Wednesday, February 18, 2015

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. 








[1] “Skepticism and Occasion-Sensitive Semantics”.

8 comments:

  1. Thanks for making this post. I'm interested in your notion of rational reconstruction but am bothered by something.

    I understand and see the usefulness of the idea of a rational reconstruction as a paradox-free way of construing some problematic discourse. And I understand this as something like giving a new but importantly related meaning to that discourse.

    But then when you mention the logical positivists/empiricists and say (approvingly as far as I can tell) that for them a rational reconstruction was 'a proposal to *give* a certain predicate an interpretation that exhibits the rationality of certain uses of that expression', this, naturally interpreted, seems to be in tension with the understanding of rational reconstruction as characterized in the last paragraph of this comment.

    The tension is: if a rational construction is not intended to be descriptive of the meaning of some discourse as it already was before the rational reconstruction, then how can it exhibit the rationality of any of it? At best, it exhibits a way of changing one's practise in order to *become* rational. Or am I missing something?

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  2. Professor, thank you for the post. I am interested in the problem of “mathematical existence” but the main idea of “rational reconstruction” is not clear enough for me.

    The given example (“there is a prime number between 17 and 34”) is a translation into a common tongue of usage of Bertrand’s postulate. Its sense can be explained in the natural language by using verbs like “there is”, “exists” etc.; in the language of logic – by existential quantifier.
    What causes the question about the character of “existence” in mathematics is ambiguity of existential quantifier. Classical first order logic requires from sentences that term variables correspond to objects (so that 19, 23… have their subject reference). This requirement is not stated in other logical systems which means that there is no need to assume the strict disjunction “either object, or fiction”. I suppose that also for this reason Professor Steven Wagner’s statement is not correct.

    Although I think we should not reject assumption that “mathematical objects are fictions”. Its criticism depends on what meaning of “fiction” we think of. It cannot be the same kind of fiction as literary fiction. Literary fiction might be constructed freely, and laws of mathematics are not free or accidental. But they are, at least in a large fragment, conventional (Russell’s Theory of Types is a good example of conventionality).
    “Conventional” does not mean “arbitral”. For example the value of money is conventional (and it is a kind of fiction called market fiction) but this convention is not arbitral (so some agreements would be senseless).

    I suppose that trouble with the Russell’s paradox is caused by realistic view on mathematical “existence” which leads to overreacting about the possibility of creating in the Sets Theory an object which is not a set. However removing this problem does not require the “reconstruction” (like Theory of Types or axiom schema of specification); all is needed is an explanation that “set” once was used as an object and the second time as predicate. The Theory of Types and some other attempts of avoiding the class paradox are awkward because they blame for the antinomy the relation of self-including. But problematical is only a certain use of a logical formula – Russell’s paradox – under no condition creates, not e.g. as a jest. (Also Wittgenstein finds such way of reconstruction redundant but for different reason, TLP: 3.33 and 3.331.)

    Is a question about “existing” of the Russell’s set (“does the set of all sets… exist?”) similar to following one: “does the verb created from the noun chair exist?”.

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