FOM: Re: Arbitrary Objects

[P.T.M.Rood@ph.vu.nl]

V. Kanovei wrote:

> Date: Thu, 31 Jan 2002 9:58:12 +0100
> From: <P.T.M.Rood at ph.vu.nl> (P.T.M. Rood)
> 
>> (begin of citation)
> 
>> Can anybody explain why
>> *change the value of the register named "x" to an object of type F*
>> is better than 
>> *let x satisfy F* ?
> 
>> The question seems besides the mark. I proposed the former
>> to be the semantic interpretation of the latter. And I explained
>> its virtues in my previous message.
> 
>> (end of citation)
 
> Example:
> 
> 1) let x be a linear operator on a Banach space
> 
> 2) change the value of the register named "x" to a
> linear operator on a Banach space
> 
> Please explain  
> how  2) can be *the semantic interpretation of* 1) ?

Well, its some sort of proposal. So there isn't much to explain,
ain't it? We need to look at the plausibility of the proposal, and its
fruitfulness. Just as in the case of an arbitrary
objects interpretation (i.e., "x is an arbitrary object
satisfying the predicate "is a linear operator on a Banach space"")
is a proposal.
I  think that my proposal is worth considering. Why? Well, as
I've said, taken at face value 1) clearly is an imperative sentence,
and not, as the arbitrary objects interpretation clearly assumes,
a declarative sentence.
Drawing inspiration from a well-known technique from computer science,
an imperative sentence can be "traced" by pre conditions and post
conditions. Ignoring much of the technical details, the idea is simply
this:
a program transforms a state (satisfying a pre condition) to a
certain state (satisfying a post condition). I propose that 1) is
very much like a program in this sense: a kind of state transformer
(I ignore the explicit formulation of pre and post conditions;
they are irrelevant for my purposes).
The general more or less philosophical intuition lying behind my proposal
is of a cognitive nature:
I view a proof as a kind of program for cognition, a program for coming
to see the truth of some proposition for my part. A similar suggestion has
also been made by von Neumann, among others: the language of
mathematics is some kind of high level
programming language running on the brain (I said this in one of my previous
messages). It seems to me that von Neumann saw a mathematician as some
sort of "interpreter" of mathermatical proofs. I am leaning very much on
this idea. And my proposal for interpreting 1) fits nicely into this picture.
On the more technical side I have drawn much inspiration from
Groenendijk and Stokhof's groundbreaking paper "Dynamic Predicate
Logic", which appeared in Linguistics and Philosophy 14 (1991).
In DPL the formula "Ex" (that is, "\exists x") is interpreted as a kind
of reset operation. (Indeed, in DPL "Ex" is considered well-formed!)
More specifically, in DPL, any formula is interpreted as a relation between
assignments. Let M = (D, I) be a model. As usual, an assignment is a
function f : VAR -> D, where VAR is the set of all variables in the
language under consideration. In DPL, then, any formula is interpreted
as a relation on D^VAR. In particular, a pair (f, g) is in the interpretation
of "Ex" iff g agrees with f except, perhaps, on x. The underlying intuition
is that variables are viewed as a kind of names of registers, and assignments
enter values in those registers.  Within such a perspective, "Ex" resets the
value of the "register" named x.

> Does 2) have any more meaning in any sense than 1) ?

I do not really see the point of this question. Again, my proposal comes
very close to this: 2) *is* the meaning of 1). Of course, 2) should be
worked out within a neat mathematical framework. And in an earlier message
I've given a suggestion as to what framework: dynamic logic.

> Where the register "x" is and how to change it to a
> linear operator ?

I am somewhat suprised by the first part of of this question. Have you
ever seen a logician wondering *where* the possible worlds are in case
of, say, the analysis of necessity? Have you ever seen a logician
wondering *where* the two players of game theoretical semantics are?
(Someone in this mailing list proposed to an interpretation of arbitrary
objects along game theoretical lines.) And I could mention many other
examples. These possible worlds and players are theoretical entities in
the first place. Of course they are "based on" some sort of intuitive
considerations, but that doesn't make them have a definite spatio-temporal
location.
Turning back to your example, 1) means the
creation (or, as Kant perhaps would say, construction) of a linear operator
on a Banach space. That is, create an object of the type "linear operator
on a Banach space". This type is a collection of entities with certain
associated operations. Indeed, a structured
data type. What enities? Well, it doesn't matter. We only need some kind
of interface allowing us to work with them. What interfaces? Well, symbols,
for example. What operations? Composition, addition, and so on.
That's all that matters. What these entities and operations "really" are
I dare not to say. This is, as they sometimes say it, "irrelevant
implementation detail". Perhaps the question is, in the end, relevant for
neuro scientists.
I think what I've said thus far also answers the second part of your
question.


> To which exactly operator we change the register "x"
> whatever it be ? 
> To "arbitrary"? 

It doesn't matter. See above. Admittedly, I've in a sense explained
away arbitrary objects.
 
Ron Rood