Mathematical notation seems to be an area where personal taste and
tradition are sometimes at odds.  Below, I will detail my, admittedly
subjective, reasons, why I prefer Reverse Polish Notation (RPN) when
composing arrows.  While I will continue to write my papers in this
style, I will also make versions with the traditional backwards
notation available on my home page.  (I've had TeX-macros for this
purpose for quite a while.  Unfortunately, they are not easy enough to
use to make them public.)

While in the commercial world (which some people seem to confuse with
the real world) the technically superior solution does not necessarily
prevail (cf., e.g., the video formats Betamax vs. VHS, or the
operating systems OS/2 vs. Windows), I would hope that this is not the
case in mathematics.  In fact, there are two recent examples where
established categorical notations or terminology have changed:

- In their book "Coherence for Tricategories", Gordon, Power and
  Street have changed the meaning of "lax natural transformation" from
  F to G by reversing certain 2-cells, so they really display this
  directional information;

- at the recent CT97 meeting in Vancouver, Robert Par\'e officially
  switched his notation for profunctors from A to B.  He now views
  them as functors from A^op x B to Set, which finally makes it
  possible to consider hom-functors directly as pro-functors.

Here, now, are my reasons why I prefer RPN:

  Having grown up in a culture that reads and writes from left to
  right, I've developed a sense of direction.  It "hurts" every time
  to see the composition
                    f       g
  (*)		A ----> B ----> C

  denoted as g o f .  So I made a conscious decision in about 1981 to
  denote this composition by f;g instead.  I've certainly seen
  examples of this in the literature.  Another "solution" to the
  problem is to draw the arrows backwards, i.e.,
                    g       f
  (**) 		C <---- B <---- A

  A recent example of this is the book "Algebra of Programming" by
  Richard S. Bird and Oege de Moor.  In my view, this attempts to cure
  the symptoms, but not the problem.

  The problem, of course, is the denotation f(a) for the value of the
  function f at the argumant a .  This might briefly suggest itself
  when adding typing information to f , e.g., when you write f:A ---> B,
  but becomes odd as soon as you take the directional aspect of f into
  account, namely to go *from* A *to* B , as indicated in (*).

  In fact, nobody appears to defend "g o f" or "f(a)" for their
  technical superiority, the only argument seems to be conformance
  with traditional or established notation.  For an example what a
  heavy burden backwards compatibility can be, you only have to look
  at the computer industry and the success stories of DOS and Windows
  as well as the Intel processors.  Even engineers have realized the
  advantages of RPN: just check out the superb HP RPN-calculators.

  Once you take the step to break with tradition and write f;g instead
  of g o f, consistency requires further changes; e.g., hom sets in a
  category A should be denoted by <X,Y>A rather than A<X,Y>.  Of
  course, this may "hurt" somebody else's eyes initially, but I don't
  think it renders a text incomprehensible.

  There are papers, where on different levels different conventions
  are followed, e.g., the composition of functors is written in a
  different order than the composition of functions, which
  traditionallly is written in a different order than the composition
  of relations.  I find this totally confusing.  Rather, I prefer to
  think that form should reflect content, and for me notation also
  serves the purpose of ``type checking'': if I cannot write down a
  mathematical concept in a satisfactory way, maybe something is wrong
  with the concept.  You may argue that I'm a formalist or use
  notation as a crutch.  But so what?  The math is difficult enough as
  it is, so I take any help I can get!

  A referee recently claimed that RPN notation frequently is employed
  in computer science oriented papers on category theory.  In view of
  the lambda calculus and the prevailing programming languages I find
  this surprising, but I won't complain :-) The fact that in
  mathematics the issue won't go away indicates that some people are
  unhappy with the traditional notation.  Perhaps there are in fact
  two kinds of personalities: ``stack-oriented'' ones like myself and
  ``applicative'' ones.  Just check the never-ending discussions on
  the HP calculator list between proponents of RPN and the sadly
  misnamed ``algebraic'' notation.  Since I based my argument for RPN
  on a culturally acquired sense of direction, I also have to accept
  the existence of a different sense of direction, e.g., in cultures
  that write from right to left.

  Finally, if I have the TeX-macros to produce ``backwards'' versions
  of my papers, why don't I publish them that way?  Besides the fact
  that I think they are more readable in RPN ;-), I want my papers to
  reflect my way of thinking about mathematics.  And here again, form
  reflects contents.  After all, aesthetics played a major role in my
  fascination with mathematics in general and with category theory in
  particular.