In the Ike era, we grew up on Wonderbread® : a sort
of Brot ohne Eigenschaften whose
edulcorated transmogrification is known as Twinkies.
Since that time, we have learned to abjure, not only such
treif, but anything not calling itself wholegrain. Or, better still, multigrain: some brands boast seven
grains, a few claim twelve; disparate mixtures full of gritty, grainy, crunchy goodness.
Now our local
upscale supermarket offers a variety of own-branded bread, that boasts (in large letters) no
fewer than
27 GRAINS
That really surprised me. It’s one of the main points of Jared Diamond’s Guns,
Germs, and Steel that
digestible, domesticable, feasibly growable grains are not to be had for the asking; there just aren’t that many of them. The explanation is that, a little
lower down and in smaller font, the label reads
AND SEEDS
In other words, this brand of bread is equally at home in
the bakery and in the bird-feeder.
Our son, scoffing at this terminological legerdemain,
inquired why the market chose “27” of all things. Unhesitating I replied, “Because it is three to the
third power -- the trinitarian pinnacle of the Perfect Cubes”. -- Said heir and offspring, himself a
nascent mathematician, appreciated the point, but doubted that your average
shopper was aware of such things.
And indeed, there is a larger point.
For number-theorists like
Ramanujan, each integer has its own flavor, its own biography and backstory --
cf. the famous incident of the
taxicab numbers (in which G.H. Hardy plays
the straight-man or fall-guy).
But
for ordinary folks, like rocket scientists (who deal with contingent analogue
quantities, rather than integral transcendent entities) or English professors
(surrounded by a midge-cloud of pre- or sub-arithmetical post-modernists), all but a very few --
small -- integers
will be
featureless.
Some will be
visually familiar:
3 and 4 (the triangle,
the square), 5 (the quincunx on dice), 6 (boxcars, ditto),
7 (a “lucky” number for the superstitious) or 23 (ditto, for the more
cerebral), 10 (count your fingers), 20 (with its portrait of Jackson).
Some will be familiar for
incidental, non-mathematical reasons, like 100 and 1000 (which owe their
prominence to the accident of base-10 notation -- a case of
decimal fetishism).
--
Those
associations are widely shared;
but there may be others
more individual;
as,
(mostly for girls), “sweet sixteen” (or in Latin America,
la quinceañera).
Indeed, any integer small
enough that you have lived that number of years (particularly those for which
you still kept track of your birthdays). As, the poem(-collection) “Now We Are Six” (by
A.A. Milne), an anniversary in memory still green (and which I teach to the
neighborhood children when they reach that delightful threshold). Or…. 27; which, even before I had attained that age, always seemed
numinous (probably, indeed, for its prime-power nature), and which was ratified
as such by my marrying at exactly that age, quite close to my birthday. Whereas, for most folks, a
number like 81 (pourtant a perfect
square, as well as 3^2^2, to boot)
tells no tale, sings no melody.
~
The preceding remarks are of psychological or
anthropological interest, but of no moment for mathematics itself: They present an external,
human-centered view of the integers.
But further considerations suggest a subtle distinction between two ways
a given integer may be “interesting” (a more bloodless equivalent of our
anthropomorphic term crunchy). We might dub these internal and
external: both times internal to
mathematics as a whole, but in one case only, internal to number theory in its most
elementary sense.
Thus, consider 5. Number-theoretically, it’s a prime and
that is pretty much that; but in the geometry of three-space, it is the number
of Platonic solids. Or 17: Number-theoretically it is both a prime and a Fermat number;
but in the geometry of two-space, it is the number of crystallographic planar
symmetries.
https://en.wikipedia.org/wiki/Wallpaper_group
Or: 230, the
number of space groups.
Here, the integer in question is not a creature or crystal
considered distinct and in itself, but a stopping-point one arrives at by
calculating and counting. There
might have turned out to be, say, 18 plane symmetry groups, without upending
the world; but the internal structures of 17 and 18 are unrelated.
~
This dichotomy of
internal
versus
external interest
chez the integers, represents ideal
poles, which are not exhaustive, but bookend a spectrum.
As, probably intermediate: the “crunchiness”
of natural numbers as viewed by students of finite groups.
(Here the masticatory
metaphor
returns unbidden, for I
always imagine finite groups along the lines of a wrinkly-surfaced walnut.
Finite simple groups -- those for which
no homomorphism can split out any subchunk as its “kernel” -- the
hardest nuts, the ones you can’t crack.)
Simon Norton seems to have had such an intimate gustatory appreciation of individual finite groups, before he
Threw It All Away and went off to ride the bus.
~
In light of our training, we know at least one thing to do, should we ever be given
an unfamiliar integer and locked in a room, with no other toys to play
with. Namely, we can probe for
primality. (An activity that
becomes, indeed, crucial, for cryptographers.)
But now imagine that instead we had been handed some
fraction like
(2 + √137) /4081
Untrained, we react with dismay.
Yet later, learning of the Golden Section, (1 + √5)/2, and
its many remarkable properties -- not the least of these being that it can be
represented as the infinite continued fraction consisting of nothing but ones
-- we conclude that the critter is crunchy indeed;
and that there are more things in heaven and earth, than are
dreamt of here below, and that we must anticipate the afterlife, before we
could begin to embrace them.
~
Our treatment focused on what the innumerate are missing,
much like what the Daltonist, unbeknownst, lacks of the hues.
But there is such a thing as
unearned crunchiness --
a bogus
significance assigned to certain numinous numbers, like 19 among the Baha’i’s;
the “
23 enigma”; 666; 1000 AD as the Millennium; the mumbo-jumbo numbers
in “
Lost” and "
Touch"; for all which, cf. Wikipedia on
apophenia.
In the face of such things (which
have snared some otherwise rational people -- a close friend of mine,
critically brilliant but unmathematical, fell for the “23” business), we are
inclined to say, along with
Nulla extra ecclesiam salus,
that
Nulla
extra mathematicam ratio.
~
For a rather recondite example of crunchiness, consider the ‘amicable
numbers’ ( الآعداد المتحابة) reported by
the medieval historian Ibn-Khaldun
in his Muqaddima
.
Apparently only two of these were known to the Arabic medievals (or:
they are the only two numbers characterized by a theorem of Thabit ibn Qurra),
and they are not much to look at: 220
and 284; but they meant something
to contemporary practitionars of the talismanic art.
The modern view is summarized here:
