But yet when considered, may help us to enlarge our thoughts towards greater perfections of it in superior ranks of spirits. … The several degrees of angels may probably have larger views.

-- John Locke,

__An Essay Concerning Human Understanding__(1690)
Man’s understanding, though allied to the angelical, operates differently. The angels understand intuitively, man by the painful use of the discursive reason.

-- E. Tillyard,

__The Elizabethan World Picture__(1942)
It is presumably not obvious to the chimpanzee (or, if this be setting his smarts too low, to the humble woodchuck) that for all

*m*,*n*in**Z**,*m*+*n*=*n*+*m*. Nevertheless, in his daily scurryings and burrowings, he will repeatedly meet up with particular instantiations of this modest truth.
For the woodchuck (at any event the southern northeastern lesser striped variety) builds a number of nests and other temporary dwellings, each of which has the framework of a variously triangulated polyhedron, built tinkertoy-fashion from a fixed number of sticks. Now, gathering them one by one would take too long, nor can the tidy woodchuck stand to have any sticks left over. So when constructing his summer dwelling -- an icosahedron, which needs thirty sticks (did I get that right? My calculating powers are not much beyond those of a woodchuck) -- he normally harvests a jubjub bush, which has twenty-two sticks of exactly the right specs and which blooms in the spring, then rounds it out with the eight-sticked glubglub bush, which sprouts slightly later.

But then one year, the blooming of the jubjub was delayed, and the woodchucks despaired. All but one, the enterprising Willie, who went doggedly (or groundhoggishly) ahead and harvested the available glubglub, supplementing this when the jubjub arrived slightly later. This remarkable exploit was recorded in the annals: for 22 then 8, one may substitute 8 then 22.

It was subsequently found that a mubmub bush (18 sticks) followed by a nubnub bush (12) would do just as well –

*und zwar*, in either order! This fact too was recorded.
The years went by, then the centuries, and the millennia, and the annals grew to seven times seventy stout volumes, densely filled with such arcana as: a cube-for-cubs may be constructed of a lublub (7) plus a rubrub (5),

*and this in either order*; and so on for billions of examples. All this was considered a branch of botany, a purely empirical science.
By this means, the woodchucks arrived at an analogue of Babylonian mathematics.

Until one day one Wisedome Woodchuck, a distant descendant of Willie, figured the whole thing out, and in a remarkable demonstration of only eighty pages (rather hard to follow, but sound), showed that

*m*+*n*=*n*+*m*was a perfectly general fact, replacing the seven-times-seventy volumes at a stroke, and freeing up his brethren for yet further architectural innovations, which previously had been shunned, as their particulars were not yet in the book. The annals were placed in a museum, which the elder woodchucks might still visit, marveling at favorite exhibits (as who could forget that remarkable winter, when 5,878 + 519 turned out to be equal to 519 + 5,878? A tour de force!). Meanwhile generations of young woodchucks (the pride and despair of their parents, who could not follow them into Canaan, with their aging brains) studied Wisedome’s proof, breaking their little heads against it.
Meanwhile in Metropolis… The humans, learning of this, politely saluted Wisedome’s modest accomplishment, and experienced a pang of sympathy for woodchuck-kind; yet felt no inclination to visit their Museum of Particular Results: for which they felt, indeed, a kind of horror. And even the general result, while true, is somehow to us not truly

*interesting*. In any case we are all too busy wrestling with the Riemann Hypothesis, to have time to look back.
Meanwhile in Elysium, where throne the angels

*sensu strictior*, the lowest order of angelic beings*sensu lato*, a mock compliment is paid to Andrew Wiles, who*finally*figured out that little Fermat puzzle, with which the angel-kind are wont to amuse the nursery. Not that the angels arrived earlier at his proof, nor any refinement thereof. They simply scoop up a few infinities of integers with their fractal fingers, twist them this way and that—and see, it doesn’t fit! Simple.
Moreover, all

