Re: Is equality evil?‏

["Mattias Wikström" <mattias.wikstrom@gmail.com>]

Toby Bartels wrote:
> Right. I would prefer to use no identity types at all.
> If there is a particular predicate (on some particular type)
> that you would like to think of as equality (on that type),
> then write down its definition and introduce a symbol for it.

If any (well-behaved) equivalence relation can be thought of as
equality then it would seem to follow that equality is relative. That
is:
1) A and B may be the same thing from one point of view and two
different things from another point of view,
2) the cardinality of {A, B} may be either 1 or 2 depending on what
point of view one takes, and
3) it may be that from one point of view there is no property that
distinguishes A from B while from another point of view there is a
property P that distinguishes A from B (for consistency I guess one
has to say that relative to the first viewpoint, not only does P not
exist, but the second viewpoint also fails to exist).

In the philosophical literature the idea that equality/identity is
relative is credited to philosopher Peter Geach (I can give a list of
references if anyone is interested). It has few adherents, but maybe
category theory can make it popular.

Best regards,
Mattias


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