  Current concerns about reproducibility in many research communities can be
traced back to a high value placed on empirical reproducibility of the physical
details of scientific experiments and observations. For example, the detailed
descriptions by 17th century scientist Robert Boyle of his vacuum pump
experiments are often held to be the ideal of reproducibility as a cornerstone
of scientific practice. Victoria Stodden has claimed that the computer is an
analog for Boyle's pump -- another kind of scientific instrument that needs
detailed descriptions of how it generates results. In the place of Boyle's
hand-written notes, we now expect code in open source programming languages to
be available to enable others to reproduce and extend computational
experiments. In this paper we show that there is another genealogy for
reproducibility, starting at least from Euclid, in the production of proofs in
mathematics. Proofs have a distinctive quality of being necessarily
reproducible, and are the cornerstone of mathematical science. However, the
task of the modern mathematical scientist has drifted from that of blackboard
rhetorician, where the craft of proof reigned, to a scientific workflow that
now more closely resembles that of an experimental scientist. So, what is proof
in modern mathematics? And, if proof is unattainable in other fields, what is
due scientific diligence in a computational experimental environment? How do we
measure truth in the context of uncertainty? Adopting a manner of Lakatosian
conversant conjecture between two mathematicians, we examine how proof informs
our practice of computational statistical inquiry. We propose that a
reorientation of mathematical science is necessary so that its reproducibility
can be readily assessed.

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