Perhaps we share the same biases, but I have been finding in my
recent work on the use of mathematics in science (part of my
studies of multimedia semiotics in scientific print articles
and computer media) that it is very important, in order to
understand how mathematics is integrated with verbal text, to
see that mathematics is itself a direct extension (mainly semantic,
but also operational) of verbal language. A look at the history
of mathematics, certainly European, but early Egyptian, Babylonian,
etc. too, shows this quite clearly. (It's harder to figure out
exactly what the nature of the key semantic extensions are. I have
some guesses.) The visual cues of a written mathematical symbolism
are meant, historically and in terms of much mathematical practice
to be interpreted in the context of a verbal, or at least a very
language-like set of syntactic and rhetorical patterns. To see
mathematical symbolism as a purely visual system autonomous from
language, either as a theoretical position or as a novice's
unconscious Ansatz, would be terribly misleading and confounding
I think. So you are surely right to try to get students to blend
the visual cues with linguistic interpretations of the meanings
of the symbols.
This is again not to say that language is dominant here. Mathematics
can say many things that verbal language (except as it is extended
in mathematical registers whose semantics and syntax are very
divergent from other registers) cannot. That is why it was invented!
And one can clearly "think" in mathematics, sometimes even with a
minimal reliance on the usual textualizations of verbal language.
But both the kernel (metaphoric, not Chomskyan) of language remains
in all mathematics, and the semantic strategies (typological categor-
ization, etc.). It is a really crucial case for study, not unlike
the key case of sign languages, though for different reasons.
(There is an interesting study in progress, using functional linguistic
paradigms, of classroom spoken and written mathematics, by Kay
O'Halloran at Murdoch University in Western Australia, not yet
published or completed.)
It is also important that mathematics evolved to enable us to
describe more continuous topological changes (beginning with
non-simple ratios) and relationships, of the sort that visual and
tactile systems deal with better than does language. But math
provides the bridge that enables us to integrate linguistic-
conceptual (typological) reasoning with visual-'quantitative'
(topological) meaning-making. Or so I believe.
JAY.
JAY LEMKE.
City University of New York.
BITNET: JLLBC who-is-at CUNYVM
INTERNET: JLLBC who-is-at CUNYVM.CUNY.EDU