Re: abundance of irrationals!)

From: W. Mueckenheim (mueckenh_at_rz.fh-augsburg.de)
Date: 02/24/05 

Date: 23 Feb 2005 23:48:22 -0800

Matt Gutting <tchrmatt@yahoo.com> wrote in message

> Why does a number need to be mappable onto a physical object in order to exist?

What is a number? Consider the set {street, green, piano, exciting,
Mary}. It contains some objects or ideas but it does not contain any
number. Why doesn't it? Because a number must be capable of beeing put
in a relation with all other numbers, and, in particular, with the
unit. A real number must be intermingled with other numbers, but must
be clearly distinguished by the relation "<" from other reals.
Therefore mathematicians like Archimedes, Wallis, Cantor tried to give
meaning to irrational proportions or numbers by determining their
value "as close as possible" for more than 2000 years. Cauchy's and
Weierstrass' and Cantor's work culminated in the definition of an
irrational number. It is defined by some construct (sequence, series,
infinite product, modular identity, continued fraction, ...) such that
for any eps > 0 there is an n_0 such that for any n > n_0 the value of
the construct X_n differs by less than eps from the irrational number,
pi, say. Of course the basis p of the p-adic expansion is arbitrary.
Important is alone the fact that the number can be approximated as
closely as one likes, i.e., with an error smaller than any positive
number. This is the definition which all mathematicians adhere to. But
they have not yet recognised that it is not the spirit of a human
alone which can accomplish the necessary calculation. By pure
imagination you cannot determine the ratio of circumference and
diameter of the circle to better than 5%, you cannot calculate more
than 10 digits of pi (few people come so far) you cannot even remember
more than 100 digits of pi (world record is about 40,000 digits, if
there are no newer results, but most people stop at 2 or 4 digits).
Hence, in order to calculate pi to an arbitrary precision, our mind
needs help from supports like pencil, chalk, computers. This help,
however, is limited by the 10^100 particles of the universe, which
could serve as CPU and memory of a big computer.

Therefore, it is never and by no means possible to approximate pi to
better than 1/10^10^100 (if it is a normal number - I think the first
10^10 digits don't show any anomaly, though this fact does not mean
anything). It is by no means possible to distinguish pi from the
number you get, when replacing the digit no. 10^100 by 5. Therefore,
pi and all other irrational numbers are called numbers without a
thought being given to the fact that they are not numbers, in fact,
because they cannot be measured in units of 1.

Regards, WM

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