FOM: signature; Boolean rings; Pratt's rewrite
Stephen G Simpson
simpson at math.psu.edu
Thu Mar 19 15:36:32 EST 1998
Vaughan Pratt 18 Mar 1998 21:55:11 writes:
> One consequence of having the notion of signature be independent of
> theory is the one we've seen, that it seems inevitably to lead to more
> than one variety of Boolean algebras, which one must then account for
> in some ad hoc way.
The notion of signature *is* independent of theory. This is accepted
in almost all math books, except possibly category theory books.
Pratt doesn't accept it. Pratt's "more than one variety of Boolean
algebras" refers to the fact that the variety of Boolean rings is
distinct from the variety of Boolean algebras. Pratt regards this as
undesirable and wants to overcome this by rewriting the entire
literature of algebra, universal algebra, and model theory.
> Those working within what they perceive as the one variety, such as
> Sikorski writing about Boolean algebras, don't have this problem
> they don't need to modify the theory let alone remove it, and so think
> in terms of a single variety, as Sikorski evidently does when he says
This is Pratt's incorrect reading of Sikorski's straightforward
treatment of Boolean algebras. Sikorski doesn't think that the
variety of Boolean rings and the variety of Boolean algebras are the
same variety, any more than anybody else does.
> "Every Boolean algebra is a Boolean ring"
Pratt is deliberately, dishonestly misquoting Sikorski. I reproduced
Sikorski's statement accurately in my posting of 16 Mar 1998 13:30:03.
Either Pratt is assuming that nobody remembers my posting or has
access to Sikorski's book, or else he just doesn't care what anybody
thinks of him.
> Naturally Steve views Lawvere theories, monads, and all other "categorical
> dys-foundations" as "incredibly complex."
Pratt appears to be misinterpreting the meaning of my phrase
"incredibly complex". What I intended was, "incredibly complex
relative to the small or nonexistent benefit to be gained from using
them in this context".
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