“Among all positive integers, the integer 1 is the largest.”

Here’s a funny anecdote about the subtleties of logic. The isoperimetric theorem is the statement that, among all planar figures of a fixed perimeter, the circle (of that circumference) encloses the greatest area.

Steiner gave five proofs of the isoperimetric theorem. Lovely as they are, he left one point open to attack: all proofs assume the existence of a solution (his strategy is always to take a figure that is not a circle and show that its area can be improved). This did not go unpunished. The analyst vultures can smell an existence assumption from miles away. […] Perron at least jokes about it:

Theorem. Among all curves of a given length, the circle encloses the greatest area.
Proof. For any curve that is not a circle, there is a method (given by Steiner) by which one finds a curve that encloses greater area. Therefore the circle has the greatest area.

Theorem. Among all positive integers, the integer 1 is the largest.
Proof. For any integer that is not 1, there is a method (to take the square) by which one finds a larger positive integer. Therefore 1 is the largest integer.

— Viktor BlåsjöThe evolution of the isoperimetric problem (2005), quoting Oskar PerronZur Existenzfrage eines Maximums oder Minimums (1913)

Posted May 19th, 2013 in Academic. Tagged: , .

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