Rational Reconstruction
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
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[1])
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.
