Divine Infinity as "Allness" vs "Without Limits"; Infinitesimals vs Scholasticism

I once made a brief note that divine simplicity or "Oneness" is never used as God's name in the Bible, nobody called God "the One" in the Bible and is only used as an attribute, "Almighty" or Shadday is frequently used directly as God's own name. If you say "the Almighty" you'll have named God directly, whereas if you said "the One", people might think you're referring to Neo in the Matrix.

I wish to extend this analysis by considering by looking much close to the concept of infinity itself and how one of the etymology suggested for the Shadday might re-orient our understanding of infinity to be closer to the mathematical concept of infinitesimals rather than the scholastic understanding of "without limit".

First the etymological suggestions. Victor Hamilton in his commentary on Genesis 17:1, where Shaddai first appears in the Bible, makes the following observations:

-The Book of Genesis, Chapters 1-17 (New International Commentary on the Old Testament)

Thus, there is the suggestion that Shaddai focuses on sufficiency, the "allness" within God, and based on this allness, it overcomes the finitude. This idea is carried over into the New Testament where in 2 Corinthians 6:18 uses the Greek term "Pantokratōr" All-Mighty, we can recognise the "pan" prefix as "all" in many of our words in English such as "pantheism", etc. The Septuagint also translates the Old Testament Shaddai as Pantokratōr.

I would like to contrast this conception with the scholastic understanding of infinity as "without limit" or as a mere negation of finitude. We can see this definition in Aquinas's Summa expressly in relation to the divine infinity: "We must consider therefore that a thing is called infinite because it is not finite." In the whole of the Old Testament this sort of divine infinity is spoken of only once, and it is in Psalms 147:5,

We must be clear that these two conceptions infinity are not contradictory of one another, but I think that there is a difference in emphasis, and I would argue that Shaddai, the "allness" conception, is logically prior and fundamental to the "without limit" conception. It is because God's "allness" includes or "overpowers" all finitude that we can speak of God as without limit. But the logical inference doesn't go the other way, things can be without limit and not include all. There are, for example, no limits to the number of positive integers, but the set of positive integers do not include all the integers. Another way to think about this contrast is that the Shaddai conception is the positive conception, allness covers and includes all, whereas the scholastic conception is the negative conception, it merely negates limits.

To appreciate this I think it would be interesting to take us on a detour through the mathematics of infinitesimals and how it defines infinity differently from standard calculus. In standard calculus we are frequently reminded that infinity is not a number, and that it is merely a convenient notation to refer to a function that increases without bound or limit, but it does not "reach" this number called infinity. This sounds closer to the "no limit/bounds" definition of infinity.

In the calculus of infinitesimals on the other hand there is a hyperreal number called infinity, but it is defined in terms of infinitesimals, the "infinitely small". To grasp this we need to look at the definition of infinitesimals:

A number h is an infinitesimal if h>0 and h < r for every positive real number r.

Thus h is not zero, but it is smaller than every positive real number. Thus, in a way, the definition of infinitesimals includes all the positive real numbers and is defined as smaller than all of them. Infinity is thus defined as the reciprocal of the infinitesimal, if you divide 1, or any finite number, by the infinitesimal, you get infinity. Thus, infinity is the number larger than all real numbers.*

Now you might wonder, what difference does this subtle difference make? It sees like a very fine-grained point. I would however suggest two reasons for this excursion. First. as already noted, we have the form of divine revelation itself and how God choses his own name. As already mentioned, God did not name himself the One, the En-Mispar ("without limit"), God named himself El Shaddai, and since this is a name God has chosen to reveal himself, we ought to take some interest in the form of his name.

Second, I think it makes a phenomenological difference in how we experience and understand God in this sense: if God's infinity as "Allness" includes or is defined in relation to how He overcomes/overpowers all finite things, then we can directly relate to how God will act or acts in our everyday finite situations. All finite problems and troubles in our lives are included in God's All-Mightiness which is larger than any such finitude, no sinful pleasure ought tempt us when we contemplate the "All-Sufficient" who can provide pleasures greater than whatever finite pleasure before us, etc. The scholastic mere negation of finitude does not connote the positive conception of God providing another positive thing greater than the finite evil before us.

As such, from these considerations, I would commend the positive "Allness" conception of infinity over the Scholastic "Without Conception of Infinity". (There is I think an argument that Duns Scotus himself had a positive conception of infinity as being all inclusive rather than the Thomistic mere negation of limits conception, but that would need a discussion in itself.)

*Actually if you look at the standard calculus precise definition of limits at infinity, the definition really isn't all that different, and in standard calculus limits at infinity is defined as the function being larger than any real number whenever the domain approaches a specific number. So there is still the "allness" sense in the standard definition, although not as emphasised as in infinitesimals.