It must be understood from the outset that it is in no way possible to demonstrate or prove conclusively whether God does or does not exist. The question relates to the reasonability of postulating one way or the other. To examine this, it is important that we first make the distinction between "God the Creator" and "God of the Torah."
I maintain that belief in a Creator of the Universe is perfectly reasonable. We do not understand how existence came to be what it is, and it is reasonable to speculate that there is an Intelligence behind it. (This is not, to be clear, an argument for Intelligent Design, which claims to offer evidence for God's existence based on gaps in nature, especially in evolution, that it proposes could not have been bridged were it not for Divine intervention. The consensus at this time in the scientific community is that Intelligent Design is a pseudo-scientific enterprise, whose arguments do not comprise anything on the order of evidence of the supernatural.) When I say that belief in a Creator is "reasonable," I mean from the standpoint of human psychology - i.e., it is a natural thing for us to believe. If we are conscious beings with the ability to create, who is to say there is not a consciousness greater than ourselves which is responsible for the universe? Does that mean that God's existence is therefore "likely"? No. Does it say anything as to the nature of God, or for that matter how many "gods" are involved? No. Might there be a race of aliens who spawned life on Earth? There may well be.
Point being, the idea of God is pure speculation, reasonable coming from the human being, but speculation nonetheless. However, the more specificity one attaches to God the Creator, often the less reasonable the speculation becomes, and the more complex theological argumentation one must formulate in order to support that belief. For instance, the notion of a "benevolent" God is one which is difficult to support. Positing such inevitably leads to the theological paradox often brought between God's omnipotence and God's benevolence (i.e., the question as to why an all-powerful God does not intervene to curtail suffering). It is precisely theological questions such as these, which appear contrived, needless, and frankly irksome, which draw many people to an atheistic position. After all, why enter into such a paradox when there is a far more straightforward way out?
Much ink has already been spilled on arguments for God's existence (cosmological, teleological/unmoved mover, Jewish-historical, etc.), and I do not intend to elaborate on these here. Suffice it to say, such arguments may highlight lapses in our knowledge, or phenomena that are unique and highly intriguing, but they in no way require that we fill in those spaces with "God," as opposed to any other natural explanation. In any case, my focus here is not on God in the abstract, ineffable sense, but on God as the giver of the Torah, God the commander, maker of miracles.
We spoke about specificity in regards to God as being inversely related to reasonability. On that count, the God of the Torah is another level of specificity altogether. To believe in this God, one has to accept that He came to Abraham, Isaac and Jacob, to Joseph and Moses, spoke to them, took the Children of Israel out of Egypt with wonders and miracles, and commanded the mitzvot of the Torah, all of which took place approximately between 3800 and 3300 BCE. In traditional Orthodox terms, one also would have to believe (despite it not saying so it the Torah) that God dictated the entire Torah letter by letter to Moses. Again, one cannot "prove" that this did, or did not, happen. The question is one of reasonability. To that end, let us ask two simple questions:
1. Where is God now?
Stated bluntly: "Show me the miracles." If a sea can split on command, if the voice of God can be heard on the top of a mountain giving over instructions to a nation, if a column of fire/cloud can follow a people's path in the desert, if manna can fall from the sky, why is it that we do not see and experience these things (or anything remotely similar) today? To one who says that some individuals do indeed hear the word of God, I say that if you examine closely there will invariably be a more reasonable explanation. It will be accountable to some combination of self-delusion, fakery, mistaken attribution, dream, trance, hypnosis or hallucination. To one who cites real-life miracles (chance meetings/synchronous events, miraculous healings/rescues, etc.), to which hundreds of thousands of individuals regularly attest, once again there will always be a more plausible explanation, such as statistical eventuality.
