Judah Halevi was a poet. Saadia Gaon and Moses Maimonides were philosophers. On the surface, their approaches to Judaism seem almost completely different. But they agreed almost completely on one point: "Why believe?"
Halevi (1086-1145 CE) is today best remembered for The Kuzari , which presents a fictional dialogue between a king and a Jewish sage. The king dreams an angel told him that his actions were not pleasing to God, so he asks a philosopher, a Christian, and a Muslim for advice.
The philosopher responds with airy abstractions, saying that God is beyond our understanding and cannot be pleased or displeased. The philosopher gives logical arguments but dismisses the king's main concern: what he should do to please God.
The Christian and the Muslim are more sympathetic, but they fail to offer adequate evidence for their religious claims. Both of them recognize the validity of the Torah, however, so the king finally turns to a Jewish sage.
The sage says simply that he believes in the God of Abraham, who revealed Himself to the entire Jewish nation at Sinai and proved Himself by miracles. Because God revealed himself publicly to a vast number of people, His existence and revelation are undeniable, as is His choice of the Jewish people as "the pick of mankind."  The sage says that by converting to Judaism and following the law, the king could make his actions pleasing to God.
Halevi was deeply distrustful of our ability to find religious truth by reason. He based his claims on the belief of Jewish people in his time that their ancestors had experienced an anthropomorphic God at Sinai.
On the other hand, both Saadia (882-942 CE) and Maimonides (1135-1204 CE) thought that reason could find any religious truth we could understand. Saadia wrote:
"There exist three sources of knowledge: The knowledge given by sense perception; the knowledge given by reason; and inferential knowledge." 
Saadia thought we could get most of our knowledge, including religious knowledge, from those sources. He later added "authentic tradition" as a fourth source, but unlike Halevi, did not mainly rely on it. Maimonides wrote similarly about the three grounds of knowledge:
"The first is a thing for which there is a clear proof deriving from man's reasoning ... The second is a thing that a man perceives through the five senses ... The third is a thing that a man receives from the prophets or from the righteous." 
Like Saadia, Maimonides recognized the validity of tradition because he believed in rabbinic Judaism, but he put it in last place. Most of his ideas about God were based on philosophical reasoning.
Saadia and Maimonides both recognized tradition, but Halevi denied that reason was a reliable guide in religion. How did the three of them agree "almost completely"?
They agreed in assuming that beliefs could only be justified by looking back. For Halevi, we looked back at our tradition about Sinai. For Saadia and Maimonides, we looked back at empirical evidence and logical arguments. We had to look backward at the reasons for the beliefs, not forward at the results of the beliefs.
As a result, they were only vaguely aware that beliefs do a lot more than make statements. Beliefs also perform moral, psychological, and social functions. They help us to lead decent, happy lives in stable, harmonious societies. Saadia and Maimonides followed the principle in their actions, even though it did not fit into their philosophy. The closest Maimonides came was his view that certain beliefs were "necessary" for social harmony.
Religious beliefs, in particular, are mainly about life rather than about logic. You could give a dozen logical interpretations of a belief that "God gave the Torah to Moses at Sinai" and they'd all be wrong. But the belief isn't meaningful and justified because of logic. Instead, it's meaningful and justified because it supports community, morality, respect for law, and reverence for the Divine.
Many centuries later, Moses Mendelssohn observed that even logically incorrect beliefs could have good results. He said that we should consider the importance of those results:
“Whoever cares more for the welfare of mankind than for his own renown will keep a rein on his opinions concerning [incorrect beliefs] … I am obliged to remain silent if these errors are accidentally connected to the promotion of the good.” 
Logic, evidence, and tradition are all valid justifications of belief -- but results are also important.
The truth of the Jewish tradition resides only partly in history. Its role in our present and future gives it on-going truth and meaning for our lives.
Gottlieb, M. (2011), Moses Mendelssohn: Writings on Judaism, Christianity, and the Bible. Brandeis University Press, Waltham, MA.
Lewy, H. et al (2006), 3 Jewish Philosophers. Toby Press, London, UK. Kindle edition.
Twersky, I. (1972), A Maimonides Reader. Behrman House, Springfield, NJ. Kindle edition.
1. The formal title is The Book of Refutation and Proof on Behalf of the Despised Religion.
2. Lewy, H. (2006), loc. 8403.
3. Ibid, loc. 3040.
4. Twersky, I. (1972), loc. 6194.
5. Gottlieb, M. (2011), p. 12.