What makes the question does god exist important, is a question all by itself. The reason it is an important question or at least one reason is its historical place in the western history of ideas. The credit for asking this question and then trying to answer it by reason rather than faith or authority, is usually given to Thomas Aquinas and the Summa Theologia, 1270.
By accident, I read a heavily abridged copy of Summa at sixteen, after an argument with my mother who was Catholic. She threw the name and work at me as an intimidating bluff---go read Aquinas and get back to me. The source of the argument which went on for weeks and caused a serious family battle, was my refusal to be baptized in my father's church (Protestant). I was surprised my mother thought I should be, and even more surprised by her adament belief in God. So, I showed her. I went to the library, checked out Aquinas, and read the first sections and other miscellaneous parts from the thinest copy I could find. And worse, I understood a lot of it. Unfortunately, I remember agreeing with all the arguments against and none of the arguments for. And, even worse still, what I learned from Aquinas was not about the existence of god, but about the existence of philosophy! I had no interest in looking up the biblical footnotes, but I was interested in finding out more about Aristotle.
Since I had gone to the philosophy section to find Aquinas, Aristotle wasn't very far away. The Aquinas book had footnoted both the Metaphysics and Physics, and I picked the Physics because it was shorter. I had almost no luck with it. But there were words like infinity, being, substance, nature, and these alone were pretty heady stuff. It was much more fun and exciting to think about being, substance, infinity and nature, than thinking about Jesus Christ and the boys.
In other words, I accidently re-capitulated the historical pivot between medieval scholasticism and rational philosophy that followed the re-discovery of Aristotle. Of course I had no idea I had experienced any such thing until much later.
Here are the five ways mentioned by E.C.Estabook (see Article 3):
``Article 2: Whether God's existence is demonstrable.
We proceed thus to the second point. It seems that God's existence is not demonstrable, for it is an article of faith. What is a matter of faith cannot be demonstrable, for demonstration allows one to know, whereas faith, as Paul says, is in "things not seen" (Heb. 11:1). Therefore God's existence is not demonstrable.
Furthermore, the central link in any demonstration is a definition; yet we cannot know what God is, but only what he is not, as John of Damascus says. Therefore we cannot demonstrate God's existence.
Furthermore, if God's existence were demonstrable, this could only be through his effects; yet his effects are not proportionate to him, for he is infinite, his effects are infinite, and there is no proportion between the two. Therefore, since a cause cannot be demonstrated through an effect which is not proportioned to it, it seems that God's existence cannot be demonstrated.
But on the contrary Paul says, "The invisible things of God are understood by the things that are made" (Romans 1:20). Such could not be the case unless God's existence could be demonstrated by the things that are made, for the first thing to be understood about a thing is whether it exists.
Response: It must be said that there are two types of demonstration. One is through the cause, is called a demonstration propter quid, and argues from what is prior in an absolute sense. The other is through the effect, is called a demonstration quia, and argues from what is prior according to our perspectives; for when an effect is better known to us than its cause, we proceed from the effect to knowledge of the cause. In situations where the effect is better know to us than the cause, the existence of the cause can be demonstrated form that of the effect, since the effect depends on the cause and can only exist if the cause already does so. Thus God's existence, though not self-evident to us, can be demonstrated through his effects.
To the first argument, therefore, it must be said that God's existence and other things about him which (as Paul says) can be known by natural reason are not articles of faith but preambles to the articles of faith. For faith presupposes natural knowledge just as grace presupposes nature and perfection presupposes something which can be perfected. Nothing prohibits what is demonstrable and knowable in itself from being accepted on faith by someone who does not understand the demonstration.
To the second it must be said that, when a cause is demonstrated through its effect, the effect substitutes for the definition of the cause within the demonstration. This is particularly true in arguments concerning God. When we prove that something exists, the middle term in the demonstration is what we are taking the word to mean for purposes of the demonstration, not what the thing signified by the word actually is (since the latter, the actual nature of the thing in question, is determined only after we determine that it exists). In demonstrating that God exists, we can take as our middle term definition of what this word "God" means for us, for, as we shall see, the words we use in connection with God are derived from his effects.
To the third, it must be said that perfect knowledge of a cause cannot be derived from an effect that is not proportionate to the cause. Nevertheless, the existence of the cause can be demonstrated clearly from the existence of the effects, even though we cannot know the cause perfectly according to its essence.
Article 3: Whether God exists.
Thus we proceed to the third point. It seems that God does not exist, for if one of two contrary things were infinite, its opposite would be completely destroyed. By "God," however, we mean some infinite good. Therefore, if God existed evil would not. Evil does exist in the world, however. Therefore God does not exist.
