Mathematical proof: a research programme 35
betray an ambition to reach a high level of generality. Mueller’s contrastive
analysis discloses how distinct means are constructed and combined for the
proofs to fulfi l this ambition. Despite their failure in modern eyes to achieve
their goal, the two sets of proofs in the texts appear to form two strikingly
diff erent, but carefully designed, architectures of arguments inspired by the
task that the authors had set for themselves. Taking the value of general-
ity into account in his interpretation allows Mueller to use fi ner tools and
describe with greater accuracy the argumentative structures and the diff er-
ences between them. Mueller thus highlights how the conduct of proofs can
bear the hallmark of epistemological values prized by the actors.
More generally, Mueller’s analysis indicates how much more there can
be to the study of a practice of proof than simply assessing whether proofs
adequately establish their conclusions or not. Th e kinds of elements the
practitioners design for their proofs, the ways in which they use them,
and other questions, all essential for a historical inquiry into the activity
of proof, will appear quite fruitful in the following chapters. In particular,
the question of how a kind of text has been designed for a certain practice
of proof – a question that the multiplicity of the proofs examined brings to
the fore – appears relevant again for the further analysis of the sources. Its
fundamental character will soon become even clearer.
Further widening: the text of a proof
In his Arithmetics , Diophantus opts for a completely diff erent style of
composition and presents solutions for hundreds of problems relating to
integers. Each problem is followed by the reasoning that leads to the deter-
mination of a solution. To formulate the problems and the kind of proof
following them, both of which involve statements relating to numbers and
unknowns, Diophantus regularly makes use of symbols. In his chapter,
Reviel Netz focuses on the question of determining the role played by this
symbolism in the development of the reasonings Diophantus proposed to
establish the solutions to the problems.
Th e fact that the symbols introduced are essential to Diophantus’ project
is made clear by the fact that they are the main topic of the introduction to
his book. On the other hand, Diophantus stands in contrast to his known
predecessors in that he makes explicit the reasoning by which he establishes
the solutions to problems. Th erefore, the question of how the former are
linked to the latter is not only natural, but also essential to an analysis of
Diophantus’ activity of proving. Such is the main question of the chapter. It
pertains, as one can see, to the text with which an argument is conducted.