The Smooth Continuum

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.

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