Saturday, December 17, 2011

On Symbols

We spoke earlier of the relative adequacy, as a “symbol”, of the traditional OT picture of Jehovah,  to the uncountably infinite trans-reality of the Godhead Him-/It-/Blorg-/self.  This symbol is, literally, infinitely inadequate.  And yet adequate to our understanding, depending upon how high we have climbed on the ladder;  and crucially more adequate than certain alternatives, proposed by various paganisms and modern heresies.  Our purpose here is broadly  to defend the very idea of using symbols, and indeed the intellectual integrity of an admittedly threadbare image, simply by pointing out that such reduction to the palpable  is no mental vice unique to religion, but may be found even within -- nay, not science merely, to show that would be child's-play:  but even within that most abstract and impalpable exercise in mental extension, mathematics.

Consider, then, one of the simplest of mathematical objects:   the torus.  You have all seen this described as the “surface of a donut”.  Set aside the humbly sensuous coffee-dunking aspect of this:  the very essence of the image, be it of inner-tube or anchor-ring (the latter metaphor  was favored by our ancesters), is already a concession to our infirmity:  specifically, to our own creaturely incarnation in three spatial dimensions (this particular Sitz im Leben being of no mathematical significance whatsoever).  It is a symbol -- and quite an adequate one -- of the embedded torus:  but not (and this is crucial) of the torus as mathematicians now understand it, as conceived within itself:  but rather as embedded (with some bending) into the cramped space in which we dwell.   Beholding the donut, you think:  How curvaceous, surely more curvy than the surface of a tennis ball.  And yet, in its essence, it is not curved at all: Its intrinsic geometry is completely flat; and this “inner flatness” manifests itself  even in the embedded object, whose Euler characteristic turns out to be zero (as against 2 for the sphere).  (For a magisterial exposition of all this, see Jeffrey Weeks,  The Shape of Space.)   A more adequate symbol would be simply a rectangle with opposite edges identified -- or better yet, the toroidal covering-space which carpets RR with infinite replications of this patch-sample.  (Already how distantly we have left behind the donut !)   This symbol is still not the torus itself, as it thrones in Platonic heaven;  but it is a symbol which, unlike that of the donut, is wonderfully productive of new ideas.  As: Identify one pair of opposite edges as before, in parallel; but the other pair, anti-parallel, “with a twist”.  The result is a Klein bottle, not embeddable in three-space at all.  (Though four will suffice.)   Or, as:  Instead of parallel-identification of opposite edges of a square, perform an orientation-matching identification of opposite faces of a cube (thus resulting in a universe that is finite though Euclidean -- something they told me was impossible, when I was but a little boy in short-pants).   Or indeed (and now the usual geometrical torus of the nursery, seems as pale a reflection of the basic idea, as does that java-sodden donut of the former):  Consider any product (finite or infinite, countable or not) of R-mod-1 with itself -- a torus let loose from the leading-strings.
Similar such exercises result in surfaces we could never previously have imagined, as the very best scientists did not, but which, looked at correctly, become almost intuitive.  One of them might be the shape of the very space we live in.

It is, I suspect (speaking under correction), in part for such intellectual   reasons, that the Church Fathers, cognizant of the very partial truth of the God-the-Father Yahweh image, attempted to fill the picture out, with the doctrine of a Trinity (a blessed doctrine, it may be:  but arrived at, not handed to us on a platter in the Bible).  The symbol is not fruitful if you press it into fetishistic ends, snorfling up triads wherever they may be found; but it is an improvement on what went before.

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