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At 13:34 +0200 2001/05/22, Lars Hellström wrote:
>The conceivable limitations this would impose (and you still haven't
>produced a single example of a published paper in which there would have
>been any limitation at all!)

If you want examples, I think it is best to inquiry in say the newsgroups
sci.math, sci.math.research. Greg Kuperberg
to the xxx.lanl.gov archive -- perhaps you can check with him.

> are negligible

The past experience with LaTeX is that mathematicians that did not like
whatever limitations there were did not use LaTeX at all. A lot of work has
been done so that mathematicians should be able to feel comfortable with
LaTeX.

> in comparison to the
>limitations posed by the blackboard as the primary medium for new
>mathematical notation and the fine motor skills of the average
>mathematician. If you don't believe this, you can try the following
>experiment:
>
>1. On a blackboard, using a piece of chalk, write down the calculations
>showing Jacobi's identity
>$$> [[\phi,\emptyset],\varnothing] + [[\emptyset,\varnothing],\phi] + > [[\varnothing,\phi],\emptyset] = 0 >$$
>where $[a,b]:=ab-ba$ is the commutator, the underlying ring is associative
>but not commutative, and using precisely those glyphs from Computer Modern
>to denote your variables (you've claimed yourself that they can be used to
>denote different quantities).
>
>2. Convince another mathematician that it is possible to see which symbol
>is which without relying on the structure of the calculations.

I have no idea what you are trying to prove here: The handwritten math and
the typeset math are entirely different media, and they are not
exchangeable.

Also, some mathematicians would today use only TeX and overhead pictures,
and would rarely use the blackboard at all.

Further, in the days of typewriters, one would use some kind of markup,
like different colors, or various forms of underlining, in order for the
typesetter to be able to select the right typeset glyph. All that is needed
is some kind of translation table.

So use whatever is legible in handwriting, or on the blackboard, and then
use a suitable translation table for the typeset output. If you give talk
in math using a blackboard, it is common that the notation is invented as
the talk is going on: One checks that the audience can follow what one is
speaking about. It is not even the case that what may work in handwriting
will work on a blackboard or vice versa.

Otherwise, if you want examples from differential geometry, I use a
different notation for the Levi-Civita connection on a Lorentz manifold (as
in general relativity) and the "del" used in physics as applied to
three-space: The latter has some fattening like an inverted uppercase delta
that the former does not have. But neither are boldface. I recall that I
designed the former as a special glyph. I made this choice in order to
conform to the traditions in the different fields differential geometry and
physics.

If you want to use \emptyset and \varnothing side by side, I have no
problem in conjuring up a possible example: Say a paper in denotational
semantics which uses math to denote \varnothing to denote the empty set.
Then \emptyset could be  used to denote a polymorphic variable pointing to
an empty object.

Or suppose one is studying grammars, where \epsilon is used as to denote
the empty transition; then it would be natural to use \varepsilon if one
say is using analysis to study complexity, or for some other use.

Whatever: When one starts to combine mathematical fields, then it suddenly
becomes difficult to find good symbols.

Hans Aberg