**The Modal Logical Interpretation and "Equivalent
Descriptions”**

*In 1976, when I delivered the John Locke Lectures
at Oxford, I often spent time with Peter Strawson, and one day at lunch he made
a remark I have never been able to forget. He said, "Surely half the
pleasure of life is sardonic comment on the passing show". This blog
is devoted to comments, not all of them sardonic, on the passing philosophical
show.*

*Hilary Putnam*

In a forthcoming
paper

[1],
Steven Wagner raises a dilemma (actually, a trilemma) for the view I defended
in “Mathematics without Foundations”. There, and in one place in my previous
post, I described the modal logical interpretation (an interpretation on which
quantification over mathematical objects is interpreted as talk of

*possible* existence of structures

[2])

and the objectualist interpretation, on which
mathematical objects are taken to be genuine immaterial entities–as “equivalent
descriptions”. Wagner asks what ”equivalent” means here, and argues (rightly)
that mathematical equivalence (interdeducibility) is too weak, and that there
are problems with interpreting my “equivalent” as “semantically equivalent”
(which he thinks is what I meant). And if my talk of equivalence really
disguised the claim that saying that a number or a set exists is talking about
a

*fiction*, and the modal logical
interpretation is a way of

*eliminating*
those fictions from our discourse, I should not have spoken of equivalent
descriptions.

Well, I agree that
I should not have spoken of equivalent descriptions. But neither am I willing to describe
myself as an eliminationist. This post and the posts that will follow will
explain the alternative I see to accept Wagner’s trilemma.

[3]
**Escaping the trilemma**

In fact, semantic
equivalence is

*not* what I have ever
claimed. I have long written papers

[4]
defending the idea that two theories that have incompatible ontologies or make
incompatible claims

*if* we take them
at “face value”, are, nevertheless, sometimes “equivalent descriptions”, and I
have frequently said that this is

*not*
a claim about “translation practice”, i.e. about semantics.

An example I have used for many years is the
following: imagine a situation in which there are exactly three billiard balls
on a certain table and no other objects (e.g. the atoms, etc., of which the billiard
balls consist do not count as “objects” in that context). Consider the two
descriptions, “There are only seven objects on that table: three billiard
balls, and four mereological sums containing more than one billiard ball” and
“There are only three objects on the table, but there are seven *sets* of individuals that can be formed
of those objects.” What it means to be a *realist
who recognizes conceptual relativity* with respect to *this* case is to believe that there is an aspect of reality which is
independent of what we think at the moment (although we could, of course,
change that reality by adding or subtracting objects from the table), which is *correctly describable either way*.

I do not think that the two sentences I just used as
examples *have the same meaning* by any
reasonable and non-tendentious standards of sameness of meaning. For one thing,
the first sentence implies the existence of mereological sums, and the second
implies the existence of sets, and I do not see that someone who accepts the first
sentence is committed to the ontology of sets at all, or that someone who
accepts the second is committed to the ontology of mereological sums at all.
These are not synonymous sentences. They are not “semantically equivalent”. And
similarly, an arithmetic sentence, or a sentence of set theory, is not
synonymous with its image under the sort of mapping of mathematical sentences
onto modal logical sentences I proposed in “Mathematics without Foundations”.
These pairs of sentences are not semantically equivalent. But, I claimed, they
are “equivalent descriptions”. But what was the criterion?

The example of the three balls was
an artificial one. In my closing lecture to the conference in honor of my 80

^{th}
birthday in Dublin

[5],
I said the following about a genuine scientific example (a case of
quantum-mechanical “duality”):

“My own notion of “conceptual relativity” (which I
originally called “cognitive equivalence” is beautifully illustrated by [the
duality example]. The different “representations” are perfectly
intertranslatable; it is just that the translations don’t preserve “ontology”.

What do they preserve?
Well, they don’t merely preserve macro-observables. They also preserve *explanations*. An explanation of a
phenomenon goes over into another perfectly good explanation of the same
phenomenon under these translations.

But who’s to
say what is a phenomenon? And who’s to say what is a perfectly good
explanation? My answer has always been: *physicists* are; not linguists and not
philosophers.”

—And I gave a similar
explanation in my *first* paper on this
sort of equivalence in 1978.

But now, I admit, a problem like Wagner’s “dilemma”
does arise. Of course, if we are to apply this criterion to mathematics, we
shall have to say that it is *mathematicians
*and not physicists who should be the ones to say. But now a number of
problems arise, problems that make me think I should not have tried to export the notion of “equivalent descriptions” from empirical science to the present
case at all. Mathematics, after all, is not about “phenomena”, but about
proofs, ways of conceiving of mathematical problems, mathematical approaches,
and much more. And it does not seem reasonable to think that the mapping
Hellman and I proposed of mathematical assertions onto modal-logical assertions
*preserves * these.
It is not just that a proof and its formalization in modal logical terms
aren’t “semantically” equivalent; if the criterion is supposed to be that the *mathematician* would regard them as the
same, the objection immediately arises that the modal-logical version, unlike
the quantum-mechanical representations I mentioned in Dublin, is not one
mathematicians are even *aware* of. (And if they were, I doubt very much that they
would regard them as equivalent, except in the sense of deducible from each
other, which is clearly insufficient.) The pragmatic criterion of equivalence I
proposed for physics has no obvious analogue here. So, though not for Wagner’s
reasons, I do see a real difficulty with what he calls “translational semantic
modalism”.

The alternative I now
propose is this: the interpretation of arithmetic and set theory as modal
statements is neither a piece of straightforward semantics nor a substitute for
something we have come to reject as false (as if numbers and sets were “fictional
entities”). The modal logical interpretation is a *rational reconstruction.*

(*to be continued*)

[1]
Steven Wagner, “Modal and Objectual”, forthcoming in

*The
Philosophy of Hilary Putnam* (Chicago: Open Court,

*The Library of Living Philosophers*, 2015).

[2]
E.g., the graphs described in the previous post.

[3]
What follows is adapted from some paragraphs in my Reply to the paper cited in
note 1, which will appear in the same volume.

and probably other papers.

[5]
“From Quantum Mechanics to Ethics and Back Again,” in my

*Philosophy in the Age of Science*, 51-71.