X-Message-Number: 26678
Date: Sat, 23 Jul 2005 00:06:10 -0400
From: Thomas Donaldson <>
Subject: Re: identity, for Mike Perry

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.

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. We might do
this, for instance, if only the behavior of the functions on
the interval from 0 to 1 matters to us.

In one previous discussion of identity on Cryonet, I used just 
such a notion of identity to argue that even a long period of
time between one version of me and a second version does not
mean that both aren't identical. Simple: we use a notion of
identity which simply ignores such time intervals.

I will also say that mathematicians very quickly found out that
if they always insisted on exact identity lots of useful 
theorems would turn out to be false. Anyone reading this who
has studied the theory of integration will know this immediately:
two functions have the same Lebesgue integral even if they
differ on all the rational numbers (a countable infinity of
numbers).

"Identity" simply isn't a fixed notion that remains unalterable
by us. True, if we choose one definition of identity we can't
go around changing it to other definitions without making that
change very clear to readers; but so long as we do not do that,
we can start with any notion of identity which satisfies the
3 conditions I listed above. And of course, once more, that is
one reason why I think that a precise copy of me 1000 years in
the future would be ME.

           Best wishes and long long life for all,

               Thomas Donaldson

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