There is a phrase within a clause within a prayer, of which I am particularly fond, and upon which I ponder ceaselessly: “…the Father almighty, Maker of Heaven and Earth,
and of all things visible and invisible…”
… visibilium omnium et invisibilium. Doubtless each line of the credo might profitably exfoliate into a tome,
but let us now, here, pause for a moment at this one.
The phrase is no mere afterthought. There are in fact a whale of a lot of invisibles out there, and it’s not just ghosts, or disembodied ectoplasm, or ethereal unstructured mush. Nor do I mean “dark matter” or “dark energy”, though apparently there are gobs & gobs of that as well: more than of ordinary matter, which now rattles about in the cosmos like spare change. (Or, to coin a phrase, like angry candy.) No, dark-whatever, doubtless all jolly stuff, and quaint in its way, but no more inherently fascinating than, say, rabbits. It’s merely the library-paste and plasticine that happened to be lying about when God got around to making this particular universe on some particular day, possibly with leftovers from some earlier practice project; and now it hangs about, drifting moodily hither and yon, like so much unemployed blancmange. Being invisible doesn’t make it significant or interesting. Let us have no fetish about the invisible. If for some reason the credo had said, “… and of all things probable and improbable,” and if the improbable had somehow been mostly ignored, yet contained most of what was of interest in the universe, then we’d be talking about the improbable; or the fantastical; or the ironical. The entities I shall be getting at here are not significant because they’re invisible; I’d be even willing to concede that they’re significant despite being invisible, that visibility would be one further and delightful perfection, one which we may someday hope to glimpse. In any event, what is meant here is the mathematical scaffolding, on which the sun and the moon and the quarks hang like so much laundry. That is, the plan of the thing, so much more permanent and pervasive than the things themselves. I mean the symphonic score from which our ephemeral melodies derive.
Properly apprehended, it is a structure of – crystalline palaces, transparent and thus largely invisible to the untutored gaze, save as the light of insight glances off them at an angle, and so catches the inner eye. These ideal edifices are as hard and as chiseled and as real, as our own makeshift dungeons of stone: nay, more real, for these intricate perfections are the prototypes, whereof our own poor earthbound shantytown is but the fallen, partial, semi-crumbled, quasi-scrambled, half-forgotten misremembered afterimage. They are, it is true, invisible: but in part (in increasing part) -- not unimaginable. Through intense and lifelong study, we may – by luck, or grace, or mental sweat – eventually acquire a glimpse of their upper ramparts, from which turrets rise, from whence pennants flutter – flutter in a plenitude, an infinitude of dimensions, one upon the other like palace halls; so that our own most swirling ballet or crashing waves are but as the slogging of an ant trapped between the narrow glass walls of the ant-farm.
These diaphanous entities, being (as we shall argue) a part of the Creation, display a different side of God, from what we customarily encounter. Or rather, as it may be, many different sides: the mystery of the Trinity becomes the mystery of the Infinity. For we must not think of “Math” as just some subject in school, or as a section at the bookstore, beyond “Gardening” and next to “Pets”. For one thing, there is just so much of it, acres of math like fields of wheat, with more unfolding with each passing day, and much which, when first met with, seems qualitatively, drastically diverse: not like different species, say a wolf and a fox, but like different phyla – a microbe and a mastodon. And even as science has discovered some of the commonalities between mastodons and microbes, in the process deepening our appreciation of each, so too does the steady, then accelerated, and finally springing advances of our collective understanding – as it might be, the Mathematical Overmind – deepen and widen and heighten and… beyonden our sense of the unitary structure of All There Is. The whole enterprise is so fantastic, with such unity-in-diversity (again, compare the Trinity) that whole new fields have evolved at a metalevel, just to keep tabs on it all: Set Theory and Proof Theory, to police our reasonings, and Category Theory, to provide display cases for all the genera of the menagerie, in the museum of the mind.
At the bookstore-cum-giftshop in Hilbert’s celebrated Hotel, you will find aisles for: History; Fiction (including Astrology and Economics); Physics ‘n’ Chemistry; Biology; Number Theory; Point-set Topology; Algebraic Topology; Algebraic K-Theory; Topological K-Theory; Real Analysis; Complex Analysis; The Riemann Hypothesis; Sheaf Theory; Topos Theory; The Poincaré Conjecture.; and Miscellaneous (i.e., gardening, computers, self-help, sports, celebrities, stamp-collecting, and all the rest). There is no separate section for Theology, since that overlaps all of them.