Furthermore, one should not needlessly multiply elements in an explanation. It seems that we can account for everything we see in this world on the assumption that God does not exist. All natural effects can be traced to natural causes, and all contrived effects can be traced to human reason and will. Thus there is no need to suppose that God exists.
But on the contrary God says, "I am who I am" (Ex. 3:14).
Response: It must be said that God's existence can be proved in five ways. The first and most obvious way is based on the existence of motion. It is certain and in fact evident to our senses that some things in the world are moved. Everything that is moved, however, is moved by something else, for a thing cannot be moved unless that movement is potentially within it. A thing moves something else insofar as it actually exists, for to move something is simply to actualize what is potentially within that thing. Something can be led thus from potentiality to actuality only by something else which is already actualized. For example, a fire, which is actually hot, causes the change or motion whereby wood, which is potentially hot, becomes actually hot. Now it is impossible that something should be potentially and actually the same thing at the same time, although it could be potentially and actually different things. For example, what is actually hot cannot at the same moment be actually cold, although it can be actually hot and potentially cold. Therefore it is impossible that a thing could move itself, for that would involve simultaneously moving and being moved in the same respect. Thus whatever is moved must be moved by something, else, etc. This cannot go on to infinity, however, for if it did there would be no first mover and consequently no other movers, because these other movers are such only insofar as they are moved by a first mover. For example, a stick moves only because it is moved by the hand. Thus it is necessary to proceed back to some prime mover which is moved by nothing else, and this is what everyone means by "God."
The second way is based on the existence of efficient causality. We see in the world around us that there is an order of efficient causes. Nor is it ever found (in fact it is impossible) that something is its own efficient cause. If it were, it would be prior to itself, which is impossible. Nevertheless, the order of efficient causes cannot proceed to infinity, for in any such order the first is cause of the middle (whether one or many) and the middle of the last. Without the cause, the effect does not follow. Thus, if the first cause did not exist, neither would the middle and last causes in the sequence. If, however, there were an infinite regression of efficient causes, there would be no first efficient cause and therefore no middle causes or final effects, which is obviously not the case. Thus it is necessary to posit some first efficient cause, which everyone calls "God."
The third way is based on possibility and necessity. We find that some things can either exist or not exist, for we find them springing up and then disappearing, thus sometimes existing and sometimes not. It is impossible, however, that everything should be such, for what can possibly not exist does not do so at some time. If it is possible for every particular thing not to exist, there must have been a time when nothing at all existed. If this were true, however, then nothing would exist now, for something that does not exist can begin to do so only through something that already exists. If, therefore, there had been a time when nothing existed, then nothing could ever have begun to exist, and thus there would be nothing now, which is clearly false. Therefore all beings cannot be merely possible. There must be one being which is necessary. Any necessary being, however, either has or does not have something else as the cause of its necessity. If the former, then there cannot be an infinite series of such causes, any more than there can be an infinite series of efficient causes, as we have seen. Thus we must to posit the existence of something which is necessary and owes its necessity to no cause outside itself. That is what everyone calls "God."
The fourth way is based on the gradations found in things. We find that things are more or less good, true, noble, etc.; yet when we apply terms like "more" and "less" to things we imply that they are closer to or farther from some maximum. For example, a thing is said to be hotter than something else because it comes closer to that which is hottest. Therefore something exists which is truest, greatest, noblest, and consequently most fully in being; for, as Aristotle says, the truest things are most fully in being. That which is considered greatest in any genus is the cause of everything is that genus, just as fire, the hottest thing, is the cause of all hot things, as Aristotle says. Thus there is something which is the cause of being, goodness, and every other perfection in all things, and we call that something "God."
The fifth way is based on the governance of things. We see that some things lacking cognition, such as natural bodies, work toward an end, as is seen from the fact that they always (or at least usually) act the same way and not accidentally, but by design. Things without knowledge tend toward a goal, however, only if they are guided in that direction by some knowing, understanding being, as is the case with an arrow and archer. Therefore, there is some intelligent being by whom all natural things are ordered to their end, and we call this being "God."
To the first argument, therefore, it must be said that, as Augustine remarks, "since God is the supreme good he would permit no evil in his works unless he were so omnipotent and good that he could produce good even out of evil."
To the second, it must be said that, since nature works according to a determined end through the direction of some superior agent, whatever is done by nature must be traced back to God as its first cause. i\In the same way, those things which are done intentionally must be traced back to a higher cause which is neither reason nor human will, for these can change and cease to exist and, as we have seen, all such things must be traced back to some first principle which is unchangeable and necessary, as has been shown.''