FOM: Boolean rings: another epiphany?
Stephen G Simpson
simpson at math.psu.edu
Sun Mar 15 15:52:47 EST 1998
Vaughan Pratt 15 Mar 1998 10:44:14 writes:
> The concept of isomorphism has nothing to do with the question. Only you
> and Steve seem to think it does.
What??? This context-dropping is outrageous.
The whole Boolean rings debate started when some category theorists
disputed my observation that Boolean algebras are *not isomorphic* to
Boolean rings. Vaughan Pratt himself disputed it in his posting of 10
Mar 1998 11:31:48:
> From: Stephen G Simpson 1/31/98 4:04PM
> >Note first that Boolean algebras are not isomorphic to Boolean rings.
>
> This is only true for those who find the distinction useful, which
> many people don't. ....
and that was the genesis of this surrealistic mud-wrestling match.
Vaughan Pratt 15 Mar 1998 10:44:14 continues:
> My promised "actual point" is that the category theoretic truth that
> every identity is an isomorphism appears not to hold in universal
> algebra.
Hmmmm .... Perhaps this is the final reductio ad absurdum of Pratt's
subjectivistic, solipsistic denial of the distinction between Boolean
algebras and Boolean rings.
> On the one hand we have Sikorski's statement that every Boolean ring
> is a Boolean algebra and conversely ....
No, you are misunderstanding Sikorski and other math textbooks. You
may think that Sikorski is saying that Boolean rings are Boolean
algebras, but he isn't. I'll try to straighten it out for you when I
get access to my copy of Sikorski.
> On the other we have the quite correct point that Simpson and Kanovei are
> making, that the official definitions of signature and isomorphism make
> Boolean rings with their standard signature not isomorphic to Boolean
> algebras with their standard signature ....
Let's state it more concisely: Boolean algebras are not isomorphic to
Boolean rings. Vaughan, do you agree????
> Together we have placed these two undeniable facts on a collision
> course which would appear to sink a particularly elementary
> assumption of category theory.
Is this another epiphany? A foundational crisis? Don't worry,
Vaughan! Keep the faith! Category theory will not let you down.
I'll straighten it all out for you when I get hold of Sikorski.
> The way I (and therefore all the sensible people in the world :)
> want to resolve this conflict is by working with clones,
> i.e. signatures that are closed under composition.
This proposal isn't correct as it stands, because a signature is only
a collection of relation and operation symbols with arities assigned.
Composition of such symbols makes no sense. It only makes sense when
you interpret the operation symbols as operations. However, I'm sure
this proposal could be made sense of, at some cost.
Fortunately, there's another way to resolve Vaughan Pratt's crisis,
namely to accept the definitions that are usual in math textbooks.
This includes the existence of signatures which are *not* closed under
composition. It also includes the generally accepted definitions of
Boolean algebras and Boolean rings, according to which Boolean
algebras and Boolean rings have specific finite signatures and are
therefore not isomorphic. Most mathematicians resolve it in this way.
> Given that signatures not closed under composition lead to the above
> inconsistency,
No, this isn't correct. What lead to the inconsistency is your
misreading of standard material in mathematics textbooks such as
Sikorski.
-- Steve
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