THE NATURE OF PROOF
First is the idea that someone is arrogant for holding to a position they believe to be true. This is a rather absurd position to take, since you are by definition holding a position contrary to theirs and you believe it to be true. This means you are not only violating your own very tenet: that it's arrogant to obstinately hold to some position and argue it, further doing so is hypocritical - one is self-awarely doing the opposite of a stated position. It is not arrogant to insist what one believes to be true, it is consistent and logical. I am no more arrogant for insisting that tax cuts to a certain point increase government revenues than I am to insist that 2+2=4. Evidence, experience, and logic dictate both are true. The arrogance can come in how I express these positions but merely doing so is not arrogance or condescension.
THE NATURE OF PROOF
Second, we are running into a modern error regarding proof and epistemology (how we know things). When someone asks for "proof" they aren't agreed upon what proof actually means so the definition and standard for proof continually shifts as the discussion goes on, depending how much they care to agree with or admit what is being argued is true. Proof has to have a defintion, an agreed upon meaning before it can be applied in any discussion. That's why structured debates are so much harder than just discussions, because the definitions are agreed upon in advance, so you can't play word games and appeal to emotions to win. You have to stick to the truth and logic.
Nothing can be proven to such a degree that it removes all possible doubt. There is always some doubt about every single argument or proof no matter what. Let's take something that seems really obvious and certain, that appears to be absolutely sure. 2+2=4. Now, we'll put aside fractions and imaginary math and focus on the basics here. Two apples plus two more apples equals four apples, right? Maybe, what if you are seeing double and there's actually only 1 apple in each group? What if you think they are apples, but one is made of wax. What if its an optical illusion and there aren't really two apples in each group? What if you are imagining some of the apples?
See how it works? You can always cast doubt on any argument, any proof and any statement. That's why courts require proof "beyond a reasonable doubt" in most cases, and never "beyond the possibility or shadow of a doubt" in any case. You cannot prove something so conclusively that there's no way to question or doubt it, ever.
All of those apple examples could be true, but it's unlikely to the point of absurdity that at least one is true every single time someone adds 2+2.
When an argument degrades to the point of being ridiculous, you've effectively won your case, even if the other person won't admit it. When your opponent has to stretch to find something irrational, silly, or immensely improbable to argue, then they have surrendered, often without being aware of it. That's where logical fallacies come in, when someone can only offer logical fallacies in response to your point, you've won the debate.
Most internet arguments fail this test, however, because the definition of proof is not agreed upon and thus it cannot be achieved. Further, in many cases, neither side understands nor can they articulate their points well enough and the debate degrades to name calling, circular arguments, attempts to distract, and a festival of logical fallacies (ad hominem being the most common - you are an idiot rather than here's how you're wrong).
Much of what passes for "debate" or logic on the internet is instead sophistry: sophisticated, important sounding things that lack logic and weight of proof. Most of this sort of writing is holding arguments to unreasonable standards of proof such as "without any doubt." It appeals to emotions or seems so weighty and sophisticated that it must be true; it is "dazzling with BS" rather than convincing and persuading. Sadly, this kind of argument has for all human time been very compelling to those without education and awareness of logic.
THE COSMOLOGICAL ARGUMENT
Let me take the initial argument and show how this works in the context of proof and logic. God exists, and taken at the most basic and fundamental definition as a theistic creator I can prove it:
First, we have to establish that at least some things actually do exist. If you are a madman, you might honestly and truthfully believe that nothing really exists and we're just figments of ... well no one's imagination, because nothing exists. However, sane and rational people admit, despite their arguments and philosophical exercises, that something exists, even if they won't admit everything around them does. So let's take something at hand: your computer. You're reading this on a computer, or at least someone did, then printed it out. That computer, at least, exists.
That computer did not have to exist, that is, there's no cosmic law demanding that your computer must be here and must have come about - it could have simply not been made, it could have not existed at some point. This is a bit esoteric, so read it through again. Your computer might not have existed.
Now, if your computer exists, and it didn't have to, where did it come from? The factory, you say? Where did those parts used to assemble the computer come from? Petroleum and so on, right? Where did that come from? If you keep pushing backward in time and complexity far enough, you reach a point where you have to have a beginning. There is a story of an old woman who believed that the earth rested on the back of a gigantic turtle Discworld style, it had to rest on something or it would fall down! When asked what the turtle was standing on to keep it from falling, she replied "why, another turtle!" What was this turtle standing on? The old woman was not to be caught, she replied "you can't trick me, it's turtles all the way down!"
This is a logical absurdity, simply pushing things back another stage does not eliminate the need for a beginning somewhere, science, common sense, and logic all point out that something cannot come from nothing. If you ever started with nothing, what would you have later? Still nothing. No matter how long you wait, it will always be nothing. So your computer (and everything that exists) came from somewhere. So here did everything come from?
In order for there to be anything there must have been something that has always been here. There simply cannot have been nothing, then suddenly everything showed up, big bang or no.
