To Let: Unsuccessful Stipulation,
Bad Proof, and Paradox
by Laurence Goldstein
Letting is a common practice in mathematics.
For example, we let x be the sum of the
first n integers and, after a short proof, conclude
that x = n(n+1)/2; we let J be the point
where the bisectors of two of the angles of a
triangle intersect and prove that this coincides
with H, the point at which another pair of bisectors
of the angles of that triangle intersect.
Karl Weierstrass's colleagues, in an attempt
to solve optimization problems, stipulated
that the minimum area for a triangle with a
given perimeter be a straight line segment
conceived as a triangle with zero altitude.
(Weierstrass complained that this obscured
the insight that some problems have no solutions.)
In mathematics applied to physics,
we let x be the temperature in Fahrenheit
corresponding to 30° Centigrade; we let v be
the velocity of the Earth through the luminiferous
ether. Before the error was spotted, the
official rules for Little League Baseball made
an inconsistent stipulation about the dimensions
of home plate (Bradley 1996).