Long, long ago, in the morning of
the world, I sat in a lecture hall listening to Gene Kleinberg talk
about elementary calculus. The hall was, for its day, rather well
equipped. The chalkboards (remember those?) consisted of a number of
panels, arranged in several layers. At the start of the lecture
all the panels would be at the bottom of
the board, superimposed on each other, at an easy height for writing
on. However, only the frontmost panel was visible at that moment
(this is an important point; keep it in mind). The lecturer would
write until the panel in front was filled, at which point he or she
would push a button. Hidden machinery would spring into action, and
the first panel – call it Panel 1 – would rise, majestically, to
a spot well over the lecturer’s head but still in clear view of the
students, while the clean surface of Panel 2 would be exposed for the
lecturer to continue his or her writing. Between classes, physical
people normally came in and washed (and lowered) all the panels.
particular morning, Professor Kleinberg filled the first panel with
something no doubt important, though what it was I have nevertheless
long since forgotten, and pushed the button. Panel 1 rose, as it was
supposed to do, but Professor Kleinberg, to judge by his expression,
was not expecting his next discovery. For the surface of Panel
2 which was exposed was not clean, not pristine, not ready to
exposed surface of Panel 2 contained the following remarkable text.
Alexander the Great Rode a Centipede
Proof that Alexander the Great Rode a
with the main proof, we shall need a Lemma:
Lemma 1: Alexander the Great’s Horse had Six Legs
Alexander the Great’s horse was a horse. Therefore, at the front, it
had forelegs, and at the back it had two more legs. But four legs plus
two more legs make six legs. Therefore, clearly, Alexander the Great’s
horse had six legs, as was to be shown.
And from Lemma 1, we shall need to
derive an Easy Corollary:
Corollary 1 to Lemma 1: Alexander the Great’s Horse
had an Infinite Number of Legs
By Lemma 1, we know that Alexander the Great’s horse had six legs.
Six is an extremely odd number of legs for a horse to have.
But six is an even number. Therefore by lemma 1 we see that
Alexander the Great’s horse also had an even number of legs.
No number smaller than infinity can be both odd and even.
Therefore Alexander the Great’s horse must have had an infinite
number of legs, as was to be shown.
And now, from the Corollary, the
theorem readily falls.
For we realize at once
number of legs is a larger
number of legs than
that possessed by any living mammal
, or any living reptile
or indeed any living insect
. In fact, if we consider any animal other than the
Centipede, it is at once obvious that such an animal possesses fewer
legs than Alexander’s horse.
From this we can easily see our way
clear to the only possible conclusion:
Alexander the Great’s “horse” was no horse at all,
but was, rather, a CENTIPEDE.
Kleinberg looked at the proof, slowly, thoughtfully, taking time to
read it carefully. And then, silently, with no comment, he erased
it, and continued with the lecture.
But it stuck
in my memory. I’m not sure it was stated exactly as I have written
it here, but it was certainly very much like the proof given here. I
also have no idea where it originated, nor who it was who crept into
the classroom before the lecture and placed the proof upon the second
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