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fom
Posts:
1,035
Registered:
12/4/12
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Re: distinguishability - in context, according to definitions
Posted:
Feb 17, 2013 1:20 PM
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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 hope that helps. It is difficult to
explain things with which one disagrees.
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