Objection 1. It seems that the existence of God cannot be demonstrated. For it is an article of faith that God exists. But what is of faith cannot be demonstrated, because a demonstration produces scientific knowledge; whereas faith is of the unseen (Hebrews 11:1). Therefore it cannot be demonstrated that God exists.
Objection 2. Further, the essence is the middle term of demonstration. But we cannot know in what God's essence consists, but solely in what it does not consist; as Damascene says (De Fide Orth. i, 4). Therefore we cannot demonstrate that God exists.
Objection 3. Further, if the existence of God were demonstrated, this could only be from His effects. But His effects are not proportionate to Him, since He is infinite and His effects are finite; and between the finite and infinite there is no proportion. Therefore, since a cause cannot be demonstrated by an effect not proportionate to it, it seems that the existence of God cannot be demonstrated.
On the contrary, The Apostle says: "The invisible things of Him are clearly seen, being understood by the things that are made" (Romans 1:20). But this would not be unless the existence of God could be demonstrated through the things that are made; for the first thing we must know of anything is whether it exists.
I answer that, Demonstration can be made in two ways: One is through the cause, and is called "a priori," and this is to argue from what is prior absolutely. The other is through the effect, and is called a demonstration "a posteriori"; this is to argue from what is prior relatively only to us. When an effect is better known to us than its cause, from the effect we proceed to the knowledge of the cause. And from every effect the existence of its proper cause can be demonstrated, so long as its effects are better known to us; because since every effect depends upon its cause, if the effect exists, the cause must pre-exist. Hence the existence of God, in so far as it is not self-evident to us, can be demonstrated from those of His effects which are known to us.
Reply to Objection 1. The existence of God and other like truths about God, which can be known by natural reason, are not articles of faith, but are preambles to the articles; for faith presupposes natural knowledge, even as grace presupposes nature, and perfection supposes something that can be perfected. Nevertheless, there is nothing to prevent a man, who cannot grasp a proof, accepting, as a matter of faith, something which in itself is capable of being scientifically known and demonstrated.
Reply to Objection 2. When the existence of a cause is demonstrated from an effect, this effect takes the place of the definition of the cause in proof of the cause's existence. This is especially the case in regard to God, because, in order to prove the existence of anything, it is necessary to accept as a middle term the meaning of the word, and not its essence, for the question of its essence follows on the question of its existence. Now the names given to God are derived from His effects; consequently, in demonstrating the existence of God from His effects, we may take for the middle term the meaning of the word "God".
Reply to Objection 3. From effects not proportionate to the cause no perfect knowledge of that cause can be obtained. Yet from every effect the existence of the cause can be clearly demonstrated, and so we can demonstrate the existence of God from His effects; though from them we cannot perfectly know God as He is in His essence.
Source: Summa Theologica (New Advent)