Re: Lambek's lemma

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

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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