Goodstein’s theorem was undecidable in the framework provided by the
Peano axioms, but admitted a solution when the axiom of infinity from
Zermelo-Fraenkel set theory was incorporated into the axiom set. This
raises an obvious question: Is the nature of undecidability simply a mat-
ter of choosing the correct axioms, or the correct tools?
Or are there genuinely undecidable propositions that do not lie within
the reach of any consistent axiom set?
A second category of insoluble problems exists because of the inability
to obtain adequate information to solve the problem. Failure to obtain
this information may occur because the information simply does not ex-
ist (many quantum-mechanical phenomena come under this heading),
because it is impossible to obtain accurate enough information (this de-
scribes random and chaotic phenomena), or because we are exposed to
information overload and simply cannot analyze the information effi-
ciently (this describes intractable problems).
We come now to the third category of problems we cannot solve: those in
which we are asking for too much. So far, the most significant problems
we have found in this area are the ones from the social sciences, involving
the quest for voting systems or systems of representation. There are in-
numerable formal problems that fall under this description, such as the
problem of covering the chessboard with the two diagonal squares re-
moved with 12 tiles, and possibly the techniques involved for analyzing
such problems will be useful in more practical situations.
Finally, there are the questions that turn out to have several right an-
swers. The independence of the continuum hypothesis and the resolution
of the dilemma posed by the parallel postulate fall into this category. It
seems reasonably safe to predict that there will be other surprises await-
ing us; questions to which the answers lie outside the realm of what we
would expect, including the possibility that there are questions whose
answers depend upon the perspective of the questioner. For example, the
theory of relativity answers the riddle of which came first, the chicken
or the egg, with the answer that it depends upon who’s asking the ques-
tion—and how fast and in what direction they are moving.
There is one last recourse when we are absolutely, completely, and totally
stymied: try to find an approximate solution. After all, we don’t need to
know the value of to a gazillion decimal places; four decimal places suf-
fice for most problems. Although it is impossible to find exact solutions
by radicals to certain quintics, one can find rational solutions to any de-
sired degree of accuracy. It is extremely important to be able to do this.
Salesmen, after all, are very likely to continue traveling even in the ab-
sence of a polynomial-time solution to the traveling salesman problem,
240 How Math Explains the World