Saadia Gaon lived in Babylonia from 882-942 CE under Muslim rule. Much of what we know about his work comes from letters and materials found in the Cairo Geniza. Saadia was apparently one of the only geonim successful in proving that world Jewry viewed Babylonia's religious leader as more authoritative than Israel's. There had been tension between Babylonia and Palestine for generations with Babylonia obviously gaining ascendency because of their Talmud scholarship.
Aaron ben Meir, the gaon of the Palestinian Jewish community, tried reclaiming some of that authority in 921 CE by introducing a new three-year Jewish calendar which changed the date of both Passover and Rosh HaShanah. This was a real challenge to Saadia's authority
Saadia wrote a civil letter asking that ben Meir not go changing things which had worked so well for the past 400 years and were not of Palestine's business anyway. Ben Meir, feeling that his authority was being threatened, restated the change in stronger language: all true Jews would celebrate Passover on Sunday in 921 CE, not Tuesday. Saadia, always ready for a fight, blasted him. Letters flew back and forth between Sura and the rest of the Jewish world, each hotter and angrier. Tension mounted, literally around the world as Passover approached.
On Sunday of the first year of ben Meir's new calendar, many Jews in Palestine and some Jews outside of Babylonia held their Passover festival. The Jews of Babylonia did not. On Tuesday, the Jews of Babylonia held their Passover Seder, and most of the Jewish world followed their lead including some Jews in Palestine. That Rosh HaShanah, the vast majority of the Jewish world accepted Babylonia's calendar.
Aaron ben Meir knew when he was beaten, and he retracted his calendar. This marked the high point of the Babylonian gaonate's world authority.
During Saadia's life, the Jewish intelligentsia of Babylonia spoke Arabic and were fairly easily accepted into the Arab culture. They found the Arab world very attractive, and Saadia had the job of keeping upper-class Jews Jewish. This was not an easy task.
Think about America today. We dress, speak, and think American Western culture. We know some Jewish tradition, but not much. Our primary concentration is on American matters.
Wealthy Jews in Babylonia, North Africa, and Spain in the tenth century had a comparable situation. Like the Jews of Alexandria, they were attracted to the rediscovered Greek philosophers and many were considering rejecting their Jewish practices.
Saadia met this cultural crisis head-on and won. The high points of Muslim culture were its beautiful use of Arabic and its fondness for the Greek philosophers. Saadia wrote a philosophic work, The Book of Beliefs and Opinions, in magnificent flowing Arabic. In it, he defended the rational underpinnings of Judaism and showed logically that every rational Jew could believe in the Torah as well as Aristotle and Plato.
By applying both the accepted philosophical methodology and the language revered by the Muslim culture, Saadia succeeded in refocusing many semi-assimilated Jews back on Torah and Halachah.
However, it was a radical change in Babylonian Jewish tradition. Previously, the gaon had limited his work to Talmud and halachic teachings. Saadia's book of philosophy caused quite a stir.
Saadia didn't stop there. He wrote the first Hebrew grammar book which explained how the holy language worked. He provided a Hebrew dictionary plus a compendium of rhyming words for Hebrew poets. He was the first to write an Arabic translation of the Bible. He included commentaries, explanations, and grammatical notes as well. His translation continues to be the authoritative Bible for Jews in Arab lands.
Saadia thus brought a new rich understanding of Jewish tradition to the Babylonian academies and the Jews living in the Muslim world. His philosophical work paved the way for future Jewish thinkers, and his approach to Torah influenced a century of Jews.
His most important accomplishment, however, was his confrontation with the Karaities.
He issued articles, letters, and responsa attacking the doctrine of the Karaites, and even declared that they were not Jews. One of his primary targets was Aaron ben Asher. The fury of his attack must have shocked the Karaites. They responded with their own letters and attacks, but their Arabic wasn't as good as Saadia's, and their defenses were less convincing. Saadia successfully defended rabbinic authority against the Karaite philosophical invasion.
Saadia didn't win all of his fights. The Exilarch during Saadia's youth was David ben Zakkai (no relation to Yochanan ben Zakkai). David ben Zakkai was involved in some (apparently shifty) land deals which came to court. He, as head of the community, was expected to judge the case. He asked for the signatures of the gaon and his colleague to make his decision appear legitimate and free from prejudice. Saadia suspected that something was not right, and, on legal grounds, he refused to sign. David ben Zakkai was furious. He wrote angry letters, threatening to have Saadia's position taken away from him. He went so far as to assign another scholar to Saadia's post.
Saadia proceeded to name a new exilarch, an incredibly chutzpadik thing to do. News of the controversy came to the ears of the caliph. Not liking to see splits in his community, he took matters into his own hands. He informed Saadia that the exilarch had right of way. Saadia quietly obeyed and was dismissed from his post.
Sources: Gates to Jewish Heritage