Jon Barwise barwise at phil.indiana.edu
Mon Nov 10 17:04:15 EST 1997

```Harvey writes:
>
>Jon, I think many of us will want to hear more details and examples about
>the distinction you draw between "interpreted" versus "uninterpreted"
>languages, in order to judge how heretical your views really are.
>
>Thank you for your contribution to fom. Hope you find the time to give us
>more.
>
>Your conservative friend, who defends the status quo --- HMF.
>
The basic point is that, contrary to Russell, I think mathematical
statements wear their content pretty much on their sleeve.  If they appear
to be talking about natural numbers, curves in R^2, or finite sets, say,
then their content is a claim about natural numbers, curves in R^2, or
finite sets, respectively.  More generally, I think there are various
mathematical PROPERTIES and RELATIONS and the things that have these
properties and stand in these relations.  The language of mathematics has
predicates to express things about these properties, relations, and the
things that have these properties and stand in these relations.

I don't think this is a heretical idea.  Godel certainly had it, though he
preferred to talk of CONCEPTS rather than PROPERTIES and RELATIONS.  He
spoke for example of the mathematical concepts of SET and MEMBERSHIP, which
I would call a property and a relation.  But let's use Godel's term.  Godel
points out that these concepts are not sets but are mathematical objects
about which we speak using the language of mathematics.

Now if mathematicians speak of these concepts, properties, relations,
etc., in the language of mathematics, they must be using a language where
the predicates are interpreted, that is, they have an intended significance
(I purposefully use the nontechnical term "significance" here to avoid any
of the technical terms like interpretation, reference, extension, etc.).
The words "real," "set." "member" etc. must signify the concepts of real,
set, and membership.

I also don't think this idea is at all deep.  It is just common sense.  But
the closest we can come to an interpreted language in first-order logic is
to speak of a language paired with an intended model.  But that model is
still extensional, there are no properties (or concepts) in the picture.
And I think that keeping this idea in mind can prevent one from making
various errors, like confusing a mathematical concept with a particular
set-theoretic model of the extension of that concept.

====

For those who are not already bored, let me approach this in a different
manner.  Here are two extreme views:

A) Mathematics takes place in an extended version of the mathematician's
native language, one where words and expressions mean whatever they mean.
The rules of inference that are valid are a consequence of the meanings of
sentences in the language.

C) Mathematics takes place in a totally formal language by using certain
purely syntactic rules of inference.

In between is a different idea which I think we often teach our students::

B) Mathematics takes place in a formal language where certain items have
fixed meanings and the rest are uninterpreted.  Usually we would be the
boolean connectives and quantifiers among the meaningful and the rest among
the meaningless.

To me, it seems pretty clear that (A) fits the facts better than either (B)
or (C).  I look at the formal languages we have developed as mathematical
models of an idealization of (A), one from which we can learn a lot about
mathematical activity.  But they are only that: models of what is going on
in mathematics, not mathematics itself.  They should be judged the way we
judge models, by how well they fit the data, make predictions, square with
phenomena, etc.

I suggest that the relationship between standard partially interpreted
first-order languages and the actual language of mathematics is analogous
to the relationship between the (mathematical) notion of Turing computable
and the (informal) notion of effectively computable.  In each case we have
a modeling relationship.  It is just that I don't see any reason to believe
the analogue of Turing's Thesis in this context.

Jon

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