  We clarify the role of the Born rule in the Copenhagen Interpretation of
quantum mechanics by deriving it from Bohr's doctrine of classical concepts,
translated into the following mathematical statement: a quantum system
described by a noncommutative C*-algebra of observables is empirically
accessible only through associated commutative C*-algebras. The Born
probabilities emerge as the relative frequencies of outcomes in long runs of
measurements on a quantum system; it is not necessary to adopt the frequency
interpretation of single-case probabilities (which will be the subject of a
sequel paper). Our derivation of the Born rule uses ideas from a program begun
by Finkelstein (1965) and Hartle (1968), intending to remove the Born rule as a
separate postulate of quantum mechanics. Mathematically speaking, our approach
refines previous elaborations of this program - notably the one due to Farhi,
Goldstone, and Gutmann (1989) as completed by Van Wesep (2006) - in replacing
infinite tensor products of Hilbert spaces by continuous fields of C*-algebras.
In combination with our interpretational context, this technical improvement
circumvents valid criticisms that earlier derivations of the Born rule have
provoked, especially to the effect that such derivations were mathematically
flawed as well as circular. Furthermore, instead of relying on the
controversial eigenvector-eigenvalue link in quantum theory, our derivation
just assumes that pure states in classical physics have the usual
interpretation as truthmakers that assign sharp values to observables.
