Re: semantics

A few quick comments ... sorry I can not be more involved.

> Mathematics is a formal science.  

Yes and no.  For researchers it is as much an art as a science.  Their
formal representation and understanding of structures is highly fluid
and tentative.  It is quite inappropriate to attach rigid semantics to
this type of work.  Note also, there are many parts of 'mathematics'
that are not developed within the logical framework of some set theory.

> The reason for putting off the problem is that we (mathematicians and
> people who employ mathematical discourse) work in a naturally
> efficient way: discursive traffic proceeds quite well without
> specifying semantics.

Again not at all true for researchers.  A common question at conferences
is, 'What did you mean by that?'.  The semantics behind a concept are
more linked to an informal intuitive model, not just a formal
statement.  In addition there may be many formal models for any formal
theory.  Do semantics reflect the principal intuitive model or the
spectrum of non-standard models?

One reason for not attempting to tie a notation to its intended
semantics is the danger of the converse: tying a semantic concept
artifically to some notation.

Does introducing semantical bindings serve the evolution of knowledge? 
No ... it artifically introduces a straightjacket around current

> So my own attitude toward semantical specification is that:
> a. Yes, it can be done to a variety of depths for individual situations;
> b. The wider the situations become, the more difficult the teasing is; and
> c. The more detail we require in our semantics, the harder it is to
   understand the analysis and the more vocabulary we have to learn.
> These attitudes toward semantics hold for all areas of knowledge, not
> just mathematics.

Thankfully not ... human thought generates a far more interesting and
ill defined mindscape.

> The job is more manageable for a
> standard text on first-year calculus.  

But again limits diversity and innovation ... math is about thinking not
just calculating.  

> The questions for our
> group members may be:
> 2. How can we leave some of this to OpenMath so that we can attend to
>    other details?

What do you see as attractive about the work of OpenMath?

> 3. How can we accommodate both lower-end texts which may use rather
>    straightforward semantics and high-end research information whose
>    semantics is unsettled and much more complicated?

Unsettled it is ... leave it alone - don't introduce any explicit
semantics.  Give researchers a mechanism for introducing their own
semantics dynamically (eg. Mathematica's InterpretationBox) and taking
advantage of the semantics of particular systems (<math
type=Mathematica>).  Give them a rich system for expressing the
notational representation.

Follow-Ups: References: