Reasoning and symbolism in Diophantus 353
My interpretation of Diophantus thus relies on two theoretical contexts
I developed elsewhere: deuteronomy , and analysis as a tool of presentation.
What is Diophantus’ project? I interpret this within the theoretical context
of what I call deuteronomy: it is to systematize and complete previously
given materials, making them all conform with some ideal standard. Th is
systematic structure is two-dimensional. Horizontally, all units should
conform to each other. Vertically, all units should conform to the ideals of
Greek elite mathematics. How does Diophantus then fulfi l his project? I
interpret this within the theoretical context of analysis as a tool of presenta-
tion. If all the units are to be the same, then the most natural format to take
is that of a problem. And to make those problems conform to the ideals of
Greek elite mathematics, the method of analysis is deployed, so as to display
the rationality of each of the moves made through the text.
Th is, fi nally, I suggest, is the function of reasoning in Diophantus: to
build a rational bridge leading from the terms of the problem, to the solu-
tion. I now need to show how Diophantus’ symbols may serve this function.
Diophantus’ symbolism and the display of rationality
My basic thesis is that the reasoning in Diophantus is designed, primarily,
to display a rational bridge leading from the terms of the problem to the
solution. Two questions arise: (1) How does symbolism such as that used
by Diophantus help with this goal? (2) Why would it help with such a goal
here, and not elsewhere in Greek mathematics?
Let me fi rst discuss the appropriateness of Diophantus’ symbolism for
his goal.
Diophantus’ goal, as I reconstruct, is in one sense limited, in another
sense ambitious. Th e goal is limited, because he does not aim at powerful,
general theoretical insight into numerical problems. He merely aims at
classifying and completing them as a system. Th e goal is ambitious, because
each solution, at each step, has to clear a high cognitive hurdle. It has to
display, step by step, its rationality.
Both the limit and the ambition explain why a general, theoretical
approach such as text 3 above would not be appropriate. It is not called
for, because of the limited ambition; and it is undesirable because, with
the prolix phrases and the diffi culty of fi xing the identity of the entities
involved, it becomes impossible to survey, step by step, the rationality of
the argument as it unfolds. Note that in a text with theoretical goals, local
obscurity can be tolerated: the reader is then expected to work his or her
way through the text. It is quite feasible to have valid arguments expressed