Wednesday, February 18, 2015
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 my post on mathematical “existence” on Dec. 13, 2014 [since then I have been posting on other topics], I summed up the difference between my previous and present views thus: In “Mathematics Without Foundations”, where I first proposed the modal logical interpretation, I claimed that conceptualism and potentialism [the position that mathematics is about the possible existence of structures, not about the actual existence of what Quine called ‘intangible objects’] are “equivalent descriptions”. In the three preceding posts I have retracted that claim. But I don’t agree with Steven Wagner that rejecting objectualism requires one to say that sets, functions, numbers, etc., are fictions, and that the mathematics student on the street is making a mistake when she says that there is a prime number between 17 and 34. I now defend the view that potentialism is a rational reconstruction of our talk of “existence” in mathematics. This rational reconstruction does not “deny the existence” of sets (or, to revert to an example I used in the Dec.12 post), of “a square root of minus one”; it provides a way of construing such talk that avoids paradoxes.
In a comment (Jan. 9), Andrei Pop asked “what the objects of rational reconstruction are, if they aren’t fictions? Vague or contradictory concepts?” and I should have answered that question earlier—anyway, I will do so now!
Let us recall that for the logical empiricists (Reichenbach didn’t like to be called a “positivist” but both he and Carnap accepted “logical empiricist”), a rational reconstruction (Rationale Nachkonstuktion) was a proposal, a proposal to give a certain predicate an interpretation that exhibits the rationality of certain uses of that expression. Reichenbach and Carnap did not understand Frege, for example, as providing a semantic analysis of the expression “natural number” (rightly not, as it happens), but as providing an interpretation of that expression that fits the work required of it in the sciences, and that avoids Frege’s “Julius Caesar” problem (the problem of providing a truth value for all expressions of the form 2=a, including ones in which a is not a mathematical term). Another example of a rational reconstruction is my account of the context-sensitivity of “knows” in “Skepticism, Stroud and the Contextuality of Knowledge” (collected in Philosophy in an Age of Science). Here the occasion for a rational reconstruction was (as I explained in another article in the same volume) that
“The reason skepticism is of genuine intellectual interest—interest to the nonskeptic—is not unlike the reason that the logical paradoxes are of genuine intellectual interest: paradoxes force us to rethink and reformulate our commitments. But if the reason I undertake to show that the skeptical arguments need not be accepted is, at least in part, like the reason I undertake to avoid logical contradictions in pure mathematics (e.g., the Russell Paradox), or to find a way to talk about truth without such logical contradictions as the Liar Paradox; if my purpose is to put my own intellectual home in order, then what I need is a perspicuous representation of our talk of “knowing” that shows how it avoids the skeptical conclusion, and that my nonskeptical self can find satisfactory and convincing. (Just as a solution to the logical paradoxes does not have to convince the skeptic, or even convince all philosophers—there can be alternative ways to avoid the paradoxes—so a solution to what we may call “the skeptical paradoxes” does not have to convince the skeptic, or even convince all philosophers—perhaps here too there may be alternative solutions.) It is not a good objection to a resolution to an antinomy that the argument to the antinomy seems “perfectly intelligible,” and, indeed, proceeds from what seem to be “intuitively correct” premises, while the resolution draws on ideas (the Theory of Types, in the case of the Russell Paradox; the theory of Levels of Language in the case of the Liar Paradox -- and on much more complicated ideas than these as well, in the case of the follow-up discussions since Russell’s and Tarski’s) that are abstruse and to some extent controversial. That is the very nature of the resolution of antinomies.”
Another example: one purpose of my (and Geoffrey Hellman’s) “modal-logical” or “potentialist” interpretation of mathematics was, as I have said in these posts, to show how “Benacerraf’s Problem (and generalizations of it to non-denumerable “totalities” like the supposed totality of all sets) can be avoided.
But in none of these cases does it seem to me that one can just say that the concept in question (the concept of number, or the concept of a square root of minus one, or the concept of knowing, or the concept of truth, or the concept of a set, is “vague or contradictory”. Indeed, there are philosophers who don’t think these concepts (with the possible exception of “square root of minus one” in the nineteenth century) need rational reconstruction at all! I would prefer to say that a concept needs rational reconstruction when we don’t want to simply give it up and it is problematic, and I would immediately add that whether a concept is actually “problematic” in cases like these is a philosophical question. There is no single universally agreed-on test for being problematic. Often philosophy, from Plato on, causes me to see that a concept is problematic that I had always felt I could just take for granted.