*facts*about all structures of ordinal type omega, whether or not deducible by any finite axiomatization, are equally transparent to the angels. They just*look*.
So, is Elysium the mathematical Paradise? Not quite…

In a remarkably lucid and accessible article*, which should be packed into every pupil’s lunchbox by a considerate mom, Gödel observes that our continuing failure to resolve Cantor’s continuum problem, left over from the previous century, is quite an embarrassment. It means that we are unable to wrap our minds around the very simplest multiplication problem possible, beyond the finite ones that these days can scarcely stump a woodchuck. Namely, two times two (times two, times two – keep going). He writes:

“It is easily proved that the power of the continuum is equal to 2^(aleph-nought). So the continuum problem turns out to be a questions from the ‘multiplication table’ of cardinal numbers: namely, the problem of evaluating a certain infinite product (in fact the simplest non-trivial one that can be formed). There is, however, not one infinite product (of factors > 1) for which so much as an upper bound for its value can be assigned. […] It is not even known whether or not

*m*<*n*implies 2^*m*< 2^*n*.”
We are so to speak staring helplessly at a pile of sticks.

Nor does the subsequent Cantor+Cohen demonstration of the independence of the continuum hypothesis from a particular system of axioms for set theory set the matter aside. Gödel had already anticipated Cohen’s result, and wrote:

A proof of the undecidability of Cantor’s conjecture from the accepted axioms of set theory (in contradistinction, e.g., to the proof of the transcendency of pi) would by no means solve the problem. For if the meanings of the primitive terms of set theory … are accepted as sound, it follows that the set-theoretical concepts and theorems describe some well-determined reality, in which Cantor’s conjecture must either be true or false.

Indeed Gödel suspects that the Cantor conjecture is actually, factually false: which means that somewhere, among the actual literal real numbers, there is hiding a set of cardinality intermediate between aleph-nought and its power set, with definite members which the angels could name. Not, however, the lowest order thereof; this lies beyond them. But at the next step up, the archangels hang these sets from mobiles over their infants’ cribs. In fact a woodchuck may somewhere inadvertantly have used one of these sets for nesting materials, and even now lies sleeping on it – a night of troubled dreams.

So much for a simple pancake-stack of omega-many deuces – the limit of the lower-angels’ ken. What about the square root of omega-to-the-omega; or cross sections of fibre bundles on toroidal cap-omega-cross-theta space? For each level of angels, there will be something beyond them that they just don’t get.

*

There are two poles of the range of approaches to the problem of infinities. One is that of the badger-like Brouwer, who simply sweeps the chessmen to the floor, folds up the board and goes home. (An only somewhat more amenable figure, says Gödel, is Weyl, who allows as how there might be something to board games, but suggests we play checkers – or Chutes ‘n Ladders – rather than chess.) The other pole says: Infinities are tricky, but they all exist, and are present to the

*Infinite Mind*. Gödel himself uses that term, e.g. noting that Ramsey’s admission of formulae of (countably) infinite length might be constructivistic for an infinite mind but not for our own. Gödel does not, however, seem to feel much need for any desperate appeal to such a mind, in the course of an ordinary day, since he -- like Badger’s amiable friend the Water-Rat-- is a thoroughgoing Realist, and comfortable as such in his own skin. For him the assumption of infinite classes “is quite as legitimate as the assumption of physical bodies, and there is quite as much reason to believe in their existence.” The outwardly gloomy Hungarian is really the jolly Dr. Johnson of set theory.
Only now there’s a problem, of a sort which did not confront the schoolmen, who never counted on the uncountable: the Infinite Mind is all very well, but --

*Which*infinity did you have in mind?
Who comprehends *everything*? God does, by definition. Yet He cannot be simply the crown on a tower of constructively ascending intelligences. He is like an “inaccessible cardinal” – and not the first. Nor perhaps ‘the last’, if there is no last. Whatever He might be, there is Cantor in the wings, grinning, waiting to perform a Power Set on God, yielding – what? -- Nothing one can begin to commence to pretend that we can approach with our sadly finite understanding.

