The students thus logically deduce that there can
be no quiz any day the following week. They were
legitimately surprised then by a pop quiz that came
Wednesday. (Sometimes this paradox is phrased in
terms of a prisoner awaiting his execution. In this set-
ting it is called the “unexpected hanging paradox.”)
The Wallet Paradox
Two men decide to play the following game:
Both men open their wallets. Whoever has the
least amount of money wins the contents of the
other’s wallet.
Each man can legitimately argue that he stands to
gain more than he can lose, and therefore that the game
is biased in his own favor. However, such a simple
win/lose game cannot simultaneously be advantageous
to both players.
Aristotle’s Wheel Paradox
Two wheels of different sizes are glued together so that
their centers are aligned. The entire system then rolls
along a double track as shown:
In one revolution, both wheels roll the same dis-
tance and so, in the diagram, x= y. But xis the circum-
ference of the small wheel and ythe circumference of the
large wheel. Thus we are forced to conclude that two
wheels of different sizes have the same circumference!
These paradoxes rely on clever
SELF
-
REFERENTIAL
statements (Grelling’s paradox, Berry’s paradox, the
barber paradox, and even the lawyer paradox), hidden
false assumptions (students assumed the teacher was
telling the truth—they do not know this; winning and
losing the wallet game are not equally likely; and could
the barber be a woman?), and disguised compound
motions (the small wheel is carried forward by the large
wheel as it rotates and so slides along the upper track).
See also
CONDITIONAL
; H
ILBERT
’
S INFINITE HOTEL
;
J
OURDAIN
’
S PARADOX
; T
RISTRAM
S
HANDY PARADOX
;
Z
ENO
’
S PARADOXES
.
parallel Two lines in a plane are said to be parallel if
they never meet no matter how far they are extended. If
two lines labeled Land Mare parallel, then we write
L
M.
The notion of parallel lines is an abstract concept:
one cannot physically draw a line infinite in length, nor
can one check the entire extent of two infinite lines to
determine whether or not they eventually intersect.
(We, as human beings, can only conceive of finite quan-
tities and local phenomena.) In order to make working
with parallel lines feasible, the geometer E
UCLID
(ca.
300
B
.
C
.
E
.) introduced an
AXIOM
, called the
PARALLEL
POSTULATE
, to describe the local behavior of parallel
lines. He asserted that, for any
TRANSVERSAL
crossing a
pair of parallel lines, alternate interior angles are equal.
(Thus, for instance, the pair of angles labeled xin the
diagram on page 376 are indeed equal in measure, as
are each pair of right angles about the lines labeled a
and b.) From this Euclid was able to establish several
properties of parallel lines that we intuitively expect to
be true. For instance, Euclid proved:
Two parallel lines always remain a fixed distance
apart. That is, in the diagram on page 376, it
must be the case that length aequals length b.
(Observe that the two triangles in the diagram share
the same angles and a common diagonal length. The
ASA principle now assures that they are
CONGRUENT
FIGURES
. Consequently, the distances aand bmust
be equal.)
Euclid also observed that the converse of his paral-
lel postulate is true and requires no special assumptions
about geometry for its proof. (It follows immediately
from the
EXTERIOR ANGLE THEOREM
.)
If two lines cut by a transversal produce equal
alternate interior angles, then the two lines
must be parallel.
This result allows one to readily construct parallel
lines. To produce a line through a given point Pparallel
parallel 375
Aristotle’s wheels