The History of Mathematical Proof in Ancient Traditions

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1 Th e Euclidean ideal of proof in Th e Elements

and philological uncertainties of Heiberg’s

edition of the text

Bernard Vitrac, Translation micah ross

Introduction

One of the last literary successors of Euclid, Nicolas Bourbaki, wrote at the
beginning of his Éléments d’histoire des mathématiques :

L’originalité essentielle des Grecs consiste précisément en un eff ort conscient pour
ranger les démonstrations mathématiques en une succession telle que le passage
d’un chaînon au suivant ne laisse aucune place au doute et contraigne l’assentiment
universel ... Mais, dès les premiers textes détaillés qui nous soient connus (et qui
datent du milieu du v e siècle), le « canon » idéal d’un texte mathématique est bien
fi xé. Il trouvera sa réalisation la plus achevée chez les grands classiques, Euclide,
Archimède, Apollonius; la notion de démonstration, chez ces auteurs, ne diff ère en
rien de la nôtre. 1

I am unsure what was intended by the last possessive, whether it acts as
the royal or editorial we designating the ‘author’, or if it ought to be under-
stood in a more general way: ‘la nôtre’ could mean that of the Modernists,
of the twentieth-century mathematicians, of the French, or formalists. All
jokes aside, the affi rmation supposes a well-defi ned and universally accepted
conception of what constitutes a mathematical proof. Th e aforementioned
conception, the citation for which is found in a chapter titled ‘Fondements
des mathématiques, Logique, Th éorie des ensembles’, is at once logical,
psychological (through a rejection of doubt), and ‘sociological’ (based on
universal consensus). Perhaps this assertion ought to be considered nothing
more than a distant echo of the Aristotelian affi rmation that all scientifi c
assertions (not just mathematical statements) are necessary and universal.
Th e following list of Greek geometers is also interesting. It contains the
classics, and the triumvirate was probably intended to follow chronological
order. Here, then, Euclid is not simply a convenient label, sometimes used to
designate one or several of the many adaptations of Euclid’s famous work,
as when one speaks about the Euclid of Campanus ( c. 1260–70), the Arab

(^1) Bourbaki 1974: 10.