All of which suggests, if nothing else does, that God is something more and other than an alternately wrathful and affectionate granddad with a perfectly enormous white beard – however much longer that beard might be, than the stubble which disfigures your chin or mine. Who one day, apparently from sheer idleness, as one might choose chocolate, chose the Jews. Who later, some say, cast a Jove-like eye on a certain Palestinian virgin. And who at present is very angry indeed with the Democrats (or the Ravens, or whomever). Yet what He in fact might be, we cannot even begin to imagine anyone’s beginning to conceive. (Cf. the suggestion of 1 Kings 8:27 that the heavens themselves have heavens (and so on up); and that the whole omega-tower of them cannot encompass God.)

* * *

~ Commercial break ~

We now return you to
your regularly scheduled essay.

* * *

We actually wind up with a sort of hamstringing of the Ontological Argument. Notoriously its conclusion does not really follow from its premise; but now even its premise limps: “Since we can imagine a Perfect Being…” But that’s just it, we can’t! Not even little infinite bits of one! Yet paradoxically (and God reportedly loves paradox – at least Chesterton does, His publicity agent on Earth), this seeming stomping on the prostrate corpse of the offspring of Anselm, this despairing cry that somehow even Infinity does not suffice, so far from opening the agora to legions of snickering atheists chanting “Toleja so!”, points somehow upward, -- outward, -- onward …. Praise Him!

Postscript:

John Locke himself, normally regarded as the Poster Boy for Empiricism, of I'm-from-Missouri common-sensicality, yet delivers himself of this (

__Essay__, III.vi.12):
That there should be more species of intelligent creatures above us, than there are of sensible and material below us, is probable to me from hence: that in all the visible corporeal world, we see no chasms, or gaps.

That is to say: The gap between ourselves, and God, must somehow be filled, according to the Principle of Plenitude.

And again (IV.iii.23):

He that will consider the infinite power … of the Creator of all things, will find reason to think, it was not all laid out upon so inconsiderable, mean, and impotent a creature, as he will find man to be; who in all probability, is one of the lowest of all intellectual beings …

Angels of all sorts are naturally beyond our discovery, and all those intelligences, whereof ‘tis likely there are more orders than of corporeal substances, are things, whereof our natural faculties give us no certain account at all.

Since theism is far from central to Locke’s Essay, it is curious to see the emphasis on this

*scala naturae*idea.
--------------

*”What is Cantor’s Continuum Problem?”, repr. Benacerraf & Putnam, eds.,

Compare further (re ascending ranks of abstraction and generality):

.

*Philosophy of Mathematics*.
~

Postscript: For the possibility that the structure of

*certain*mathematical truths relating to an infinite domain might resist any but a case-by-case “Babylonian” approach, cf. the quotation from Michael Dummett towards the end of this post:Compare further (re ascending ranks of abstraction and generality):

.

This is one of the most important essays I have read in a long time.

ReplyDeleteAnselm would likely have insisted that what he meant was that a being, conceived of as infinite in knowledge and power, albeit by a mind that cannot itself grasp infinity, was still conceived of. And the rest of his argument would then continue.

But what you've asserted here is ultimately more Orthodox, both in terms of the Eastern Ecclesiastical definition, and the broader sense of fidelity to the Great Tradition.

You've gathered the abundant evidence of our intellectual inadequacies, albeit through a delightfully erudite treatise which speaks to mathematicians as one of their own, and drawn the conclusion that all the delight of a universe in which even one person can expound on such things for an audience who can be intrigued...is a universe which points to the puzzling possibility of...order.

And in light of such order, as the Psalmist said, the heavens declare the glory of God (Ps. 19:1). And so do the electrons. And so does 2x2.