Mathematical proof: a research programme 33
Th e lines of inquiry just outlined illustrate some of the issues that more
generally have imposed themselves as central issues in the following chap-
ters of the book. To begin with, these issues are taken up from diff erent
perspectives in the next two chapters of the book, both also devoted to
Greek sources.
Th e issue of generality in relation to proof directs Ian Mueller’s analysis
of marginalized Greek writings dealing with numbers, albeit from a diff er-
ent perspective. Because they have been overshadowed by the treatment of
arithmetic in Books vii to ix of Euclid’s Elements , the techniques of proof
used by Nicomachus in his Introduction to Arithmetic and by Diophantus
in On Polygonal Numbers have not yet been the object of detailed analysis.
Ian Mueller chooses to focus on them because they deal with numbers –
polygonal numbers – in a singular way, approaching them through the
prism of confi guration and procedure of generation. Th ese features raise
the problem of defi ning the polygonal numbers as general objects, making
general statements about them, and proving such statements in a general
way. Th e challenge is to reach generality not only with respect to all polygo-
nal numbers of a specifi c type, such as triangular or square numbers, but
also to defi ne and work with n -agonal numbers.
Both Nicomachus and Diophantus attempted to meet with this chal-
lenge, by composing treatments of these numbers in general, stating propo-
sitions about them, and accounting for the validity of these statements. In
particular, both authors set themselves the task of establishing the value of
the n th j -agonal number. Th e conclusion of Mueller’s analysis is that both
attempts equally fail to establish the conclusion aimed at with full general-
ity. Nonetheless, the diff erences between the ways the two authors shape
textual elements to approach polygonal numbers, formulate statements
about them and design modes of proving to deal with the topic raise con-
siderable interest. Th is is what emerges from Mueller’s detailed description
of the diff erent techniques of reasoning by which both authors address
these numbers and try to establish their properties.
Nicomachus makes use of specifi c diagrams that iconically represent the
numbers as confi gurations of units. In addition, Nicomachus introduces
a key tool – sequences of numbers – in a way that will be characteristic of
his approach. To begin with, he constructs arithmetical ways of generating
these sequences. He then strives to establish relationships between these
sequences and the fi rst sequences of polygonal numbers (triangular, square,
pentagonal and so on). It is for this task that Nicomachus’ diagrams are
brought into play. Because of their features, these diagrams can be used
to indicate the reason of the correctness of the relationship only for the