**Authors:** Elemér E Rosinger

One is reminded in this paper of the often overlooked fact that the geometric
straight line, or GSL, of Euclidean geometry is not necessarily
identical with its usual Cartesian coordinatisation given by the real
numbers in **R**. Indeed, the GSL is an abstract idea, while the Cartesian,
or for that matter, any other specific coordinatisation of it is but
one of the possible mathematical models chosen upon certain reasons.
And as is known, there are a a variety of mathematical models of GSL,
among them given by nonstandard analysis, reduced power algebras,
the topological long line, or the surreal numbers, among others. As
shown in this paper, the GSL can allow coordinatisations which are
arbitrarily more rich locally and also more large globally, being given
by corresponding linearly ordered sets of no matter how large cardinal.
Thus one can obtain in relatively simple ways structures which
are more rich locally and large globally than in nonstandard analysis,
or in various reduced power algebras. Furthermore, vector space
structures can be defined in such coordinatisations. Consequently,
one can define an extension of the usual Differential Calculus. This
fact can have a major importance in physics, since such locally more
rich and globally more large coordinatisations of the GSL do allow
new physical insights, just as the introduction of various microscopes
and telescopes have done. Among others, it and general can reassess
special relativity with respect to its independence of the mathematical
models used for the GSL. Also, it can allow the more appropriate
modelling of certain physical phenomena. One of the long vexing issue
of so called "infinities in physics" can obtain a clarifying reconsideration.
It indeed all comes down to looking at the GSL with suitably
constructed microscopes and telescopes, and apply the resulted new
modelling possibilities in theoretical physics. One may as well consider
that in string theory, for instance, where several dimensions are supposed
to be compact to the extent of not being observable on classical
scales, their mathematical modelling may benefit from the presence of
infinitesimals in the mathematical models of the GSL presented here.
However, beyond all such particular considerations, and not unlikely
also above them, is the following one : theories of physics should be
not only background independent, but quite likely, should also be independent
of the specific mathematical models used when representing
geometry, numbers, and in particular, the GSL.
One of the consequences of considering the essential difference between
the GSL and its various mathematical models is that what appears to
be the definitive answer is given to the intriguing question raised by
Penrose : "Why is it that physics never uses spaces with a cardinal
larger than that of the continuum ?".

**Comments:** 31 pages.

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[v1] 18 Apr 2011

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