Now, Carl Sagan would say that the universe has always existed, everything has just always been here. The idea is that the universe at least in one form or another has always existed and thus the raw materials have always been there for everything, in one form or another. The problem is, science simply does not support this concept. The second law of thermodynamics states:
If the universe has existed for eternity, then it would eventually become a vast open space filled with the most simple form of existence, whatever that may be. There would be no complexity, no forms, no energy, just that simple gray expanse of sub-dust. "But wait!" you cry, "maybe that will be true eventually but we're early in this process!" The problem with this objection is that it requires a greater level of absurdity than the proof given: eternity means forever, that means there has been an infinity of time before this point. That means we're not early in the process, it is by definition impossible to be early in a process that has no beginning.
OK, so how about the fluctuating universe? This theorizes that the universe has always existed, but not always in this form. The idea is that the big bang occurred at some point in the distant past, and in the distant future, the universe will slow and begin to contract because of mutual gravitational pull. That collapse causes everything to eventually contract to a single point in space, immensely dense and compact, then explode at some later point. This contract/explode process is said to have gone on for eternity, and because of that, the system "resets" and thus entropy is avoided.
There are four basic problems with this. First, the second law of thermodynamics would require each successive big bang be, well, smaller and smaller. As pointed out above, this process would eventually cease because of the loss of energy out of the system and thus we would be reduced to zero complexity.
Second, observation and testing shows us at present that the universe is not actually slowing, it is accelerating apart. The planets, galaxies, and stars are pushing apart faster not slower.
Third, in order for there to be enough matter to actually generate the required gravitational pull to contract these distant bodies into one central point again, there has to be a certain amount of matter in the universe. At present, we can find about 10% of that matter. 90% of the required matter is either hiding, in some form we cannot detect... or simply doesn't exist.
Finally, there does not appear to be any one central point everything is moving away from. The Big Bang would have a point of origin, the center of the explosion that everything is exploding away from and that has yet to have been even remotely discovered.
Now remember what I said above about proof. It is possible that the second law of thermodynamics could simply cease to apply at a certain point and thus no energy be lost in successive explosions. It is possible that 90% of the universe's matter is out there hiding behind a tree. It is possible that there's a central point of this vast explosion that is so difficult to figure out that we can't tell by careful observation. All of those things could be true, just like it could be true that 2+2 does not actually equal four. The problem is, its more absurd and violates science and logic more to presume those things than to recognize the problems and reject them.
So what are we left with? Something outside the universe created it, something we'll call "God" because it's a familiar term. This "God" has no beginning, and is not part of the Universe, but rather it's external creator. Logic, common sense, and science demand this be true, there's no other option. The existence of God is thus proven - it requires greater absurdity and illogic to believe otherwise than to admit this truth.
NOT ENOUGH FAITH
Now, you can begin to make arguments against these things, and by all means do - but you'll be demonstrating the proof above rather than countering it because your arguments will be more absurd and less logical. At least, that's been the case for centuries since the first thinkers came up with the Cosmological Argument (that's what the case I just laid out is called). As Dr Geisler says, I don't have enough faith to be an atheist.
Now I haven't proven that the Christian God is that creator, or Allah, or Brahma or what have you. All I've proven is that a creator must exist - what manner of creator is another discussion, but there's no basis for true atheism.
COMPLEXITY AND KNOWING
So what can we know about this "God" then? What by observation and logic can we determine is true about this theistic creator?
God would be more complex than any of us, by nature of power and intelligence, at the very least. And it is true that the more complex something is, the harder it is to understand. I can understand a rock fairly easily (when you get to the molecular and atomic levels, you've gotten things complex again, but then we aren't examining a rock any more we're examining molecules and atoms). A human being is more complex and thus more difficult to understand.
The more complex something is, the more help you need from them to understand them. I need help to really know Sally, although I can know many things about her, I don't really know her until she helps me out intentionally or no (this is the great danger of internet dating and to a lesser degree ordinary dating - you learn only what they choose to reveal).
For something as vastly complex as God, it would be vastly difficult to understand because of the level of complexity. One could even say by nature God is infinitely complex, because the creator would be so much more vast and intelligent than we. This would make God infinitely impossible to understand... except what that God has revealed to us. It is through this revelation (the world around us, to begin with) that we can learn about this God.
God is by definition the most powerful being in existence, because the creator made everything else - the creator is more potent than the creation by nature of being able to make the thing (don't argue that humans are weaker than bombs - we didn't create the bombs, they assembled them from pre-existing materials). God by definition must be more intelligent than anything created in the universe, so intelligent that this creator was able to make everything in its vast complexity. Further, God is the final arbiter and origin of justice and morality, because that God created such in us. If there is any higher authority, then that authority is the creator; you are playing games with turtles again by trying to claim someone or something ordered or dictates to God.
This is the place where Kierkegaard's "leap of faith" slams into: we can actually know some things about God because of that revelation, because of the things that we have been told about. It does not require some blind belief based upon no logic to believe in God, it merely takes admitting the logic and science of it. This is where the attempt to divorce religion and science makes no sense on either side of that equation: science proves God, religion embraces science. The "leap of faith" assumes God is completely unknowable and impossible to understand, and thus it is merely a personal belief that you cannot say others ought to have or agree with. The existence of God is just as reasonable as 2+2=4, it just carries a lot of baggage that many people would rather avoid, and it's easier to just deny that proof.
Now if only God had written a manual, some sort of book to help us all understand that complexity a little better...