Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

From: Ralph Hartley (hartley_at_aic.nrl.navy.mil)
Date: 01/13/05

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    Date: Thu, 13 Jan 2005 13:35:39 -0500

    tchow@lsa.umich.edu wrote:
    > Ralph Hartley <hartley@aic.nrl.navy.mil> wrote:
    >
    > I wasn't sure to what extent you were defending the
    > earlier posters in this thread

    To be honest, neither was I.

    I have spent way more time on this thread than I should have, and learned
    more than I expected. But I can't continue much more.

    > But in any case, the real issue again is your assumption that
    > mathematicians use an effective procedure to determine mathematical truth.
    > As I've argued elsewhere in this thread, CT theses of whatever flavor
    > don't yield this assumption; CT theses will only say that if mathematicians
    > are indeed using an effective procedure to find, or are *computing*,
    > mathematical truths, then the theory of [partial] recursive functions
    > applies and lets us deduce further conclusions. But it doesn't say
    > whether mathematicians are indeed using an effective procedure.

    This argument is not without merit. I don't *think* I totally agree, but I
    don't have time to decide for sure, or to explain my problems with it much
    more than I already have.

    It may not be enough, for your argument, for the procedure to not be
    effective, it needs to be *better* than any effective procedure. In
    particular it needs to be both sound and complete.

    Otherwise you will come to "know" false things, or there are true things
    you will never know.

    If a mathematician in a particular universe can correctly answer all
    members of a class of questions (by any means, effective or not), and there
    is any recursive process to determine what she has concluded, and my
    version of the physical CT thesis holds for that universe, then the class
    of questions is recursive.

    > I know of nobody---atheist or believer---who thinks that
    > theological doctrines are generated by an effective procedure.

    I'm not sure I would go so far as to say that I *think* that, but I see no
    reason to conclude that it isn't. There are effective procedures for
    producing nonsense.

    Some believers consider reference to a particular text to be the first,
    last, and only way to obtain truth. Looking up the answer in a book seems
    pretty effective to me.

    > You've argued that if something is obtained by a procedure that is not
    > effective, then it's not *knowledge*. Perhaps that's true, but the
    > the corpus of what is *generally called* "mathematical knowledge" is
    > not generated by an effective procedure in any obvious way. "Every
    > vector space has a basis" is generally considered to be a mathematical
    > truth. It can be proved using various axioms, including the axiom of
    > choice. But how did we come to accept the axiom of choice?

    I'm not sure I would call "Every vector space has a basis" a mathematical
    truth if that is what you really meant when you said it. I think I *would*
    call "In ZFC every vector space has a basis" a mathematical truth.

    But you are presumably a mathematician, and when mathematicians make
    unqualified statements, with no other context, they usually *mean* "In ZFC
    ...".

    In a sense, mathematical axioms are not knowledge because they are not
    statements that can be true or false in an absolute sense. They can be
    viewed as being more like definitions.

    The axioms of group theory are not facts that are "known", they describe
    what we *mean* when we talk about a group.

    There was once quite a bit of fuss over the truth of the Parallel
    Postulate. Nowadays, we would call it a property of a space. It isn't true
    or false in an absolute sense. Some spaces have it and some don't.

    Similarly, accepting AC can be viewed as being more specific about what you
    mean by "sets".

    I am not sure I am willing to follow this line of reasoning to its ultimate
    destination. I like to think I know what I mean by *the* integers, and
    there are some statements that I might have trouble viewing as "a matter of
    definition", even though they are independent of all the axioms.

    I imagine people used to feel that way about points and lines.

    I really am quite fond of that "little white lie".

    > There were heated arguments, and through a complex sociological process, the axiom
    > of choice won out. Was this an effective procedure? Was it even objective?

    Which is exactly what one would expect if it was a matter of definition,
    not of truth. Truth isn't normally considered a matter of consensus, and is
    not considered negotiable, but definitions are.

    Definitions can be produced by an effective procedure or not, because there
    is no need for them to be objective.

    There are good definitions and bad ones, but it is mostly a matter of
    utility. Asking if a definition is *true* is nonsense (we can, and should,
    ask if a definition is consistent).

    > What's to stop some similar mess from happening again---say, with the axiom
    > of projective determinacy, which Woodin and others have been advocating?

    It most certainly will, if not in that case, then in some other.

    > If these things aren't knowledge, or aren't mathematical truths, then
    > your assumption comes at the cost of discarding a lot of what most people
    > consider to be mathematical truth.

    In the case of set theory, that would be less than one page of text.
    Including definitions might require a small font.

    The axioms themselves are a very small part of mathematics. Most (one could
    argue all) mathematical knowledge is in the theorems.

    One way to evaluate a mathematical theory is to look at its "theorem to
    axiom ratio". All else being equal, bigger is better.

    A big bunch of complex axioms, from which little additional can be proven
    is barely mathematics. We like small a set of simple axioms with an
    enormous number and diversity of theorems (e.g. set theory).

    "Good" theories are more useful. The small set of axioms mean they apply
    more often, and the large set of theorems mean you get a lot of answers.

    I suspect that one reason that AC is accepted is that it is simple and
    produces many diverse theorems.

    It is unreasonable to expect to get an answer to *every* question. Some of
    the *best* theories allow one to express questions that they cannot answer.

    You can always *define* answers to questions that your axioms don't answer,
    but it is unclear in what sense that is the same as knowing the answer, and
    there in no sure (or even effective) way to avoid making inconsistent
    assumptions.

    If you keep adding axioms, without proving then consistent, you will surely
    add an inconsistent one eventually.

    Some statements cannot be proven "safe" to use as axioms (even if they
    are). (The safe axioms are the complement of an r.e. set) If you only add
    safe axioms, there are some statements you will never decide.

    AC is a not the best example, since it *was* proven safe.

    Ralph Hartley

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