Our physical experience of the world about us offers the illusion of a smooth continuum – as Achilles catches up with the tortoise, one has but to examine the story minutely enough and their movements may be broken into as many episodes in the story as one may wish, throughout each of which, no matter how brief, the same illusion of continuity remains so faithfully echoed as to seduce the expectation that such subdivision may be carried on arbitrarily.

The standard mathematical tool for modelling the continuum is the real
numbers

, often via the complex
numbers: the reals form a continuum
which lives up to those expectations (and may be found among the surreals, a far
richer continuum which lives up to those expectations even more convincingly).
It suffices, for my purposes, to allow that the physical domain has such
continuity; I intend to describe comparatively naïve mathematical tools
(the reals are significantly more cosmopolitan) enabling a discussion of that
physical domain which describes its continuity without feeling obliged to
construct a full and faithful model of its continuity. In so far as a scalar
continuum (such as the classical real or complex numbers) does model the
physical continuum, I intend to express the inevitability of its success in
terms of its relationship with
the positive rationals. (For the
present, I'll ignore the possibility that quantum
mechanics might complicate that.)

Physically, we intuit that we can move an object from any position to any
position

subject to inevitable clauses connected with practicality, such as
that the positions not be far apart – the important issue being that we
can make sufficiently small movements in all possible directions

,
wherever we may be, from which we can build up, albeit subject to the same
practicalities, arbitrary movement. Gauge theory turns this symmetry under
translation

into conservation of momentum

, in which physics
encourages us to believe. In the flat universe of Newtonian physics it was easy
enough to identify positions in the universe with the points of a Euclidean
continuum and do arithmetic with vectors to characterize the in various
directions

aspects of the matter; but in the Einstein's legacy, we inhabit a
curved continuum in which life is less simple.

None the less, the idiom of the smooth manifold

presents itself as
the natural mathematical tool to describe the curved continuum: and is rich
enough to allow us to salvage what we need

to serve as the physical
isometries at the heart of the translation/momentum gauge effects. What physics
needs to describe is the experimenter's ability to move an object about: on a
smooth manifold, one describes smooth deformations which only move anything
around in some topologically simple neighbourhood of the laboratory, while
taking the form of (at least more-or-less isometric) translations of position on
the experimenter's bench. One can then describe movements within the
laboratory as if

they were movements in a flat Euclidean continuum.

Now that's qualified by topologically simple

: on a smooth manifold,
every position has a topologically simple neighbourhood (consequently hordes of
them), so this shouldn't be a problem; but a position's simple neighbourhoods
may be very small, so one does have to ask whether space-time is topologically
simple in any neighbourhood of a real physicist's laboratory. If it turns out
that matter isn't topologically simple, albeit clearly its complexity is on a
scale much smaller than the laboratory, we have to do some more work: but,
again, there's no huge problem. One easy approach is to use transformations
which leave out

little patches of the manifold, within which the
topological complexity is contained; one's transformations are then obliged to
leave out, also, the corresponding patches which would have been
transformed

to positions left out; but I have no problem with partial
functions

, which can still survive to describe the continuity of the
continuum within which the knots move about.

One way or another, one obtains a continuum of translations

one may
apply to the laboratory, modelled as a group of transformations of the physical
continuum. We can arrange for this group to look like

the additive group
structure of a vector space over a scalar continuum (in classical terms); we can
use repeated addition

to beget integer multiplication, whence naturally
division (subject to our intuitions about continuity), hence
rational scaling

of translations

. Meanwhile, we can consider
mappings from translations to translations (notably such are may, more-or-less,
deserve to be described as isometries): among these, such as respect
addition

– i.e., f(a+b) = f(a)+f(b); among those, the ones which
commute with every mapping which respects addition. These last constitute the
natural real scalings

of the physical continuum: they'll form a
continuous field (or similar), and they'll include the rational scalings.

So I'll suppose there's a real number continuum

proper to the
physical continuum. Within it, I'll pay particular attention to the positive
rationals. I reserve the right to appeal to properties of the classical reals,
especially when the surreals share those properties, when I can't think of a
better way to describe the physical continuum; but I do intend to avoid the
need.