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.

The orthodox legacy

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.

Characterising the continuum

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.

Description using relations

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.
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