Mathematical proof: a research programme 45
numbers. 51 Once the computation of a reciprocal has been recorded on
the tablet, the same algorithm is applied to the result and shows that
one thereby returns to the starting point of the original algorithm. In
fact, more accurately, this structure is typical of only one type of tablet
devoted to the algorithm computing reciprocals, precisely those tablets
that contain only numbers. Th ese tablets record successive numbers pro-
duced through the fl ow of computations according to a determined and
highly specifi c layout until the result is yielded, and then record numbers
obtained through applying the algorithm to the result. However, as Proust
emphasizes, another type of text also refers to the same algorithm. In these
other tablets, the algorithm is expressed in words and the text prescribes
the operations to be carried out in succession. Among all the tablets con-
taining either formulation (the two never occur on the same tablet), Proust
chooses to concentrate on two tablets (Tablet A and Tablet B), one for each
kind of expression. In fact, she selects the two texts that repeat the same
pattern in a signifi cant number of sections.
Th e verbal expression of the algorithm had been essential for Sachs to
interpret the purely numerical expression for it. However, once he had
established that the two tablets relate to the same algorithm, a key question
remained, which Proust addresses: why do we have two expressions of the
same algorithm? What are the specifi c meanings conveyed by each of them?
And, especially in her case, what does the numerical tablet say?
To answer these questions, Proust combines several methods. She restores
the practices of computation to which both tablets adhere, bringing to
light that they relate to the fl ow of computations in diff erent ways. She also
compares the tablets to other parallel specimens. Lastly, she examines every
detail of the numerical tablet (Tablet A): the layout, the numbers chosen, the
way of conducting the algorithm in the direct and the reverse computations.
Th rough sophisticated reasoning, Proust can establish that the second part
of each section – the one containing the computations in the reverse direc-
tion – did most probably not play the part of checking the results of the direct
algorithm. She further demonstrates that the layout designed to record the
numbers, as evidenced in Tablet A, was created for such kind of texts and
introduces a way of managing the space of the tablet that was artifi cial.
Th is conclusion leads her to suggest that the spatial elements of the layout,
like columns, are precisely those which convey the meanings expressed by
Tablet A. We see here at its closest how the composition of a kind of text
relates to the work carried out with a text. In her view, the columns may be
51 For greater technical detail, I refer the reader to Proust’s chapter.