SUO: Re: Policy On Substitutions
Matthew West wrote:
> Dear Jon,
>
> I share your concern about the nature of variables,
> but fail to see from what you right below how they
> can be eliminated. Please could you explain with
> a non-abstract example?
>
> Regards
>
> Matthew
>
<...>
>
> [Jon Awbrey wrote:]
> >
> > Once upon a time, a very long time ago, back when
> > I was deep in the muddle of my very first, and my
> > most extenuous "crisis of foundations" (COF, COF),
> > I was very sorely afflicted, among the variety of
> > other affectations that flesh and grass are heirs
> > to, with an anxious "ontological insecurity" (OI),
> > focussed on the "ontological status of variables",
> > the eventual resolution of which difficulty, over
> > too long a time to believe for the obviousness of
> > its ultimate form, was simply to banish variables
> > altogether from the realm of our finely canonical
> > & our most finally, commonly, sans-critical koine.
> >
> > Henceforward, and for the rest of time, let us then say
> > that there are actually no "open sentences" at all, not
> > in our most wholy sanctioned languages, but that such a
> > common nickname or vulgar form of words as "Bla, bla X"
> > is really just a general name, a pleonastic periphrasis,
> > or perhaps just a colloquial circumlocution for a whole
> > collection of signs or sentences of a similar configure,
> > except with the "X" respliced by a fully concrete piece
> > of a fixed signal, or a fitful token of a sentence part.
> >
> > It needs to be noted that the specification of
> > this collection of signs in no way relies on the
> > particular artifices or the specialized devices of
> > anybody's rude mechanics for performing substitutions,
> > but since the collection of signs is conceived to exist,
> > in some sense of "existence", prior to its secular and
> > timely indications by the likes of me and thee, well,
> > then we note, ours is not to question why, ours is
> > but to specify.
> >
> > The point is that any way of specifying the desired collection of signs
> > is equally as good as any other, to any of which indiscernible outcomes,
> > assuming that they do indeed meet any one of many reasonably equivalent
> > specifications, one might affix any convenient nickname for the species.
> >
> > So, for example, the so-called "open expression" of the form "x + x",
> > where the values that one associates with the variable token "x" are
> > the elements of the non-negative numbers N = {0, 1, 2, ...}, is just
> > a nickname for the set of constant expressions that begins like this:
> >
> > "0 + 0", "1 + 1", "2 + 2", ...
> >
> > And it is species of perfectly constant signs like this
> > that are the actual subjects of our generic discussions
> > in this regard, thereby removing every last trace of my
> > onetime "ontological insecurity" (OI) about "variables".
> >
> > And that, my friends, is something to celebrate!
> > So leave me to my illusions, at least for a day!
> >
> > Cheers,
> >
> > Jon
> >
> > ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
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Matthew,
Maybe I need to say that I mean "eliminating variables"
in the sense of "explaining them away", in other words,
more like "taking them off the list of basic concepts".
My particular worries here, to the extent that I can even remember
what they were all about from twenty or thirty years down the line --
though I do recall the appropriation of various phrases that I had
probably learned in numerous other contexts, where I described the
core feeling as "ontological insecurity" or a "semantic equivalent
of free-floating anxiety" -- but I think that all of that business
just meant that, whenever it came to "abstract algebraic subjects"
as abiding over and above "concrete arithmetic subjects", I simply
had no determinate sense of what all the equations and expressions
and formulas were ever supposed to be denoting, in the first place.
So I am not exactly sure how to come up with a less abstract example,
as the whole problem only came up in the relatively abstract domains
of algebra and what we were once accustomed to call "symbolic logic",
as if it were some brand of purely and thoroughly modern methodology.
The satisfaction that arises from finding a relatively concrete denotation
for "open expresions", "patterns", "schemata", "templates", or whatever you
want to call them, whether in algebra or in logic, is simply the feeling of
being able to point to a definite "meaning", in the sense of an extensional
reference that abides, moderately contentedly, in the realm of instantiated,
constant-saturated expressions, each of which has, if evaluated, a constant
value in some ordinary domain of "arithmetic" values, say booleans or reals.
The sense in which one is successfully "explaining variables away"
is just this, that one comes to regard them as incidental devices
for getting down to the actual references of the open expressions.
This is and ought to be a fairly unsurprising solution to the
worry about variables, but it does involve a subtly different
attitude toward the subject matter than any of the approaches
I had previously taken up, as it takes regarding the level of
signs or syntactic entities, for instance, the above sorts of
constant expressions, as a distinct reality or real dimension
in and of itself. And treating these significant expressions
as significant in their own right, not simply as something to
be cashed out as quickly as possible for the indicated values,
this marks the first stirring of a sign-theoretic sensibility.
It also means that one has to regard a whole set of instances,
not just a few of them, as being the real bearer of the sense.
Speaking of sense -- did this make any at all?
Jon Awbrey
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