The Definitions of Number in Boethius’s Introduction to Arithmetic

Abstract

The paper enquires into the reasons why in his loose translation of Introduction to Arithmetic by the Neopythagorean philosopher and mathematician Nicomachus of Gerasa Boethius altered the passage addressing the definition of number. While Nicomachus’s text contains three definitions of number, Boethius lists only two. However, he also pays attention to the definition he omits, even though he does not regard it as a proper definition. In his view it fails to embody the substance of number and is to be understood as a description of the components constitutive of the substance of number. Although this is a possible explanation of Boethius’s dismissal of the definition provided by Nicomachus, the description also occupies an important place in relation to the general characteristic of number, because Nicomachus’s definitions fully correspond to the three basic topics which were central to contemporary arithmetic, viz. the science of number: number as discrete quantity, referring to the properties of numbers and their classifications; number as collection of units, leading to the topic of figural numbers; and number as quantity emanating from unit and subsequently returning to it, corresponding with numerical ratios, sequences and their transfers.