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.