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

poopdeville_at_gmail.com
Date: 02/04/05 

Date: 4 Feb 2005 00:35:04 -0800

Keith Ramsay wrote:
> I hope nobody minds if I shorten the lines in the quoted
> text to keep Google from wrapping them itself.
>
> poopdeville@gmail.com wrote:
> | Jeffrey Ketland wrote:
> [...]
> | > Your arguments (correct me if I'm wrong!) for this
> | > classification go roughly as follows:
> | >
> | > 1. The notion of "set" is indeterminate (or model-
> | > relative). This is because the "syntactic rules" (a
> | > notion you accept as kosher) for ZF don't "pick out
> | > a unique interpretation".
> | > 2. The notion of "truth simpliciter" is meaningless.
> | > This is because the *only way* that truth simpliciter
> | > could be a kosher notion would be via reductive analysis:
> | > i.e., if the "syntactic rules" of the language "picked
> | > out a unique interpretation", then truth simpliciter
> | > would be truth in this unique model.
> | > 3. The notion of "natural number" is indeterminate (or
> | > model-relative). This is because there are non-standard
> | > models of the "syntactic rules" of Peano arithmetic.
> | >
> | > Is that it?
> |
> | Yes. Though 1 and 3 are intances of the same phenomenon.
> |
> | The essential point is that we each have intuitive notions
> | of what numbers are. And we've agreed to talk about them
> | using the rules of inference of the FOL and the Peano axioms
> | (these are the "syntactic rules" I speak of.)
>
> No, we haven't. I'm not just being flip here-- to assume
> that we start from such a formal system is a rather huge
> assumption, and I think it's a mistake.
>

Obviously not. Arithmetic has been done for centuries without
formalization, and Peano drew from centuries' worth of inspiration when
formalizing PA. PA, however, is the de facto language of arithmetic,
in the sense that if one makes an arithmetical claim, it is assumed to
be formalizable in PA unless the speaker says he is referring to a
different form.

> As Tim Chow has nicely pointed out, you can't apply this
> formal interpretation to *everything*-- you have to start
> out with at least some concepts that aren't relativized
> in the same way. Otherwise one has an indefinite regression
> of interpretations.

And we *do* have an indefinite regression of explanations. However,
explanations must come to an end somewhere. That happens when you
understand what a speaker has uttered.

>
> Assuming axiom-system and/or model relativity helps to
> cloud the real issues, by leading people to worry about
> the part of mathematics that they interpret in such a
> way, when what makes the crucial difference is how much
> of mathematics they regard as meaningful without resorting
> to such a dodge.

I find this obscure. What are the real issues? Why is
"model-relativism" a dodge?

>
> | The language
> | we speak admits *very* different interpretations, as Godel
> | showed.
>
> We don't speak a formal language, but leaving that aside,

But we speak about "numbers." Our talk of numbers admits wildy
different interpretations.

> the different interpretations of the language of PA exhibited
> (implicitly at least) by Goedel are mostly irrelevant. They
> are interpretations whose existence is demonstrated assuming
> that one has a meaningful interpretation of the language of
> arithmetic to begin with.

Please explain. From my understanding of it, Godel's theorem is purely
syntactic, though it implicitly shows that non-standard models of PRA
(is that what it was called? -- I don't mean PA) exist.

<snip>

> | Because of this, you and I cannot be sure we have
> | to same (mental or logical) model in mind when we speak
> | about numbers.
>
> This is a crucial and rather huge step taken, apparently,
> by most proponents of "model relativity" for arithmetic
> on usenet. I disagree with it.
>
> On a very basic level, the mere possibility of exhibiting a
> nonstandard interpretation of someone's utterances surely
> can't be enough to prevent communication. There are just
> too many possible ways of doing it. It stands as a kind of
> standard philosophical problem how to account for this.

The existence of non-standard models does not prevent communication so
long as one fixes a context in which an utterance is to occur. In
mathematics, this is done by (1) finding an axiomatization we'll agree
to work with (that may not capture all the features of your private
model) or (2) finding an axiomatization which picks out your private
model exactly.

>
> It's akin to the Goodman "grue" paradox. Another case of the
> same question is known as the Kripke-Wittgenstein paradox.
> They ask, how is it that we follow rules at all? By following
> some rules for how to follow rules? Not generally; that would
> be a regression.

Good catch. "Kripkenstein" (among others) is the reason I espouse
"model-relativism." However, your understanding of the Paradox is a
bit flawed.

>
> One difference between those cases and the mathematical
> case is that it's much _less_ likely that anybody always
> intends the nonstandard interpretations (exhibited using
> Goedel's theorem and the like) when using mathematical
> language. Again, they are all or nearly all defined IN TERMS
> OF an assumed usual interpretation. When I make a number-
> theoretic statement, it's possible but unlikely that I'm
> quietly making it intending to mean that its Goedel number
> satisfies the predicate T as described above. It is *not*
> possible that I mean it that way and am *unaware* of the
> usual interpretation. And if there is no such thing as the
> usual interpretation, the whole construction is itself
> meaningless, which would make it impossible for me to be
> coherently intending this second interpretation in the first
> place.
>
> | So we each have two options: we prove
> | theorems that are true in each model of PA, or we find a
> | way to axiomatize our intuitive model and prove theorems
> | about *that* -- this may even be impossible.
>
> "Axiomatization" isn't the only way to describe a structure,
> unless you take "axiomatization" in such a broad sense that
> it automatically includes all legitimate means of describing
> a structure, or we place our means of description onto a
> kind of arbitrary Procrustean bed.
>

Please explain how else one might describe a structure if not by giving
a method to generate a set of relations the structure is to satisfy.

>
> I think the term "ideology" applies much better to the kinds
> of thinking that tell us that so much of our usual discourse
> (like in mathematics) is "meaningless" because it fails to
> satisfy this or that arbitrary criterion.

Mathematical discourse is clearly not meaningless. Just look at
sci.math. However, voodoo realist talk of truth of propositions
without fixing a context in which to evaluate them *is* meaningless.
(This has actually been dealt with -- Tim et al actually *meant* to fix
a model, V, the universe of sets, in which to evaluate the truth of AC.
 I was not familiar with this usage of "true.")

'cid 'ooh

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