Re: Lambek's lemma

[Vaughan Pratt <pratt@cs.stanford.edu>]

Prof. Peter Johnstone wrote:
> But I think there is still some point in drawing the second
> square in A1.1.4, at least in pedagogical terms: until you've seen
> (or at least visualized) the second square, it's hard for the mind
> to accept the argument that says af = 1.

Agreed 1000% (UK: 100%).  In fact it could serve as a real-world example
of the difference an extra square can make in a diagrammatic argument
whose verbal counterpart gains nothing from it.  As such it would be
interesting grist for the mill currently being ground more finely lately
by those interested in diagrammatic reasoning, represented here by Sol
Feferman who's been taking quite an active interest in it lately.

(Even though I'm more of a visual thinker, quite the opposite of Gordon
Plotkin for example who considers himself as verbal as I am visual, for
some reason I tend to view commutative diagrams as more verbal than
visual, perhaps because my ability to visualize makes it clearer to me
that they are depicting equations between words and are therefore really
verbal entities, appearances notwithstanding.  This would help explain
my lack of enthusiasm for the second square.  But I would still draw it
explicitly if teaching the lemma, just like you.)

> And, as someone (I forget who, but it may have been Mike Barr) pointed
> out long ago, one can (well, almost) define the variety of groups
> as the variety defined by a single binary operation satisfying a
> single equation; 1 < 3, but no sane group-theorist would do it
> that way.

Substitute "Boolean algebra" for "group" and one finds in Stephen
Wolfram's ANKS the same claim: one binary operation, satisfying one
equation; in this case 1 < 12 or thereabouts when defining a Boolean
algebra as a complemented distributive lattice.  I'm sure Stephen is
under no delusions as to the pedagogical benefits of his definition.

Unfortunately neither definition is correct because both varieties so
defined have as their initial object the "empty group," resp. "empty
Boolean algebra," one, resp. two, elements too few.  Or did you not
count e when you said "one binary operation?"

Vaughan


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]