A LAST LOOK AT RECIPROCITY 235
mathematicians must think in patterns. If we can describe some
phenomena carefully enough as a pattern, then mathematics may
be able to give information about that kind of pattern.
There is a big debate as to whether logic is part of mathematics
or mathematics is part of logic. We use logic to think. We notice
that our thinking, when it is valid, goes in certain patterns. These
patterns can be studied mathematically. Thus, logic is a part of
mathematics, called “mathematical logic.” There are some amazing
theorems here, such as G
and the theorem on
algorithmic unsolvability of Diophantine equations mentioned near
the end of chapter 6. On the other hand, because logic in some
sense encompasses all valid thought, you may prefer to say that
mathematics is a part of logic.
Another example of patterns: What is a group? It is just a pattern
that certain things can exhibit when you have a composition law
for always getting a third thing by combining any two others. Then
the same group-pattern can show up in pure mathematics, particle
physics, crystallography, and so on.
A generalized reciprocity law is the bringing together of two
patterns. One pattern is the set of traces, or more generally the
characteristic polynomials (if you know what these are), of Frob
acting in a Galois representation. The other pattern comes from
the black box—another mathematical object of some different type.
The law is like a mirror in which you can see the pattern better
than in the original. But it is a two-way street. You can consider
the Frob’s as the original problem and the black box as the easier
thing to get a handle on, or vice versa. Sometimes both sets of
patterns are well understood and the reciprocity law boils down to
a very beautiful symmetry of numbers—as in the case of quadratic
We suggest (Hofstadter, 1979; Nagel and Newman, 2001; Smullyan, 1992) as possible
starting points to learn about this.