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From another viewpoint, Diophantus’ text can be contrasted with other
types of problem texts, which also attest to mathematical work on and with
operations or computations. Several of the following chapters consider
types of writing of the latter kind. Both the use of operations on statements
of equality and the introduction of symbols to carry out these operations
found in the Arithmetics contrast with what other traditions formatted
as algorithmic solutions to problems, for which the correctness needed
to be, and was, established. Even if these other writings bear witness to
other means of proof, via other techniques and in pursuit of other goals,
many parallels can be drawn between the Arithmetics and these other texts.
Th ese texts all deal with operations and operations on operations, illustrat-
ing how diff erent modes of manipulating mathematical operations were
devised in history. Most of these texts reveal an ideal of writing sequences
and combinations of operations in such a way that the meaning becomes
transparent. However, despite the fact that they shared a common feature,
in what follows we shall see that how this ideal was achieved depended on
the context and the type of text constructed. Lastly, these writings all raise
the question of what was meant by a problem and the procedure attached to
it. Was a particular problem representative only of itself, or was it read more
generally as a paradigm for all problems in the same class? Netz develops
an interpretation of the way in which Diophantus conceived of general-
ity. Whether or not this interpretation is accepted, it stresses an essential
point: the symbols used by Diophantus were not abstract. Th is feature
sheds an additional light on how these symbols diff ered from modern ones.
Moreover, it implies that if they had a general meaning, it was conveyed in
a specifi c way, requiring again a specifi c reading.
Th is chapter thus leads to two general conclusions, essential for our
purposes. Firstly, Netz’s article analyses how diff erent groups of mathemati-
cians created diff erent kinds of text in relation to the practice of proof they
adopted. Note that this approach off ers one of the ways in which one could
systematically develop the programme suggested by Lloyd and account
for the variety of practices of proof in ancient Greece. More generally,
Netz foregrounds the fact that proofs have been conducted in history with
various kinds of texts, each being shaped in relation to the operations spe-
cifi c to a given kind of proof. Th e text of the proof is correlated to the act
of proving. Th e general question raised by Netz in his approach to Greek
sources may be phrased as ‘What types of text were shaped for the conduct
of which kind of proofs?’ and has already proved relevant above. Clearly,
this question opens a fi eld of inquiry into proof that could be – and will
prove so below – extremely fruitful. In particular, we can expect that the