The school of philosophy called Antipodianism briefly flourished on the fringes of the Hellenistic world more than 2,000 years ago. The sect held that every person has an opposite number, a mirror image who inverts all our beliefs, feelings, actions and attitudes. If I smile, my antipodian counterpart frowns; when I wake, she sleeps. If I’m a Mac, she’s a PC. For every liberal Democrat there’s an antipodian Tea Party Republican. In this way the universe is held in balance. It’s an enforced equilibrium, which none of us has the power to upend, hard as we might try.
The earliest of the Antipodians believed that every such matched pair (commonly designated A and ∀) live at diametrically opposite points on the surface of the earth. This arrangement ensures that A and ∀ can never meet—thereby averting a cosmic catastrophe. A later quantum-field version of Antipodianism relaxed the geographic constraint by allowing for the creation and annihilation of A∀ pairs, but that idea never really caught on.
One day an Antipodian master was teaching an exchange student from New Zealand. The child was crafty.
“Is it not true,” she asked, “that A and ∀ always do the opposite thing?”
“Yes, antisymmetry demands it,” the master replied.
“If A walks north, ∀ must walk south?” the child asked.
Again the master assented.
“If A goes east, ∀ must go west?”
“If A turns to the right, ∀ must turn to the left, no?”
The master agreed, although he sensed trouble coming.
“I’m afraid the universe is out of joint,” said the child. “If A goes north and turns to the right, while ∀ goes south and turns to the left, afterwards they are both walking east. They are doing the same thing.”
Needless to say, this was a moment of crisis in Antipodian doctrine. The Pythagoreans, you may recall, resolved a similar impasse by resorting to violence. When some upstart challenged their precept that “all is number” by showing that no known number can be the square root of 2, the Pythagoreans tossed the troublemaker out of the boat. But in this case the Antipodian master kept his calm.
“Ah my little Kiwi,” he said to the student. “You are clever but not wise. Your own statements refute your claim. Did you not begin by saying that A and ∀ always do the opposite thing? When A walks 10 paces north, ∀ walks 10 paces south. When A turns right, ∀ turns left. But now you would have us believe they both take a step forward, contradicting the most basic law of their nature. What really happens is that A walks forward and ∀ walks backward. Thus A goes eastward and ∀ westward, and all is well with the world.”
Through this brittle sophistry the master extricated himself from the classroom—though he may have had to walk backwards to make good his escape. He never taught again. The Kiwi student went on to a brilliant career studying the weak interactions of neutral K mesons. As for Antipodianism, it vanished without a trace.
Or maybe it left a tiny trace. I’ve never visited the antipodes, but I hear that corkscrews Down Under turn the other way.