Mathematical proof: a research programme 37
problem. To this end, Diophantus shaped, primarily thanks to his symbols,
a kind of text that would enable the reader to survey in the best way pos-
sible the method followed. Th is is how Netz argues in support of his thesis
that the expressions formed with the specifi c symbols introduced are con-
substantial with the project and the kind of proof specifi c to Diophantus’
Arithmetics.
Note that here again, as in Mueller’s chapter, the examined proofs
proceed through operating with statements of equality between numbers.
However, in the Arithmetics , the symbols developed helped carry out such
operations in a specifi c way, linked to the peculiar features of Diophantus’
reasonings. Since they were allographs, they allowed the reader to keep
the meaning of the computations in mind. On that count, these symbols
diff er from modern symbolism. Th is diff erence in nature possibly echoes a
diff erence in use: Diophantus’ symbols do not seem to have been used for
proving through blind computations, as is the case with modern symbol-
ism. Instead, they helped form a kind of writing transparent with respect
to the meaning of the statement. Since the symbols were abbreviations,
they enhanced the surveyability of the expressions, in the same way as
the technical writing of a number helps understand the structure of the
number. 43 Th is conclusion raises a general question. Th e surveyability of
a procedure or a proof depends on the kind of text constructed to write
down and work with the proof or procedure. Which resources did various
groups of practitioners create, or borrow, to this end? Netz’s contribution
can be viewed as a step towards a systematic inquiry in that direction. We
shall soon meet with further evidence that can be fruitfully analysed from
the same perspective.
To create this form of writing, Diophantus made use of possibilities
available in the written culture of his time, but used them in a way specifi c
to his project. As Netz stresses, Euclid’s Elements also exhibits evidence of
creating a specifi c language, characterized by distinctive formulaic expres-
sions. Th us we meet with the same phenomenon already emphasized on
several occasions above from yet another perspective: the kind of text used
is correlated to the type of proof developed. Given the fact that the kinds of
proof and the project embodied by the Elements diff er from the objectives
of Diophantus, the kind of writing employed in the Elements diff ers from
those used by Diophantus.
43 Neugebauer also interpreted some features of Mesopotamian ways of writing mathematics as
making statements surveyable. Høyrup 2006 quotes at length the passages by Neugebauer on
this point and discusses them, with respect to Mesopotamian, Greek, Latin, Arabic and Indian
sources as well as sources written in vernacular European languages.