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Hans Aberg 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.

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

For example, the usage of upright Greek uppercase letters in
standard English and American math is -- in my opinion -- a bad tradition,
as it isn't consistent with the rules (see below) and is, as far as I know,
due to an English punchcutter/typographer who cut upright uppercase Greek
to go along with his italic lowercase Greek (maybe it was Baskerville,
I read that story once, but I can't remember neither the source nor the name
any more, so I will be grateful for any hints).
Another bad tradition in this sense is the use of (a,b) for the
greatest common divisor of a and b, as this notation is heavily
overused and could be easily confused. Even worse, there is no standard
notation for the least common multiple. This seems to be due to lack
of appropriate type (and/or of an appropriate symbol, of course).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The rules for the use of upright and italic type are as follows,
according to DIN 1338 and the recommendations of IUPAP
(taken from Satz- und Korrekturanweisungen, Duden-Taschenbuch Band 5',
Bibliographisches Institut, Mannheim, 5th ed. 1986):

Letters denoting variables should be set italics,
especially vectors in (semi-)bold italics.
Also in italics: functions and operators without fixed meaning.

Upright type should be used for digits, parentheses and all
symbols with fixed meaning. That goes for all multiletter symbols as
sin, lim etc., for \sum and \prod, then for constants as e, pi, i
and Euler's constant (gamma or C), but also for all differential
operators like d and \partial, and for operators like
\nabla and the Laplace operator (an upright Delta), etc.

The Duden book her comments that the last few symbols mentioned which
only consist of a single letter are often set in italics, especially
in books intended only for mathematicians
(rein mathematische Fachliteratur' -- I couldn't think of a better
translation for that at the moment).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In my opinion, this shouldn't be the rule but the exception, as
typesetting according to these rule greatly enhances the readability
of mathematical texts, even more so as most math books aren't read
as a whole -- jumping into a text is much easier with good typography,
which relies as little as possible on the context for the understanding
of the used symbols.

Setting function names in upright type isn't that important (or could
be dropped at all) as functions are recognizable as such in most
cases by the following parentheses and arguments (thus usage of
e.g. \mu(...) as the Moebius function only througout a text will
always be clearly understandable, whether set upright or italic),
but single letter symbols for operators or constants could be
easily confused with other symbols, if not set upright
(e.g. the imaginary unit with an index, e with a unit or eigen vector etc.)

Also, typesetting function names in upright may hamper the good
effects of the upright-type-rule, e.g. pi(x), the function counting
the primes upto x, may be confused with the circle number pi,
when both are set upright.

Fixed meaning' as mentioned in the rules above means (in my opinion)
a meaning fixed throughout a whole text or even fixed in a whole
branch of mathematics.

>   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.

And as for TeX: TeX/LaTeX should have the needed glyphs (e.g. upright
lowercase Greek) and should have standard control sequences to follow
the above rules (in some cases even more than one standard way to
access a symbol). Examples and propositions:

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

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.

BTW, what about the TeXnical working group on extended math font
encoding? Haven't heard about it since that article of
Justin Ziegler.

Johannes Kuester

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