Download Modeling math language

Modeling math language
Charles Wells email
This document is a preliminary discussion of issues concerning the possibility of modeling
math language using a categorical approach.
Mass nouns and count nouns
Mass nouns seem to be rare in math writing. I have done a little poking around the math
journals in JStor and have several observations to make. All are tentative observations based on
a small amount of evidence (and thinking).
“Space” as a mass noun
“Space” as a mass noun was common before WWII but is rare now. A search for “in space” (in
quotes to make it the phrase that is searched for) gives mostly references to outer space and to
very old papers, mostly before 1930. Conjecture: The disappearance of mass nouns in math
writing is a consequence of the rise of structural thinking in math.
One recent paper where “space” occurs seemingly as a mass noun, in the title no less, is: F. W.
Lawvere, Categories of Space and Quantity, in J. Echeverria et al. eds. The Space of
Mathematics: Philosophical, Epistemological and Historical Explorations, DeGruyter, Berlin
(1992), 14-30. However, the word “space” appears as a mass noun only once in the body of the
paper (according to my hasty scan) and many times as a count noun. Anyway I am not sure it is
being used as a count noun in the title. It is paired with “quantity”, which is surely an abstract
noun, not a count noun.
Areas of math as mass nouns
Areas of math are commonly used as mass nouns, for example, “Using calculus, we see that the
function has one maximum”, or “the result follows by straightforward algebra”. The language of
math contains several sublanguages with different uses (symbolic language, rigorous language,
“rich” language) and one of them is the metalanguage used for talking about doing math, as
those examples surely are.
A search for “in Boolean algebra” found only three citations since 1990 and they all seemed
to be references to the area of math. A search for “a Boolean algebra” found 45 references since
2000.
Mass nouns and plurals
In the paper La Palme Reyes M., Macnamara J. and G. E. Reyes (1999). Count nouns, mass
nouns and their transformations: a unified category-theoretic semantics, in Language, logic and
concepts: Bradford Book, MIT Press, Cambridge, Ma, 1999, pp 427-452, the authors say that
plural nouns are mass nouns, in fact they are the free mass nouns corresponding to count nouns
under the adjunction developed in that paper. (The Wikipedia article on mass nouns doesn’t
seem to regard plurals of count nouns as mass nouns.) Now plurals are mass nouns with atoms
(like “furniture” rather than like “water”). Of course, plurals occur all over the place in math writing.
Conjecture: In rigorous math prose the only mass nouns that occur are plurals, or at least are
mass nouns with atoms.
I am suspicious of the way Reyes, Macnamara and Reyes smush together mass nouns with
atoms (furniture) and mass nouns without atoms (water). (“Atom” means in the lattice of parts.
“Some of my furniture” can include a bed and two tables, but not the leg of a table. “Water” is
treated in language as if it were infinitely divisible. Of course it really does have atoms in the
physical sense.)
These two kinds of mass nouns behave differently in many ways. The most important is that
plural nouns can refer to either distributive plurals or collective plurals. (“All groups have
identities” is distributive, “the voters were in favor of the proposition” is collective.) I doubt that
these different kinds of mass nouns constitute a natural grammatical class.
Languages of math, registers, etc
Conjecture: Mathematical English (ME) and the symbolic language of math (SL) are two
distinct languages, not dialects of the same language.
I have asserted this in several places (Handbook, abstractmath.org) but I am not a linguist
and it could be that linguists would disagree with this conjecture, or that the study of a
mathematical corpus would reveal that another theoretical take on the situation would be more
appropriate.
I have listed some relevant points below. I intend to expand on them in later posts.
 Is ME a dialect of English or a register of English? Or does it have some other relationship
to English?
 ME appears to have several dialects or registers. One register is that used for what
mathematicians call “formal proofs”. These are not formal in the sense of first order predicate
logic, but their language is constrained, with the intent of making it easier to see the logical
structure of the argument. Another register is that of “intuitive [ or informal] explanations”. This is
more like standard English.
 The SL is clearly not a spoken language. It is a two-dimensional written language using
symbols from English and other languages and some symbols native only to math. People do try
to speak formulas aloud occasionally but this is well known to be difficult and can be done
successfully only for fairly simple expressions.
 There are other non-spoken languages such as ASL for example. I don’t know whether
there are other non-spoken languages that are written. I don’t think dead languages count.
Math English
Symbolic Language
These are two distinct languages.
Math english has dialects or registers or whatever.
Rigorous proof
Writing ABOUT proof
Rich discussion
Conjectures
The disappearance of mass nouns in math writing is a consequence of the rise of structural
thinking in math.
Examples
 People used to say, “
Sketches
Representations and total structure
Rich and rigorous
References
La Palme Reyes M., Macnamara J. and G. E. Reyes (1994). Functoriality and Grammatical
Role in Syllogisms. Notre Dame Journal of Formal Logic 35(1): 41-66 (1994)
La Palme Reyes M., Macnamara J., G. E. Reyes and Zolfaghari, H. (1994). Reference, Kinds
and Predicates. In Macnamara J. and G. E. Reyes (eds.) (1994), The Logical Foundations of
Cognition. New York: Oxford University Press. 91-143.
La Palme Reyes M., Macnamara J. and G. E. Reyes (1999). Count nouns, mass nouns and
their transformations: a unified category-theoretic semantics. In Language, logic and concepts:
Bradford Book, MIT Press, Cambridge, Ma, 1999, pp 427-452.
Wells, Charles, Sketches: Outline with References. Charles Wells, 8 December 1993
Wells, Charles, abstractmath.org
Wells, Charles, The Handbook of Mathematical Discourse. Infinity Publishing Company,
2003.
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