16 1. THE ORIGIN AND PREHISTORY OF MATHEMATICS1.16. Logic enters the mathematics curriculum in high-school geometry. The rea-
son for introducing it at that stage is historical: Formal treatises with axioms,
theorems, and proofs were a Greek innovation, and the Greeks were primarily ge-
ometers. There is no logical reason why logic is any more important in geometry
than in algebra or arithmetic. Yet it seems that without the explicit statement
of assumptions, the parallel postulate of Euclid (discussed in Chapter 10) would
never have been questioned. Suppose things had happened that way. Does it follow
that non-Euclidean geometry would never have been discovered? How important
is non-Euclidean geometry, anyway? What other kinds of geometry do you know
about? Is it necessary to be guided by axioms and postulates in order to discover or
fully understand, say, the non-Euclidean geometry of a curved surface in Euclidean
space? If it is not necessary, what is the value of an axiomatic development of such
a geometry?1.17. Perminov (1997, p. 183) presents the following example of tacit mathematical
reasoning from an early cuneiform tablet. Given a right triangle ACB divided into
a smaller triangle DEB and a trapezoid ACED by the line DE parallel to the leg
AC, such that EC has length 20, Ε Β has length 30, and the trapezoid ACED
has area 320, what are the lengths AC and DEI (See Fig. 3.) The author of
the tablet very confidently computes these lengths by the following sequence of
operations: (1) 320 -Ξ- 20 = 16; (2) 30 · 2 = 60; (3) 60 + 20 = 80; (4) 320 4- 80 = 4;
(5) 16 + 4 = 20 = AC; (6) 16 - 4 = 12 = DE. As Perminov points out, to present
this computation with any confidence, you would have to know exactly what you
are doing. What was this anonymous author doing?
To find out, fill in the reasoning in the following sketch. The author's first
computation shows that a rectangle of height 20 and base 16 would have exactly
the same area as the trapezoid. Hence if we draw the vertical line FH through the
midpoint G of AD, and complete the resulting rectangles as in Fig. 3, rectangle
FCEI will have area 320. Since AF = MI = FJ = DI, it now suffices to find
this common length, which we will call x; for AC = CF + FA — 16 + χ and
DE ~ EI - DI = 16 - x. By the principle demonstrated in Fig. 2, JCED has the
same area as DKLM, so that DKLM + FJDI = DKLM + 20x. Explain why
DKLM = 30 • 2 • x, and hence why 320 = (30 • 2 + 20) • x.
Could this procedure have been obtained experimentally?
1.18. A famous example of mathematical blunders committed by mathematicians
(not statisticians, however) occurred some two decades ago. At the time, a very
popular television show in the United States was called Let's Make a Deal. On
that show, the contestant was often offered the chance to keep his or her current
winnings, or to trade them for a chance at some other unknown prize. In the case
in question the contestant had chosen one of three boxes, knowing that only one
of them contained a prize of any value, but not knowing the contents of any of
them. For ease of exposition, let us call the boxes A, B, and C, and assume that
the contestant chose box A.
The emcee of the program was about to offer the contestant a chance to trade
for another prize, but in order to make the program more interesting, he had box
Β opened, in order to show that it was empty. Keep in mind that the emcee knew
where the prize was and would not have opened box Β if the prize had been there.
Just as the emcee was about to offer a new deal, the contestant asked to exchange
the chosen box (A) for the unopened box (C) on stage. The problem posed to the