A Midrashic Reading:
Abraham lived in a culture that accepted the concept of a god as a given, just not the concept of one god. It was within that framework, (where the concept of deities was undisputed), that he logically presented monotheism. This is his argument, seen in Gen Rabah 38:13:
נסביה ומסריה לנמרוד א"ל נסגוד לנורא א"ל אברהם ונסגוד למיא דמטפין נורא א"ל נמרוד נסגוד למיא א"ל אם כן נסגוד לעננא דטעין מיא א"ל נסגוד לעננא א"ל אם כן נסגוד לרוחא דמבדר עננא א"ל נסגוד לרוחא א"ל ונסגוד לבר אינשא דסביל רוחא א"ל מילין את משתעי אני איני משתחוה אלא לאור הרי אני משליכך בתוכו ויבא אלוה שאתה משתחוה לו ויצילך הימנו
... He took him and gave him to Nimrod. He said to him, "Let us bow to fire." Abraham said to him, "Rather we should bow to water that puts out fire." Nimrod said to him, "Let us bow to water." He said to him, "If so lets us bow to clouds that hold water." He said to him, "Let us bow to clouds." He said to him, "If so let us bow to wind that disperses clouds." He said to him, "Let us bow to wind." He said, "Rather we should bow to man that holds wind." He said to him, "You are speaking words. I only bow to fire, see I will throw you into it and let the Gd that you bow to come and save you from it!"
The עץ יוסף says that Nimrod's protest of "You are speaking words" was that Abraham was making a cyclical argument, and according to him, no local aspect of creation could be worshiped.
In fact, Nimrod's decree to kill Abraham in fire completes the cycle. Fire beats man beats wind beats clouds beats water beats fire! So Abraham couldn't accept any of these gods, because each one was inferior to the other in an endless loop, (and this concept could be applied universally in creation, either through one big circle, or many smaller ones with shared aspects, if enough time was spent).
In addition, Abraham was perhaps met with another problem. How could all of these factors exist with one another in the same creation? As soon as the fire god, the water god, the cloud god, the wind god, and man came together, they would simultaneously destroy one another and cease to exist!
The עץ יוסף also says that Nimrod refused to step outside of the physical plane that he could see and hear and smell and touch to consider something more.
Abraham, on the other hand, stepped outside of this endless cycle. He concluded that only One Gd, who controlled all of these conflicting aspects and was subject to none of them, could have woven them together harmoniously in to the creation that he saw.
A Logical process from Duties of the Heart:
A possible breakdown of Abraham's argument is presented in The Gate of Unity in Duties of the Heart, (translated and explained in exhaustive detail here, thank you ray!). It is much more exhaustive and detailed than the above source, and it seems to encompass the Midrashic argument of Abraham in its scope. I will attempt a cogent paraphrase, but the reader should bear in mind that it is just that and maybe not address everything that the original work does.
It is important also to bear in mind that logic, being a human construct, is always subject to the limits of humanity. This is why A person's faith is so important. A perfect argument certainly exists for Gd, but we may not have yet grasped it or codified it.
Regardless of arguments and logic, one must know Gd as a fact. Otherwise logic, being human-made, can lead a person astray6.
The Translator himself says, in his remarks: "This approach is not an attempt to answer what must be but rather to escape into endless and inconclusive speculation as to what might otherwise be."
The proof begins with three premises that once logically proven assert that the universe has a Creator that created it from nothing:
A thing cannot make itself: If something exists, when before it did not, either it must have created itself, or been created. If it created itself, it must have done so before it existed, in which case it was absolutely nothing1, or after it existed, in which case it did nothing. Both of these concepts are impossible. Thus it must have been created.
Beginnings are limited in number: Whatever has an end must have a beginning, because something which has no beginning (i.e. existed eternally) has no end (i.e. is indestructible). Given the finite nature of all of the world's causes, there must have been an absolute starting point2 3 4.
Anything composite, (made up of components or parts), Must have been created: It cannot consequently be eternal. Anything composite is composed of more than one thing, and these things which it is composed of must precede it by nature. Likewise, whatever assembled the compound must also precede it by nature and by time5 6.
The work proceeds to formally establish the existence of a Creator by way of the above premises. The union of conflicting elements into a coherent harmonious creation is mentioned, which seems to touch on Abraham's argument as understood above:
We do not have the capability to join the four elements, in the natural way we find them compounded in nature because they are different or even repel each other, and if we attempt to artificially combine them, the result rapidly changes and disintegrates, while the synthesis brought about through nature is complete and endures until the (appointed) time of its end...
Since all existing things that we find are from the elements, and composed of them, and we know that they were not combined on their own, and by their inherent nature do not join together because of their repelling characteristics, it is clear to us that something else must have joined them and bound them, and fused them together against their nature, by force - this is their Creator, who joined them and established their union.
1. With regard to virtual transient particles, which appear and disappear for extremely short periods of time in vacuum space and are claimed by Atheists to allow for something from nothing: 'The virtual particles "borrow" for a very short amount of time the ground energy that is already available from time/energy uncertainty principles and converts that to virtual particle with E=mc2. This does not violate the conservation laws because the kinetic energy plus mass of the initial decaying particle and the final decay products is equal. Furthermore, a space-time vacuum governed by quantum mechanics is still something. It is not nothing. We would need absolute nothingness here, as stipulated by the Author'.
2. Also, anything which has parts must have a whole, since a whole is merely the sum of parts. Something infinite can't be comprised of parts, because a part is an amount separated from another amount. Something which is infinite and has a part taken from it, will be less than what it was before. If it is still infinite, then one infinite will be greater than another infinite, which is impossible. Alternatively, if it is now finite, and we put back the part that was taken, then the whole will be finite, but it was originally infinite. The same thing being finite and infinite is a contradiction and impossible. If we take out a part of all things that have ever existed, the total number of individual things of this part is finite, therefore the whole together is also finite. if you have something non-eternal in the present, you cannot explain its existence by saying that it was the effect of something else non-eternal (its "cause"), etc. endlessly, in an infinite regress, because this is an attempt to get something from nothing, that depends ultimately upon the nonexistent.
