42 karine chemla
paradigms that they were, they also provided a semantic fi eld for interpret-
ing the operations of the algorithm or the operations required for the proof
of correctness. Secondly, just as problems provided particular numerical
data, geometrical fi gures displayed simple dimensions, and they were used
in the same way to make explicit the meaning of operations. 46 In a moment,
we shall come back to this comparison, but note that exactly the same situ-
ation holds true for Sanskrit sources analysed by Colebrooke. 47
In addition, the Susa texts that Høyrup analyses formulate the explana-
tions by describing the result of each operation in two ways: on the one
hand, a numerical value is provided and, on the other hand, an interpre-
tation of the magnitude which is determined is made explicit in the geo-
metrical terms of the fi eld of interpretation. Such a kind of ‘meaning’ for
the eff ect of operations recalls what is found in Chinese texts. In the latter
sources, a specifi c concept ( yi ) is reserved to designate that ‘meaning’, and
the meaning is made explicit by reference to the situations introduced in
the statements of problem. In my own chapter on early China, I discuss the
interpretation of this concept and provide cases where it is used in Chinese
sources. In correlation with this parallel, in early Chinese mathematical
writings we also fi nd algorithms that are transparent regarding the reasons
of their correctness: the successive operations are prescribed in such a
way as to simultaneously indicate their ‘meaning’, which can be exhibited
directly in the context of the situation described by the problem. 48
Th is parallel shows that the early mathematical cultures which worked
with algorithms developed partially similar techniques for ‘understand-
ing’, even though they did so in diff erent ways, as we shall make clearer
below. More broadly, these remarks raise a general issue. Th ey invite us
to study systematically the devices, or dispositifs , that various human col-
lectives constructed for ‘understanding’ and interpreting the ‘meaning’
of operations, or conversely, the kind of ‘interpretation’ that was rejected.
Interestingly enough, this question enables a perspective from which we
may cast a new light on the ‘Method’ described by Archimedes in the text
devoted to this topic, which Lloyd discusses in his chapter. Indeed, what
Archimedes off ers with his ‘mechanical method’ is a way of ‘interpreting a
fi gure’ in terms of weight – specifi cally, an interpretation from which he can46 See chapter A, in CG2004: 28–38 and Chemla 2009 , which presents a fully developed analysis
of these issues.
47 See p. 6.
48 Chemla 1991. Chemla 2010 analyses more generally the two fundamental ways in which the
text of an algorithm can refer to the reasons for its correctness. Both can be recognized in the
way in which texts for algorithms were recorded in the tablets discussed by Høyrup.