Monday, December 20, 2010

Theologia Mathematica: I


            Beginning with a parsimonious outset of only two Postulates,
            (1) Die ganzen Zahlen hat der liebe Gott gemacht … [Kronecker]
            (2) …visibilium omnium et invisibilium [the Credo]

we conclude to the Realist position in mathematics, associated with Cantor and Gödel. We note the nice fit with theism.]


The great Laplace, having presented his Mécanique Céleste, and having been asked by the Emperor, "Mais où est Dieu dans tout cela ?", notoriously replied:
"Sire, je n'ai pas eu besoin de cette hypothèse."

The cheese-eating atheist, feeling pleased with himself

(I am reminded of the austere style of his countryman Lagrange, who boasted that his Mécanique analytique contained no pictures to help make things plain; and later their compatriot Dieudonné, who in the preface to his celebrated Foundations of Modern Analysis, warns his audience that the tome will contain no such sweetmeats as pictures or diagrams, as that would only encourage the reader.)

            Perhaps people read too much into that oft-quoted remark (or in a sense, too little).  It is often taken, I suspect, as a dismissive, not to say smart-alecky reply:  a snook cocked at theists.  But really the context is both richer and more narrow.  The famous remark is more austerely analytical, I believe, and comparable to Newton’s celebrated “hypotheses non fingo”.
            Newton discovered the basic clockwork of the planetary scheme, but it was not at the time apparent, whether that system were ultimately stable under the various perturbations that mass is heir to.   And if it turned out in fact not to be so, then how to explain its evident stability over all geological time (itself of a vastness only recently appreciated – long enough to let *us* evolve, for instance)?  Before inertia was discovered (or, again rather, in a way more like posited, but still: based upon a more systematic survey of the phenomena), a  traditional perspective had angels impelling the planets in their paths by constantly puffing on them from behind (a pleasant thought); now it seemed as though these same angels might have to be called out of retirement, not to impel the planets, but to herd them from time to time, lest they wander off like lost sheep.  It was Laplace’s great achievement to prove stability by sheer mathematical means – as noble a use of the imaginative faculty  as ever graced the Sistine Chapel, and by no means a snub to the Watchmaker.  Laplace, in demonstrating that planets (‘wanderers’ in Greek) were (like the a-toms) misnamed,  proved we may dispense with the hypothesis of shepherding angels –  for this.
            But there is more beneath the firmament than the placid planets.  Beyond the deductive system of classical mechanics, there is the inductive panoply of actual life.  This too Laplace addressed, in his Essai philosophique sur les probabilités.  And again, he made no use of the supernatural.  But perhaps he took too much comfort from the pleasing and positive example of the stability of the solar system.  There was an overconfidence, a sunny bumptiousness, taken to task at length by Keynes in his Treatise on Probability (a work too little known, and which I commend to your attention).  Keynes descries a witless wizard behind the mathematical manipulations that pretend to deduce so much, and who ultimately throws us back -- surreptitiously -- on human intuition to make sense of events.  (Pay no attention to the man behind the screen.)

            Since Keynes’ time, the problems of induction and prediction  have only grown worse – or rather, they remain the same as ever, but our awareness of their depth and paradox has grown.  We have now become familiar with examples of dynamical chaos, even within the heart of the classical theory; and learned that such systems are, theoretically, rather the rule than the exception.  Even that simplest paradigm of all, the fabled billiard table, affords examples.  There are tables so shaped that, given a desired degree of knowledge of the trajectory by some future time, one may attain it by sufficiently precise initial conditions, whose precision is in some sense reasonably related to that of the required precision of prediction;  and there are tables so shaped that one may not.  (That is, the relation of the intransigent epsilon to the hapless delta, is in one case that of a banker, with perhaps rather stiff interest rates, and in the other, that of a highway robber.) This directly refutes the Laplacian determinist vision, which once extended to the universe as a whole, and which now fails right in the pool hall.

[A note, since this essay is going out to a diverse audience:  The time is past, when an author need pull his punch, for fear some member of his audience may not have met this or that notion.  You are online, or you wouldn’t be reading this.  So if “dynamical chaos” is an unfamiliar term, simply Google it up in Wikipedia.  Wikipedia knows all that can be known to Man.  – Though incidentally this does *not* mean that we should actually *pray* to Wikipedia.  Just so you know.]

            So we are back to a paradox.  System after system is either itself chaotic, or is in contact with, and thus influenced by, chaotic systems:  The parson is a placid man, but he lives within the weather.  Why then does the center apparently hold?  Why does it not all eventually – spin out, or bubble down to a sort of porridge?  Does it, like the imagined planets, need a nudge from time to time, to get it back on track?  -- Well, whether it needs them or not, it gets them, but: from us.  For the nonce, let us leave God and the seraphim aside.  [Note: In what follows, we assume the three-level system of natural description which C.S. Lewis expounded in Miracles. Roughly: subnatural = quantum; natural = classical (including relativistic etc.);  supranatural = involving free will.]  For supra-natural intervention, we need look no farther than our own fingers – righting the wineglass that had started to topple, or pulling the baby back from the edge of the stage. (The paradigm case of supra-natural intervention is supernatural intervention, by ghosts or by God; but as a logical problem, this differs little from the intervention of human will, and thus may be dispensed with where parsimony suggests.)

            Most of what goes on the the universe is (let us call it) naturalistic – whether ruled by the (classical, relativistic) laws of the natural, or the (quantum, aleatory) laws of the subnatural, or some complexus of both. But most of what *we* experience, day to day – that is, experience in consciousness, as opposed to this or that enzyme oozing about – is generously admixed with the supra-natural.  I mean this in its familiar, almost ho-hum sense (except that, by dint of our ongoing ho-humming, we have become dulled to the fact that it is, strictly, miraculous.  Our simplest Saturday afternoon crackles with miracles like a fourth-of-July sky.)
            Now, this in itself need not point to God (let alone prove Him), any more than this or that billiard-ball impact proves classical mechanics, or some passing photon yields Maxwell’s equations.  Indeed, in so far merely as itself, it does not so much as indicate that this supra-natural capacity in ourselves is even rational or good:  picture (though only for a dreadful moment) a universe peopled exclusively by madmen and sociopaths, the free-willed equivalents of scorpions.  That is to say, an outside observer of our universe might detect its departures from plain (non-quantum, non-noetic) determinism, whether from sub-natural or supra-natural inputs; but lacking internal access to the lived experience, he could not tell whether these departures made sense.  Indeed, our hunch now is that the subnatural inputs do not “make sense” – that is, no moral sense, no sense beyond themselves.  It will all (within its own world) dutifully trot along in the path laid out by the Schroedinger equation, while its inputs to our world look to us like clowns piling out of an infinite Volkswagon: but it will not, pace philosophers from Protagoras to Penrose, supply or even heighten our humanity, our morality, our free will.  Indeed, from our present vantage, the subnatural is somehow even more alien to the noösphere, than is the shadow play of Newton, or the passion play of Darwin.  He who would seek the key to our humanity there, seeks the stars in a mudpuddle.
            Whereas we, in our priviliged observatory of our own shared experience (though of nothing else), can report:  Yep, it makes sense.  It’s often in practice too f***ed-up for words, but we’re not just ensouled scorpions, we’re … possibly fallen, anyhow substantially tattered angels.


 A distant kin to the imagery of the ushering angels, impelling the planets and keeping them on course, cropped up again in 1925, with de Broglie’s idea of “pilot waves”, guiding the electrons in their rounds while orbiting the proton; revived again in another context by David Bohm, in his hidden-variable theory of quantum mechanics.


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