Re: Fractions v natural numbers in

From: Bob Silverman (anonymous_at_mathforum.org)
Date: 07/23/04 

Date: Fri, 23 Jul 2004 18:12:22 +0000 (UTC)

On 22 Jul 2004, The Last Danish Pastry wrote:
>Roger Penrose's new book "The Road to Reality" arrived today.
>
>In the preface (p. xviii) he is talking about fractions. We find:
>
>"The magic is that the idea of a fraction actually works despite the fact
>that we do not really directly experience things in the physical world that
>are exactly quantified by fractions - pieces of pie leading only to
>approximations. (This is quite unlike the case of natural numbers, such as
>1, 2, 3, which do precisely quantify numerous entities of our direct
>experience.)"

I presume that you mean one can't have 1/2 of a pie because it is
impossible to measure EXACTLY 1/2 pie. But if you talk about
measurement, it is also impossible to have exactly 1 pie if (say) a
whole pie is defined by some physical characteristic (e.g. weight).

According to this viewpoint one might argue:

Suppose you have a pile of wood; ostensibly a collection of 6 2 x 4's.
How many 2 x 4's do you REALLY have? Each of those pieces of lumbar
is only an APPROXIMATION to a 2 x 4. How close the approximation is
depends on how much care was taken in cutting them. However, with
enough granularity of measurement, even 1 inch is an approximation.
In the real world, EVERYTHING is an approximation.

:-)

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