The Parable of Achilles and Charon, or Quantum Logic

The Anger of Achilles 1810 Michel Martin Drowling

Achilles, great Grecian warrior, has recently perished on the Trojan battlefield, achieving for himself eternal glory and a home for himself in the blessed fields of Elysium. He has, however, earned himself the ire of one of the gods for his desecration of Hector’s corpse — Apollo, the guardian god of Hector has sworn revenge against Achilles. He has devised a devious plan to foil the entrance of our hero Achilles into Elysium. While the Greeks, as is their custom, placed a coin on the eyes of Achilles’ corpse, to pay for the journey across the River Styx, Apollo by means divine and secret has arranged that the coin placed on Achilles’ eyes has been erased of its adornments — where once this coin proudly boasted the image of Zeus for its head-side and his awesome lightning bolt for its tail-side, now only two completely blank, smooth surfaces survive. Apollo has also promised Charon the fairest of his next-born daughters, if he only allow those whose coins are perfect, to cross. Charon accepts the arrangement, but on the less stringent condition that anyone who can correctly point out which side is heads and which side is tails. The two divine beings shake hands. The trap has been set!

Achilles, weary of war and conflict, and now destined for the lofty abodes of the underworld, stumbles and rages at the ferryman’s request. “Is it not enough, Charon, that I should achieve fame and glory by being the best of Greeks in battle, that now you block my passage to my rightful home?” He pauses and reflects: “ Charon, if I give you the best possible answer, will you allow me travel to Elysium?” The graven, sombre ferryman nods his skeletal head.

Achilles thinks again to himself: “If I tell him that this side is heads and that side tails. I have only an even chance of either proceeding or stumbling. If I tell him that that side is tails and this heads, I still only have an even chance.

At this exact moment Zeus discovers the treachery of his son. Zeus fuming on his throne on top Olympus orders Urania, the muse of exact sciences to fill the mind of Achilles. Urania dancing and inhaling the magical fumes of Olympus, throws her arms into the air and from those ethereal arms inspiration trickles down into the underworld.

“A-ha! Master of the Underworld, and master of my universe for all intents and purposes, tell me this first how will you judge which side is which?”

Charon becomes anxious and agitated.

“Even if I asked Zeus, master of the universe, he would not know; even if I asked Cronos, father of the master of worlds and god of time itself, he would not know.”

“Nevertheless Charon, I will anser you such: This side is heads-tails and that side is heads-tails.”

“This is no answer my lord, and most honoured warrior. What do you mean?”

“Let me explain. There is only one objective answer to this question; either this side was heads or tails, and the other the correspondingly inverse of the first dilemma. That, however, has been lost to history: it is impossible to know with certainty. But in the absence of certainty, we must seek refuge in another kind of certainty. I intend to receive my honours in the underworld and nothing shall stop me now. The answer I give you is certainly correct, given our criterion of the best possible answer.

When I say heads-tails, I mean to say that this side of the coin is potentially both simultaneously. Similarly, the second is potentially both simultaneously. In this way I cover all possibilities, of heads-tails and tails-heads. Only by allowing indeterminate choice between the two options can I guarantee that both possibilities are, well, possible.

“But my lord, you have given me two answers where objectively there should only be one as you yourself said.”

“Some things Charon will forever be lost to time. The best we can do in the absence of definite knowledge is to cover all the bases. FUrthermore my answer includes the objectively correct answer, but includes the wrong “answer” as well.

“I would like to further elucidate why this is objectively the best decision given the impossible dilemma foisted upon me by the scheming gods.

Let us assume the objectively correct answer is HT, but I have no idea that that is the case. My chances are even whether I am absolutely correct (HT) or absolutely wrong (TH) . The middle answer H-T, H-T, gives me 100% chance of being both correct and wrong at the same time. This approximate correctness is a better option because it is always correct-and-wrong, thus producing a result that is never wrong (just as much as it is never right). That is, I would rather always be correct-wrong then sometimes correct and wrong. In fact it is exactly twice as good an option (if we take the expected value of the options)

“My lord! Perhaps you should have dallied with the philosophers instead of the warriors. This is all to my liking. You may proceed!”

“One more observation ,dear friend, imagine we played a game where for every time I got the correct answer I won a point and every time I got it wrong I lost a point. This game is a coin toss where I have to guess the side once the coin has been tossed into the air. One more caveat, however, if you go below zero you lose the game completely.

What you will notice is that this game is on a razor edge every time you play. To make one mistake at the beginning is disastrous. Even the best strategy of randomly guessing heads or tails is risky. The only method to ensure a stalemate would be to take the mean, and guess heads-and-tails every time. You would never win, because the points would cancel out, but you also wouldn’t lose.

If we imagined a perfect simulation of this game, the heads and tails should cancel out when added transcendentally, that is excluding the time in which the actual appearance of the coins came. We simulate this perfect outcome in the individual part of the system by choosing both simultaneously. Incidentally, I think this may be the answer to a riddle yet to be solved in the millennia ahead.

“Ah my lord, I don’t want to be rude but I have other dead people to attend to.”

“Yes, of course, take me to Elysium dear dead-and-alive ferryman.

Achilles boards the ferry, singing a charming ditty all the way across the river. He is greeted in Elysium with broad hugs and big smiles!