X-Message-Number: 26682
Date: Sat, 23 Jul 2005 18:53:03 -0700
From: Mike Perry <>
Subject: Identity vs Equivalence
References: <>

Thomas Donaldson writes in part,

>For Mike Perry, a mall, short point:
>
>In your message on Cryonet #26667, you say, along the way, that in
>mathematics identity must be exact.
>
>That's quite wrong, and I've ctually raised that fact when answering
>questions about identity on Cryonet. Identity is a relation among
>different objects (mathematical or other) which satisfies 3
>conditions: 1. if X is identical to Y, then Y is identical to X
>2. if X is identical to Y, and Y is identical to Z, then X is
>identical to Z, and 3. X is identical to X. We may claim that
>X "is identical to" Y WHENEVER ANY relation holds that satisfies
>these 3 conditions.

It's a quibble over terminology, perhaps, but what you've described is an 
equivalence relation, not necessarily identity. X and Y would be 
equivalent, but (I would say) not necessarily identical. (Identity is a 
special case of an equivalence relation, but not the only possible case.)

>I could get quite mathematical here, but here is an example which
>I think should be simple. Suppose we have two functions which are
>identical only on the interval from 0 to 1 (including its end
>points) but differ wildly from one another elsewhere. We can say
>that these two functions are "identical" and in doing so, are
>using the postulates for identity quite correctly.

Again, I would call the functions equivalent, based on a particular notion 
of "equivalence" (that they agree on the interval [0,1]) rather than 
identical, unless they also agree elsewhere. One place it seems to make a 
difference is in set theory. Two sets are considered one and the same, that 
is, to share identity, if they both have the same members. That is, X is 
identical to Y, and we write X = Y, if and only if, for all z, z is a 
member of X if and only if z is a member of Y. We then consider that X and 
Y are one and the same thing entirely, and not merely "equivalent" in some 
sense. So the point I would make is that (at least as I perceive it) 
"identity" in mathematics does indeed mean that two entities are actually 
one and the same thing in all respects (we have just assigned different 
names to the same thing). But, in various ways this strict notion of 
"identity" does not always apply in the real world, though we still find 
reasons to use the term.

Mike Perry

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