Thursday, December 11, 2014
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.
In a forthcoming paper, 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) 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.
Escaping the trilemma
In fact, semantic equivalence is not what I have ever claimed. I have long written papers 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 80th birthday in Dublin, 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)
 Steven Wagner, “Modal and Objectual”, forthcoming in The Philosophy of Hilary Putnam (Chicago: Open Court, The Library of Living Philosophers, 2015).
 E.g., the graphs described in the previous post.
 What follows is adapted from some paragraphs in my Reply to the paper cited in note 1, which will appear in the same volume.
 My first paper about cognitive equivalence of theories which are incompatible if simply conjoined was “Equivalenza,” Trans. P. Odifreddi. Enciclopedia, vol. 5 (Torino, Italy: Giulio Einaudi Editore, 1978), 547-564; English version published as “Equivalence” in Realism and Reason, 26-45. After that came my 1987 Carus Lectures, The Many Faces of Realism, in which I used the mereological sums example for the first time; “Reply to Jennifer Case.” Revue Internationale de Philosophie 55.4 (December 2001): 431-438; Lecture 2 in Ethics Without Ontology (2004); “Sosa on Internal Realism and Conceptual Relativity” in J. Greco (ed), Sosa and His Critics (Oxford: Blackwells, 2008);
and probably other papers.
 “From Quantum Mechanics to Ethics and Back Again,” in my Philosophy in the Age of Science, 51-71.