language, math, structuralism

June 24, 2011

Just a few thoughts.

I am, at present, fascinated by what I will call the role of language
in mathematics. In using this phrase I wish to call to my own
attention the similarities between [natural language, and how it is
used to imbue theory with structure] and [language in mathematics]. I
want to make the point that mathematical innovations are linguistic
innovations, and that this observation is important, though frequently
overlooked. Let us draw an analogy between the material reductionist
and the mathematical formalist. In the former, all utterances are
understood to be rough shorthand for statements in terms of atomic
particles or some finer physical stratum. In the latter, all
mathematical theorems and their proofs are understood to be precise
shorthand for statements and derivations in some atomic, axiomatic
system of symbols. While it is sufficient (and often necessary) to
offer a definition of a mathematical term by way of previously defined
terms, (i.e., on a tree of definitions rooted in the formal base language),
one immediate question following such a definition is why this
definition at all? We are lead to believe that “mathematics”
encompasses not only the naming of natural and relevant formal
constructions, but that assigning arbitrary definitions, and remarking
with arbitrary observations, is also mathematics, provided the
definitions are formal and the observations are proved. That is, the
essential in mathematics is its absolute precision, and not in the
particular choices of structure in the theory. And why not? after all,
who can account for particular choices? We call these *tastes*, their
justification resides in a wholly different realm than the
justification for a line in a proof. They depend on the whims of
complex and erudite connoisseurs, not on reason. But when a student
asks why the property of a subgroup H in G, that gHg^{-1}=H, is given
a name (let alone why the particular name), what can they be answered
with? The usual answer would go along the lines of “normal subgroups
are extremely important, ubiquitous, characterize kernels of
homomorphisms, provide for series which classify groups, (etc.)”. That
is, the student is given a structuralist justification, and not “it is
a matter of taste to consider this property important and not, rather,
some other arbitrary property,” which would constitute a very weak answer. I
argue the reason normal subgroups have a name, (versus, say, subgroups
which have order any of {7, 48, 184}) a) is because there are multiple
contexts where normalcy is referenced, and b) has as much to do with
the essence of mathematics as does the precision of the definition of
normal. Indeed, it is the appreciation for concepts such as “normal”
that leads people to mathematics, and not merely the precision, as is
typically perceived.

Another typical perception is that mathematicians invent words and
generate new questions along the path of solving a given problem.
Hence the endless pursuit that is mathematics. Indeed, this perception
is accurate and many a mathematician corroborates it to the outside
world. Related is the myth that generalization=improvement. What is
less often discussed, in the same discussion, is that *any*
philosophical inquiry leads both to new words and questions and to a
generalized view. This is because the vocabulary introduced today
becomes naturalized in tomorrow’s theory. And generalization, when it
happens naturally, is a sign of sophistication. You might work with a
group for some time, because it is the symmetries of some aspect of
this problem you are working on. At some point you identify some
interesting properties of this group. A collection of these properties
fit together nicely, and so you christen the collection of properties.
Now you cannot help but wonder a few things about ****** groups, can
you? The construct of being a ****** group not only has a formal
definition, but it has semantic value to you now. It has an identity,
one that can and should be explored. When I think of generalization I
think of the fact that the Library of Babel has volumes on various
theories (albeit mostly inconsistent and fictional). For every
coherent theory or novel or case study book, there are books which
reference these (although referencing one book from another presents
its own problem, doesn’t it?) so that the theory can be built up to
unimaginable structures (let’s just allow ourselves to adhere labels
to a few of the bindings). Already in the humanities you have a
running gag about names of papers such as the famous “Transgressing
Boundaries, Toward a Transformative Hermaneutics of Quantum Gravity”
(or make up your own). Yet, with enough time, one could very well be
interested in various hermaneutics of quantum gravity, transformative
and otherwise. Why not? But the important thing is not whether some
particular esoteric hodgepodge of theory *has* meaning, but whether it
has interest to you, at your current level of sophistication. As the
levels of sophistication are unbounded, at some point there are
volumes of theory which will be utterly inaccessible to any human. We
have no choice but to call such a text “meaningless.” On a similar
note, one does not begin group theory by studying a subsuming theory,
such as the theory of magmas, not only because it is difficult to
handle such generalization from the beginning, but because groups are
interesting on their own.

Finally, on a somewhat related note, I wish to point out that computer
scientists are frequently fluent in many programming languages and are
frequently at work designing new ones, to improve structural issues of
those existing. I wonder why we do not do this more actively with our
natural language. We evolve our natural language like the brain has
evolved, by packing on new to old, and by tiny discrete changes here
and there. Could our language be more expressive or better organized?
Here is an example: suppose I want to connote a relatively small
amount of some unit, given some context, and I want to be accurate but
not precise (meaning I want to say exactly what I mean, while
remaining vague). It would be ideal if I could attach the same
diminutive suffix to any dimension, so for example, lengthette, timette,
massette, forcette. Not only does no such suffix exist, but if I begin
an essay by “let ette be the suffix which connotes relative smallness
of the linear dimension to which it is appended,” well, let’s just
say this would be unorthodox. I imagine if enough people did this
language would organize itself a little more according to deliberate
design, as is arguably the case in mathematics.