There are a couple of different ways in which that blandly
even-handed conjunction,

*Language and Math*, can be desymmetrized. As:(1) The subject-matter is language, seen in a mathematical light. This leads to ‘mathematical linguistics’, about which we have little to say.

(2) The subject-matter is
mathematics, seen from a linguistic perspective.

[For a similar symmetry-breaking or more properly ‘duality’,
compare the title of the engaging recent book by Edward Frenkel:

__Love and Math__. The book is mostly about the love*of*math, but with an impish cinematic excursus about the mathematics of love… ]The latter endeavor again splits, along the familiar fault-line of the

*actio/actum*distinction :

(2a) We examine

*the way actual mathematicians talk*in their subject, quite the way francophone linguistics must be based on the way actual Frenchmen talk.

(2b) We consider the sheer

*subject-matter*of mathematics, in complete independence from the quirks and foibles of present-day mathematicians, and examine the ways in which either any ideal approach to this matter, or perhaps even the structure of the matter itself, can or must be seen in a perspective of ‘language’ in some relevantly extended sense.(2a) is a lot of fun; I have been collecting examples over the years, of syntactic and semantic phenomena peculiar to this field of discourse, which perhaps can be shared someday. There is more intellectual substance to this exercise than in the usual dialectological study (that is to say, inventorying the predictable quirks of this or that regional patois; that of a

*Geistesgemeinde*is another matter entirely, and forms the backbone of my own corpus-based

__Dialect Notes__, available on the high side), but it is of no importance either to linguistics or to mathematics as theoretical disciplines.

(2b) has grown in core importance, beginning
peripherally with the obtention of clarity on non-Euclidean geometries, and
becoming foundational with Russell-Whitehead and later GÃ¶del.

Thus, a typical example, from the introductory paper in a symposium volume commemorating GÃ¶del’s 60

^{th}birthday:

From the viewpoint of a realistic philosophy of mathematics, the incompletability theorem can be regarded not as calling into question the independent reality of mathematical entities such as sets or numbers, but rather as indicating an essential limitation in

**the expressive power of symbolism**: the limitation being that no symbolism can fully succeed in characterizing a system of objects as rich as the natural numbers.

-- Stephen Barker, “Realism as a
Philosophy of Mathematics”, in: J.
Bulloff et al, eds.

__Foundations of Mathematics__(1969), p. 4That paper is disappointingly brief and even shallow, and will not be considered further. But it does raise a semiotic issue -- only, one not restricted to mathematics. Our natural language is, after all, inadequate for discussing

*anything*that really exists out there independently of ourselves: such as (to take one plump example), a

*penguin*. Philosophically incurious beings that most of us are, we do not notice how tongue-tied we really are, when it comes to expressing anything beyond a few platitudes; only in new technical areas do we become semantically self-conscious (“Is it a

*wave*? Is it a

*particle*? Is it neither? Is it both?”) But try as we might, we shall never manage to express the essence of The Quintessential Penguin.

[Note: The paragraph quoted above is Platonist, in that it assumes that something can exist, even if we cannot name it -- here, to be sure, in a more sophisticated version of that truism. For essays relating to mathematical Platonism, click here:

http://worldofdrjustice.blogspot.com/search/label/Realism

For a use of the notion “can exist even if we cannot name it”,
compare Quine on substitutional versus objectual quantification. He personally is inhospitable to
the real existence of things we cannot name: such is the impoverished moonscape of the

**Nominalist**. ]
*

For further musings
from this pen,

check here:

*

~

Some tidbits re (2a):

**The way mathematicians use language**.Obviously, mathematicians coin new terms (like “tensor” or “Hilbert space”), and use extant words in specific new ways (“function”, “space”, “point”, etc. etc.); these are explicit, and are not our concern, for they differ in no respect from the

*termini technici*of any other field, be it biology or rock-and-roll. Rather we focus on the semantic

*Akribie*characteristic of the mathematical mind. That is to say: While mathematicians are utterly at liberty to posit their own jargon, just like folklorists (“oikotype”) or stamp-collectors (“tÃªte-bÃªche”), what is striking is

*their fussing and fretting*over their own patois. They are aware of subtle semantic pitfalls, and are at pains to be properly understood.

For example: Here
a mathematician writing carefully for the general public, manages to make points usefully accessible to both
laypersons and professionals:

The term “

**Lie algebra**” is bound to create some confusion. When we hear the word “algebra”, we think of the stuff we studied in high school, such as solving quadratic equations. However, now the word “algebra” is used in a different connotation: as part of the indivisible term “Lie algebra” … Despite what the name suggests, these objects do not form a family in the class of all algebras, the way Lie groups form a family in the class of all groups.
-- Edward Frenkel,

__Love & Math__(2013), p. 119
These “

**quantum fields**” have nothing to do with “number fields” … This is another example of confusing mathematical terminology, though in other languages there is no confusion: the French, for example, use the word “champs” for quantum fields and “corps” for number fields …
-- Edward Frenkel,

__Love & Math__(2013), p. 269
Or again:
A mathematician writing a textbook, not for the general public to be
sure, but still to an audience wider than that of professional mathematicians
(his Introduction states that the book is aimed at engineers and physicists) :

The procedure we have followed is typical of Cartan’s method of the
moving frame.

