FOM: Re: Arbitrary Objects

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

On Januari 28 Charlie Silver wrote:

>    It seems that mathematicians do not to want to scrutinize exactly what
>they're doing when they say "let z be...", where, even if it's not
>explicitly mentioned, z is intended to be "arbitrary".   They just reason
>with arbitrary objects as a matter of course. 

And he wondered:

>  What
>*are* those arbitrary objects that figure in so many mathematical proofs?
>

In saying things like "let x be an integer" and the like, mathematicians
frequently suggest that x was meant to be arbitrary. Kit Fine, in his
well-known book, proposes to take this literally: besides ordinary objects,
there are also arbitrary objects. And Fine has achieved some highly 
interesting
technical achivements on this point. Nonetheless, I have to admit that I find
myself somewhat unconmfortable with the idea of an arbitrary object.

One might wonder whether in saying "let x be an integer", x is meant to be an
arbitrary object or an arbitray integer. It seems that in this respect James
R. Brown touched upon an interesting point in his FOM message from january
29. Contrary to Fine, Brown seems inclined to think that we should choose
for the second option.

I take it for granted that mathematicians do say things like "let x be an F".
How should such sentences be understood? In terms of arbirtary objects? 
I would like to propose to change our perspective on the aforementioned
sentences a bit. More specifically, I would like to propose to change our
semantic perspective on such sentences. As a consequence, I feel that the
questions which seem to have arisen on behalf of them will also be put in a
different light. Moreover, I think that on the basis of this change in 
perspective
we do not need arbitrary objects. Still, I believe that we can make perfect
sense of sentences of the aformentioned type. I do not no whether and to
what extent what I am going to say makes sense and whether it can be
maintained if worked out in more detail. Nonethless, it seems to me worth
trying it. Let me explain what I have in mind.

I take it that Kit Fine holds that, in case of a sentence of the form "let x 
be an F"
we should think of "x" as referring to an arbitrary object satisfying the 
predicate
"is an F". In other words, Fine interprets such sentences in terms of truth
conditional semantics. However, by allowing for arbitrary objects he does so
in a "nonstandard manner", so to speak.

Upon closer inspection, however, it seems that Fine wants to think so because
he attempts, among other things, to understand some specific inferential moves
mathematicians are supposed to make in practice. For example, inferential 
moves
like elimination of the universal quantifier, that is, an inference from 
"(Ay)(y is an F)"
to "x is an F" . The latter sentence would then be the first-order counterpart 
of "let
x be an F". I take it that Fine thinks that if we understand "x" as referring 
to an
arbitrary object, then we can in some way illuminate or increase our 
understanding
of some inferential practices adopted by mathematicians.

I won't go into Fine's reasons for thinking so; these can be found in his 
well-known
book on the matter.

Let me first say that I find it hard to understand *what* inferential 
practices the
aformentioned inference types (e.g., elimination of the universal quantifier) 
are
supposed to illuminate. Taken at face value, Fine is concerned with an 
alternative
semantics of a specific class of formal languages. In terms of such formal 
languages,
it can be rigorously specified what a proof is, and with the help of an 
associated
semantics it can be exactly specified what "logical validity" means. It is not 
obvious,
however, to what extent and in what sense such proofs increase our 
understanding
of the proofs resulting from mathematical practice. On this point, let me 
suffice to
say that it seems to me that the logician's proofs are much more concerned 
with the
systematic organization of mathematical results--that is: theorems--than to 
actual
mathematical proofs.

This relates to my second point. Notice that the original sentence "let x be 
an F" is
interpreted by Fine as "x is an F", where, again, "x" is taken to refer to an 
arbitrary
object. However, this reading is far from obvious. For notice that "let x be 
an F" is
an imperative, while "x is an F" is a declarative sentence. It is not clear 
that the
semantic interpretation of the former should proceed along the lines Fine 
suggests.
Indeed, it is not even clear that the semantic interpretation of "let x be an 
F" should
be formulated in terms of classical truth conditions at all.

I propose that we do not react by saying that the imperative nature of the
former sentence is simply due to "misleading grammatical surface structure";
its "genuine" logical form would then be such that, in some way, it can be
interpreted truth conditionally along familiar lines. Why? Wait and see.

Notice, by the way, that actual mathematical proofs are often full of 
imperative
language: besides "let x be an integer" we also have, for example: "take x 
from
the set A", "let C be the union of A and B", "assume that \phi holds", and so 
on.

If taken seriously, it seems to me that this suggests that actual mathematical 
proofs
may very well be thought of a some kind of programs, written in what often 
seems
to come close to a kind of imperative programming language.  More 
specifically,
we could think of proofs as a kind of programs for cognition. I am not yet 
sure
whether and to what extent all this makes sense. I am very well aware that 
this
leaves open a whole lot of questions, and I am surely unable to answer them 
all.
Nonetheless, I cannot help but finding it a promising idea. (There have beem 
others
who have related mathematical proofs to programs. For example, Martin-Lof.
Interestingly, I recently found John von Neumann, in his little book "The 
Computer
and the Brain", making a comparison between the language of mathematics and
a higher-lever programming language running on the brain--this perhaps 
suggests
that proofs should indeed be interpreted cognitively).

If we think of proofs as a kind of (imperative) programs, then it seems 
appropriate
to wonder whether they should be interpreted in terms of a dynamic semantics,
that is, a semantics which interprets "statements" (in the sense in which 
computer
scientists use the term) in terms of pairs of states, i.e., pairs consisting 
of an "input
state" and an "output state". Classical, "static" semantics may very well be
considered inappropriate here.

If we pursue this semantic line of thought then we could try for "let x be an 
F".
It seems promising to interpret such a sentence in terms of a "reset 
operation":
change the value of the register named "x" to an object of type F. Thus 
viewed,
we need no arbitrary objects, only  the objects we are long familiar with, 
together with their associated types. Moreover, by interpreting this sentence 
in
terms of a kind of operation, I take it that we are on a way to taking its
imperative character seriously.

But what, then, is the object to which the value of the register named "x" is 
set?
It seems to me that we don't need to answer this question in full detail. The 
value
is unknown to the "user" of the variable "x", and he need not know it. We 
perhaps
might think of this value as "irrelevant implementation detail".

Ron Rood