Friday, March 6, 2015

Response to one more comment
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 his comment, Stanislaw Jedrczak thinks that the problems in philosophy of mathematics that I have been talking about in recent posts are due to what he calls “the ambiguity of the existential quantifier”. But ambiguity is a semantical notion, and many – probably a large majority – of philosophers of mathematics deny that “exists” is  ambiguous. It is precisely to avoid problematic semantical claims that I say that potentialism is a rational reconstruction of certain uses of “exist”, rather than a meaning analysis.


7 comments:

  1. Professor,
    Thank you for your reply.

    As shortly as I can I will try to make clear what I had in mind mentioning “ambiguity” of the existential quantifier.
    By this I understand a problem connected with the structuralist reasoning.
    If certain mathematical objects, not being sets, are isomorphic with sets, then - in order not to fall into a vicious circle - defining this isomorphism we have to use non-set-theoretic notion of “isomorphism”.
    Analogically: Shall we not need other than strict set-theoretic notion of existence, if it is assumed that also non-sets may exist (as potentialism claims)?
    (Otherwise the situation would be as follows: we agree that non-sets may exist but their existence is always claimed in the language of the set theory.)
    Thus I thought the quantifier notion of existence seemed to be ambiguous, because cases in which this notion is, or is not, set-theoretic are not clearly separated.
    Is it not a problem for the structuralist statement?

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  2. This comment is interesting. I am interested in the problem of ambiguity in the formal languages used in mathematical reasoning, but I have not yet read Prof. Putnam's posts on the modal logical interpretation, mathematical existence and equivalent descriptions. I think I shall have to read them now. Are you suggesting that the expression "there exists an x" and the expression "x is a possible instance of the definitional principle of A" play equivalent roles in their respective systems? In what respect exactly are they equivalent, and in what way do the differ? I have always wondered why mathematics seems to feel the need to assume the existence outside of the mathematical language of a (maybe preexisting) universe of objects analogous to things numbers could be used to count, when all you need is the notion of the unbounded iteration of a generative principle making possible a perhaps open-ended set of possible instances (where this set can be structured, as in the system of natural numbers). Also, in linguistic semantics it is necessary to recognize a distinction between what I would call the extension of a generative category, i.e., the set of possible instances determined by the rule, which are purely logical objects, and on the other hand what I call the denotata of the category, i.e., the perhaps open-ended set of all possible objects that uses of the category may refer to, out there in the world of intentional objects, such as things to count.

    (BTW, with regard to the notion of reference, and this relates to Prof. Putnam's Tarski posts (which I did read), isn't it the case that "truth" of statements in mathematical language systems (e.g. in proofs) is evaluated with respect to the inclusion structure determined by the generative rules of the definitions and axioms ("consistency") and not with respect to a relation of reference to a world of intentional nonlogical objects independent of the system? Isn't the adequacy of the intended referential relation between a descriptive sentence such as "This snow is white", i.e., the descriptive, words- to- world truth, to be evaluated by a description, using a metalanguage in principle different from that of the objectlanguage sentence, of the referred to situation (what if that snow is in fact "yellow") and of the generative principles of the categories used (e.g., the category 'white')? Tarski's approach begs this question, and involves no reflection on the generative principles of the objeclanguage categories used.)

    Well, I am only a linguist interested in descriptive linguistic semantics, so if what I've said is complete rubbish in the context of the way philosophers view things, please correct me. I would appreciate a philosopher's view, since, although I am a philosophile, their ways often puzzle me. I just had these thoughts in response to this comment, but I will now read Prof. Putnam's posts.

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