Re: Lambek's lemma

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

Dear Vaughan,

Of course I agree with you that, logically, there is no point in
drawing a commutative square to prove that x = x. I also agree that
5 < 7. 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. (I feel strongly about this,
having spent two hours this afternoon in an examples class for the
students attending my first-year graduate course on category theory.)
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.

Peter
-------------------
On Fri, 13 Nov 2009, Vaughan Pratt wrote:

> Prof. Peter Johnstone wrote:
>> Vaughan's argument appears in the Elephant
>
> Oops, I keep forgetting to check there for these things, sorry Peter!
>
>> (but with the second square,
>> which he has indicated by dots, drawn in, so that there are seven arrows)
>
> Actually my dots were not to indicate the second square but merely to
> prevent mail forwarding programs from deleting initial spaces on lines.
> I don't know why they do so, but it screws up formatting of ASCII
> diagrams.
>
>> Though it's not credited there, I learned it from
>> Peter Freyd --- I can't remember when, but that part of the Elephant
>> was written well before 1998.
>
> The reason it came up in 1998 is that I was preparing a lecture then for
> my algebraic logic class and was trying to reconstruct the proof I'd
> seen Peter F. give some years earlier (for all I know PTJ and I heard
> PJF give it at the same talk).  I came up with the five-arrow diagram
> and sent it to PJF asking if that was his proof.  He replied "I don't
> see where you proved that fa = 1. Here's the way I'd present it," and
> sent me the seven-arrow diagram as per the Elephant's Lemma A1.1.4.
> Some discussion ensued, the outcome of which was that he agreed I'd
> proved fa = 1 after all.
>
> I thought no more of this until a couple of days ago when I suggested to
> Mikael Vejdemo-Johansson, who is teaching a CT course here at Stanford
> this quarter, that he present Lambek's lemma.  Reviewing my
> correspondence with PJF, it occurred to me that people ought to know
> that the second square could be suppressed for the sake of two fewer
> arrows in the diagram, FWIW as they say, whence my message.
>
> Mathematically speaking this observation is a triviality (which is why
> PTJ is comfortable calling the 5-arrow and 7-arrow diagrams "the same").
>
> But by the same token the Reidemeister moves are a triviality inasmuch
> as they relate "the same" knots.  (As a meta-Reidemeister move let me
> remark that the first time I saw the Reidemeister moves was when I was
> writing my fourth year honours thesis in Pure Maths at Sydney in 1965,
> in a class of 14 that included Ross Street and Brian Day, for which I'd
> chosen knot theory after grinding to a halt trying to write about
> Riemannian manifolds without any supervision--I found I could at least
> read the knot theory literature without supervision!  My knee-jerk
> reaction to the Reidemeister moves was "how is this mathematics?" and I
> moved on to the Alexander polynomial and other algebraic techniques,
> about which I wrote some ninety pages of typical undergraduate
> misunderstandings, none of which mentioned the Reidemeister moves.  This
> was well before either Conway's reworking of the Alexander polynomial
> (when I was just starting Berkeley's CS PhD program) or, 14 years after
> Conway, Vaughan Jones' polynomial (when I was working at Sun
> Microsystems).  My real interest in 1965 was theoretical physics, for
> which I hoped honours math would be good preparation for honours
> physics.  But then in 1967 computers happened along as yet another
> career option.)
>
> Getting back to whether 5 is any smaller than 7, there is something
> disconcerting about the righthand square in A1.1.4 (page 6 of the
> Elephant), since it asserts that aTa = aTa.  Why should the equation x=x
> have to take up fully 50% of the diagram proving Lambek's lemma?  My
> reconstruction of PJF's proof did not deem x=x worthy of such a large
> share of the proof.
>
> This is the sort of argument only a proof theorist could love.  PTJ is
> quite right when he says these are the same proof.  Being no less a
> Platonist than anyone on this list, I said as much in my reply to PJF in
> 1998.
>
> On the other hand, what sort of Platonist would reject 5 < 7?
>
> Vaughan Pratt
>
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>


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