Sunday, May 13, 2012

Infallible: A Parable

[a continuation of this]

<Bi-smi llâh :>

This mini-essay concerns, not the unfailing truths of mathematics, which predate us, but the human practice of mathematizing, which (like everything else situated within History) is in principle contingent, and thus eminently fallible. (For more on the distinction, click here.)

As a matter of curious historical fact, our mathematical progress has been much less contingent than you might expect.  There have been no real false-starts, and many simultaneous independent discoveries.  Why that should be, would be a subject for a dissertation in itself, except that no human could write it, as it would require a God’s-eye view of the mathscape, across which we mortal/mental Flatlanders  grope our way …

And while we are on -- or rather off the subject, not having yet commenced the actual mini-essay itself, nor so much as hinted at its topic (today is Sunday, you see, and we are at leisure): 
A God’s-eye view -- truly?  Possibly not.  Perhaps an angel could write it:  much as a three-dimensional Fatlander can tell the Flatlanders what’s what, without requiring yet higher dimensions.  But if an angel were to write such -- could we understand what he wrote?  (“Wenn der Löwe sprechen könnte, wir könnten ihn nicht verstehen.”  Angeli, a fortiori.)   More generally:  Could we ever understand something that  in principle  we ourselves could never have discovered unaided? 
One is inclined to think, not.  -- Which in turn suggests a very depressing prospect of the Afterlife, to anyone who envisions the latter  more in terms of Topological Pataphysics than in terms of harps, and gowns, and tennis played on clouds (the New Yorker-cartoon school of eschatology.)   The Resurrected Body -- a shimmering mystery, but --  soit.  Yet what of the Resurrected Brain?

Yet it may be, that we have already had a taste -- here below -- of how we might handle this.  For we have had Revelation, both Scriptural and mathematical (though the mathematicians modestly call it Intuition).   The result?  Some of us understand some of it, somewhat, some of the time.

[End excursus.  Or precursus.  Whatever.]

<Ammâ ba`d:>

And so to the Parable.

(1) The closest we come to innovative external guidance towards what we think or hope might be the Truth, in the moral sphere, is from a pronunciamento from our spiritual leader, be he Pope, or imam, or (for Babarites) Babar.  Otherwise we rely on traditional community consensus -- ijmâ`, to use the useful Muslim term for this.

(2)  In mathematics, there is no Pope or mufti, nor could there be.  There is usually a “dean” of mathematics at any one time, and occasionally a “Prince” (Gauss).   The latter we admire;  but we wouldn’t take their word for anything -- anything at all.
(Andrew Wiles proved Fermat's Last 'Theorem';  I grovel in the gravel should his shadow pass.
Yet let Wiles opine, that Diet Pepsi is the beverage of choice;  and I reply,  "Oeww.... reahhhhlehhh....")

Thus, the proposition erroneously known as “Fermat’s last theorem” would doubtless never have received all the effort and attention it did, had it been known merely as “Schlumpfnerd’s Conjecture”.   But for all that, the prestige of Fermat did not in itself incline mathematicians to reckon that the proposition was probably true, and they soon came to the conclusion that Fermat had been deluding himself as to possessing a(n unwritten) proof.   The real reason the proposition is so celebrated is because of that endlessly retold anecdote about his having a proof but the margin was too small to contain it.
Outsiders might imagine that Proof is Pope, but that’s not quite right either.  Though we  from time to time  do use explicit Proof,  here too  consensus is the ultimate arbiter  -- e.g., re whether a purported proof is actually valid.  This is not to say that mathematical truth is socially constructed, merely that our growing/groping understanding of it  is partial, and historically conditioned.
Occasionally -- although, in general,  mathematics has been blessedly bereft of fitna -- occasionally  consensus is lacking,  not merely on the technical status of a purported proof (think:  de Branges), but about the philosophical status of a means of proof:  thus, Intuitionists rejecting the non-constructive.

(3) Both theological and mathematical understanding may be revised and enlarged  in light of subsequent discovery or revelation:  and indeed, in such a way that the previous understanding is not so much refuted, as sublated -- that is, transcended, as a saint transcends the ovum.
Such a state of affairs does not mean that the concept of Truth is no more than: “true for our time”, on the slippery slope down to:  True for the Trobrianders;  True for Solipsistic Intersexuals;  True for J.Q. Ortcutt, sole member and founder of Ortcuttism, a new religion.  Truth, more than beauty, is aere perennis.

Now:  I have mentioned that the historical course of mathematical invention (presumably because it is largely a matter of mathematical discovery) has run more smoothly than one might have anticipated, given the repeated futilities of all human activities else.   Yet let us, in this Parable -- it is almost a sotie -- imagine that the muse of mathematical history, Clio -- or as it might be, Clotho:  for we are in that dim time, before the gods -- had willed things differently (wise, yet wilful, is She):  and that initially, there had been no Parallel Postulate.   That is to say, mathematicians began with a slightly different Euclidean Creed.   The postulate was by no means denied;  it simply was not generally noticed;  and by those (and they were most of them) who tacitly took it for granted, they took it for granted as well, that others too took it for granted.
And indeed, one can prove enormously much without ever invoking the thing -- all the truths that go over to the general Gaussian case.   Yet for certain particular propositions -- as gradually became clear -- the thing is needed.
At which point (so our parallel parahistorical fantasy runs), mathematicians fell to quarreling, dividing into warring camps;  and discord reigned in the land.   Practitioners fell into sects:  the Parallelists;  the Intersectionists; and the Diversionists (the latter prefiguring Riemann resp. Lobachevsky).  Yet none could convince the other, and the acrimony waxed hot.   Many the bloody broil  that ensued:  and many the skald, lyre-strumming atop the hill, who sang brave deeds of battle.

Now:  among the sectarians, none could  by any means  point to any concrete grounds for his belief -- or call it, an intuition -- yet only the more stubbornly, and with an inmost light, held fast to what he took as Truth.

At length -- the wide world  weary of the fray, the fields in ruins, sheep scattered amid the tares -- a great Conclave was called.  And after hearing-out each side, Pope Euclid the Great stood up at last, and, spreading his robes before the multitudes, proclaimed, “infallibly”, that the Parallel Postulate should hold sway.  It was duly added to the Credo.

Plain men rejoiced, since this axiom, after all, matches our everyday intuitions.  And so the matter remained for over 2,000 years.
Then lo, there arose in the West, not one prophet but two:  Saint Riemann, aye, and e’en Saint Lobachevsky (known to the faithful of another pastorate as Saint Bolyai), each proclaiming a new truth, axiomatized and elaborated with startling clarity;  and the scales fell from our eyes.  Then the Church of Mathematics welcomed this advance; and the Riemannian Postulate, and the Lobachevskian Postulate, took their honored places in the geometric tryptich, alongside that of Saint Euclid:  which remains, in our own day, in its own sphere and as far as it goes, absolutely True.

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