Oriens - Occidens 7:113-154 (2010)

Ioannis Vandoulakis
Hellenic Open University
The style of arithmetic in the treatises the Neo-Pythagorean authors is strikingly different from that of the "Elements". Namely, it is characterised by the absence of proof in the Euclidean sense and a specific genetic approach to the construction of arithmetic that we are going to describe in our paper. Lack of mathematical sophistication has led certain historians to consider this type of mathematics as a feature of decadence of mathematics in this period [Tannery 1887; Heath 1921]. The alleged absence of originality in these works has also given grounds to believe that “the arithmetic presented in these works derives substantially from an ancient, primitive stage of Pythagorean arithmetic” and to use them as “an index of the character of arithmetic science in the 5th century” [Knorr 1975]. In this paper, we take Nicomachus’ Introduction to Arithmetic for point of departure, because it is the richest and most well organised treatise representing this tradition. However, we also take into account the works of other Neo-Pythagorean authors. We are going to show that the Neo-Pythagorean arithmetic might have been developed in a natural, self-contained manner as a simple theory of counting over a domain of concrete initial objects, designated by fixed signs. This approach can be comfortably realised without appealing to assumptions of axiomatic character, but relying upon some ‘genetic’ constructions intented to be carried out by means of the designated entities.
Keywords Greek number theory  Neo-Pythagorean number theory  Nicomachus of Gerasa  Greek mathematics
Categories (categorize this paper)
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 71,410
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

No references found.

Add more references

Citations of this work BETA

Add more citations

Similar books and articles

Problems in the Interpretation of Greek Number Theory: Euclid and the 'Fundamental Theorem of Arithmetic'.W. Knorr - 1976 - Studies in History and Philosophy of Science Part A 7 (4):353-368.
Mathematical Aspects of the Periodic Law.Guillermo Restrepo & Leonardo Pachón - 2007 - Foundations of Chemistry 9 (2):189-214.
Number Theory and Elementary Arithmetic.Jeremy Avigad - 2003 - Philosophia Mathematica 11 (3):257-284.
Pythagorean Heuristic in Physics.Sorin Bangu - 2006 - Perspectives on Science 14 (4):387-416.


Added to PP index

Total views
21 ( #537,400 of 2,519,862 )

Recent downloads (6 months)
1 ( #406,012 of 2,519,862 )

How can I increase my downloads?


My notes