An excerpt from H. Petard, "A contribution to the mathematical theory of big game hunting," The American Mathematical Monthly, vol. 45, no. 7, pp. 446-447, 1938:
The Hilbert, or axiomatic, method. We place a locked cage at a given
point of the desert. We then
introduce the following logical
system.
- Axiom I. The class of lions in the Sahara Desert is non-void.
- Axiom II. If there is a lion in the Sahara Desert, there is a lion in the
cage.
- Rule of Procedure. If p is a theorem, and "p implies q" is a
theorem, then q is a theorem.
- Theorem I. There is a lion in the cage.
The method of inversive geometry. We place a spherical cage in the desert, enter it, and lock it. We perform an inversion with respect to the cage. The lion is then in the interior of the cage, and we are outside.
The method of projective geometry. Without loss of generality, we may
regard the Sahara Desert as a plane.
Project the plane into a line, and
then project the line into an interior
point of the cage. The lion is
projected into the same point.
The Bolzano-Weierstrass method. Bisect the desert by a line running N-S. The lion is either in the E portion or in the W portion; let us suppose him to be in the W portion. Bisect this portion by a line running E-W. The lion is either in the N portion or in the S portion; let us suppose him to be in the N portion. We continue this process indefinitely, constructing a sufficiently strong fence about the chosen portion at each step. The diameter of the chosen portions approaches zero, so that the lion is ultimately surrounded by a fence of arbitrarily small perimeter.
No comments:
Post a Comment