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.
Interlude: A physicist depicts the arithmetical state-of-play in a
papyrus from Egyptian/Babylonian times:
It records the resolution of a
great number of fractions into a
sum of aliquot parts, the original
numerator always being 2: as, for
instance,
2/97
= 1/56 + 1/679 + 1/776
But no rules are given for
effecting such resolutions, and the whole treatise seems to be a mere
compendium of results obtained by repeated trials.
-- James Jeans, The Growth of
Physical Science (1947 [posthum.]; 2nd edn. 1951), p. 11
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 question
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 Austrian 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.,
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):
.
