semantics

[[
   I sent a copy of this to the html-math list earlier today, but
   have not received a copy in return.  Be aware that you may
   receive duplicate copies, and can afford to trash at least
   one of them.  Apologies for the clutter, if it occurs.

   -Ron
]]



Ping writes:

> It is a fairly fundamental difference in approach, yes.  I think
> that opinions vary about how difficult the semantic problem is
> (i think it can be handled a piece at a time and by people in
> the appropriate areas of expertise, but someone on www-html has
> dismissed it as virtually impossible).
>
> ...
>
> I'm curious to know in more detail what you mean by your last
> sentence.  In your mind, what have you "ruled out" and "not ruled out"
> among ideas from MINSE?

Discussing pros and cons about semantical markup is a Difficult Thing.
Those who think it's an intractable problem are in the awkward
position of saying something "can't" be done, but lack a formal proof
for the assertion and, as a result, are almost always unconvincing to
those who intuitively feel that semantical markup is the proper way to
go.  It may be that the Can'ts approach the problem from Without
while the Cans approach it from Within.

This said, I'll take a little walk around the rim of the Abyss.
I think long-windedness will not help at all.  I do have the feeling,
though, that, much as we'd like to finesse the issue, it does divide
us to some degree as a group since we're all trying to accommodate
semantical markup, but probably don't have common notions as to what
we're trying to accomplish.

Mathematics is a formal science.  One can take various attitudes
toward mathematical meaning, but a recurring theme is that the
meanings of terms lie in their positions within formal theories.  This
is meant to imply that "meaning", whatever it is, attaches
significance to a wide variety of instances and that meanings don't
come alone, but rather within collections of other meanings.

"+" might "mean" the addition operator of Peano arithmetic (considered
as a first- or second-order formal system or a part of set theory or
category theory).  It might also "mean" the group operation in a class
of commutative groups, a situation wherein we eschew talk of formal
systems and rather speak of a class of Platonic models.  Now Platonic
models might only be considered real insofar as they're specified or
instantiated within some more fundamental foundation (such as set
theory), and formal foundational systems might only be considered as
abstract means of discussing collections of models.  With the advent
of computer algebra systems, we can also arrive at a semantics which
says "+ means whatever Mathematica `thinks' it means".  My point here
is that we have a number of ways of closely (or not-so-closely)
specifying notions of semantics, but I suspect we all adopt a rough
and ready attitude of putting close specification off until we really
have to use such a thing.

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.

  You'll enjoy reading the book.
  What do you mean "read"?  How do you regard this as a "book"?

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.

So what does one expect in attaching semantics to HTML-Math?  We
clearly must deal with what the OpenMath folks have called "contexts".
I don't claim to know what the word means in detail, but I believe the
idea includes both "areas of discourse" (such as Algebraic Geometry
and Real Analysis) and "formal theories" (elliptic curves over C,
...).  The collection of "contexts" covers the subject of mathematics
as neighborhoods on a manifold in that contexts sometimes contain one
another, sometimes overlap, sometimes are disjoint.  We expect that
there are a great many of these, and that to be manageable, software
will have to help in telling us what's available, how we've named
them, what their relations are.

Contexts are also not enough in themselves since we work with
"objects" within these contexts and "understanding" (be it by a human
reader or by software) must discern the meanings or designations of
the objects.  Objects may be at the level of the "+" in a commutative
group or "an analytic function from C to C" or a "map from X to M
representing an nth cohomology class of X with coefficients in M".
Generally, one might expect that the collection of semantical
indicators of a given math notation snippet will contain both
identifiers of "classically understood" mathematical objects and
identifiers of "ad hoc" or "bound" objects ("bound" by a quantifier
such as "all" or "some"; one might employ an argument about all
continuous functions, f, and then use the f as an indicator of any
such function) peculiar to the snippet.  In widening our area of
interest from a snippet to a paper to a book, the number of "ad hoc"
semantical specifications also increases, the degree depending upon
the specific language of the paper or book.  Research mathematics
tends to include many ad hoc objects, as well as objects which might
become classically understood (i.e. part of some formal theory) at
some point, but are not now.  A K-12 text probably contains only
classically understood operators and some relatively simple type
specifications (e.g. "n is a natural number", "p is a polynomial over
Q" (which may well turn out to really be "p is a polynomial over any
field")).

It is clear, I think, that the job of doing semantics in full (to the
point of naming each object's context and attaching a type-theoretic
specification) is large for a 20 page paper published in one of
the AMS research journals.  The job is more manageable for a
standard text on first-year calculus.  The job of generating
specifications and software to handle semantics over all types
of mathematical discourse is also large.  The questions for our
group members may be:

1. How can we bite off something we can handle?
2. How can we leave some of this to OpenMath so that we can attend to
   other details?
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?

My own feeling is that we will do best by making our semantical
bindings as late as possible, when semantics is needed and when the
target is clear (why worry about whether p is a polynomial over Q or
some other field until we have to?).  Semantics disambiguates (by
definition), but at the cost of enlarging the notational base and
therefore disturbing the efficiency of abstract discussion.

This view must be moderated by recognizing that there are other
situations (such as producing a calculus text with Scientific Word and
Maple) where it is quite valuable to have semantics attached.  And
more generally, while the view expressed in the previous paragraph may
be appropriate for free-form discussion on objects which lie at the
frontiers of realms whose calculable aspects have not yet been
implemented, it misses the case where discourse is still original,
albeit within mathematical regions whose structure is conventionalized
and where software tools exist.

I do urge that, as each of us discusses need for semantics, we keep
in mind the various uses and difficulties for different types of
mathematical discourse, and that we comment on these targets as
appropriate.


-Ron