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Hans Aberg wrote:

> >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.

Okay, but there are constant function names as \mu for Moebius function,
\phi for Euler function (totient) etc., but I think these shouldn't
be set upright. May be the best example for this is \pi(x), when
denoting the prime counting function (thus constant or with a fixed
meaning) versus \pi, the circle number. Here \pi should be set
upright in both cases, according to Hans Aberg, whereas I suggest not
to use the set in upright rule' to functions, whether constant or variable.

There is another rule here to which most authors/typesetters don't obey
but which reduces the confusion between functions and other symbols: i.e.
using a little space left of a left parenthese, if multiplication is meant;
in TeX notation e.g.
$f(x)$   function f with argument x
$f\,(x)$ f times x
(Silly example, but imagine a  more complicated expression instead of x'

>   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.

Yes, but may be TeX shouldn't just stop in trying to obey to old traditions.
Of course one has to know the traditions, but one also has to consider
how they developed, and I don't think that the TeX community should
confine itself to obey to rules which were mainly established because
of lack of appropriate type. The fear expressed by DEK in the METAFONTbook,
that mathematicians might go out and create their own symbols, hasn't
come true at all. Mathematicians are even too lazy to think of new
symbols in cases where they are badly needed; rather they tend to stick
to old traditions, whether mathematically necessary or not.

Of course, for new symbols, there should be some qualifications,
e.g. they should be international, mnemonic, consistent with the
rules for typesetting, consistent with old symbols in the sense of
not being too different from them (may be) etc.

>   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?

Excuse me? I don't think that they ever were variables.
Rather mathematical typography wasn't that developed at that time.
Most of the traditions of mathematical typesetting aren't that old.
(may be we all should go to the library now and have a look and
Euler's Introductio in Analysin Infinitorum'...)

>   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... :-) )

Setting in bold italic is consistent to the rules (as most vectors are
variables), vectors and their components are more closely tied together
typographically (both in italic, bold and normal), and uppercase
bold italic could be used for matrices then (again consistently).
Bold upright couldn't, as this could be confused with number set symbols.
I think the usage of bold upright is just due to lack of bold italic,
nothing else!
And as for the pure mathematicians: It just depends on context.
I would use bold italic only for vectors in \mathbf{R}^n or \mathbf{C}^n
(n-dimensional real or complex spaces), may be in geometry, too.
It's not useful/needed in e.g. algebra, when all scalars are written in greek
and all vectors in latin: most occuring variables are vectors and
easily recognizable as such, so why treat them specially?

>   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.

Yes, of course, but standard control sequences do help (e.g. exchanging
files via e-mail, reading other people's TeX sources, composing a
book of articles from many differt authors,...). I think there should be
standards, for math, it could be a package math.sty', may be with
a lot of options to include control sequences for different branches
of math and with language-specific switches, may be with options
to short some control sequences (silly example: the standard could
be \numbersetN' to get the symbol for natural numbers, by the option
numsetshort' one could use \N, and by another option the actual typesetting
of this symbol is determined).

[Upright Greek]
>
>   I think this is already covered by NFSS (at least in text mode): You only
> have to find the fonts.

As they don't exist yet...
It could be treated be NFSS then, of course.

>   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.)

I've prepared fonts with some of the needed symbols for my private
use, but there still should be this TeXnical working group...

[Slanted Fraktur/Upright Script]

>   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.

Okay. But I still think usage of italic/slanted and upright type
should be confined to roman type, as it might be confusing rather
than helping with other styles. And that there is no such thing
as slanted Fraktur is IMHO a good tradition, as it would look
disgusting. But may be it could be useful in math...

>   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?

Well, may be that would go against the traditions again...
Besides, the Euler fonts were designed for the concrete' font family,
and I do dislike them only because of their lack of slanted/upright
distinction. This is badly missing in the concept of the whole
concrete' math fonts.

Johannes Kuester

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