As precursor to the notion smooth
one needs the notion of
continuous variation
which, in turn, depends on a continuum in which to
vary. This will require a characterisation of indefinite indivisibility
while avoiding the paradoxes Zeno used to disuade Greek mathematics from
discussing variation: the model must allow Achiles to overtake a tortoise.
One way or another, classical (i.e. post-Gauss but pre-Gödel)
foundations give us a continuum
of numbers - known as the real
line
: Conway can give it to us raw as the surreal numbers
(which handle infinities
and infinitesimals systematically, alongside the plain reals: but one can
extract the plain reals from them) and the orthodox analysis I was taught as an
undergraduate obtained it by filling in
gaps between rationals with
limit points
wherever sequences of rationals converge
. One then
classically uses the real continuum to embed notions of continuity in other
contexts.
The classical continuum has the odd property that it's a continuum of
discrete points
; I'll nick-name it the pointilist continuum
as a
result. The space-time continuum of physics might conceivably be pointilist,
but we now know for certain that we can't deal with it as such; at best, we can
deal with very small regions of it, but modern physics quietly abandons belief
in the individual indivisible discrete points
which classical analysis
presumes.
Furthermore, the classical tradition has hit a wall with the pointilist
continuum: it obliges one to use reductio and the axiom of (trans-finite)
choice all over one's proofs which, aside from offending the constructivists
(who can readily throw taunts based on Gödel's theorem), leaves one with
such perversities as the Banach-Tarski construction, in which a ball is cut into
seven pieces which can then be re-assembled into two balls, each of which is as
big as the original, with no gaps or holes in either. This hinges on the pieces
being non-measurable
(one cannot sensibly discuss their volumes
)
and starkly obliges one not to ignore the perversity of: nearly all subsets of
the geometric 3-sphere are non-measurable, yet all exhibitable subsets of it are
measurable. Such perversities are the natural consequence of using the axiom of
choice and reductio, which enable one to prove that things exist
without
obliging one (even to show how) to construct
them.
None the less, the pointilist continuum provides a formally tractable model
of the physical continuum, which has enabled us to identify structural
(categoric) approaches to modeling the continuum; these inspire me to an
exploration in search of a characterisation which will suffice to describe the
continuum without making direct presumptions about its implementation
details
.
Smooth is all about the continuum. The world of relations is discrete
mathematics
: it concerns itself with values
which a context is
supposed to be capable of exhibiting and comparing; it can model a continuum via
the classical charade of the finely-balanced infinities of the real
numbers, or via the elegant spectacle of Conway's surreal
ones; but
always it believes in discrete
values and the ability to
distinguish
them - even when they are persuaded to form a continuum, it
is a continuum of distinguishable points
. The real continuum of the
physical universe isn't like that: it's a continuum. You cannot pin down a
single point
in it; but, to borrow language from the classical description
of the pointilist continuum, you can identify continuous
transformations
between neighbourhoods
within the continuum; and you can compose
these in ways which mimic the composition of relations (typically, indeed, that
of mappings; and the subset hierarchy
of the neighbourhoods mimics the
composition of collections).
Yet, once we begin to deal with the continuum, we must discuss a binary
operator, construed as composition, which is, none the less, not the composition
of relations; yet, by the standard embedding of any binary operator in the
domain of relations, the composition
of continuous
mappings
between neighbourhoods
of the continuum may be represented by the actual
composition of relations among the continuous
mappings thus
composed
. This begets an approach owing more to category theory than to
Gauss.
Intuitively, I assert that the (underlying) continuum of the physical
universe is everywhere the same
when examined on small enough
scales
; that is, leaving aside such intrusions as the tensor fields
describing the metric, electromagnetic field and other inhabitants
of the
continuum, you can chose an arbitrary pair of locations (albeit, in the absence
of points
, I can't yet be clear about what a location
is) and
exhibit a one-to-one correspondence between a neighbourhood of one and a
neighbourhood of the other; this correspondence must preserve the underlying
structure of the continuum, in terms of which we are to describe notions such as
variation
which are needed if we are to discuss smoothness
,
without which we cannot introduce the tensor bundles needed to describe the
continuum's inhabitants.
Furthermore, the one-to-one correspondences between neighbourhoods of
space-time admit of continuous deformation
- to picture which, consider
two identical mugs, one of them hanging still on a hook in my kitchen, while I
use the other for my morning coffee. Because they are identical, one can use
them as the skeleton of a correspondence between the regions of space they
occupy; the mug I'm using moves about so its end of the correspondence moves
continuously from place to place while the idle mug's end remains where it is.
Stripping out the temporal
elements of the description, one sees a
continuum of correspondences between the stationary mug's neighbourhood and the
various locations to which the active mug can be carried.
At each location (at least once I've finished my coffee) I can rotate the
active mug; in any orientation, I can move it from place to place. The former
will beget a group of local rotations; the latter, a group of local
translations. The latter is rich enough
that one can introduce
correspondences between any small enough neighbourhood within the space-time
continuum and neighbourhoods within the continuum of transformations of
space-time.
One can also combine the correspondences between neighbourhoods: given two
correspondences, sharing one neighbourhood as one of their ends but having
different other ends, one obtains a correspondence via the shared end between
these other ends. For example, returning to my two mugs, the idle mug serves as
a common end-point for each of the correspondences the two provide me with as I
move the active mug about. While I'm drinking my coffee I get a correspondence
between a neighbourhood in my kitchen and a neighbourhood near my mouth; while
I'm typing this, the mugs identify the same neighbourhood in the kitchen with
one just above my desk. Combining these two, I can obtain a correspondence
between the neighbourhood above my desk and the neighbourhood near my mouth -
which is, indeed, the same correspondence I could obtain by bringing the idle
mug upstairs and sitting it on my desk while I'm drinking out of the active mug.
Thus one can compose
correspondences; the composite is again a
correspondence.
A naïve pointilist model would allow me to describe the continuum as a collection of points and the correspondences as actual one-to-one relations (i.e. monic mappings) whose end-collections are neighbourhoods within the continuum. Composing correspondences would really be the same thing as composing the relations encoding them. None the less, within a pointilist model, I can still characterise each correspondence by the mapping, from correspondences to correspondences, which encodes the effect of composing with the given correspondence.
Even without points, I can still use the composition of correspondences to identify each correspondence with the mapping composition induces from correspondences to correspondences; this represents each correspondence as a mapping, hence as a relation, and models correspondences from within the universe of discrete mathematics - i.e. relations.
Written by Eddy.