]> The Orderly Continuum

# The Orderly Continuum

The tools I build describe relationships among such external entities as some discourse using my tools wishes to discuss; I use the same tools to describe relationships among relationships, in terms of the properties of the relationships thus related. The physical universe isn't a relationship, but we can model it using suitably-characterized relationships among its parts; we can then manipulate those relationships using the generic tools, based on the abstraction of the relation characterizing a relationship with two participants.

Quite straightforwardly, I accept that space-time forms a continuum; and endeavour to abstract a suitable characterization of the continuity thus alluded to. For now I'm discussing the matter with myself out loud, so shall tend to presume terms from classical topology, in so far as they fit reasonably well with comfortable enough intuitions, until such time as I've worked out what notions I want to use in their place; which I'll endeavour to specify before using, though I shan't for the classical presumptions.

One possible approach (2008/05/12) to characterizing the continuum is in terms of measure-theoretic indistinguishability from what the reals would give. So instead of real numbers, true quantities are things that can be modelled by likelihood distributions on the (sur)reals and linear spaces derived from them; and a mapping's only meaning is what it does to such distributions. This should induce a kind of continuity, by ignoring isolated discontinuities (Conway's base thirteen function, in these terms, is the constant zero function, for example) and refusing to discuss the things not distinguishable by probing with likelihood distributions. The fiddly part's going to be working out how little of the usual limit/continuity idioms I'm going to have to mimic, how closely, to be able to describe the model reals well enough to make the system loop back on itself and yield the true numbers as the fixed-point of a cycle; that starts with something real-enough; builds its (piece-wise polynomial ?) smooth enough distributions; discusses what can be probed by them and isolates the reals modulo what we can't probe – to find that these are the model reals we started with. This will need something very like category theory (and Yoneda).

## The problem with the reals

As a physicist I have problems with the real numbers: they are infinitely precise and reality isn't. Any actual physical measurement that we presently represent by a number is better described by a distribution, over the values we orthodoxly could record, indicating the likelihood that the parameter (of our model of reality) that we were trying to measure takes each of those given values. I say better described by such a distribution, but such a distribution is itself expressed in terms of the reals, hence flawed for the same reasons. So what I really want is something enough like the reals to let me build such distributions, but not so much like the reals that it has the infinite precision that isn't faithful to reality.

## Local Equivalence

Slogan: anywhere you look, if you look at suitably small neighbourhoods, the continuum looks the same.

One may characterize this by a discussion of suitable one-to-one relations between neighbourhoods within the continuum; the notion of neighbourhood needs to describe a patch of the continuum with a definite interior; the suitability of relations needs to encode the sense in which they respect the structure of the continuum, locally - i.e. anything true in one neighbourhood is transformed, by the relation, into a corresponding truth about the other. The structure in question will certainly involve the topology and may involve, say, a notion of distance. I'll refer to such a relation, r, as a mutual encoding of the two neighbourhoods (|r:) and (:r|).

Furthermore, one can continuously deform a mutual encoding of U and V into a mutual encoding of U and W, at least when W and V are near one another, but to discuss that properly I'll need scalars, which I want to leave for a while. That would have embedded some interval of the scalars in the space of mutual encodings, holding the left side, U, fixed and moving the right side from V to W. Absent the scalars I can, none the less, embed a neighbourhood of the continuum in the space of mutual encodings, holding one side fixed; which I'll now take as the right values; so I get ({mutual encoding (continuum:|P)}: f |Q) for some neighbourhoods P, Q within the continuum, with f(q,p) being a position on the continuum which varies continuously with each of the positions q and p within Q and P respectively; f's transpose is then naturally ({mutual encoding (:|Q)}: |P) and has similar form.

For given q in Q, we have (:f(q)|P) and (|f(q):) is the neighbourhood mutually encoded with P by f(q); this neighbourhood varies continuously - in the inferred continuum of neighbourhoods - with q; likewise, the neighbourhood (|transpose(f,p)|Q) various continuously with p; indeed, (: c← (P: c = f(q,p), p←q :Q) :{relations with non-empty interiors}) specifies a relation whose right values should be mutual encodings of neighbourhoods within P with neighbourhoods within Q.

Now what I really want is to make f a one-to-one correspondence between Q and (|f:), with the latter understood in terms of the sub-topology induced on it by the topology on {mutual encoding (continuum:|P)} induced by that on the continuum itself. Ideally, (|f:) will provide a full and faithful model of Q and the continuum's structure within Q (possibly including the metric).

Consider (P: reverse(f(x))&on;f(y) :P) for any x and y in Q; this will give me a mutual encoding of two neighbourhoods within P; next, given z in Q, consider f(z)&on;reverse(f(x))&on;f(y) which will be a mutual encoding between some neighbourhood within (|f(z)|P) and some neighbourhood within P = (:f(y)|); so it becomes interesting to ask whether there is some w in Q for which f(w) subsumes f(z)&on;reverse(f(x))&on;f(y). Clearly, if f(z)&on;reverse(f(x))&on;(y) is empty (i.e. the neighbourhoods involved don't overlap enough), this will hold true for arbitrary w in Q; otherwise, the interesting case is going to involve there being a unique such w.

I'll describe ({mutual encoding (continuum:|P)}: f |Q) as a displacement chart of Q in {mutual encodings with P} precisely if its transpose is ({mutual encodings with Q}: |P) and, whenever f(a)&on;reverse(f(x))&on;f(c) has an interior (i.e. both it's (|:) and its (:|) are non-empty neighbourhoods), there is a unique f(b), with b in Q, which subsumes it. I'll describe f as a mutual chart of Q from P precisely if it is a displacement chart of Q and its transpose is a displacement chart of P; in particular, I'll describe its transpose likewise as a mutual chart of P from Q.

