When quantities are amenable
to addition, one can use repeated
addition to build up a notion
of scaling by the positive
naturals. In so far as such scalings are invertible, one may construe the
inverses (division by positive naturals) as scaling by reciprocals of naturals;
and composites of natural scalings and reciprocal scalings as
scalings by rationals (general
fractions). [Where the quantities involved form a continuum, we can look to
extend scalings to a topologically complete

continuum, having the form of
orthodoxy's real number continuum. I
would like, however, to avoid building the reals per se
as far as I can, except in so far as they arise naturally as scalings.]
Orthodoxy describes such structures in terms of **vector
spaces**.

Orthodoxy's vector spaces are, by definition, additive groups: this implies
(at least) that they have to have negatives; yet some of the quantities with
which physics wishes to deal (e.g. temperatures and masses) are intrinsically
positive, with zero only as a notional or theoretical limit, yet support
meaningful addition – and scaling by *positive* values. Orthodoxy
also (mirroring its general progression from sets to functions) begins by
defining a vector space, then introduces linear maps among them (without any
thought to relations in general); I would prefer, instead, to start by
specifying linearity as a property of general relations, then to illustrate the
property with special cases that are mappings (linear maps) and collections
(identity linear maps, construed as **linear spaces**). However,
it seems I need to start by specifying an additive structure, whose domain is
the linear space; and this amounts to the same thing as specifying the linear
space at the outset.

There are spaces – of positions

– which don't formally
support an addition but do support a vector space
of **translations**, differences or **displacements**
whose action on position-space has enough of the character of an addition that
if one (arbitrarily) chooses a position to treat as origin

, one can
identify each position with a translation – which maps the selected origin
to the given position – and thereby induce an additive structure on
positions. None the less, this additive structure is not intrinsic to the space
– it is an artifact of our choice of origin – although the addition
on our space of displacements is intrinsic and so is their action (in the guise
of translations) on the members of the underlying space.

In general, an addition
is a combinator and can be
expressed as a mapping which, given one of the things the combinator knows how
to add (to other things), maps that to a translation among the other things. In
the case of a space of positions, any choice of origin provides *an*
embedding of the space in its associated space of translations; in the case of a
true addition, it so happens that there is a canonical one of these (which uses
the additive identity as origin). We may thus unify the two cases by discussing
the space of translations, anchored by: a requirement that there
be *some* suitable embedding of the underlying space in its space of
translations, and; a characterization of the action of translations and its
relationship to such embeddings.

The principal problem of constructing only rational scalings is that a real
vector space of modest dimension can be construed as a vector space over the
rationals of infinitely larger dimension; however, if the space itself provides
its own topology, I suppose we can specify scalings as those *continuous*
automorphisms of the addition which commute with all others. Continuity of the
additive structure ensures the positive rationals show up among the scalings;
the topology of the space itself should then do topological completion for us,
at least if we can manage to establish an ordering.

Once one has linear spaces, one can discuss simplices (which generalize 2-D
triangles and 3-D tetrahedra to all dimensions); in a linear continuum, one is
particularly interested in non-empty open simplices, since their
edge-displacements span the space of translations (i.e. for a simplex with any
interior, the edges out of any vertex form a basis). Suitably non-degenerate
open simplices then open the door to bases; they also make it possible to
specify chords of continuous (but not necessarily linear) maps from their
neighbourhoods. Such chords necessarily imply linear maps we can construe as
gradients, from which we can induce a specification of differentiability and of
derivatives in linear continua. My hope is that this may be achieved without
resort to any scalings but rationals, by exploiting the structures of the linear
continua themselves; that is, topological completeness of the space of scalings
of a linear space can flow from the algebra and topology *of that space*
without having to be induced from (a topology on) the scalings we are obliged to
construct in order to study the space.

Note that a given collection of values, even a given topology thereon, may
support more than one binary operator which we can (albeit perhaps by violating
our normal understanding of the values in question) interpret as addition

and, from this, infer an additive continuum. This, in turn, provides for the
possibility of a linear

map from one additive structure to the other. On
the space of linear maps from a linear space to itself, or indeed of scalings of
the linear space, one has both the usual addition induced from point-wise
addition at each input – u+v = (: u(x)+v(x) ←x :) – and their
composition as mappings; if we construe the latter as an addition

, the
linear maps from the usual additive structure to this modified structure have
the form of exponential functions. I'm fairly sure this is worth
exploring.

The deep relationship between solving polynomial equations and selecting bases with respect to which assorted linear maps take canonical forms (diagonal for symmetric cases, a little more intricate for anti-symmetric) is also apt to be worthy of study.

- Scalars, linearity and consequences (which see for further links).
- Newer work on linearity and simplices (also providing further links).
- The positive rationals, the reals, complex numbers and other kinds of scalars.
- Vector spaces
- Antilinearity and the alternating algebra
- The Fourier transform
- Measuring cuboids
- Some thoughts on colour perception.
- Binary operators, associativity and distributivity.
- Group theory