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 Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=26682