Now, when the identity on P is f(q) for some given q in Q's interior, f's outputs for inputs near q will be close to the identity, so their left neighbourhoods will overlap substantially with P, enabling us to compose f's outputs, producing mutual encodings between neighbourhoods within P. If f(x)&on;f(y) agrees with f(z) where defined, for some given x, y and z in Q, it's going to be natural to think of x+y=z, with q given as an identity for the given addition. I'll encode that by asking f(z) to subsume f(x)&on;f(y), though it may be worth asking for: f(z) and f(x)&on;f(y) have an intersection which has an interior and their union is a mapping (i.e. they agree on their intersection).

Even without an identity in (|f:) I can chose a q in the interior of (:f|) and replace f with ({mutual encodings (P::P)}: reverse(f(q))&on;f(x) ←x :Q), whose (:|) is a neighbourhood, within Q, of q. We'll then be asking for f(x)&on;reverse(f(q))&on;f(y), when it has an interior, to be subsumed by a unique f(z), enabling us to construe z as x+y.

So consider a mutual chart, f, of Q from P and suppose f relates the identity on P to some q in Q's interior; f(q) = (: p←p |P). At least within some neighbourhood of q, for at least some natural N, we can define scaling by members of N using: f(n.x) subsumes repeat(n,f(x)). We can likewise define inverse scalings, at least for some natural M, by: repeat(m,f(x/m)) = f(x) for m in M, x near q. The apparent arbitrary divisibility of the continuum may be encoded by saying that this works with M = {naturals}; though, since Q always subsumes (: x/m ←x |), one may expect (| x/m ←x :Q) to be highly localized when m is large. In particular, intuition about the physical continuum says that, at least on small enough neighbourhoods, such constructions should give us (: x/m ←x :Q) as a mutual encoding of its (|:) nested firmly in the interior of its (:|), at least for some m.

Any sufficiently small displacement may be arbitrarily finely sub-divided ? Must be, if any m>1 allows (U:x/m←x|U), with U a neighbourhood, to be a mutual encoding of its (|:) and U; it composes cleanly with itself as often as we like, enabling us to shrink by any power of m.

What's effectively asked for is a continuum of three-entity relationships for each of which: specifying the value for one entity, the thus-restricted relationship among the remaining two entities is a one-to-one correspondence which respects the structure of the space-time continuum. Values taken by the entities are positions in space-time, so each such reading of the relationship is an embedding of a space-time neighbourhood (the range of the specified entity) in the mutual encodings of neighbourhoods within the continuum. From any such embedding and any position in its (:|)'s interior, one can construct an addition, induce scalings from this and build up linear structure (albeit the carefully worded linear structure of a neighbourhood of the origin in some linear space). Linear structure and the intuition of finite dimension suffice to provide us with a concrete topology in the classical sense; whence one may characterize differentiation in linear domains; whence one may pull back, through the embedding used to induce linearity, to characterize the differential structure of the physical manifold.

For a manifold M, we get a collection, C, of relations ({(M::M)}: :M) which subsumes each of

• {reverse&on;f: f in C}
• {transpose(f): f in C}
• {f&on;g(x): f, g in C, x in (:g|)}
• {(: f(x)&on;g(x) ←x :M): f, g in C}
• {(: f(p)&on;g(x) ←x :M): f, g in C, p in (:f|)}

and most of what we need to discuss really concerns U = unite(: (:u←u:f) ←f :C), the collection of relations (M::M) that appear as left values of members of C; we may well want to insist that U's members are mutual encodings. U then subsumes each of

• {reverse(u): u in U}
• {u&on;v: u, v in U}
• {(|u:): u in U} and {(:u|): u in U}

the final pair of which should suffice to generate an open topology on U, via finite intersections and arbitrary unions. Our liberty to intersect comes from u&on;v - when u and v are identities, composition and intersection are the same operation. To unite members of U, we need them to agree wherever they have common values - thus, x in (|u:) and in (|v:) must imply ({x}:u|) = ({x}:v|) and, likewise, y in (:u|) and in (:v|) must imply u(y) = v(y). Then, if U subsumes some V whose members agree on their overlaps, pairwise as just described, we need unite(V) to be in U. If V's members are all identities, as arises when building our topology, the agreement constraint becomes trivial - x in u = (|u:) and in v = (|v:) implies ({x}:v|) = {x} = ({x}:u|), likewise for the right constraint using y.

The symmetry under reversal is characteristic of mutual encodings, so I'll leave it out of account while considering, first, the topological structure. So how much of topology can I resurrect by considering a collection, U, of relations with: U is closed under composition; the left and right equivalences of U's members are in U; as is any union of members of U which agree (pairwise) with one another (in so far as they have any overlap). From composition we obtain, on the end-equivalences, finite intersection; from unitable when compatible we obtain arbitrary unions of end-equivalences (though I may need to insist that, for some equivalence M, the end-equivalences of U's members are all restrictions of M - i.e. if e is (|u:) or (:u|) for some u in U, then (e:M:e) = e).

This might plausibly leave room (with reversal still pending) to constrain U to be closed under composition and impose only the left- or right- half of some of the constraints above: U would then include all left (or possibly right) equivalences of its members and any union of U's members which, pairwise, satisfy y in (:u|) and in (:v|) implies (|u:{y}) = (|v:{y}). What's being built here is, naturally, a category (in other clothes) whose morphisms are relations in U and whose left and right identities carve out a well-behaved sub-relation of subsumes.

Written by Eddy.