QUESTIONS AND PROBLEMS 217Frontispiece of De arte combinatoria, from Vol. 5, p. 7 of the
collected works of Leibniz. @ Georg 01ms Verlag, Hildesheim.It was for that reason that combinatorial problems were used as the mathematical
background of the film Good Will Hunting.Questions and problems
8.1. We know a mathematical algorithm for computing as many decimal digits of
y/2 as we have time for, and \/2 has a precise representation in Euclidean geometry
as the ratio of the diagonal of a square to its side. It is a provable theorem of Eu-
clidean geometry that that ratio is the same for all squares, so that two observers
using different squares should get the same result. To the extent that physical
space really is Euclidean, this definition makes it possible to determine \/2 empir-
ically by measuring the sides and diagonals of physical squares. In that sense, we
could theoretically determine \/2 with arbitrarily prescribed precision by physical
measurements. In particular, it makes perfectly good sense to ask what the 50th
decimal digit of \f2 is—it happens to be 4, but rounds up to 5—and we could try
to get instruments precise enough to yield this result from measurement.