### The Concept of Number and Magnitude in Ancient Greek Mathematics

#### Abstract

The aim of this text is to present the evolution of the relation between the concept of number and magnitude in ancient Greek mathematics. We will briefly revise the Pythagorean program and its crisis with the discovery of incommensurable magnitudes. Next, we move to the work of Eudoxus and present its advances. He improved the Pythagorean theory of proportions, so that it could also treat incommensurable magnitudes. We will see that, as the time passed by, the existence of incommensurable magnitudes was no longer something strange. Already in the period of Plato and Aristotle, their existence was common place: up to the point of being considered absurd that all magnitudes were commensurable. Aristotle criticized the Pythagorean program and defended that, though belonging to the same category (quantity), number and magnitude are of distinct species: number is discrete and magnitude is continuous. We finish by presenting briefly how the concept of number was amplified throughout the centuries until it came also to include the notion of continuity.

#### Keywords

#### Full Text:

ABRIR PDF/OPEN PDF FILE» (Español (España))#### References

Barnes, J. (1984). The complete works of aristotle: the revised oxford translation. Princeton University Press.

Bell, J. (2005). The continuous and the infinitesimal in mathematics and philosophy. Polimetrica.

Euclid. (1956a). The elements (T. Heath, Ed.). Dover Publications. (volume 2)

Euclid. (1956b). The elements (T. Heath, Ed.). Dover Publications. (volume 1)

Fritz, K. V. (1945). The discovery of incommensurability by hippasus of metapontum. The Annals of Mathematics(2), 242–264.

Heath, T. (1981). A history of greek mathematics: From thales to euclid (No. v. 1). Dover Publications.

Kirk, G. S., Raven, J. E., & Schofield, M. (1994). Os filósofos pré-socráticos. Fundação Calouste Gulbenkian. (Trad. Fonseca, C. A. L.)

Morgan, A. (1836). The connexion of number and magnitude: an attempt to explain the fifth book of euclid. Taylor and Walton.

Neugebauer, O. (1969). The exact sciences in antiquity. Dover Publications.

Newton, I. (1769). Universal arithmetick: or, a treatise of arithmetical composition and resolution. Printed for W. Johnston. (Trad. Raphson, J. and Wilder, T.)

Plato. (1952). The dialogues of plato. William Benton. (Trad. Benjamin Jowett)

Smith, D. (1958). History of mathematics. Dover Publications. (Vol. 2)

DOI: https://doi.org/10.22370/rhv.2017.9.848

Copyright (c) 2017 Diego P. Fernandes

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.