Simple enough question, and very much related to the topic.
According to traditional Jewish sources, on what basis should people believe that there's a God?
How does a Chasid know that G-d exists? He has Emunah.
How does a Litvak (a.k.a. Yeshivish Jew) know that G-d exists?
The Rambam says so, and the Raavad doesn't argue. :o)
Assuming we all exist, think, know and interact with our actual surroundings etc.
The Ontological Proof
The first class of Divine Proof is the Ontological proof. It goes basically like this:
This doesn't really convince anyone of anything.
Many philosophers have spent a lot of time tweaking this proof, but in the end I don't think anyone else really cares about it.
The Cosmological Proof
First off, you could just say that there can be an infinite regress of events.
Secondly, in the current proof what can God tell you to do? Move? Not so powerful...
Two avoid the second problem, some have changed "move" to other powers and given those to God.
Alternatively, some focus on the fundamental nature of the proof which is that there exists a being external to the general rules of science. They will argue something like this:
This doesn't really get exactly the God we want, but it is better if you assume Morality/Rationality/etc. is eternally True independent of circumstances.
The Teleological Proof
It might not be. Everyone agrees this is not a strict proof. [BTW this is why people are so opposed to Darwin, not the textual issues from Genesis 1]
Err.. Find something REALLY REALLY complex.
There is no strict philosophical proof for God; however, there is no strict proof against the existence of God either. You need to evaluate the evidence and BELIEVE.
Sources: Philosophy classes and personal research on the subject.
To my knowledge, the only "argument" for the existence of God given in the Torah itself is that He directly revealed Himself to us at Sinai:
In other words, direct experience does not require philosophical proof.
Rabbi Jacob Emden expanded this argument into our own time, saying that, "When I consider these wonders [of the survival of the Jews in exile], they appear greater to me than all the miracles and wonders that God did for our ancestors in Egypt, and in the wilderness, and in the land of Israel."
All philosophical arguments for the existence of God made in traditional sources are only intended to reinforce this basic experiential knowledge that is the heritage of the Jewish people. While these arguments can serve to shore up our beliefs against challenges, many sources see these arguments as helping us acheive a more personal, immediate connection to God.
The most common such arguments found in Jewish works are:
In my personal opinion, the various philosophical arguments for the existence of God are mainly useful for countering Hume's arguments against miracles. Briefly stated, he argues that no testimony of a miracle should be believed unless the falsehood of the testimony would be more improbable than the miracle itself. It follows, therefore, that one's ability to accept the testimony of the Jewish people's historical experience of miracles has an inverse relationship with the degree to which you think miracles are improbable.
All of the classical arguments for the existence of God are, fundamentally, arguments that we can perceive an element of the supernatural in the natural world itself. Thus, each such argument makes the possibility of miracles more plausible. At some point, it becomes more likely that Sinai Revelation really occurred than that it was made up (which, per the Kuzari argument, is very unlikely). Once you reach that point, then you have the Sinai Revelation to rely on for everything else.
I'm a terrible one for sources, however as far as I am aware there are four ways in which Judaism knows that Gd exists.
** Edit: It seems that the original midrash talks about man being better than spirit because of a wordplay, so I'm not sure where I heard this version from, but the basic concept is the same.
Rabbi Hirsch says a "proof is in the pudding" type of argument, namely if you keep the mitzvos it will become clear to you through your experiences that you are involved in something higher. It need not be explained - you will feel it, if you are keeping the mitzvos properly. This experience is an experience of G-dliness, and is the best "proof" of His existence.
This is quoted in Dayan Grunfeld's intro to Horeb.
Intelligent Design is the classic and most powerful argument for the existence of God as scripture says "from my flesh I shall see God" (Job 19:26)
(see the the treatise shaar bechina of chovos halevavos which delves into this at depth or the book "The Universe Testifies" by Rabbi Avigdor Miller. These works not only address the intelligent design of life forms but also at the structure of the universe as a whole - everything has a function. see there.)
The rise of the random evolution theory last century caused the argument to lose some impact but the tide has been turning.
With the advance of microbiology the argument has become much more powerful as the famous philosopher Anthony Flew, who for many decades flew the flag for atheism and then reversed said:
There is the "cosmological" argument of King David in Tehilim 19:1-4, where creation itself "bears witness" to the existence of G_d:
I'll admit it now, I am biased in my belief that the traditional scriptures (i.e., the 24 books of the Hebrew Bible) trump any and every other source of traditional authority.
To be sure, virtually every other traditional source of authority has good and helpful things to offer, and those who are familiar with such sources should always feel free to mine them for nuggets of truth that can benefit both them and the larger community of faith (and even the community of non-faith!).
There is a hierarchy, as it were, of authority, and God's Word trumps all other authorities.
Rambam Hilchos Yesodei Hatorah Chapter 1
Halacha 5 This entity is the God of the world and the Lord of the entire earth. He controls the sphere with infinite and unbounded power. This power [continues] without interruption, because the sphere is constantly revolving, and it is impossible for it to revolve without someone causing it to revolve. [That one is] He, blessed be He, who causes it to revolve without a hand or any [other] corporeal dimension.
Here a few of the proofs I collected a few a years ago (all from major scientists and professors by the way)
Of course, I have many stored in my computer somewhere.
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