3. Something that is infinite in actuality doesn't have quantity because infinity is not a number, it is a description. In mathematics it is possible for an infinity to be comprised of a set of units. This is possible in the realm of mathematics, but is nevertheless impossible in reality. This is because mathematics is pure human logic, which is merely a tool which we use to investigate topics, but anything it has to say on the subject is from premises which we supply. Even in mathematics, you can sometimes run into trouble if you treat infinity as a number.
4. A rock can exist forever in the past and still not be infinite, theoretically, but at some point it must have not existed as a rock since a rock cannot make itself, therefore something else caused its existence, such as energy, which also cannot make itself, etc. until you reach something of infinite nature, without beginning.
5. elementary subatomic particles can be converted to radiation whose energy can be combined and divided, and are thus composite. (Also, in kabala anything physical is viewed as a composite of physical and spiritual forces).
6. The limitations of human logic and still present attacks of these premises are addressed in the article.
(This portion of my answer is somewhat peripheral to the topic at hand as it does not deal with the existence of Gd directly, but rather with Unity of His existence).
The work then brings seven arguments for Gd's unity7:
- Causes are always fewer than their effects, namely, the higher up one ascends into the chain of causes, the fewer the number of causes, and the more and more one ascends this chain, the fewer and fewer will be their number until eventually one reaches one Cause, which is the Cause of causes.
- We find the world's roots and foundations to be similar in their derivatives and equal in their parts, and the signs of wisdom manifested in the smallest of the creatures as well as the biggest testify that they are the work of one Wisdom. It is also interdependent for its maintenance and welfare, like links in a coat of armor. If this world had more than one Creator, the form of wisdom would be incongruous in the different parts of the world, and change in its general nature and parts.
- It is impossible for the existence of the world without at least one Creator. And if it were possible to conceive that the world could exist with less than one Creator, we would consider this. But since we cannot conceive that something less than one can bring anything into existence, we conclude that the Creator must be one. In the case of things which were established through logical proofs, and the proof of their existence is impossible to deny, we do not need to assume more than what is necessary to account for the phenomena which the proof demonstrates.
- Multiple creators must either be of one essence, in which case they are one being and are one, or of different essences, in which case there is some distinction between them due to their difference and non-similarity. If so, whatever is distinct is limited/bound. And whatever is limited/bound is finite. And whatever is finite is composite, and anything composite was brought into existence, and anything brought into existence must have a Creator.
- The idea of plurality is that of a sum of unities. Plurality therefore cannot precede unity of which it has been formed. If we discover something plural whether intellectually or through our senses, we will know with certainty that unity preceded it just like when counting things, the number one precedes the rest of the numbers. And whoever thinks the Creator is more than one, should nevertheless concede that there was a preceding unity, just as the number one precedes all of the other numbers, and just like the notion of unity precedes that of plurality. Therefore, the Creator is absolutely One, and Eternal.
- The Creator made essence and incident. Plurality is an incidental property ascribed to the essence, and comes under the category of quantity. Since He is the Creator of essence and incident, none of these attributes can be ascribed to Him. When it is clear through scripture and reason that the Creator is above and beyond all comparison with, and similarity to, any of His creations, plurality cannot be fittingly ascribed to the Creator. If He cannot be described as plural, He must certainly be One because there is nothing in between the two possibilities.
- If the Creator were more than one, then either each one of these supposed creators is capable of creating the world by itself, in which case the other Creator is superfluous, since the first is capable without him and does not need of the other, (and since he is not needed, and there is no evidence for more than one Creator, why should we be in doubt or be concerned for his existence?), or each is not capable without the help of the other, in which case each one of them is not all-capable and omnipotent since each lacks the necessary power and ability and was weak. And whatever is weak has a limit to his strength and power, and anything which is limited is bound, and anything which is bound is composite, and anything composite is brought into existence, and anything brought into existence has a Creator.
Lastly the works distinguishes between incidental unity and true unity and ascribes the Latter to the Creator7:
- Incidental unity occurs in two ways. In one of them the character of plurality is apparent in it, such as one genus which includes many species or like one species which includes many individuals, and like one man which is comprised of many parts or one army which includes many men. The second way is the unity attributed to a single individual thing which does not appear to be plural, and is not a collection of several things, but it is essentially plural from the perspective of its being composed of matter and form, essence and incident, it is susceptible to "creation" and "destruction", division and combination, separation and association, change and variation.
- True unity also has two kinds. Conceptual true unity is numerical, the symbol and representation for a beginning with no previous beginning. this is an idea, not a reality. Actual true unity exists. It is neither plural nor susceptible to change or variation, is not described by any corporeal traits, nor subject to creation, nor subject to destruction or limitation, nor motion, is not comparable to anything nor can anything be comparable to it, cannot be associated with anything. True unity and the root of everything plural and thus it foils all perspectives and perceptions, since as we demonstrated unity is the cause of plurality.
- In essence, unity precedes plurality, just like the number one precedes the rest of the numbers, thus the First cause of everything that is plural, which was at the head of all beginnings is itself not plural. Since causes must reach a limit at their beginning, and it is not possible for a thing to make itself, it is not possible for the cause of unity and plurality to itself be of unity and plurality like them. And since the First Cause of the creations cannot itself be plural nor a combination of plurality and unity, it must necessarily be that the Cause is a true (absolute) unity7.
7. See the article for more detailed explanations, as well as questions and answers.