what exactly is the difference between the rationalist approach and the non-rationalist approach in Judaism. Have heard the Rambam as belonging to the rationalist side.
I assume both believe in miracles.
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Rationalism in philosophy is more about the method used to achieve conclusions than the underlying conclusions themselves.
Rationalism in religion tends to forbear supernaturalist thinking for scientific style reasoning.
Rationalist Judaism as a philosophical school is a relatively modern and loosely defined construct. "Jewish Rationalists" lay claim that Rambam and Rav Hirsch are antecedants to their school of thought, though whether they were actual rationalists or just wrote works absent mystical reference is a matter of some debate. A Jewish "rationalist" would, however, be inclined toward the style of those two Ba'alei Mesorah rather than, say, Tanya or the Zohar for handling things like metaphysics and miracles.
Bottom line: a rationalist would tend to ignore metaphysics as not empirically observable by humans, so it can't really be ascertained or meaningfully discussed. Historical miracles would be handled in the fashion of the Rambam - as engineered "coincidences" of a grand scope that utilize the rules of nature for uncanny results.
Rationalism in Jewish sense is in not philosophical rationalism but, rather a tool. In analyzing Jewish tradition as there have been many different forms Conservative, Reform, Orthodox, Modern Orthodox, Haredi, and Hasidic. The rationalist in any of these is more of trying to separate falsehoods from truths with the use of reason.
"The biggest stumbling block to love of God is the belief that the only way to remain true to the Bible is to interpret it literally. The result of literal interpretation is a material conception of God, which, in Maimonides’ opinion, amounts to idolatry." (Maimonides Stanford Encyclopedia)
For example, concerning G-d's incorporeality Rambam talks about how some Jews can make the simple mistake of G-d's being corporeal because taking the Torah literally in that sense.
"The biblical texts suggest that God is corporeal and the interpretation of these texts is not a very easy task"(The Guide for the Perplexed). From this, we can conclude that some Jews may be confused, led astray, or be susceptible to conversion to other religions when they misinterpret the Torah. If they can't understand G-d's incorporeality they might lose their faith. This is why Rambam wants to guide Jews to pure monotheism and rationalism, being using reason as a tool is helpful.
Also, to note Rambam, the famous rationalist had created the 13 principles of faith. three of which are like miracles.
The belief in divine reward and retribution.
The belief in the arrival of the Messiah and the messianic era.
The belief in the resurrection of the dead. (Chabad)
The word miracle definition in the Dictionary.com "is an effect or extraordinary event in the physical world that surpasses all known human or natural powers and is ascribed to a supernatural cause."
As it said in the Torah "for dust you are, and to dust you will return." and along with the notion that people can be resurrected is a miracle as itself as people do not see their dead relatives coming back to life.
Also, the Rambam states "The existence of G-d and his unity were admitted as unquestionable by all Jews; all Jews as Jews know that G-d exists and that He is one, and they know this through the biblical revelation or the biblical miracles."(Guide of the Perplexed PG XXI,Volume 1)
Both faith and reason can work together and in Judaism are not conflicting but rather complementary each other. For rationalism it is simply a tool to help us understand G-d and how we are live in a proper state to worship G-d. This proper state should be clear of misunderstandings from superstition to G-d's incorporeality.
The difference between the rationalist and non-rationalist is the focus on reasoning. Rationalist use of reason to help them while, non-rationalist do not use any form of philosophical techniques from the Greeks, Western, British, and etc. All non-rationalist forms of Judaism tend to be mystical rather than philosophical.