In the modern era, we associate Islam mainly with terrorism, barbarism, and opposition to science. But it was not always so. Long ago, early encounters with Islam influenced the later development of modern Judaism.
Truth to tell, I don’t know why Medieval Islamic civilization fell apart and became the house of horrors that it is now. Some people study it. I hope they don’t see the same things happening in our civilization.
Around 762 CE, Baghdad became an intellectual center as Islamic scholars translated ancient Greek works of philosophy, science, and mathematics into Arabic. They wanted to learn “falsafa” (philosophy) so that they could argue effectively against Jews and Christians. They translated and discussed the available works of Aristotle and Plato, as well as Euclid (mathematics), Archimedes (mathematics and science), Galen (medicine), and Ptolemy (astronomy). 
Based on what they learned from the Greeks, Islamic sages published religious apologetics called “Kalam” (“speech”). Jewish thinkers were influenced by Kalam’s methods of argument, as well as by its assumption that Divine revelation must be entirely compatible with reason and science.
Our sages’ response to Kalam changed what many Jews thought it meant to be Jewish.
Prior to our confrontation with Islamic Kalam, our ancestors conceived of Judaism simply as trust in God. With the exception of the Hellenistic Jewish philosopher Philo (25 BCE - 50 CE), they didn’t give much thought to metaphysical questions, such as God’s nature. Nor were they terribly interested in epistemological questions, such as how they reached and validated their beliefs. Menachem Kellner writes that historically:
“Loyalty to God, Torah, and Israel, therefore, is the hallmark of the Jew: loyal behaviour, not systematic theology, is what is expected and demanded.” 
Islamic Kalam changed that. No longer could Jews live by a simple faith without thinking much beyond what it required of them. Now, they had to answer arguments and define terms. They had to ask, “What do I mean by ‘God’? How do I know that the Torah and the rabbis are correct?” As Kellner remarks:
”With the rise of Islam from without and of Karaism from within, Judaism was confronted by challengers that it could not ignore. Islam was an aggressively proselytizing religion, and Karaism denied the *Jewish* legitimacy of Rabbanite Judaism.” 
Based on Greek philosophy, Aristotle in particular, Kalam texts all have pretty much the same structure:
“[They] … follow a set pattern of discussion, which starts from universal issues (epistemology, the creation of the world, God’s unity and justice) to issues that are more narrowly tied to the specific religion of the author.” 
It’s worth comparing that to the structure of Saadia Gaon’s (882-942 CE) Book of Beliefs and Opinions, which gave a Jewish response to Islamic Kalam:
* Chapter 1: Concerning the belief that all existing things have been created
* Chapter 2: Concerning the belief that the Creator is one
* Chapter 3: Concerning command and prohibition
* Chapter 4: Concerning obedience and rebellion
* Chapter 5: Concerning merits and demerits
* Chapter 6: Concerning the soul and death
* Chapter 7: Concerning the resurrection of the dead
* Chapter 8: Concerning the redemption
* Chapter 9: Concerning proper conduct in this world
Just like Islamic Kalam, Saadia’s book starts at a universal level, then progressively drills down to more specific issues of Jewish religious faith and morals. Saadia’s work also shares many stylistic features with Islamic Kalam, such as dialectical arguments (“As for those who assert …, to them I say …”).
As a result, Jewish Kalam texts, such as those by Saadia, are “so closely akin to Muslim kalam that, at first sight, only the prooftexts appear to be different.” 
However, Kalam’s influence was more than methodological and stylistic. Yoram Hazony’s book The Philosophy of Hebrew Scripture shows that the Torah contains epistemology, but only of a primitive kind, such as “look at evidence.” For the first time, Jewish thinkers were challenged to examine the basis of their knowledge and beliefs. Their answers corresponded almost exactly to Islamic answers: sense perception, reason, and “authentic tradition.”
Grounding belief on empiricism and logic was not a content-neutral change. Instead of simply accepting certain beliefs about God because the Torah or rabbis said so, Jewish thinkers were forced to ask how they could square the beliefs they professed about religion with the beliefs they professed about knowledge.
That process led them to a new view of Judaism: It was no longer simply our people’s way of life, but it was a set of more-or-less provable beliefs. Thus was born Jewish systematic theology and, Kellner laments, a certain amount of dogma.
Maimonides (1135 - 1204 CE) stated the new viewpoint starkly in Chapter 13 of the Mishneh Torah:
”"He who repudiates the Oral Law is ... classed with the epicureans (whom any person has a right to put to death). As soon as it is made public that he has repudiated the Oral Law, he is cast into the pit and is not rescued from it, he is placed on a par with heretics, epicureans, those who deny the divine origin of Scripture, informers, and apostates -- all of whom are ruled out of the community of Israel." 
Frank, D. and Leaman, O. (2003), The Cambridge Companion to Medieval Jewish Philosophy. Cambridge: Cambridge University Press.Hazony, Y. (2012), The Philosophy of Hebrew Scripture. Cambridge: Cambridge University Press. Kindle edition.
Kellner, M. (2006), Must a Jew Believe Anything? Portland: Littman Library of Jewish Civilization.
McGinnis, J. and Reisman, D., editors (2007), Classical Arabic Philosophy: An Anthology of Sources. Indianapolis: Hackett Publishing Company. Kindle edition.Rosenblatt, S., translator (1948), Saadia Gaon: The Book of Beliefs and Opinions. New Haven: Yale University Press.
Twersky, I. (1972), A Maimonides Reader. Springfield: Behrman House. Kindle edition.
 McGinnis, J. and Reisman, D. (2007), loc. 107.
 Kellner, M. (2006), p. 18.
 Ibid, p. 49.
 Frank, D. and Leaman, O. (2003), p. 68.
 Ibid, p. 71.
 Twersky, I. (1972), loc. 2733.