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Johannes Kuester <[log in to unmask]> wrote:

>>   In mathematics, variables are usually typeset in some kind of slanted
>> type (like italics), whereas contants (usually function names, and the
>> like) are usually typeset upright, even though tradition provides many
>> exceptions (like the numbers e and pi, which are contants usually typeset
>> in italics).
>
>No, at least most function names shouldn't be typeset upright,
>whereas e, pi and i (the imaginary unit) should be.
>I would not call it a tradition if they aren't,
>rather that is due to the laziness of most mathematicians
>and their lack of knowledge about mathematical typography.

  Actually, Johannes Kuester say exactly the same thing as I, except that
he has misunderstood the terminology I use: I used "variables" to indicate
anything that may vary, including the "f" in the function "f(x)"; so here
"f" is not a "function name", but the name of a variable that happens to
refer to a function.
A "function name" is then "sin", "cos", and the "Hom", in constructs like
"Hom(A,B)"; I call these "constants", and Johannes Kuester call then "names
with fixed meaning".

>Maybe one has to distinguish here between `good' and `bad' traditions,
>i.e. traditions that really developed out of a mathematical necessity
>and/or are helpful in making a mathematical text more readable,
>and traditions which are due to lack of appropriate type, lack
>of knowledge about either mathematics or typography etc.

  The mathematical typesetting traditions are very old, and the typesetters
substituted fonts and symbols for the mathematicans handwritten symbols; in
addition, it was costly having many sets of font styles. So it is only
natural that tradition comes with many simplifications.
  When it comes down to names of constants like "e", "i", "pi", these
really were "variables" from the beginning, when they were discovered, and
therefore should be typeset slanted. Nowadays, this is no longer the case,
being regarded as "constants", and further, any choice of typesetting can
most easily be achieved using TeX, so why not change it?
  For the same reason vector "variables" are likely to be set in upright
bold, but why not change it to bold italic, as suggested by the Duden rule?
(Of course a very pure mathematician would never use bold to indicate a
vector... :-) )

  -- This is one reason I brought this topic up: I do not think there is an
absolute way of describing say what is "variables" and "constants", this is
a tradition, but also, there are good reasons for a change to better
traditions.

>Maybe \PI for upright pi (the circle number),
>      \I  for the imaginary unit and
>      \E  for e (=\exp(1)),
>as these are used frequently and should be short (besides, quite similar
>names are used in Maple V),
>maybe \df for the differential operator d
>and then \pdf for the upright \partial
>etc.

  There is no reason for always finding short names for such common
symbols, as such choices are likely to conflict, and as any writer can
always define new short-hand macros (or using "\let") for any given
manuscript.
  For example, I use "\iu" for the imaginary unit, and one might use such
long names as "\diffop" and "\partdiffop", or even
"\partialdifferentialoperator", for other symbols. (Or using the idea of
computer "objects" or "modules" for a classification.)

>Upright lowercase greek by \ualpha, \ubeta etc/ (u for upright),
>maybe a set of \uGamma and \sGamma for uppercase (upright/slanted)
>which could be used to get explicitly a glyph, whereas \Gamma could
>be set by an option to slanted or upright, according to language and/or
>tradition etc.

  I think this is already covered by NFSS (at least in text mode): You only
have to find the fonts.
  Anyway, there seems to be a need for full set of fonts/styles also in
math mode for all fonts.
  (Perhaps some expert can help here.)

>>   It would in fact be a good idea of having a good set of upright and
>> slanted (both upppercase/lowercase) of fraktur and script styles for
>> mathematical purposes (the AMS-Fonts package does not provide it). For
>> example, when speaking about categories C, D, one would use say slanted
>> script, but when indicating the functor category Fun(C,D), the name "Fun"
>> would be typeset in upright script.
>
>I disagree. Such things as slanted fraktur do belong to the Typographical
>Chamber of Horrors. And upright script... I don't know, but I would set
>`Fun' in upright roman, as all multiletter symbols. Script alone is
>special enough, upright script alongside with slanted script would
>confuse rather than help.

  Again, there are several subquestions involved here, and proper analyzing
requires them to be treated separately:

  First, there is the question whether there is a mathematical use for it:
It fits the idea of "variables" and "constants", and I found that I had a
use of it, so a I mentioned. So from this point of view, the idea is worth
to be investigated.

  Then there is the typographical question, and this is in fact two
different questions: The use say fraktur in text and in math; the "math"
design problem probably gives the fellow who designs the font more
flexibility (as it is less important forming good looking words in math). I
would think this question could only be resolved by consulting an
experienced font designer, which may then even have to experiment with it
during a period of time.
  The AMSFonts "Euler script" font is upright, and the TeX "calligraphic"
font is slanted (but none are very "scripty"), so why not designing a new
font?

  Hans Aberg