I think during the Mackie episode I mentioned that proving the existence of God through Reason seemed to me to be a decidely Western and Christian undertaking. I speculated that it wasn't an issue for Eastern religions (those that have a concept of God or gods) and declared that it wasn't one for Judaism.
It occurs to me that I should stop speaking on behalf of the religion with which I affilate and yet do not practice. This past Thursday was Rosh Hashanah, the Jewish New Year. (The Jewish calendar is lunar, so the new year comes at a different time relative to our standard calendar each year.) Rosh Hashanah kicks off a 10-day period with Yom Kippur - the Day of Atonement - known as the 'High Holy Days'. It is the most deeply religious period for Jews for which there is, I think, not a direct correlation in Christianity (I'm not sure about Islam or other non-monotheistic religions).
Jewish tradition teaches that during the High Holy Days God decides who will live and who will die during the coming year. As a result, during Rosh HaShanah and Yom Kippur (and in the days leading up to them) Jews embark upon the serious task of examining their lives and repenting for any wrongs they have committed during the previous year. This process of repentance is called teshuvah. Jews are encouraged to make amends with anyone they have wronged and to make plans for improving during the coming year. In this way, Rosh HaShanah is all about making peace in the community and striving to be a better person. --About.com
So to make amends to you, fair readers, I apologize for any unjustified and baseless claims about Judaism I have made over the last year on our site or in our podcasts and will endeavor to be more responsible and scholarly in the coming year. To that point, please to enjoy a brief digression about Judaism and proofs for the existence of God.
It would sound facile, perhaps, to say that Judaism requires God. Religion needs God (or gods), no? Well, let's take a look. You may have heard Jews referred to as 'the people of the book'. This is because Jews believe that the Bible - what y'all call "The Old Testament" - and more specifically the Torah (the 5 books of Moses) were given to the Jews by God and that this book contains a large number of commandments (mitzvot in Hebrew). The book is a covenant between God and the Jews: if we obey the commandments, God will reward us with a little piece of land in the Mediterranean and maybe, if we're especially good, end all war, make the land flow with milk and honey and raise the dead so they can read Torah.
There is a convention that there are 613 mitzvot in the Bible. What's important in the practice of Judaism is to fulfill the mitzvot. Rabbinical scholars spent much intellectual energy over the last several thousand years carefully reading the bible and arguing about what all the mitzvot are, who can do them, when we should do them, how we should do them, etc. Just like with Medieval Christian scholasticism, they raised it to an art form in the era of Maimonides, a 12th century Rabbi.
Maimonides's enumeration of the 613 mitzvot can be found here. The thing that should be striking in even a casual perusal is the fact that none of the mitzvot require that you believe in God. You must love and fear God, sanctify his name, emulate his ways, but you don't have to believe that he exists. Can you not believe in the existence of God and yet completely fulfill the mitzvot? Does that even make sense?
If you read the long list of remaining mitzvot, it seems like you could do most if not all of them without taking a stance one way or the other on the existence of God. As long as you could justify commitment to the covenant one way or the other, you can be observant without explicit belief. Opinion: It is this subtle disentanglement of faith and moral observance that made possible Jewish humanism that gained so much traction in the Western intellectual tradition beginning in the 18th and 19th century.
The key to answering the seeming possibility of Judaism without God is in the first mitzvah (singular) to 'know there is a God'. It's important to understand that in this context, 'to know' is not what we understand as knowledge vs. belief. The first mitzvah comes from the section of the Bible containing the Ten Commandments, Exodus 20: "I am the LORD thy God, who brought thee out of the land of Egypt, out of the house of bondage." The Rabbinical scholars that Maimonides was following interpreted this as establishing the following things: 1) God exists, 2) He alone is God, 3) His authority to pre- and proscribe actions is absolute.
Maimonides interpreted it as requiring belief in God. I.e. the first mitzvah - the first moral commandment - is to believe in God. That commandment is in turn a statement by God that he is, and is your God. So you could say the first article of Jewish faith is to believe that God exists, based on the fact that God revealed through his divine prophet Moses that He exists. This is reinforced in the most important prayer in Judaism, the Shema, which is recited twice daily: "Hear, O Israel, the Lord is our God, the Lord is One."
To net this out, in Judaism, belief in the existence of God is not a subject of epistemic speculation or in need of rationalist justification. It is a moral imperative - in fact it is the first moral imperative that grounds all of the other moral requirements of the religion. Now, again, it may seem an obvious point that belief in God is necessary for religion, but a key distinction here is that doubting the existence of God is not indicative of a crisis of faith, it is a direct contravention of moral commandment. In this way, proving the existence of God is not only not necessary (belief is commanded), but also, if undertaken because one needs the justification, is a violation of a central moral tenant.
So you don't find a tradition of trying to prove the existence of God in Judaism. What would be the point? Either you are a practicing Jew and you follow the commandment to believe just like you do keep kosher, act justly towards others and pray or you don't. It doesn't mean people didn't think of it: Maimonides himself trotted out a few versions of the Cosmological Argument and did some other work in negative theology and the problem of evil. But it's a strictly intellectual exercise by someone already in the faith and not motivated either by a desire to reconcile faith and reason, justify faith rationally or provide compelling arguments pro-theist or anti-atheist.