X-Message-Number: 26734
Date: Fri, 29 Jul 2005 21:32:02 -0700
From: Mike Perry <>
Subject: Further researches on "identity"
References: <>

Thomas Donaldson says:

>However it's still important to remember that there is NO sense
>of identity at all. We may apply the symbol "=" as an abbreviation
>for the sense of equivalence we want to talk about.

In a general context, this may be true. At least "identity" is used in 
different ways (not all of them referring to "equivalence" even, I would 
say). As for mathematics itself, I checked on the Web to see if there might 
be a special convention in use (as a general rule, maybe not always), such 
as "identity means equality." The situation seems somewhat confused; the 
equality symbol, for one thing, does not always mean "equality" in the 
sense that two things are one and the same. (T(n) = O(n^2) describes the 
growth of function T(n) as n-> infinity, as being like the second power of 
n, rather than saying that "T(n) is the same thing as O(n^2)".) However, my 
basic position about "identity in mathematics" is more or less captured in 
the following (found at http://encyclopedia.laborlawtalk.com/identity):

In logic, the identity relation is normally, (by definition), the 
transitive, symmetric, and reflexive relation that holds only between a 
thing and itself. That is, identity is the two-place predicate, _=_, such 
that for all x, y, "x=y" is true iff x is y.

except that maybe I should have said "in logic" rather than "in 
mathematics." (Albeit the two are closely connected; set theory was 
developed from logic and in turn served as a foundation of mathematics.) 
Here it is saying that identity is the equivalence ("transitive, symmetric, 
and reflexive") relation that holds only between a thing and itself (or 
between a thing and something that "is" that thing). So not just any old 
equivalence relation, unless you want to argue over the meaning of "is". 
(Note that here the equality symbol does have the meaning of "one and the 
same".)

Anyway, what started this whole exchange was my statement in #26667: "In 
mathematics and some other settings, 'identity' holds between two objects 
if and only if they are the same in all respects."  Again, maybe I should 
have said "in logic" instead of "in mathematics"--would anyone find that 
objectionable?

Speaking of persons, though, I'm inclined to think one should not overuse 
the term "identity," in view of the confusions and disagreements it often 
causes. Be careful to explain what you mean if you do use it--and expect 
opposition!

Mike Perry

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