If A denotes the set of all positive integers n such that the sum of the
first n odd integers is n^2 , we have shown that A contains 1, and if n be-
longs to A, then n1 belongs to A. By Axiom 5, A contains all the positive
integers.
A substantial number of deep results use mathematical induction as a
key proof technique. Demonstrating the inconsistency of arithmetic
would make a lot of mathematicians very unhappy—including David
Hilbert, whose basis theorem (an important result in both ring theory
and algebraic geometry) was proved using mathematical induction. It
seems fairly safe to say that Hilbert definitely hoped that someone would
prove that the Peano axioms for arithmetic were consistent; after all, no
one wants to see one of his most famous results put in doubt.
So there is a lot riding on establishing that the Peano axioms for arith-
metic are consistent, and Hilbert was well aware of this—that’s why it’s
Problem Number 2, ahead of some truly famous problems like the Gold-
bach conjecture (every even number is the sum of two primes) and the
Riemann hypothesis (a technical result with immense potential, but
which requires an acquaintance with complex variables and infinite se-
ries to understand it). Suffice it to say that the Clay Mathematics Institute
will pay $1 million to anyone who manages to demonstrate the consist-
ency or inconsistency of the Peano axioms.
A Postdoc Shakes Things Up
There is a belief that mathematicians do their best work before they are
thirty. Possibly forty would be a more reasonable estimate—the Fields
Medal is awarded only for work done prior to that age. Nonetheless, some
of the most important results in mathematics have been the work of
graduate and postdoctoral students.
There is a good deal of debate on why this should be the case; my own
belief is that to some extent, work on a particular problem sometimes
becomes ossified, in the sense that the leading mathematicians have
blazed a trail that most others follow—and sometimes that trail leads just
so far and no further. Young mathematicians are less likely to have been
indoctrinated—I recall vividly Bill Bade, my thesis adviser, handing me
reading material that would bring me up to date, but not suggesting what
line I should pursue after I had finished reading the papers.
Kurt Gödel was born six years after Hilbert propounded his twenty-
three problems, in what is now the Czech Republic. His academic talents
were apparent from an early age. Gödel initially debated between study-
ing mathematics and theoretical physics, but opted for mathematics be-
122 How Math Explains the World