיא מַחֲלֹקֶת שֶׁהִיא לְשֵׁם שָׁמַיִם, זוֹ מַחֲלֹקֶת הִלֵּל וְשַׁמַּאי. וְשֶׁאֵינָהּ לְשֵׁם שָׁמַיִם, זוֹ מַחֲלֹקֶת קֹרַח וְכָל עֲדָתוֹ:
(17) Every argument that is for [the sake of] heaven's name, it is destined to endure. But if it is not for [the sake of] heaven's name -- it is not destined to endure. What is [an example of an argument] for [the sake of] heaven's name? The argument of Hillel and Shammai. What is [an example of an argument] not for [the sake of] heaven's name? The argument of Korach and all of his congregation.
It was taught in a baraita that a non-Jew came before Shammai and asked him, "How many Torahs do you have?" Shammai responded, "Two, the Written Torah and the Oral Torah." The non-Jew said, "I believe in the Written Torah, but not in the Oral Torah. Convert me and teach me only the Written Torah." Shammai berated the non-Jew and chased him out The non-Jew came before Hillel and Hillel converted him. One day Hillel taught him the alef-bet, and the next day he switched the names of the letters. He said to Hillel, "Didn't you teach me the opposite yesterday?" Hillel responded, "Just as you trust me in the instruction of letters, so to you must trust me in the instruction of the oral Torah." Another time a non-Jew came before Shammai and said, "I will convert if you can teach me the entire Torah while I stand on one foot." Shammai pushed the non-Jew aside with the ruler that was in his hand. The non-Jew came before Hillel and Hillel converted him saying, "What is hateful to you, do not do to your neighbor, that is the entire Torah, the rest is just commentary, now go and study." Another time a non-Jew was passing behind the Beit Midrash and heard the voice of the sofer who was saying, "These are the garments you shall make , a breastplate, and an ephod" (Exodus 28:4) The non-Jews asked, "Who are these garments for?" The sofer responded, "For the Cohen Gadol." The same non-Jew said to himself, "I will go and convert so that they will make me the Cohen Gadol." The non-Jew came before Shammai and said, "Convert me on the condition that you make me Cohen Gadol." Shammai pushed him aside with the ruler that was in his hand. The non-Jew came before Hillel, and Hillel converted him. Hillel said to him, "We cannot anoint someone unless they learn the ceremonies of kingship, go and learn the ceremonies of kingship." The new convert went and learned the ceremonies of kingship. When he arrived at the verse, "and the common man that draweth nigh shall be put to death" (Numbers 1:51), he asked, "Who is this verse speaking about?" Hillel answered him, "It's even relevant to David, King of Israel." The new convert made himself a "kal v'chomer": Israel are called Children of God, and out of God's love for them, they are called the first born children of God. If even in regards to them it is said, "and the common man that draweth nigh shall be put to death..." how much more so for me who comes to the Jewish people with his staff and bag. He came before Shammai and told him: "How could I have ever qualified to be a high priest? The Torah states: 'and the common man that draweth nigh shall be put to death'" He came before Hillel and told him: "You are humble Hillel, may blessings rest on your head for moving me under the wings of the divine presence." Eventually the three [converts] happened to meet somewhere and said: "Shammai's strictness would have killed us. Hillel's humility brought us under the wings of the divine presence."
Marc-Alain Ouaknin, The Burnt Book, p. 84
The Talmud says of Talmudic discussions: "The words of one and the words of the other are the living words of God. This statement should be seen as conditional: If there are words of one and words of the other, then they are words of the living God, and as a result, are living words.
Yevamot 14b
Come and hear: Although Beit Shammai and Beit Hillel are in disagreement on the questions of rivals, sisters, an old bill of divorce, a doubtfully married woman, a woman whom her husband had divorced and who stayed with him overnight in an inn, money, valuables, a perutah and the value of a perutah, Beit Shammai did not, nevertheless, abstain from marrying women of the families of Beit Hillel, nor did Beit Hillel refrain from marrying those of Beit Shammai. This comes to teach that they showed love and friendship towards one another, thus putting into practice the Biblical text, "Love truth and peace." (Zecharia 7:19).