Mathematical proof: a research programme 39
development of this inquiry provides means for interpreting these texts
more accurately.
In conjunction with the fi rst point, Netz’s treatment yields insight into
how diff erent the purposes for developing proofs may have been. Th is
brings us back to the programme suggested above for our historical inquiry
into proof, namely, the restoration of the motivations behind the develop-
ment of proofs and the description of the diversity in their conduct accord-
ingly. However, before we go further in widening the set of sources to be
considered with these issues in mind, a last point must be emphasized.
Netz’s discussion illustrates how the resources Diophantus introduced
for a given type of proof were adopted to design the text of another kind of
proof, i.e. algebraic proofs, in modern times. More precisely, Netz’s analysis
highlights why Diophantus’ proofs are not algebraic in nature. Nonetheless,
the shaping of the modern algebraic proof made use, for the conduct of
a reasoning, of symbolic resources similar to those designed within the
framework of the Arithmetics. Th is conclusion off ers our fi rst insight into
the history of algebraic proofs. What are its other components and how did
they take shape? Th ese are some of the questions to which we shall come
back below.
Proving the correctness of algorithms
Th e ideal of transparency, which Netz interprets as informing the symbol-
ism used by Diophantus, is also the main force driving the way Babylonian
practitioners of mathematics composed the text of algorithms, according
to the interpretation of Jens Høyrup. Before explaining this point further,
let us fi rst recall some basic features of the writings to which we now turn.
Th ese documents are, for the most part, composed of problems followed by
algorithms which solve them. Th e fact that the algorithm correctly solves
the problem is the statement to be proved, in contrast to what we fi nd in
Euclid’s Elements , where proofs mainly deal with the truth of theorems. 44
In such contexts, proving means establishing that the procedure carries
44 Th e claim here takes into account the fact that the statement of a problem in the Elements does
not include the statement of how to carry out a task. Interestingly enough, except for some
specifi c cases, the scholarship devoted to Euclid’s Elements has paid much less attention to
problems than to theorems. Th ere are exceptions like Harari 2003. However, the problems still
await further study qua problems. How was the solution written down as text and how did the
proof relate to the formulation of the solution as such? Th ese are questions that seem to me to
be promising for future research. It may well be that aft er these problems have been studied
more in depth, the statement contrasting proofs in Euclid’s Elements with those of algorithms
may have to be made more precise.