sentences

Here is the latest entry submitted to PlanetMath as of this moment.

http://planetmath.org/encyclopedia/UniversalNetsInCompactSpacesAreConvergent.html

Anaphora seems to comes up a lot in math. This is in the usual
"textual" way --

E.g. in this definition, ($(x_\alpha)_{\in \alpha \mathcal{A}}$
refers to the same thing whereever it appears, and also sets
notation like \alpha or \mathcal{A} which will be used seperately
later.

and also in a "hypertextual" way

What the heck is a "universal net" anyway, and what are its
properties? -- properties of these and other defined terms will be
needed by a reader who wants to follow this theorem and proof.

On PlanetMath, hypertextual ultra-long-range references are dealt with
in interface by hyperlinks to entries that define or discuss the terms
being linked. In typical mathematics writing, these links aren't
hard-coded but are "implied" (i.e. the reader is expected to be able
to find or just know the definition on their own).

Combining these two features, the author feels free to introduce terms
like \alpha_0 which have not been explicitly defined anywhere.
Understanding what this sort of conglomerate symbol is supposed to
mean requires some "domain knowledge". Really, this goes for what
happens when any locally scoped definition is made (as in "for every
$x\in X$ we would find neighborhoods $U_x$" -- the reader is assumed
to know what a "neighborhood $U_x$" has to do with an element $x$).

These are probably the main linguistic issues. Math text is written
in a "compressed form" and can't be understood without background
information.

Finally, after texts like this are translated into a computer-readable
form, knowledge management-style inference will become possible and
important. This can feed back into the linguistics level, e.g. for
example some author might use the property that "a universal net in a
compact space is convergent" without explicitly stating that they used
this property! The reader would have to infer that this theorem was
applied.

This reminds me of one more feature. Generally, words (like
"convergent") mean different things in different parts of mathematics.
We can have convergent sequences, series, sets, and now (apparently),
nets. Also graphs, etc. -- but you'll see that PlanetMath's
autolinking software linked the term "convergent" in this theorem to
the page "convergent series" via an alias Converges Absolutely. Point
being: there is no mention on this page of what it means for a
"universal net" to converge. An autolinking program that was enhanced
with more semantic (domain) knowledge would perhaps indicate this gap.

Certainly, if semantics matter for understanding math (e.g. we want to
know what it means for a universal net to converge, so we either have
to be told, or we have to figure it out from whatever we've been
given), an ongoing process of identifying and filling gaps in the
knowledge base will be necessary. The extent to which this process
can be bootstrapped remains to be seen. (Are easier mathematical
texts any easier to translate into a nice computer-ready form? I'm
not sure.)