[FOM] When is it appropriate to treat isomorphism as identity?
Arnold Neumaier
Arnold.Neumaier at univie.ac.at
Fri Apr 24 03:28:35 EDT 2009
Vaughan Pratt schrieb:
> My recent complaint that uniqueness is too often taken for granted when
> dealing with questions of existence brought to light the ease with which
> people identify isomorphism with identity: a number of people wrote to
> remind me that uniqueness is common in many mathematical contexts, e.g.
> the cyclic group of order 5, the least upper bound of two elements of
> a lattice, etc.
>
> Operations of course enforce uniqueness by definition of "operation":
> the sum of two integers, the concatenation of two lists, the union of
> two sets, etc. However in a good many of the examples, "the" was not
> enforced by some operation but instead only meant "up to isomorphism."
>
> Sometimes it is necessary to treat isomorphism as identity, sometimes it
> is merely convenient, sometimes it is inconvenient, and sometimes it is
> inconsistent or paradoxical. Here are examples of each.
I think identity is _always_ up to disregarding things deemed inessential.
A person today and the same person tomorrow are different in many
respects - mood, physical composition, etc.. But they are treated as
being identical.
We believe that there are 26 letters in the alphabet. Although
these appear different in different fonts, cases, etc., two letters
A are considered identical. When sending emails, one regards the
text sent and the text received as identical although they are
physically distingushable by their location.
Thus there is no concept of identity that can be unambiguously
distinguished from ''identity up to some natural equivalence
relation''.
The identification of isomorphism with identity is just a
particular case of this general phenomenon.
Arnold Neumaier
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