Some form of description of God needs to be given if we are to even try to understand how to demonstrate His Existence. God is not finite or corporeal. He does not possess any deficiencies, is not timebound, and created and supercedes everything. Evidence for such a Being by definition must be qualititavely different than anything that exists in the physical, natural world.
Given the descriptions above, from an empirical perspective, you can only look for God in the effect that He has on other things, i.e., on what He does. This is not necessarily because God desires to hide Himself, although that is a Jewish tenet described in many sources. It is rather because the world He created cannot contain Him. Nevertheless, a cursory examination of the laws that govern the physical world reveals intelligence and order. It is as if a person tries to build a computer to which he can download all of his brainpower. To put this into perspective, it is currently believed that a human brain contains more information than all of the libraries in the world combined. It would be an unfruitful search to shop for a computer capable of holding such a vast amount of information. With God we are speaking about an Infinite Being, which means that the world is infinitely incapable of containing Him in any capacity. We have just provided evidence that an Intelligence beyond this world is vested within the confines of administering this planet.
By nature, beings can only create things that resemble them in some way. Every created thing possesses properties of its creator. Computer processing, for example, is in some respect modeled on the processing of the human brain. Even though computers can process information at a faster rate than the human brain, they are limited by human imperfection (hence computer crashes, bugs, fried circuit boards, etc.) As an aside, computers can process information faster than the human brain because that is their sole purpose and because it does not confuse them. Human beings must register and comprehend their thoughts to function properly. If a human being's brain processed thoughts at the magnitude of a computer, he might have a nervous breakdown. There are actual mental conditions whereby a person cannot slow down or stop thoughts from entering his mind. One purpose of meditation is actually to slow down the mind so that a person can isolate his thoughts.
Since God is free of all deficiencies, it reasons that in some way the world reflect His Perfect Nature. While the world is a finite place and clearly not free from imperfection, it is as close to being perfect as it can be for a created environment with limitations and restrictions. Nature functions seemingly without continual orchestration and the living beings within it provide for each other, even if this means that they consume each other. In another way, limitations are part of perfection because they force compassion. We have just provided evidence that the limitations of the world indicate God's Perfection.
Regarding God's non-temporeal nature, scientific inquiry into the mechanics of time produces some bizarre results. Admittedly this is an area that I do not completely understand, but a few different theories seem to propose that at any level smaller than the currently smallest known distance (Plank length), "the structure of spacetime becomes dominated by quantum effects, and it is impossible to determine the difference btween two locations less than one Plank length apart." Since space "ceases to exist" at this tiny level, time also ceases to be relevant. Unless in the future humans discover a way to measure an even smaller unit, it seems time "does not exist" at any more elementary level than the Plank length.
Another theory, which as well I do not fully understand, also points to the conclusion that time does not exist. This theory is actually an equation known as the Wheeler-DeWitt equation, in which "utterly lacked a time variable. The symbol t denoting time had simply vanished. Taking the result literally, time doesn't exist." (word search: Does time exist?)
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Does God Exist? (Part 1 - Definition of Proof)
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