exploiting similarities and analogies

["Al Vilcius" <al.r@vilcius.com>]

Dear Categorists,

Has anyone explored, either formally or informally, the connection between
the Melzak Bypass Principle (MBP) and adjoints?

The MBP (aka "the conjugacy principle" which embraces and generalizes
Jacobi inversion) Ref MR696771   
http://www.ams.org/mathscinet/pdf/696771.pdf  (and no, it does not appear
in either Wikipedia or PlanetMath, yet) is somewhat heuristic in
character, suggesting : Transform the problem (T), Solve(S), Transform
back(T^1), as a "bypass" given by  (T^1)ST, which looks like conjugation.
Melzak himself refers to adjoints (quite tangentially) as "being bypasses,
though dressed up and served forth exotically" p.106 ibid. (I do recall
that adjoints were generally seen as pretty exotic in the early 1970's
when I was a graduate student at UBC, to my great chagrin). The MBP is
acclaimed in MM vol.57 No.3 May 1984 as "a device for exploring
analogies", or as "a dazzling attempt to comprehend complexity".  Perhaps
"bypass" could also be seen in the words of W.W. Tait (1996) "the
propositions about the abstract objects translate into propositions about
the things from which they are abstracted and, in particular, the truth of
the former is founded on the truth of the latter".
http://home.uchicago.edu/~wwtx/frege.cantor.dedekind.pdf . For me,
"bypass" is a kenning for quasi-inverse or ad joint pairs.

Motivation for seeking such a connection is not really to revive a 25 or
30 year old idea (as brilliant as Melzak's insights were, of course), but
rather "to facilitate invention and discovery" (in Melzak's own words),
and to find additional (as well as interdisciplinary) sources of instances
of adjoints, possibly as a way to make adjoints more immediately relevant
in any introductory discussion of categories, since, of course, adjoints
are undoubtedly one of the most successful concepts within category
theory. Further inspiration could be found in the Brown/Porter "Analogy"
paper http://www.bangor.ac.uk/~mas010/eureka-meth1.pdf and others, along
with a passion for  invention and discovery through the continued pursuit
of Unity and Identity of Opposites (UIO) -  obviously referring to Bill
Lawvere. Furthermore, I would also like to see this connection developed
for practical reasons,  applied to various situations, in particular to
the structure of the www as an anthropomorphic creation that could benefit
from further categorical perspective, given by the learned categorists I
respect the most.

I look forward to your  thoughts and comments. ..... Al

Al Vilcius
Campbellville, ON, Canada