Posted by: **Alexandre Borovik** | July 31, 2008

## How did Classic Greeks understand natural numbers?

A commentary to Plato’s *Charmaides* by unknown scholar describes a clear distinction between *ideal *natural numbers and numbers used for mundane everyday counting:

Logistic is the science that treats of numbered objects, not of numbers; it does not consider number in the true sense, but it works with 1 as unit and the numbered object as number, *e.g.* it regards 3 as a triad and 10 as a decad, and applies the theorems of arithmetic to such cases. It is, then, logistic which treats on the one hand the problem called by Archimedes the cattle-problem, and on the other hand *melite* and *phialite* numbers, the latter appertaining to bowls, the former to flocks; in other types of problem too it has regard to the number of sensible bodies, treating them as absolute. Its subject-matter is everything that is numbered; its branches include the so-called Greek and Egyptian methods in multiplications and divisions, as well as the addition and splitting up of fractions, whereby it explores the secrets lurking in the subject-matter of the problems by means of the theory of triangular and polygonal numbers. Its aim is to provide a common ground in the relations of life and to be useful in making contracts, but it appears to regard sensible objects as though they were absolute.

Quoted from

I. Thomas, Selections illustrating the history of Greek mathematics with an inglish translation by Ivor Thomas. Vol. 1: From Thales to Euclid. Harward University Press, Cambridge, Ma. 1938 (reprinted 1980). ISBN 0-674-99369-1.

pp. 17–19, with thanks to David Pierce who brought this quotation to me.

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