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
and informative identity
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
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.