[FOM] Re: real numbers

[John Pais]

Hartley Slater wrote:

> ............  For instance, I can ask my greengrocer for 2 1/2 pounds
> of cabbage, but he would be non-plussed if I asked for the complement
> of 2 1/2 pounds of cabbage, or for members of 2 1/2 pounds of cabbage.
>
> How does my greengrocer enter the sophisticated realm of debates on
> the foundations of mathematics?  Because he clearly knows better than
> many theorists in this area what sort of thing, in the first place, a
> number is: it has to number things.

You say: "it has to number things". Why do you think so?

It is clear from mathematical practice that number concept has evolved in such a
way that it now extends to provide a precise and optimally useful domain of
discourse (i.e. real analysis) in which your greengrocer can, in addition to
counting his cabbage, determine its center of mass, its instantaneous velocity when
it's rolling off the stand, it's angle of deflection when it hits the ground, and
when consumed, the absorption rate of its nutrients.

One of the primary purposes for the development of the real number system (i.e. the
appropriate first-order theory) is to provide a rich and robust framework for
easily employing certain mathematical tools such as limit, derivative, integral,
and much more. So, the real number system is not merely a 'number' system, but an
optimal framework (and tool) for doing a certain type of mathematics.

> It has been well said,
> 'remember, you can convince anyone of anything so long as they are
> clever enough'.  It takes a clever person to miss what is patent, and
> missing what is patent has been endemic in logic and the foundations
> of mathematics for quite some while.
>
> As readers of my postings may, by now, have realised, it is a full
> time job, pointing out the crashingly evident to those whose minds
> are elsewhere - even if it only takes two lines, in each case.

That was good (smile)!

However, I just don't see the point in digging ones heels in, ignoring current
mathematical practice and its purposes, and maintaining essentially (merely) that
mathematicians aren't using certain words correctly, e.g. 'number' or 'the'. Who
should determine how, and whether or not, the concept of 'number' should change and
evolve, mathematicians or greengrocers?

Best wishes,
John Pais

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E-mail: pais at kinetigram.com
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