[Club2] Talk by Jeremy Avidad, July 17, 14:00, Turing

[Andrei Popescu]

Dear All,  

In mid-July, Jeremy Avigad will be visiting us and will be giving a talk on the origins 

of higher-order reasoning in classic mathematics.  

Best regards, 
  Andrei 


Dirichlet's theorem and the evolution of higher-order reasoning in mathematics
Jeremy Avigad
Departments of Philosophy and Mathematical Sciences
Carnegie Mellon University
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Wed. July 17, 14:00, Alan Turing

In 1837, Peter Gustav Lejeune Dirichlet proved that there are infinitely many
primes in any arithmetic progression in which the terms do not all have a common
factor. This beautiful and important result was seminal in the use of analytic
methods in number theory.

Contemporary presentations of Dirichlet's proof are manifestly higher-order. To
prove the theorem for an arithmetic progression with common difference k, one
considers the set of "Dirichlet characters modulo k," which are certain types of
functions from the integers to the complex numbers. One defines the "Dirichlet
L-series" L(s, chi), where s is a complex number and chi is a character modulo
k. One then sums certain expressions involving the L-series over the set of
characters.

This way of thinking about characters, which involves treating functions as
objects just like the natural numbers, was not available in the middle of the
nineteenth century. Subsequent presentations of Dirichlet's theorem from
Dedekind to Landau show a gradual evolution towards the contemporary viewpoint,
shedding light on the development of modern mathematical method.  
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