ST. THOMAS AQUINAS, SUMMA THEOLOGICA, 

[In the First volume of the Summa, Second major section (a "question"), and Third topic (article)  in that section (I, 2, 3)  Aquinas offers a quick review of some traditional arguments for the existence fo God.  These are a very famous set of 'proofs' for the existence of God.  You will find that the first three are just a variation on a theme.  Aquinas uses all of them because he is repeating arguments that others have used before him, though this is his formulation of them.
       In the first article of question 2 he has asked whether God's existence is self-evident.  If so, then no proofs for God's existence are necessary. This addresses St. Anselm's claim that just by the definition of "God," God must necessarily exist.  Aquinas does not think that Anselm's answer works.  
      In the second article of question two, Aquinas has asked whether God's existence can be established by reason.  Is not faith alone enough, and is not faith the only valid way?  He says that faith may be the only way for those not adept at philosophy, but even the Apostle Paul says in Romans that the human mind is able to arrive at the reasoned conclusion that God exists.
      You will find a summary but different version of the first three arguments in chapter 11 of In the Presence of Mystery.  It may help you understand the arguments here below.  You can look at arguments 4 and 5, but do not worry too much about them now.  They are more relevant to the deism discussed in chapter 12.] 
      The medieval Summas (or summaries) have been compared to Gothic cathedrals of the middle ages, because the structures of both are right out in the open rather than hidden behind walls or rhetoric.  The outline here -- of a topic, then some objections, then a single contrary idea, then a "I respond that" in which the writer gives some relevant analysis, then a resolution of the objections at the beginning.  This whole article will then lead to the next logical step in summarizing all of theology.

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 Part I, Question 2, article 3.
 We proceed thus to the Third Article:— Whether it can be proven that God exists.

   Objection 1.  It seems that God does not exist; because if one of two contraries be infinite, the other would be altogether destroyed.  But the name God means that He is infinite goodness.  If, therefore, God existed, there would be no evil discoverable; but there is evil in the world.  Therefore God does not exist.
   Obj. 2.  Further, it is superfluous to suppose that what can be accounted for by a few principles has been produced by many.  But it seems that everything we see in the world can be accounted for by other principles, supposing God did not exist.  For all natural things can be reduced to one principle, which is nature; and all things which happen by intention can be reduced to one principle, which is human reason or will.  Therefore there is no need to suppose God's existence.

   On the contrary, It is said in Exodus iii.14 in the person of God:  I am Who am 

    I answer that, The existence of God can be proved in five ways.  The first and more manifest way is the argument from motion.  It is certain, and evident to our senses, that some things are in motion.  Now whatever is in motion is moved by another, for nothing can be in motion except that it have a potential for that towards which it is being moved; whereas a thing moves inasmuch as it is in act.  By motion we mean nothing else than the reduction of something from being potential to becoming actual.  But nothing can be reduced from potential to actuality, unless by something already in a state of actuality.  Thus that which is actually hot, like fire, makes wood, which is potentially hot, to become actually hot, and thereby moves and changes it.  Now it is not possible that the same thing should be at once in actuality and potentiality in the same way, but only in different ways.  For what is actually hot cannot simultaneously be potentially hot; though it is simultaneously potentially cold.  It is therefore impossible that in the same aspect and the same way a thing should be both mover and moved, or that it should move itself.  Therefore, whatever is made to move must be made to move by another.  If that by which it is made to move be itself moved, then this also must needs by made to move by another, and that by another again.  This cannot go on to infinity, because then there would be no first mover, and, consequently, no other mover-- seeing that subsequent movers move only inasmuch as they are made to move by the first mover; as the staff moves only because it is moved by the hand.  Therefore it is necessary to arrive at a First Mover, made to move by no other; and this everyone understands to be God. 

     The second way is from the nature of efficient causality.  In the world of sensible things we find there is an order of efficient causation.  There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible.  In efficient causality it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the latest cause, whether the intermediate cause be several, or one only.  To take away the cause is to take away the effect.  Therefore, if there be no first cause among efficient causes, there will be no latest, nor any intermediate, cause.  But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false.  Therefore it is necessary to posit a First Efficient Cause, to which everyone gives the name of God. 

      The third way is taken from possibility and necessity, and runs thus.  We find in nature things that are can either exist or not exist, since they are found to be generated and to be corrupted, and consequently they are able to exist or not to exist.  It is impossible for these always to exist, for that which is able not to exist, at some time does not exist.  Therefore, if everything is able not to exist, then at one time there was nothing in existence.  If this were true, even now there would be nothing in existence, because that which does not exist begins to exist only through something already existing.  Therefore, if at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus even now nothing would be in existence--which is clearly false.  Therefore, not all beings are merely possible, but there must exist something the existence of which is necessary.  But every necessary thing either has its necessity caused by another, or not.  Now it is impossible to go on to infinity in necessary things which have their necessity caused by another, as has been already proved in regard to efficient causes.  Therefore we cannot but posit the existence of some being having of itself its own necessity, and not receiving it from another, but rather causing in others their necessity.  This all men speak of as God. 

     The fourth way is taken from the gradation to be found in things.  Among beings there are some more and some less good, true, noble, and the like.  But `more' and `less' are predicted of different things according as they resemble in their different ways something that is the maximum, as a thing is said to be hotter according as it more nearly resembles that which is hottest; so that there is something which is truest, something best, something noblest, and, consequently, something which is the maximum in being, for those things that are maximum in truth are maximum in being, as it is written in [Aristotle's] Metaphysics, ii.  The maximum in any genus is the cause of all in that genus, as fire, which is the maximum of heat, is the cause of all hot things, as is said in the same book.  Therefore there must also be something which is to all beings the cause of their being, goodness, and every other perfection; and this we call God. 

     The fifth way is taken from the governance of the world.  We see that things which lack thought, such as natural bodies, act towards a purpose, and this is evident from their acting always, or nearly always, in the same way, so as to obtain the best result.  Hence it is plain that they achieve their purpose not fortuitously but designedly.  Now whatever lacks thought cannot move towards a purpose, unless it be directed by some being endowed with intelligence and knowledge, like the arrow by the archer.  Therefore some intelligent being exists by whom all natural things are directed to their purpose; and this being we call God.

    Reply Obj. 1.  As Augustine says:  Since God is the highest good, He would not allow any evil to exist in His works, unless His omnipotence and goodness were such as to bring good even out of evil.  This is part of the infinite goodness of God, that He should allow evil to exist, and out of it produce good.

    Reply Obj. 2.  Since nature works out its specific purpose under the direction of a higher agent, whatever is done by nature must be traced back to God as to its first cause.  Similarly, whatever is done purposefully must be traced back to some higher cause other than human reason or will, since these are subject to change and failure; for all things that are changeable and fallible must be traced back to an immovable and self-necessary first principle, as has been shown.