**In terms of the jargon**, what we have done is to reduce the structure group of the tangent bundle of M restricted to N in a “natural” way to a subgroup that is small enough to enable one to define an induced affine connection on N.
-- Robert Hermann,

__Differential Geometry and the Calculus of Variations__(1968), p. 384
This self-deprecating passage occurs, note, almost
four hundred pages into a dense text -- a bit late to be worrying about whether
your readers are scratching their heads at the unfamiliar lingo.

Actually the language here is no more “jargon” (in any
negative sense) than the technical expressions of any discipline, from biology
to dentistry. What does partake somewhat of the
in-group

*patois*that interests us here is the use of the word*natural*; that the use here is special in a way that that of*tangent bundle*is not, the author signals by putting the word in quotes: he is using it proto-technically, intuitively, informally. Now, by our own day, a refinement of this special use of the term*natural*has received something like a strict characterization in the framework of**category theory**: such, however, is beyond the horizon of the man in the lab. (Or it was in 1968; the whole field has been developing dizzyingly.)
A large subset of mathematical terms (and grammatical terms,
for that matter) arose in just this way, starting out as words taken from
everyday discourse, and extended in a semi-metaphorical sense, whose outlines
would become clear only with further thought and the passage of years.

**(I) Lexical Semantics**

**(Ia) Explicit**

Characteristic of some is a particular care for precision,
for laying underlying vagueness bare; we gave some examples in our appreciation
of our revered late teacher

__Andrew Gleason__. Here a noted German mathematician (I give the English translation) gives at first a couple of words that some people use interchangeably in this context, but then footnotes the use, by way of a comment with genuine mathematical content:
[This] is known as the problem of
the

**solution**or**integration**of the system of differential equations. -- [Re the latter, the first being unproblematic:] This word is used because the solution of such differential equations may to a certain extent be regarded as a generalization of the process of ordinary integration.
-- Richard Courant,

__Differential and Integral Calculus__(translation of__Vorlesungen Ã¼ber Differential- und Integralrechnung__, 1924), 1936, vol. II, p. 414
“To a certain extent”, “be regarded as”: typical conscientious caveats. (For similar examples, seen from a
lexicographic perspective, try this:
“

__What is mathematics__?”)“Man soll ganz klar darÃ¼ber sein.” |

There, the author was defending the extended usage, while
first critically noticing it. In
the following, by contrast, the author first notes the current usage, then
throws up his hands:

The

**torsion tensor**of a connection is a vector-valued function that [blah-de-blah].
(Note: As far as we know, there is no nice motivation for the word “torsion”
to describe the above tensor. In
particular, it has nothing to do with the “torsion of a space curve.”)

-- Noel Hicks,

__Notes on Differential Geometry__(1965), p. 59
(Nor, we might add, with the notion of a “torsion group” in
algebra.)

~

There is an action of

*G*on the underlying vector space of**G**that is also called the*adjoint action*of*G*. (Strictly speaking, it should be called the infinitesimal version of the adjoint action of*G*on**G**, but it is customary to confuse this point.)
-- Robert Hermann,

__Differential Geometry and the Calculus of Variations__(1968), p. 90
This expression “confuse this point” is a nonce equivalent
of the traditional French “par abus de langage”.

~

The functional < ,
> is called the

*inner product*, the*metric tensor*, the*Riemannian metric*, or the*infinitesimal metric*. Notice that the word “metric” in the preceding sentence is not referring to a metric function (distance function) in the topological sense.
-- Noel Hicks,

__Notes on Differential Geometry__(1965), p.**(Ib) Implicit**

In this section we consider words not restricted to mathematical use, and whose use by mathematicians resembles the ordinary employment of the word, but in certain characteristic ways, peculiar to the subject. We are not concerned with everyday words which happen to have a technical mathematical use (of tenuous, or null, semantic connection to the extramathematical), such as “function, normal, regular, group, connection, delta, derivative, manifold, category, set, sheaf, kernel” (etc etc etc): for, linguistically and intellectually, these are of the same status as such purely mathematical terms, with no use outside the field, as “homotopy, Wronskian, Jacobian” etc.: and these in turn are of no general interest, any more than the termini technici of any field whatever (“phoneme" for linguists, “cantus firmus” for musicologists). Rather, we wish to bring gently to light, in the spirit of a philologist or literary critic dealing with some archaic or Delphic text, usages that we might term

*crypto-mathematical*: usages (especially of verbs and adverbs) which mathematicians themselves might not recognize as being special to themselves, and which laymen would puzzle at but not assume to be some sort of technical term which they could look up in a mathematical dictionary. That concept is sociolinguistic, not mathematical or narrowly lexicographic: common-coin inside the community, unfamiliar or misunderstood outside.. Every coterie has such things.

