It seems pretty much the same—unless the first statement was made by
Epimenides! If so, is Epimenides lying with the first statement? After all,
a liar is one who lies some of the time, but not necessarily all the time. If
he is a liar, then the first statement could be a lie—and so some Cretans
might not be liars, and we cannot legitimately make the deduction.
There’s some wiggle room here; what exactly characterizes a liar? Does
he or she need to lie with every statement, or is someone a liar if he oc-
casionally lies? After some refining, the liar’s paradox, as this sequence of
statements is often called, was condensed to a four-word sentence: this
statement is false. Is the statement This statement is false true or false?
Assuming that true and false are the only alternatives for statements, it
cannot be true (if so, it would be true that it is false, and would therefore
be false), and it cannot be false (if so, it would be false that it is false, and
would therefore be true). Assuming that it is either true or false leads to
the conclusion that it is both true and false, and so we must place the sen-
tence This statement is false outside the true-false realm. You may be able
to detect in this argument the faint echo of the classic odd-even proof that
the square root of 2 is irrational, which proceeds by showing that a
number simultaneously has two incompatible characteristics.
Some might place the liar’s paradox under the heading of “snack food
for thought”; on the surface it may seem little more than a curiosity- pro-
voking, but pedantic, point of linguistics. But Kurt Gödel, a talented
young mathematician, looked more deeply at the liar’s paradox, and used
it to prove one of the most thought-provoking mathematical results of the
twentieth century.
The Colossus
At the summit of the mathematical world in 1900 perched a colos-
sus—David Hilbert. A student of Ferdinand von Lindemann, the
mathematician who had proved that was transcendental, Hilbert
had made brilliant contributions to many of the major fields of mathe-
matics—algebra, geometry, and analysis, the branch of mathematics that
evolved from a rigorous examination of some of the theoretical difficul-
ties that accompanied the development of calculus. Hilbert also submit-
ted a paper on the general theory of relativity five days prior to Einstein,
although it was not a complete description of the theory.^1 By any stand-
ard, though, Hilbert was a titan.
During the International Congress of Mathematicians in Paris in 1900,
Hilbert made perhaps the most inf luential speech ever made at a mathe-
116 How Math Explains the World