The Mathematical Experience |
by Philip J Davis & Reuben Hersh |
3.Abstraction and Scholastic Theology (p113)
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.
Among the many fruits of abstraction of a mathematical type can be listed
systematic scholastic theology. In the view of
Bertrand Russell (History of Western Philosophy,
p. 37), systematic scholastic theology derives directly from mathematics.
It is especially interesting to trace this in the writings of Sa'id ibn Yusuf
(882-942). Sa'id ibn Yusuf (Saadia Gaon), philosopher, theologian, prominent
leader of Babylonian Jewry, was born in the Faiyum district of Egypt. In
922 he moved to Babylonia and was appointed head of the Pumbidita Academy.
His major philosophical work, Kitab al- Amanat wa-al Itqadat (The Book of
beliefs and opinions), makes ample references to Biblical and Talmudic authority,
but in addition draws on medicine, anatomy, mathematics, astronomy and music.
In the views of contemporary mathematicians, Saadia (we now use this more
common spelling) had thoroughly mastered the mathematical sciences, and it
is this aspect of the Kitab that we shall examine. Saadia is fascinating
because in him can be seen not only the mathematics of his day, but in his
systematic theology there were already present the methods, the drives, the
processes of thought which characterize nineteenth and twentieth-century
mathematics. The mathematics of the tenth century is present. Thus Saadia
says (p. 93),
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.
[This appears to be the source of this view posited by Xtians.]
In Treatise II: concerning the belief that the Creator of all things . .
. is one, Saadia begins his Exordium by saying (p. 88)
the data with which the sciences start out are concrete, whereas the
objectives they strive for are abstract.
This is certainly spoken like a modern scientist, and one wonders whether
this spirit is the imposition of a modern translator who renders "big" as
"concrete" and "fine" as "abstract." But I think not, for in the example
which Saadia then gives, it is clear that what is "fine" is an explanation
which is less specific, more general, and therefore an explanation which
is capable of dealing with groups of collateral phenomena, i.e., an abstract
theory. Further on he says (p. 90)
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.
[Which is not to say "he becomes blind" - but looks for the features which
exemplify the essential characteristics,or "filters" out misleading or irrelevant
information,this has the effect of rendering a mystifying view to
nomathematicians,I recall the joke of the farmer who asked a physicist to
help with his milk yield and the physicist supposed that the cow was a
sphere,which to the lay person is laughable,but topologically speaking as
far as holding milk is concerned,the essential feature of a cow is that it
has an inside and outside.The abstract cow then,is a pared down view of a
cow that can be worked on without dealing with the nature of the cow as a
creature,but it's essential feature of being able to hold milk on its
inside.Being "blind" to the other features of a cow,enables work to be done
without complicating the main issue -LB]
Saadia arrives at his concept of God in very much the same way. He has inherited
a backlog of thousands of years of theological experience, and these he proceeds
to abstract:
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.
Though there is in the corporeal something of God, God is not corporeal.
Though there is in motion, in the accidents of space and time, in emotions
or in qualities something of God, God is not identical with these. Though
these attributes may pertain to Him, for He is (p. 134) eternal, living,
omnipotent, omniscient, the Creator, just, not wasteful, etc., he has been
abstracted by Saadia out of these attributes. The Deity emerges as a set
of relationships between things some of which are material, some spiritual,
these relationships being subject to certain axiomatic requirements. When
Saadia seeks to know God through the process of abstraction, he finds a very
mathematical God. Having gone through this program of abstracting the Deity,
Saadia asks (p. 131),
How is it possible to establish this concept in our minds when none of
our senses have ever perceived Him?
He answers this by saying,
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."
[In fact there are other areas like fuzzy logic and
quantum physics
where strict Boolean logic does not necessarily apply -LB]
One might amplify Saadia's answer by pointing out that it can be done by
the process of abstraction just as an abstract graph is not a labyrinth,nor
a simple arithmetic or geometric representation of a labyrinthine situation,
but the abstracted essence of the properties of traversing and joining.
Conversely, a labyrinth is a concrete manifestation of an abstract graph.
(See Chapter 4, Abstraction as Extraction.) With respect to the current trend
of extreme abstraction, the mathematical world finds itself divided. Some
say that while abstraction is very useful, indeed necessary, too much of
it may be debilitating. An extremely abstract theory soon becomes
incomprehensible, uninteresting (in itself), and may not have the power of
regeneration. Motivation in mathematics has, by and large, come from the
"coarse" and not from the "fine." Researchers carrying out an ultra-abstract
program frequently devote the bulk of their effort to straightening out
difficulties in the terminology they have had to introduce, and the remainder
of their effort to reestablishing in camouflaged form what has already been
established more brilliantly, if more modestly. Programs of extreme abstraction
are frequently accompanied by attitudes of complete hauteur on the part of
their promulgators, and can be rejected on emotional grounds as being cold
and aloof. The same limitations are present in Saadia's conception of the
Deity. By its very nature impossible to conceptualize, it appeals to the
intellect and not to the emotions. Even the intellect has difficulty in dealing
with it. There is a story about a professor of mathematics whose lectures
were always extremely abstract. In the middle of such a lecture- he was proving
a certain proposition- he got stuck. So he went to a corner of the blackboard
and very sheepishly drew a couple of geometric figures which gave him a concrete
representation of what he was talking about. This clarified the matter, and
he proceeded merrily on his way -in abstracto. Saadia's concept of the Deity
suffers from the same defect. It requires bolstering from below. As part
of religious practice, it must be supplemented emotionally by metaphors.
Saadia himself seems to have been aware of this and so spends much time
discussing the various anthropomorphisms associated with God. He then makes
a statement that all proponents of ultra-abstract programs should remember!
(p. 118)
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 whole development in this section has a surprisingly mathematical flavor.
One of the standard mathematical activities is the proving of what are called
"existence and uniqueness theorems." An existence theorem is one which asserts
that, subject to certain restrictions set down a priori, there will be a
solution to such and such a problem. This is never taken for granted in
mathematics, for many problems are posed which demonstrably do not possess
solutions. The restrictions under which the problem was to have been solved
may have been too severe, the conditions may have been inherently
self-contradictory. Thus, the mathematician requires existence theorems which
guarantee to him that the problem he is talking about can, indeed, be solved.
This kind of theorem is frequently very difficult to establish. If Saadia
had been a theologian with the background of a modern mathematician, he would
surely have begun his treatise with a proof of the existence of God. Even
Maimonides (1135-1204) does this (to a certain extent). Thus in Mishneh Torah
Book 1, Chapter 1, he says,
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.
[Obviously this is profoundly naive by today's standards,it is amazing then
that so many religious people utilise it as a main stand by! -LB]
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.
[The "improbable universe" delivering God,suffers from this lack of
uniqueness,there are other answers besides "Jehovah God did it" -LB]
Uniqueness implies a well-determined situation, wholly predictable. Nonuniqueness
implies ambiguity, confusion. The mathematical sense of aesthetics loves
the former and shuns the latter. Yet there are many situations in which
uniqueness is, strictly, not possible. But the craving for uniqueness is
so strong that mathematicians have devised ways of suppressing the ambiguities
by the abstract process of identifying those entities which partake of common
properties and creating out of them a superentity which then becomes unique.
This is no mere verbalism, for the ambiguities are far better understood
by this seemingly artificial device of suppressing them. The drive toward
deistic uniqueness might be explained in much the same terms. Browsing still
further in Saadia, consider the following quotation which occurs as part
of his uniqueness proof,
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.
And later,
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.
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