Re: Name the thesis: "Formal sentences capture informal ones"

From: Aatu Koskensilta (aatu.koskensilta_at_xortec.fi)
Date: 01/29/05

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    Date: Sat, 29 Jan 2005 22:32:57 +0200

    tchow@lsa.umich.edu wrote:

    > The Church-Turing thesis is familiar to many people, largely because it
    > has been widely discussed both in textbooks and in popular science writing.
    > Having a name helps, too.
    >
    > There is an analogous thesis that is relevant to logic and the foundations
    > of mathematics:
    >
    > (*) Formal sentences (in PA or ZFC for example) adequately express
    > their informal counterparts.

    Erhm. What is the informal counterpart of a randomly chosen formal
    sentence, in the language of arithmetic or in the language of set
    theory? Do arbitrarily complex formal sentences have informal
    counterparts? How do we know this?

    It seems that the following very vaguely formulated thesis is more
    strongly supported by the evidence we have, and more relevant to
    applications:

      (*) Every informal mathematical statement can be given a faithful
          formal rendition in a suitably chosen formal language and relevant
          background theory

    I'm not at all convinced that all sentences of, say, the language of
    arithmetic have informal counterparts if by informal counterparti we're
    supposed to mean anything more substantial than "for every natural
    number x there exists a natural number y such that for all natural
    numbers z there exists a natural number w such that for all natural
    numbers u ..." 

    -- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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    • Re: Name the thesis: "Formal sentences capture informal ones"
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      (sci.logic)