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

¨

odel’s Theorems

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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.

Reciprocity

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

p

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

reciprocity.

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We suggest (Hofstadter, 1979; Nagel and Newman, 2001; Smullyan, 1992) as possible

starting points to learn about this.

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