Date: Dec 6, 2012 4:22 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 172

Matheology § 172

Wallis in 1684 [?] accepts, without any great enthusiasm, the use of

Stevin's decimals. He still only considers finite decimal expansions

and realises that with these one can approximate numbers (which for

him are constructed from positive integers by addition, subtraction,

multiplication, division and taking nth roots) as closely as one

wishes. However, Wallis understood that there were proportions which

did not fall within this definition of number, such as those

associated with the area and circumference of a circle:

Real numbers became very much associated with magnitudes. No

definition was really thought necessary, and in fact the mathematics

was considered the science of magnitudes. Euler, in Complete

introduction to algebra (1771) wrote in the introduction:

"Mathematics, in general, is the science of quantity; or, the science

which investigates the means of measuring quantity." He also defined

the notion of quantity as that which can be continuously increased or

diminished and thought of length, area, volume, mass, velocity, time,

etc. to be different examples of quantity. All could be measured by

real numbers.

Cauchy, in Cours d'analyse (1821), did not worry too much about the

definition of the real numbers. He does say that a real number is the

limit of a sequence of rational numbers but he is assuming here that

the real numbers are known. Certainly this is not considered by Cauchy

to be a definition of a real number, rather it is simply a statement

of what he considers an "obvious" property. He says nothing about the

need for the sequence to be what we call today a Cauchy sequence and

this is necessary if one is to define convergence of a sequence

without assuming the existence of its limit.

[J.J. O'Connor and E.F. Robertson: "The real numbers: Stevin to

Hilbert"]

http://www-history.mcs.st-and.ac.uk/HistTopics/Real_numbers_2.html

Regards, WM