Any text is given meaning by its context. A text, or body of texts, might
plausibly contain enough information for the diligent reader to infer, from
the text, enough about the context presumed by the text to be able to make
sense of the text. A sufficiently careful text (perhaps the Algol 68 Report)
may even aspire to contain enough structure, in the ways it refers to itself,
that it can be viewed as having meaning out of context
– but
still each part of it is read in the context of the rest, the whole being read
in the notional empty
context which tells us nothing (but, in fact,
assumes at least the reader's diligence).
I shall presume that you are reading this text: if you are hearing it read
to you, please interpret read
as hear
; if it is coming to you in
some other form, I trust you to interpret read
appropriately. At the
start of a text, your context and mine differ: the text introduces notions
which we understand in terms of those contexts, along with forms of expression
for those notions and ways of manipulating truths about them; from these, each
reader builds up a new context in which to read subsequent text; if the
notions and etc. have been introduced with care, our contexts soon enough have
enough in common that the text can presume tight overlap of our contexts, at
least in so far as they address the notions being discussed by the text.
Text need not be wholly linear (at least, given hyperlinks): and even
where it is, the reader may well gain by re-reading a text with a
re-considered prior context adopted in the light of a first read-through. In
any case, the context of one part of a text can make reference to another;
which may be in another text (e.g. via a hyperlink); if in the same text it
will usually be earlier, but see below
has its place. Care is always
needed in such references, at least where proofs are concerned, to
avoid proof by circular cross-reference
; with definitions, similar care
is also needed for circular
definitions to be well-behaved, where they
arise (e.g. a natural number is a finite collection of natural numbers,
each of which it subsumes
can make sense, but defining exp as the inverse
of log and defining log as the inverse of exp would fail to say enough about
these functions).
A text can introduce notions, statements and namings (to which I'll
return) to be included in some subsequent contexts. These may be for the
rest of
the text or only for some definite portion (e.g. when a proof
names some party to the statement being proved, the better to manipulate it,
but the name is only bound to that party for the duration
of the proof;
it may subsequently be used, outside the proof, to name some other entity;
similar remarks apply to dummy variables
when specifying
mappings).
I think I'm writing my pages in a manner for which the relationship between text and context revolves around:
which may have been supplied by context or may
be constructed
during the discourse, possibly from other values. I
assume that context also provides a notion of sameness
for which any
value is equal to
itself and everything true of a value is also true of
any value equal to it: equal values are indistinguishable.
The purpose of this ground-work is to discuss what a discourse may do
with values (once it has introduced any, by whatever means satisfy its
criteria of explanative adequacy) and I see no need to constrain your freedom
to use the tools I build to only such values
as I know how to
characterize: the same applies to much else in this discourse.
which are fragments of the text which the context interprets as values.
which are statements in the text in which a fragment
of text, called a name
, is given meaning as a value: it may retain that
meaning for the duration of some sub-text only or for the rest of the text; it
may even be exported
by the text for use in other contexts. Generally,
the broader the expanse of text over which a name is given meaning, the longer
the name will be: but some large-scope names are short, e.g. π and ∘
(sometimes visible as &on; in my pages; I'm slowly converting pages using
that to XHTML where I can coax this into displaying as intended). I shall
generally use single characters, English words and &…; tokens (some
of which will be recognized as HTML character entities and displayed as single
characters) for names.
which are fragments of the text which (in their
contexts) introduce terms and denotational forms for use by contexts, notably
my basic denotational forms for
plaintext. Namings are a particular kind of definition; another common
kind has the form I shall describe a donkey which has no
tail as an eyeore
in which a new
notion is introduced in terms of some previously defined
notions.
Beyond this, I must trust to your ability to make such sense as you can of the things I write when I try to express the ideas which form the context in which I'm writing this.
Written by Eddy.