Now, none of this is exactly new: it is paleo/retro-NeoPlatonism. A retread, you may say, and twice-refried. Yet there is now much more concrete substance to the view, than was available to Plato or Plotinus. They might imagine the cube and the icosahedron, and Kepler might attempt to stuff the planetary orbits into such homely and visible (risible) boxes, but they never encountered a Riemann manifold – or rather, they did, because we live in one, but they couldn’t see it so they couldn’t imagine it, any more than they imagined fibre bundles or E8.
The Atheist, viewing a world charged with the grandeur of God -- which is difficult to ignore, since it will flame out, like shining from (to coin a figure) shook foil -- sniffs and dismisses it as tinsel: seeing the shook foil but not the Shaker. So too the Nominalist, beholding or rather failing to behold the serried ranks of theorems rising like seraphim beyond sight, regards these as a mere medley of contingent things, simply frothed out of someone’s brain, and which might, like a limerick or a pop-tune, have frothed out into something quite different. (This is if anything the more charitable of contemporary dismissals, vice the dismissal of math and science as being merely the Eurocentric patriarchal dogma of the racist sexist agist ablist ruling class…)
The nominalist viewpoint was given epigrammatic utterance (1886) by Kronecker, thus:
“Die ganzen Zahlen hat der liebe Gott gemacht; alles andere ist Menschenwerk.”
(“Ganzen Zahlen” means integers, but he may have meant only the non-negative integers, or “natural numbers”.)
|Kronecker, rueing the day that ever he denied the transcendence of higher mathematics|
(Randbemerkung: A delightful swipe at Kronecker, well below the belt, and delivered with punch by a pugnacious Platonist, can be savored here: The Continuum. )
Stephen Kleene translates, “God made the integers, all the rest is the work of man”, and glosses:
We cannot expect that the cognizance of the natural number sequence can be reduced to that of anything essentially more primitive than itself.
(Introduction to Metamathematics, p. 19.)
The natural numbers – non-negative, non-zero whole numbers -- were, indeed, the only numbers recognized by the Pythagoreans. One reads somewhere of a Pythagorean casting himself into the sea in despair, upon encountering the proof of the non-rationality of the square root of 2; in fact, such self-drowning might just as well have been prompted by the sight of half an apple, since once you accept fractions, you are heading straight for E8.
For in admitting the integers as being in no wise contingent – the work, indeed, of the Necessary Being – while desiring to dismiss the rest (“Menschenwerk” sounds even more like a kindergartner’s art project than the more Biblical “the work of man”), Kronecker has left the castle of mathematical agnosticism unguarded, by leaving open its postern gate. Suppose we were – setting aside centuries of other riches – to begin by restricting ourselves to the laws of the natural numbers. We would notice (as Euclid noticed) the primes, and require, for their adequate handling, great heaps of Number Theory. Now, this already is no small thing. You could fill a succession of lifetimes with nothing but Number Theory. New discoveries emerge daily – some of them with such grave implications that they are actually classified, and at compartmented levels well beyond Top Secret, which does well enough for the design of an airplane or the movement of troops. But the wealth is not just quantitative. For to adequately handle nothing more than the primes, you need all of Number Theory, including even Analytic Number Theory, which brings in the continuum and the charmingly designated “imaginary” numbers (now much easier to imagine, though they remain as invisible as the number “23”), including indeed the Riemann Hypothesis, which already brings us to the frontier of knowledge with its outstanding unsolved problems. Soon the whole of mathematics would come tumbling in through the unguarded door.
[Note: It’s never that simple. I am aware that Kronecker himself was even more nominalistic that the famous quotation might suggest, as he did not accept the integers as a finished totality. He went to some effort to derive results in a way that makes no use of such a totality. Sort of neat if you can pull it off, like building a castle entirely out of toothpicks. But if the idea of an actual infinity is problematic, that of the integers somehow running out of breath is even moreso. I shall accept the natural numbers as given, and shall, for polemical purposes, portray, as our foil, a sort of idealized Kronecker (who may even now be repenting of his nominalism, in some warm place) – the Kronecker of the quote – as accepting them as well.]
[As to the actual forked-radish of that name, Joseph Dauben remarks (Georg Cantor, p. 66):
No-one could have been more opposed to Cantor’s ideas, nor have done more to damage his early career, than Leopold Kronecker.
Really, were it not for his key concession that the natural numbers are God-given, Kronecker might well be the villain of the piece. As George Szpiro wrote of him (in Poincare’s Prize): “Kronecker would not accept anything if it had not been invented by himself.” Actually that can’t be quite accurate, since on that account he would not accept the integers… But anyhow, a perfect summary of the solipsist/nominalist epistemology.]
[The essay continues here.]
[The essay continues here.]