FOM: terminological foolishness? substantial issues

Martin Davis martind at
Mon Aug 24 19:52:49 EDT 1998

At 04:43 PM 8/24/98 -0400, simpson at wrote:
>Martin Davis says that he and others are frustrated about my
>interminable arguments over the precise meaning of various terms.
> > In each case, Steve has apparently determined a unique correct
> > useage with which to pound those quilty of wrong-think into
> > submission.
>Have I pounded anyone into submission yet?  Evidently Martin himself
>is still in the defiant or "bloodied but unbowed" stage.  :-)
>Seriously Martin, I want to thank you for your many FOM postings,
>which have unfailingly exhibited good sense and judgement.

Thank you.

>However, I'd also like to point out that there may be more to these
>terminological disputes than meets the eye.


>Martin, what do you have to say about these substantial issues?  You
>have already commented on 3, but what about 1 and 2?
>-- Steve

I think in both cases the terminological dispute had only the most tenuous
connection with the substantive issues yyou raised. 

"Golden opportunity": this is a strange notion of how mathematicians operate.
Would you tell someone working on complex variable theory in many dimensions
that they have missed or are missing a golden opportunity to work on
numerical methods for PDEs?

The issue of Boolean algebras vs Boolean rings had IMHO very little to do
with the question of what have been called "categorical foundations". The
theorem relating these structures is clear and pretty trivial. Algebraists
haven't had the need in talking about structures to distinguish between the
set on which operations and relations are defined and the full structure in
the way a model theorist would indicate it. This is (in Bourbaki's
inimatible phrase) is an abuse of language, but ordinarily a harmless one.

When an algebraist says that the set of permutations on a given set of
objects forms a group under composition, it is not thought necessary to
specify which of the the following structures is intended by the word "group":
<G,*(multiplication),e(left-identity,I(left inverse)>
<G,*(multiplication),/(left division),\(right division)>
The difference in the Boolean case is the the algebra was defined by
abstracting from Boole's algebra of sets, whereas the ring was defined by
specializing the notion of ring, a notion that arose from an entirely
different direction.

But all this was made clear long ago, and I regret my part in reopening this
Pandora's Box. These terminological issues do not hide vicious plots. They
are normal variations in useage.

Steve: On computability,I made a number of substantive points which I don't
believe you ever answered. In particular that the word has been used in the
computer science community for decades with no confusion.


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