The Jacobian which occurs in the
denominator of both fractions is
one whose

**nonvanishing**will be sufficient to ensure that the equations really do have a solution …
-- Creighton Buck,

__Advanced Calculus__(1956, 3^{rd}edn. 1978), p. 417
Note that “nonvanishing” is here grammatically a noun. Its adjectival use is very common
in mathematical writing, and mildly jargonic; this substantivation really is special.

Related to this:

Lemma. If f and g are linearly dependent differentiable functions,
then their Wronskian

**vanishes identically**.
-- G. Birkhoff & G-C. Rota,

__Ordinary Differential Equations__(1962), p. 29
There might be some lay use of the adverb
“identically”; can’t think of one
offhand (probably a mere intensive);
so that the casual non-mathematician, reading that passage, might find
it oddly phrased, but not be in a position to place his finger upon the
oddity. What is it doing
here? It is

*kind o*f an intensive, but in a precise (and extremely interesting) sense. To say that a function (be it the Wronskian, or the Penguinian) “vanishes” at some point, is simply to say that it equals zero at that point. To say that it vanishes “identically”, means that it vanishes throughout its domain, that it is "identically zero" or “everywhere zero” (as a synonomous piece of cryptomathematical patois has it).
A subtle parallel is exemplified here:

Two such manifold structures that give rise to the same topological
structure must

**coincide**.
-- Robert Hermann,

__Differential Geometry and the Calculus of Variations__(1968), p. 81
This means, not that they must intersect

*somewhere*(be equal at some point or other), but that they must intersect*everywhere*-- be equal “identically”.
What is of general interest here, beyond the lexicographic
facts (fun enough in themselves for wordlovers), is the rich intellectual world
that underlies such talk, and is presupposed thereby. In the present case, compare the notions of “pointwise”
versus “uniform”.
Mathematicians will know immediately what I mean; nonmathematicians will have no clue,
nor can any three sentences explain it.
But at the base of it lies as conception -- itself a particular instance
of the very rich (even

*linguistically*rich!) subject of the scope of quantifiers (“there exists … such that for all” versus “for all … there exists”) -- which is very much worth your while adding to your cognitive armamentarium,*messieurs les poÃ¨tes et ingÃ©nieurs et Ã©crivains*!
~

The definition of a tangent
vector

**generalizes**the “directional derivative”.
-- Noel Hicks,

__Notes on Differential Geometry__(1965), p. 5
Such a map

**induces**a linear transformation on each tangent space.
-- Noel Hicks,

__Notes on Differential Geometry__(1965), p.
The above computation

**exhibits**the chain rule and a multiplicative behavior of Jacobian matrices.
-- Noel Hicks,

__Notes on Differential Geometry__(1965), p. 10**Trivially**, a connexion-preserving map is geodesic-preserving.

-- Noel Hicks,

__Notes on Differential Geometry__(1965), p. 60**(II) Syntax**

Mathematicians are given to chiseled concision.

Thus, in a proof-by-contradiction, we arrive at the final

*modus tollens*step:
… [implying that] the sequence

*T*(f_{i}) can have no convergent subsequence, contradicting the compactness of*T*.
-- Lynn Loomis and Shlomo
Sternberg,

__Advanced Calculus__(1968), p. 265
I.e., contradicting the statement “T is compact”, which was
the proposition taken-as-true which launched the proof (as opposed to the
temporary contrafactual assumption that such&so, which has just been
refuted).

**(III) Notational Nicety**

Many writers of math textbooks take great care with their
expression, not only lexical but symbological. The result is intellectually hygienic.

We shall use ∂Î£ as a notation for Î“, rather than bdy(Î£), to emphasize the
fact that we are dealing with both curves and surfaces as mappings rather than
sets of points.

-- Creighton Buck,

__Advanced Calculus__(1956, 3^{rd}edn. 1978), p. 417
It is customary to use the same
symbol, say,

*A*, for the matrix as for the transformation. … We do not follow this custom here, because one of our principal aims, in connection with matrices, is to emphasize that they depend on a coordinate system (whereas the notion of linear transformation does not).
-- Paul Halmos,

*Finite Dimensional Vector Spaces*(1958), p. 65[Update 19 Jan 2014] Try further the latest essay:

Bonus quote:

Alpha: While you are increasing content, you develop ideas, do
mathematics; after it you clarify
concepts, do linguistics.

Mu: Not mathematics versus linguistics again! Knowledge never profits from such
disputes.

-- Imre Lakatos,

__Proofs and Refutations__(1976), p. 99
## No comments:

## Post a Comment