Boundary Troika

          Boundary Troika

The Three Stooges surveyed the boundary between Athens and Sparta.

Moe said: point A belongs to Sparta, point B belongs to Sparta, and there is no boundary between the two points.

Larry said: point A belongs to Athens, point B belongs to Athens, and there is no boundary between the two points.

Curly said: point A belongs to Athens, point B belongs to Sparta, and there is a boundary between the two points.

Therefore they decide by 2/3 majorities:

* Point A belongs to Athens

* Point B belongs to Sparta

* There is no boundary between the two points.

Yet they all agree that Athens is not Sparta!

Dilemma of the Boundary

Dilemma of the Boundary

One version of the Paradox of the Boundary is: which country owns the border? Attempts to impose a binary solution on this inherently non-binary problem have proven to be expensive and ineffective.

The topological paradox of the boundary translates to a game-theoretical dilemma. For each country must make this choice: shall we attempt to take the boundary, or not? If neither does, then there is peace, and both share the boundary. If both do, then there is war, and the boundary becomes no-man’s-land; and if only one does, then the aggressor wins.

The boundary can be advancing, retreating, deadlocked, or porous, in a prisoner’s-dilemma context. The boundary is negotiated.

Achilles’ Torch

          Achilles’ Torch

This paradox combines the Paradox of Dichotomy with the Paradox of the Parity of Infinity.

Achilles, swiftest of mortals, opens the Olympic Games by bringing a torch to light the Olympic flame. Unfortunately he is too swift for his torch, and it flickers due to the speed of his passage. For the first 1/2 stadia of his run, his torch is lit; for the next 1/4 stadia of his run, it is out; ; for the next 1/6 stadia, it is lit; for the next 1/16, it is out; and so on indefinitely.

Achilles’ torch flickers infinitely many times by the time he reaches the Olympic flame. By that time, is his torch lit or out?