50 karine chemla
transformed. We would then have a form of algebraic proof in an algorith-
mic format.
Th e second part of my claim relates to the concern for the validity of such
a method of proof. In fact, an analysis of the commentaries reveals that the
exegetes considered the question of the validity of the operations applied to
an algorithm as such.
A close inspection shows that the exegetes linked the validity of their
operations to the set of numbers introduced in Th e Nine Chapters , which
includes not only integers but also fractions and quadratic irrationals. Th e
key point, in their eyes, was that these quantities allowed the expression
of the results of divisions and root extraction exactly, thereby allowing the
inverse operation of multiplication to cancel the eff ect of these operations
and restore the original number. Th is point recalls the Mesopotamian
tablets described by Proust, which demonstrate the same concern. Why
was this fact important for practitioners of mathematics in Mesopotamia?
Further inquiry into that question could prove interesting for our topic.
At the same time, I argue that it is when the commentators establish the
correctness of algorithms for carrying out the arithmetic with fractions
that they address the validity of applying some of the operations to lists of
operations. Several points must be stressed here.
Firstly, the analysis of this dimension of the practice of proof preserved
from ancient China brings to light an essential point, which allows us to
capture a key feature of algebraic proof: the validity of such kinds of proof
is essentially linked to the set of numbers with which one operates and how
one operates with them. Th is point, I argue, was understood in ancient
China, but it is a point of general validity regarding algebraic proof.
Secondly, the question arises whether dimensions of algebraic proofs as
we practise them today may have historically taken shape within practices
of proving the correctness of algorithms.
Th is brings me back to a point raised at the beginning of this introduction.
I insisted on the fact that the standard account of the history of mathematical
proof had nothing to say about the history of how the correctness of algo-
rithms was established in the past. At this point, I am in a position to sum-
marize our fi ndings on this question. We now see even more clearly that this
was a lacuna which contributed to the marginalization of sources that were
‘non-Western’ and sources that bore witness to practices of proof related to
computations. In addition, we also see that this lacuna may also prevent us
from providing a historical account of the emergence of algebraic proof.
Last, but not least, if the answer to the previous question proves positive,
a new historical question presents itself quite naturally: one may further