Character and Object: 2012 and 2013

(Joint work with Jeremy Avigad)

In 1837 Dirichlet proved that there are infinitely many prime numbers in an arithmetic progression whose first term and common difference are coprime, a result known as “Dirichlet’s theorem”. However, although there were no questions over the legitimacy of Dirichlet’s proof, various mathematicians published their own presentations, including Dedekind in 1863, de la Vallee Poussin in 1895/6 and 1897, Kronecker (whose work was edited and published by Hensel) in 1901, and Landau in 1909 and 1927. The central ideas invoked in, and the general line of argument of, these various presentations are essentially the same as Dirichlet’s original.

What we today recognize as particular functions called characters play a central role in each of the proofs. These are, in modern terms, homomorphisms from a finite abelian group to the multiplicative group of non-zero complex numbers. However, the treatment of the characters in the various presentations are strikingly different both to fully modern presentations and to each other.

In our work, Avigad and I examine the presentations of Dirichlet’s theorem by Dirichlet, Dedekind, de la Vallee Poussin, Kronecker, and Landau, as well as fully modern presentations. In particular, we attempt to identify, in a precise way, a number of axes along which the treatment of characters vary in the different presentations. We then explore the advantages and disadvantages of treating the characters in these various ways and bring these considerations to bear on questions as to why mathematicians gradually moved away from the original treatment to the modern one we use today. More specifically, we argue that the differences in the way the characters are treated influence how successful the presentation of Dirichlet’s theorem is at satisfying the following goals:

1) To ensure that only clear, well defined rules and norms are used and that these are able to cope with complex and subtle reasoning.
2) To promote an “efficiency of thought”; for example, to ensure that the cognitive burden on the reader is reduced by suppressing irrelevant information in the presentation, drawing attention to important features, and exploiting similarities to other, more familiar domains.

Moreover, we suggest that an attempt to satisfy both of these goals is what drove mathematicians to gradually alter their approach.