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Re: small questions regarding integrals



Thanks for your response, Dave.

> Multiple integrals can be handled in several ways:
> ...

I see now that "integral" is indeed a symbol and not an operator.
I was taking it for the latter, but should have read more carefully.


>  > 2. Is there a specific reason we don't want to use the TeX-like
>  > notation: \int_{...}^{...} ... ?  (because the _ and ^ operators
>  > become "too" overloaded?)
>  
>  The benefits of an extensible language is that we don't need to
>  decide this now. One could specify `_' as a lowering operator
>  and `^' as a raising operator that when applied to integral signs
>  denote limits of integration. This is a matter for the rules that
>  map the stream of lexical tokens into a semantic description of
>  that input. These rules can be extended by authors.

I think my question was unduly imprecise.  You suggest using the
"from" and "to" operators as primitives of the language, I believe,
and say also that they are "examples of operators that are only
applicable in limited contexts".  I take the "only applicable" to mean
that the semantical parser will discard certain possible parse trees
as a result, not that "misuse" will cause display to fail.

I'm also using the TeX _ and ^ operators as the generic "lower" and
"raise" operators (as you do in the case of ^).  My feeling is that
mathematicians cast their various Rorschachs onto a rendering medium,
and through complex and subtle cues they are able to communicate with
one another.  In this sense, "lower" and "raise" are the primitives,
and "from" and "to" are overlying conventions which apply in some
cases.  So I would allow someone to specify "from" as "lower" and "to"
as "raise", and require the semantical parser to generally interpret
lowered and raised indices on integral signs as limits of the
integration.  I think this is the reverse of what you're suggesting,
but I may be misinterpreting again.  My question meant to be asking
whether it would be ok to use ^ as you do with "(a+b)^2" and also as
the primitive mechanism for denoting the upper limit of integration.
I take your words above to be saying that one *could* extend the
language so that ^ had both "meanings", but that the primitive
language does not conflate the two.

Or perhaps we're saying the same thing.  If display is never an issue
with "from" and "to" (i.e. the display comes out with side-saddle
subs and sups no matter what), the only issue becomes the semantical
interpretation, and I understand that the two notations fail in the
same situations (i.e. wherever the semantics engine doesn't accommodate
embellishments of this type around a symbol).  I do think the "raise"
and "lower" language is easily understood and is more neutral, so I find
it preferable as the primitive.  And in general, that juxtaposition of
symbols in a discrete rendering space should be the primitive "semantics"
of the language, with higher-level semantics laid over that.

And I don't mean in this to be heavily visual.  The Rorschach tests can go
on in any medium.

-Ron


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