Monday, January 13, 2014

Language and Math

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 Gödel.

Thus, a typical example, from the introductory paper in a symposium volume commemorating Gödel’s 60th 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. 4

That 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: 
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 (“tête-bêche”), 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 über 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 darüber 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, 3rd 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 of 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 poètes et ingénieurs et écrivains!

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(fi) 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 ∂Σ as a notation for Γ,  rather than bdy(Σ), 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, 3rd 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

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