It’s often said that proofs serve as the criterion for truth in mathematics: we prove things in order to establish that they are true. This is certainly true, but it doesn’t explain something else we do, namely, provide new proofs of old results. We already know those theorems are true, so in giving new proofs we are not seeking to establish that. What we are seeking is understanding. We want to know why the theorem is true, and a proof can (sometimes) tell us that. We also want to know about connections between different parts of mathematics, which can lead to new insights and perhaps new theorems.
A case in point is the quadratic reciprocity theorem. First conjectured by Euler and Legendre and first proved by Gauss, it is a staple of elementary number theory courses. It relates the answers to two yes-no questions about two (distinct) odd prime numbers \(p\) and \(q\):
(1) Does there exist a whole number \(n\) such that \(n^2\equiv q \pmod p\)?
(2) Does there exist a whole number \(m\) such that \(m^2\equiv p \pmod q\)?
There is, a priori, no reason to expect these two questions to be related. In fact, the Chinese Remainder Theorem strongly suggests that “life mod \(p\)” and “life mod \(q\)” are completely independent, since it tells us that for any choice of \(a\) and \(b\) one can always find \(x\) such that both \(x\equiv a \pmod p\) and \(x\equiv b\pmod q\). Nevertheless, questions (1) and (2) do turn out to be related:
Quadratic Reciprocity Theorem: If either \(p\equiv 1\pmod 4\) or \(q\equiv 1\pmod 4\), then questions (1) and (2) have the same answer. On the other hand, if \(p\equiv q\equiv 3\pmod 4\), questions (1) and (2) have opposite answers.
In particular, if we know the answer to one of the questions, then we know the answer to the other.
The proof that is usually given in elementary courses goes like this: relate the answers to the questions to counting, in such a way that an even count means the answer is “yes” and an odd count means the answer is “no.” (This is known as “Gauss’s Lemma.”) Then set up a way to relate the two counts, and show that the difference between the answers is odd or even as required by the theorem.
The proof works, but it is remarkable in the fact that it gives us no insight at all into why the theorem is true. In particular, it does not yield any direct connection between “life mod \(p\)” and “life mod \(q\).” Every time I present the proof to students, I point out the feeling that yes, it comes out right, but it comes out right because the theorem is true. It’s hard to claim (and I do not believe) that counting points in a rectangle explains why the theorem is true.
Gauss felt the same about his original proof, so he ended giving six different ones (only 3 and 6, in the numbering given in this book, used “Gauss’s Lemma”). It’s not clear that this satisfied him, and it certainly didn’t satisfy others: this book counts 314 different proofs since Gauss’s time. Of course, many of these are quite similar to others, and a few are incorrect, but that’s still an impressive number.
The book is a translation by Franz Lemmermeyer of Baumgart’s thesis from 1885. An appendix by Lemmermeyer updates the story by giving an annotated list of all known proofs of the theorem. (That’s how I know there are 314.) Both Baumgart and Lemmermeyer classify proofs according to the main tools used, and Lemmermeyer, in his appendix, tries to trace which proofs are dependent on which. So this is much more than a mere catalogue of proofs!
The editor has provided double service: he offers English-speakers access to Baumgart’s account and provides a summary of what has happened since then. The result is a very useful book.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.