Sunday, June 12, 2016

Achilles and the Tortoise: the inside dope

Famed indeed is the fable of Achilles and the Tortoise, known since Antiquity to every schoolchild.  And yet the telling of it has steadily decayed, until contemporary versions have become equivalent to the old Irish Bull about how you could never walk from your table to the bar  for another pint, since in doing so, you would first have to traverse half the distance, and then half the remaining half, and then half of that remaining segment, and so on “to infinity”.    Anyone who has every enjoyed a cold, foaming pint of Guiness ™  knows how absurd that is.  -- Bartender!  Same all round!

Anyhow, the tale of Achilles racing the tortoise, as told by the mathematician Zeno in ancient times, recalls an actual contest, held on the plains of Troy ca. 1278 B.C. (Historians differ as to the precise date.)  For a bar bet, the wily Thersites bet Achilles than he couldn’t outrun a tortoise, if he gave the tortoise a head start.  The Achaean hero snorted and said, “Ha!  I’ll grant the creature a thousand cubits.  Gentlemen, place your bets!”

Achaeans and Trojans alike foregathered at the appointed time, Achilles arrogantly lounging on his shield, the tortoise waiting humbly, a thousand cubits in front.   Unfortunately, Achilles was better famed for his fleet foot and his ferocity, than for brains (you had to go to his countryman Odysseus  for that), and had neglected to inquire the length of the race.  In fact, it had been set at precisely … one thousand and one cubits.

Achilles, realizing belatedly that he’d been had, set off like lightning at the sound of the bell:  but when he finally reached the finish-line, there stood the tortoise, contentedly munching on grass.

Thahhhhh ...  Winnnnahhhhhhh !!!!

Zeno’s arithmetical point was that, although Achilles was much faster than the tortoise, he was not a thousand times as fast, and thus lost.  Indeed, even had he been a thousand times as fast, he’d have been vanquished: since in the time he covered the first thousand cubits, the tortoise would have traveled one cubit, to the finish-line.  (Left as an exercise for the reader:  How fast would Achilles have to be, to beat the tortoise under these conditions?)

In the event, Achilles had the last laugh, since he supped on the tortoise, in the form of soup.

The tale then takes a darker turn, as arcane mathematico-philosophical disputes led many who had lost the wager  to refuse to pay up;   a free-for-all ensued, which led directly to the Trojan War.
[Note:  You were probably told some foolish tale in school, about how the cause of it all was a woman.   As if.  We shall not deign to refute  that account.]


As the centuries went by, the lessons of that fateful day  were lost, finally issuing in a folkloristic version for children, “The Tortoise and the Hare”.  Once again, the tortoise wins;  but the mathematical substructure has been completely discarded.  Proving once again that the level of geometrical sophistication has sadly declined since the Age of Troy.

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