Euclid’s definitions: a typology A study of the ancient mathematical definitions has not yet been provided. This is surprising given the overall interest in definitions among scholars working on ancient philosophy and mathematics, as well as among contemporary philosophers. My paper fills – or rather opens – this gap in the existing scholarship by furnishing a typology of all definitions in Euclid’s works. The typology is based on various logical, grammatical, and textual criteria, such as: – What kind of term is defined? – What is the logical form of the definiens? – How is the definition formulated? What word forms are used? Which verb (if any) is used to connect the definiendum with its definiens? Does it include any sentential connectors? Do the definiendum and the definiens carry an article, respectively? Are they formulated in the plural or singular? – Does the definition mention any extra-mathematical terms? – What is it used (useful) for? Is it used (usable) in subsequent argument, in definitions of other terms, or to fix the reference for terms? – Is the definition accompanied by a proof or justification of some other sort? – How is the definition presented? In first-order or second-order discourse? Is it explicitly labelled as a definition? What indicates that it is a definition? – Is it stated before, within, or independent of mathematical proofs, in mathematical or non-mathematical (con-)texts, in a list or in some other sort of text? – Are there any alternative definitions, and if so, how is the alternative marked? On the basis of this typology, and of the descriptive statistics that emerge from it, various insights about methodological background assumptions of Greek mathematicians are inferred. Some of these consequences are surprising insofar as they reveal that Greek mathematicians, especially Euclid, were not only aware of the previous Platonic and Aristotelian philosophies, but that they expressly (though implicitly) encoded traditional philosophical theories and distinctions into their mathematical texts.
Benjamin Wilck
(Humboldt-Universitaet zu Berlin)
Lyceum Classics Community Seminar 