ABSTRACTION IS the life's blood of mathematics,and conversely, as P. Dirac points out, "Mathematics is the tool specially suited for dealing with abstract concepts of any kind. There is no limit to its power in this field." But abstraction is ubiquitous. It is almost characteristic or synonymous with intelligence itself.
I do not demand of Reuben 100 drachmas, but I demand of him the square root of ten thousand.
This is a turn of phrase which would probably not have occurred to the man on the street in Pumbedita, but it is surely not the most exciting thought that tenth-century mathematics could have dreamed up. But there it is, in a religious context. There is a discussion of time which might remind one of a reversed paradox of Achilles and the tortoise, not put to a destructive purpose as with Zeno but to the positive purpose of proving The Creation.
If the world were uncreated, says Saadia, then time would be infinite. But infinite time cannot be traversed. Hence, the present moment couldn't have come to be. But the present moment clearly exists. Hence, the world had a beginning.
the data with which the sciences start out are concrete, whereas the objectives they strive for are abstract.
the last rung in the ladder of knowledge is the most abstract and subtle of all.
This is all prelude to his insistence that God must be understood, in fact can only be understood, through the process of abstraction. Saadia's Deity is accordingly highly abstract,highly intellectualized. One of the major programs of modern mathematics is the abstract program. This may come as a surprise to the nonmathematician for whom such things as numbers, points, lines, equations are already sufficiently abstract. But to the mathematician whose profession has been dealing with these objects for three thousand years, they have become quite concrete, and he has found it essential to impose additional levels of abstraction in order to explain adequately certain common features of these more prosaic things. Thus, there have arisen over the past hundred years such abstract structures as "groups," "spaces," and "categories" which are generalizations of fairly common and simple mathematical ideas. In his role of abstractor, the mathematician must continually pose the questions "What is the heart of the matter?," "What makes this process tick?," "What gives it its characteristic aspect?" Once he has discovered the answer to these questions, he can look at the crucial parts in isolation, blinding himself to the whole.
the idea of the Creator . . . must of necessity be subtler than the subtlest, more recondite than the recondite, and more abstract than the most abstract.
How is it possible to establish this concept in our minds when none of our senses have ever perceived Him?
It is done in the same way in which our minds recognize the impossibility of things being existent and nonexistent at the same time, although such a situation has never been observed by the senses.
That is, we recognize that "A" and "not A" cannot coexist despite the fact that we may not ever have experienced either "A" or "not A."
Were we, in our effort to give an account of God, to make use only of expressions which are literally true . . . there would be nothing left for us to affirm except the fact of His existence.
Saadia also speaks (p. 95) of a proof of God's uniqueness .
The basic principle is that there is a First Being who brought every existing thing into being, for if it be supposed that he did not exist, then nothing else could possibly exist.
To the mathematical ear, this sounds like proof by contradiction (a much-used device);the fact that the mathematician might be inclined to label Maimonides' syllogism a nonsequitur is irrelevant here. But Saadia does not, as far as I can see, proceed in this way. The existence of God is given, i.e. is postulated. His uniqueness is then proved, and later, the properties which characterize him are inferred through a curious combination of abstraction and biblical syllogisms. Here the method of the Greeks is fused with Jewish tradition. This brings us now to the question of "uniqueness theorems."Just as an existence theorem asserts that under such and such conditions a problem has a solution, a uniqueness theorem asserts that under such and such conditions a problem can have no more than one solution. The expression "one and only one solution" is one which is frequently heard in mathematics. Much effort is devoted to proving uniqueness theorems, for they are as important as they are hard to prove. In fact, one might say that there is a basic drive on the part of mathematicians to prove them.
For if He were more than one, there would apply to him the category of number, and he would fall under the laws governing bodies.
I say that the concept of quantity calls for two things neither of which can be applied to the Creator.
It appears, then, that God cannot be quantized. Yet God can be reasoned about, can be the subject matter of a syllogism. This may strike one as analogous to the fact-which is less than 150 years old-that mathematics can deal with concepts which do not directly involve numbers or spatial relations. In sum, in Saadia's chapter on God, one finds the process of abstraction, the use of the syllogism including some interesting logical devices as "proof by contradiction." There are also certain logical concepts which have become standard since Russell and Whitehead such as the formation of the unit class consisting of a sole element. Furthermore, there is the realization of the central position that existence and uniqueness theorems must play within a theory.
See Bibliography Saadia Gaon,J. Friedman.