FOM: Boolean algebra vs Boolean ring
Colin Mclarty
cxm7 at po.cwru.edu
Tue Mar 10 19:48:28 EST 1998
Reply to message from simpson at math.psu.edu of Tue, 10 Mar
Vaughan Pratt had said that many people do find it useful
to distinguish Boolean algebras from Boolean rings. Simpson replied
>Seriously, my statement about Boolean algebras vs Boolean rings is
>true (for everybody!), and the distinction is useful for all
>mathematicians. If we adopt a "what's true for you isn't necessarily
>true for me" standpoint (i.e. a solipsist philosophy), we will never
>be able to have a rational discussion.
People used to understand a "theory" to be a collection
of axioms--so that, for example, there were many "theories" of
Euclidean geometry which differed only in which version of Euclid's
parallel axiom they used. All gave exactly the same theorems.
A major advance in logic was the decision not to
distinguish "basic" truths of a theory from "derived" ones, That
is, to understand a theory as a body of theorems--which may be
presented by one or another axiomatization but it remains the same
theory.
Now Vaughan, like many other logicians and mathematicians,
understands an algebraic theory without distinguishing "basic"
operations from "derived" ones--that is primitive from defined
operations. There is nothing arcane about this, and on this
basis Boolean algebras are exactly the same things as Boolean
rings.
Colin
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