40 karine chemla
out correctly the task for which it is given, that is, that the algorithm yields
the desired result. In this framework, the ideal of transparency that the
Mesopotamian tablets embody consists of the fact established by Høyrup
that the texts for procedures were simultaneously prescribing computations
and indicating the reasons underlying their correctness. 45 Since we have no
second-order comments by Babylonians explaining how these texts should
be interpreted, it took some time before this property was recognized.
Once again, like the previous examples, this case shows how given collec-
tives of practitioners shaped specifi c kinds of text to work with operations
and establish their correctness. It also highlights how this formation and
standardization of texts invited problems of interpretation. Th e technical
character of the texts hindered their interpretation by historians, who failed
to identify how proofs were expressed and hence drew derogatory conclu-
sions, such as M. Kline’s.
In this case, however, recognizing the proof in the text required
understanding something with respect to proof as well, that is, that the
rationale of a procedure can, at times, be given in the description of the
procedure itself and not as a separate text – this is precisely the manifesta-
tion of the ideal of transparency in this context, which demonstrates that
the same ideal can appear in various ways. More accurately, when we
examine Mesopotamian texts such as those with which Høyrup establishes
his point from this perspective, we observe that the texts of algorithms do
not only contain specifi c prescriptions for operations that achieve transpar-
ency, but also contain elements of the reasoning that develops along the
statement of the algorithm. Again, widening the corpus of proofs under con-
sideration leads us to deeper insights into how a proof can be formulated.
Th is expansion of the corpus also broadens our understanding of the
motivations for writing down proofs in the ancient traditions. In Høyrup’s
45 One speaks of the ‘correctness’ of the algorithm. On this theme, it may be helpful to clarify two
points about which I oft en read misleading comments. Firstly, the text of an algorithm is the
statement to be proved and not its proof. It is on the basis of this distinction that one can make
the point that in Mesopotamian tablets, the two texts (the statement of the algorithm and the
formulation of its proof ) merged with each other. Moreover, to perceive this requires a specifi c
reading, whereby two layers of meaning are discerned in the statement of the algorithm.
Secondly, the aim in proving the correctness of an algorithm is not only to show that the
algorithm yields an exact value – or to establish how accurate or inaccurate the value is – but
also to establish that the sequence of operations prescribed yields the desired magnitude. So
the depiction of algorithms only in association to approximations is doubly misleading. Th ese
basic misconceptions lie at the root of what most commentators who have been discounting
computation have claimed. Th e section entitled ‘Th e unpuzzling character of calculation’ in
Hacking 2000 : 101–3 comments on the text of an algorithm, overlooking the fact that this
is the statement to be proved and not the proof. Th e same pages make other claims that are
contradicted by the conclusions reached here.