At 19.17 +0200 2001-05-17, Hans Aberg wrote:
>At 18:59 +0200 2001/05/17, Lars Hellström wrote:
>>>The \epsilon and \varepsilon are definitely semantically different in
>>>(pure) math,
>>
>>Are they? Please give a verifiable example!
>
>What do you mean? I started with grad math in the middle of the seventies,
>and I have never myself or met any other thinking of them as being the same.
>

Perhaps you having started at that particular time (during the
typographical night when the typewriter reigned) is where the problem is to
be found; we've been talking about different things. Considering that:

At 19.36 +0200 2001-05-17, Hans Aberg wrote:
>
>Here are two examples:
>  $\epsilon\varepsilon R$
>  $\varepsilon R$
>
>Today one would probably write one of
>  $\epsilon\in R$
>  $\varepsilon\in R$
>(perhaps with $R$ in black-board bold).

With TeX, the set membership relation symbol has _always_ been \in, not
\epsilon, but appearently you haven't used it enough to take that
distinction for granted. I'm not arguing that \in and \epsilon should be
identified (as they are clearly semantically distinct), I'm arguing that
\epsilon and \varepsilon (which are both greek letters called "epsilon"
when a formula is read aloud) should be identified in the internal LaTeX
representation of math characters.

It is true that they have distinct code points in Unicode, but so does many
other glyphic variants of math symbols (such as U+2208, ELEMENT OF, and
U+220A, SMALL ELEMENT OF) for which there are no distinct representations
in LaTeX today. Furthermore the only other symbol which appears under the
same heading as the second Unicode epsilon (U+03F5 GREEK LUNATE EPSILON
SYMBOL, which looks like TeX's \epsilon in the code charts) is U+03F4
(GREEK CAPITAL THETA SYMBOL), whose only difference to the proper Theta
U+0398 (GREEK CAPITAL LETTER THETA) is that the horizontal line goes all
the way to the ring. Is that a reason to give this variant \Theta glyph its
own internal representation in LaTeX? I don't think so.

Recall that the Symbol font (which should rank as one of the more important
sources of mathematical symbols after Computer Modern) only contains one
epsilon glyph; thus you cannot in a math font set-up based on that font
provide visually distinct renderings of \epsilon and \varepsilon. Then it
is better have one command \epsilon and use some other mechanism for
selecting how it should be rendered, if the font set-up provides
alternative ways of rendering it.

Finally, a theory about the origin of the two epsilons in Computer Modern.
While doing some checks in pre-TeX literature on mathematical typography, I
came across the following piece of text ending a paragraph that discussed
how one distinguishes between (what would be TeXified as) \varepsilon and
\in:

  An additional complication is that, particularly in manuscripts in
  English, the sign $\epsilon$ does not necessarily mean $\in$, instead
  it can just as well mean $\varepsilon$, since this letter has a more
  ``grotesque'' shape in English typefaces.

[My translation and TeXification, assuming CM math.] With such a
traditional difference between English and, say, continental European
typographical traditions, it wouldn't be surprising if Knuth included
\varepsilon to please mathematicians who were accustomed to having the
epsilon letter and set membership symbol more distinct than \epsilon and
\in are in Computer Modern. Perhaps someone who has a copy of the book
Computer Modern Typefaces can verify or dismiss this theory.

Lars Hellström