That is, with the countless events and decisions a person encounters every day, it is to be expected that once in a while, one of those would have a serendipitous character. What happens is that such an event is taken not only with great surprise and joy but as also portending supernatural involvement, i.e., God's hand in the world. But for every one of those events are countless others which were decidedly less than astonishing. There were all the times you turned a corner just a split second before running into a friend from childhood. Similarly, for every person who makes a miraculous recovery, there are thousands more who do not, and their stories generally remain untold. Understandably, we prefer to relate the miraculous, happy endings, and therefore we lose sight of the fact of their statistical inevitability. Moreover, even if one were to posit supernatural explanations to things such as chance encounters, they are still miracles on a different order of magnitude than, say, water standing up like a wall, or all the firstborn males of Egypt dying overnight.
One argument made by believers is that God's "appearance" in Biblical times was due to its being a special era, whose specific needs required a more direct Divine engagement with humanity. Now that the revelation has taken place, God's face as it were has been "hidden," no longer given to open miracles, prophecy and other fantastic encounters. Ours is now a test of faith... Granted, this is an explanation, and a rather convenient one at that, but the far more reasonable explanation is that such miracles and other direct encounters with God never in fact occurred at all. And that is why we never see them today. It should be especially obvious given the fact that at the time that Torah came onto the scene, all peoples had their gods, and the powerful acts of these gods and their interrelations with humans are likewise interwoven as part of the national story. Should the Israelite story be any exception?
2. Are the mitzvot Divine?
It follows from the belief in a Divine Torah, wherein the mitzvot are commanded by God Himself, that such commandments are invested with an "eternal" quality. Their laws and principles are etched into the very fabric of the cosmos as part of the Divine will for Creation. Yet with only a cursory examination of history, it is clear that the mitzvot, far be it from being "timeless," are in fact a clear reflection of Bronze Age civilization. Israelite civil law bears a close resemblance to other codes of law from the ancient Near East. Other peoples gave tithes to their priests, and gave food and incense offerings to their gods, some in temples strikingly similar to the structure of the Mishkan. Some, including the Egyptian priesthood, performed circumcision, exercised dietary limitations, and immersed in ritual baths.
I ask then what is more likely, that everyone was closer to the Divine will at the time, that Bronze Age Mesopotamia just happened to strike upon the perfect "cosmic cocktail," such that all subsequent generations must now follow its particular set of rituals and norms as the living word of God, or that the ancient Israelites were simply a product of their time and place, albeit with their own unique "spin" on it all?
Would Moses, were he to first come on the scene today, possibly make us painstakingly (and at great cost) write out the Torah, letter by letter, on strips of animal hide attached together with sinews, because there is something "Godly" or "holy" to this specific set of materials and processes? Or was that simply the technology available at the time? These questions are obviously rhetorical, as it is certain that Torah is no more, and no less, than a product of its time. Yes, we have adapted it in many ways which make it "timeless," but to say that the Torah is God's eternal law is completely untenable. Therefore, the Biblical God, commander of mitzvot, is completely untenable.
People looking for "proofs" may wish to draw your attention to the millions who stood at Sinai and watched the revelation with their own eyes, and that in no way could this kind of story be "made up." Remember however that every ancient people had its origin story, in which the gods played a role. Myth and history were regularly interwoven as one. It is only in modern retrospect that we cannot fathom such a thing. Therefore we assume that our Biblical forebears would not have knowingly retold myths. But they were not "lying" by telling of miracles, wonders, prophecy, wrath, and Divine conquest in the desert; rather they were giving honor to their God and to the people. To tell a story (remember that as much as we are now called the "People of the Book" it was a primarily oral tradition for the entirety of the Biblical era and then some) without God, without miracles, would be at the very least dry and not terribly memorable, and further it would be unseemly, not befitting to any people, let alone a "holy nation." We had to tell our story in a supernatural, super-memorable fashion.
Just because someone is able to put words skillfully together, so as to appear to make a good case, does not make it "true." One still needs to consider common sense plausibility. And given the absence of open miracles and Divine intervention today, and the fact that the mitzvot and God-invested stories of the Torah are very much a product of their time, maintaining a belief in the God of the Torah, the commander, the miracle-worker, defies common sense. It defies reasonability.
Someone may wish to believe nonetheless, which is fine. But better to be honest and call it "faith."