FOM: Re: Arbitrary Objects
P.T.M.Rood at ph.vu.nl
Wed Jan 30 07:27:31 EST 2002
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:
>*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
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
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
"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
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
x be an F". I take it that Fine thinks that if we understand "x" as referring
arbitrary object, then we can in some way illuminate or increase our
of some inferential practices adopted by mathematicians.
I won't go into Fine's reasons for thinking so; these can be found in his
book on the matter.
Let me first say that I find it hard to understand *what* inferential
aformentioned inference types (e.g., elimination of the universal quantifier)
supposed to illuminate. Taken at face value, Fine is concerned with an
semantics of a specific class of formal languages. In terms of such formal
it can be rigorously specified what a proof is, and with the help of an
semantics it can be exactly specified what "logical validity" means. It is not
however, to what extent and in what sense such proofs increase our
of the proofs resulting from mathematical practice. On this point, let me
say that it seems to me that the logician's proofs are much more concerned
systematic organization of mathematical results--that is: theorems--than to
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
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
semantic interpretation of the former should proceed along the lines Fine
Indeed, it is not even clear that the semantic interpretation of "let x be an
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
language: besides "let x be an integer" we also have, for example: "take x
the set A", "let C be the union of A and B", "assume that \phi holds", and so
If taken seriously, it seems to me that this suggests that actual mathematical
may very well be thought of a some kind of programs, written in what often
to come close to a kind of imperative programming language. More
we could think of proofs as a kind of programs for cognition. I am not yet
whether and to what extent all this makes sense. I am very well aware that
leaves open a whole lot of questions, and I am surely unable to answer them
Nonetheless, I cannot help but finding it a promising idea. (There have beem
who have related mathematical proofs to programs. For example, Martin-Lof.
Interestingly, I recently found John von Neumann, in his little book "The
and the Brain", making a comparison between the language of mathematics and
a higher-lever programming language running on the brain--this perhaps
that proofs should indeed be interpreted cognitively).
If we think of proofs as a kind of (imperative) programs, then it seems
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
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
It seems promising to interpret such a sentence in terms of a "reset
change the value of the register named "x" to an object of type F. Thus
we need no arbitrary objects, only the objects we are long familiar with,
together with their associated types. Moreover, by interpreting this sentence
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
It seems to me that we don't need to answer this question in full detail. The
is unknown to the "user" of the variable "x", and he need not know it. We
might think of this value as "irrelevant implementation detail".
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