# [FOM] origin of "real"

Luigi Borzacchini gibi at dm.uniba.it
Fri Sep 23 09:03:09 EDT 2011

```   As far as I know the term 'real' (substantially in its today usual
meaning) was introduced by Descartes, to denote the real solutions of a
geometric problem discovered by the algebraic procedure (and
distinguished from the imaginary ones).
However it could be useful for a list on the foundations of mathematics
to underline that the idea had also an earlier origin outside algebra.
In Simon Stevin the thesis of an actual 1-1 correspondence between
geometrical and numerical continuum is explicitly given: the former is
the usual geometric line, the latter is the set of numbers that can be
expressed with a (potentially) infinite list of decimal ciphers.
This 'numerical' and not 'algebraic' origin of the idea of real number
was forced at the end of the 16-th century by the development of longer
and longer decimal expressions which appeared in the tables of
trigonomentric ratios and logarithms (at the beginning written as
integer numbers measuring an interval on a circle with a very great
radius, afterward as decimal numbers on a circle with unit radius).
Last but not least, it could be mentioned also a 'physical' origin in
Galileo's analysis of the motion of a body thrown upward: it had to pass
through any degree of speed before its stop, where from the context is
clear that these degrees (even though the term 'degree' denoted a finite
length) are almost a sort of infinitesimals, and the stop lasts for only
an instant, which appears equivalent to a point in time.
In addition, there are two astonishing historical facts: the first is
that in few decades this multiform emergence of a new concept became a
sort of commonplace, the obvious root of Newton's (and Leibniz')
analysis and of new mechanics as well, the second is that there was no
attempt to establish this thoroughly new idea for more than two
centuries, because regarded as obvious.

Best wishes

Luigi Borzacchini
Dipartimento di Matematica
Università di Bari
via Orabona, 70125 Bari (Italy)
http://www.dm.uniba.it/~psiche

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>    1.  origin of "real" (Alasdair Urquhart)
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> ----------------------------------------------------------------------
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> Message: 1
> Date: Tue, 20 Sep 2011 10:28:32 -0400 (EDT)
> From: Alasdair Urquhart <urquhart at cs.toronto.edu>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: [FOM]  origin of "real"
> Message-ID: <alpine.DEB.2.00.1109201018320.11114 at apps0.cs.toronto.edu>
> Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed
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> The origin of the term lies in the 17th century,
> I think, in connection with the solution of
> algebraic equations.  For example, here is
> Descartes in "La Geometrie" (1637):
>
>  	Neither the true nor the false roots are
>  	always real, sometimes they are imaginary;
>  	that is, while we can always imagine as
>  	many roots for each equation as I have
>  	assigned, yet there is not always a definite
>  	quantity corresponding to each root we
>  	have imagined.
>
> Thus, the term "real" was introduced as a contrast
> to "imaginary."  This does have some connection
> with the explanation offered below, since
> Descartes no doubt thought of "real solutions"
> as being solutions in the "real world."
>
>
>
> On Mon, 19 Sep 2011, Thomas Lord wrote:
>
>>
>> I have a history and semantics question that this
>> group might easily be able to answer (but elsewhere
>> is hard to find an answer for):
>>
>> Who coined the term "real" as in "real number"
>> and *why* did they pick that word?
>>
>> I ask because in a discourse on the programming
>> language theory website "Lambda the Ultimate"
>> someone wrote:
>>
>> "The reason real numbers have the name 'real' is
>> because they correspond to measures of the real
>> world."
>>
>> Retrospectively that sounds plausible but is it
>> in fact the origin of the term?
>
>
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Luigi Borzacchini.
Dipartimento di Matematica.
Università di Bari (Italy)

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