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Re: distinguishability - in context, according to definitions
Posted:
Feb 17, 2013 9:40 PM
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In article <x_-dnZNsYePggrzMnZ2dnUVZ_rydnZ2d@giganews.com>,
fom <fomJUNK@nyms.net> wrote:
> On 2/17/2013 9:10 AM, Shmuel (Seymour J.) Metz wrote:
> > In <WvKdnStB4bTi9YDMnZ2dnUVZ_oWdnZ2d@giganews.com>, on 02/14/2013
> > at 04:42 PM, fom <fomJUNK@nyms.net> said:
> >
> >> Here are descriptions of the received paradigm
> >> for use of the sign of equality
> >
> > They don't clarify the sentence I asked about. How are two distinct
> > sequences ontologically the same, even if both are eventually
> > constant? They can certainly have the same limit, but that is a
> > different matter.
> >
>
> I am sorry. Your objection to the statement is
> clear to me now. My statement badly expressed
> what was intended.
>
> There is a distinction in identity statements
> between
>
> trivial, or formal, identity
>
> x=x
>
> and informative identity
>
> x=y
>
> In the latter case, there is a distinction between
> when it is stipulative and when it is licensing
> epistemic warrant.
>
> The algebraic proof licenses the epistemic
> warrant for the substitutivity of the
> symbols.
>
> But, in the received paradigm for identity taken
> from first-order predicate logic, all instances
> of
>
> x=y
>
> are stipulative.
>
>
> This is not how I understand mathematics. It
> is something I strive to reconcile with my
> understanding of matters -- as meager as that
> may be.
>
> Almost every reputable mathematics department is
> giving courses in "mathematical logic," presumably
> based on this received paradigm.
>
> There is nothing the matter with the deductive
> calculus. So long as the semantic unit is a proof
> with quantificationally closed assumptions and
> quantificationally closed conclusions, one may
> speak of faithful representation in the algebraic
> sense.
>
> But, in the "logical" sense,
>
> 1.000... = 0.999...
>
> is merely a stipulation of syntactic equality
> between distinct inscriptions that is prior
> to any mathematical discourse.
I don't see how. That equality is *proven* from axioms for real
numbers, so how can it be a prior stipulation? The prior stipulations
are that place-value notation represents an infinite series, and that
"..." indicates all subsequent digits are the same.
Clearly the strings "1.000..." and "0.999..." (or "1.(0)" and 0.(9)")
are themselves not equal, but when taken to represent real numbers they
are:
Real("1.000...") = Real("0.999...")
Just as
DecimalInteger("4") = RomanInteger("IV") = RomanInteger("IIII")
Regarding ontology, there need not be any Platonic integers that are the
range of these representation functions; the equivalences, independent
of any range, suffice for all mathematical purposes. I don't know if
there is an established philosophy of mathematics that takes this view;
informally I think of it as "representationalism".
> I hope that helps. It is difficult to
> explain things with which one disagrees.
[added sci.philosophy.tech, comp.ai.philosophy]
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