Cryptonyms : “up to” (re-upped)

A more descriptive title for this essay would be “multiword idioms, typically not recognized by lexicographers as deserving own-place headword entry paralleling that of any other multimorphemic lexeme”.   But cryptonym sounds snazzier;  and the meaning you’ll find online (“secret word”) is not actually used by those who deal daily with codewords and cover-terms, so we here use it as we see fit.

Our lesson for the day concerns:  up to.

As anyone learning a foreign language is well aware, and as speakers considering their own native language are not aware at all, the hardest words in any language -- the most difficult to characterize definitionally, and the hardest for a nonnative speaker to use with real Sprachgefühl -- are the shortest ones.  (This practically follows from Zipf’s Law.)  And when you have two monosyllables cheek-by-jowl :  Trouble!

~

One of the things that used to irk me when I worked at Merriam-Webster dictionaries, was their characteristic disinclination to deny own-place entry as a lexicalized, bold-face headword, to multi-word idioms, provided these were written with spaces between the words.   If written solid or with hyphens, they were welcomed as own-place entries:  nevertheless, devil-may-care.    This meant that some phrases whose senses were not predictable from their parts, and which permitted only very limited variation in their parts   (cf.  nonetheless, the-devil-may-care) would be defined only as run-ons to one of their constituents (kick the bucket at kick, and not at bucket) -- or, far worse, received no boldface status at all,   but had to be found at some postulated and by now rare or obsolete sense, not called out typographically, and indeed disguised by the practice of replacing the headword in angled-bracketed exemplifications with a tilde:

to function with vitality and energy <still alive and ~ing at 75 years>

-- Webster’s Third New International Dictionary (Unabridged)

From this, the reader would have no idea that alive and kicking is about a million times more frequent than  breathing and kicking or raising begonias and kicking or what have you.

Now, in this case, the M-W Collegiate actually does give own-place entry to our quarry of the day:  You can find up to as a boldface headword, at its own alphabetical place:  right after uptime, rather than as a run-on s.v. up.   So far so good.  But the problem is, they basically give it only one meaning (split unimportantly into two subsenses):  that of reaching as far as a certain limit, that limit being the complement of up to.   There are, however, several other senses, which that dictionary does not there record.

As:

Are you úp to it?

Meaning:  Are you capable of doing it.  (Contrast:  “Are you up for it?”, meaning:  are you in the mood to do it;  are you game.

And:

That’s up to yóu.

Meaning:   The decision is yours to make.   -- Oddly, British English apparently uses down to in the same or similar sense.  (“It’s down to me, the change has come, she’s under my thumb.”)

Note, b.t.w., the accent marks, which are inherent in these idioms, and which scarcely any dictionary thinks of noting.    That the patterns are here different suggests that we are actually dealing with subtly distinct entities, cryptic even within the class of cryptonyms (for that is but a proper class, and not a well-defined set); but that is a refinement with which we need not grapple here.

And:

What is he up to?

The usual meaning of this would be yet another meaning:  What (sketchy, perplexing) thing is he doing?  What is he about?

There is another possibility here, the very same that the Collegiate uniquely lexicalizes, and which requires a special context to evoke.   Namely, suppose that our person is engaged on a stepwise-graduated task (like our diligent meso-hero in the fable Tales of Frontier Times).   Here the phrase would mean:  How far has he gotten?  (Answer: four thousand and seventy-two.)

~

Finally we arrive at a further, delicious use, which is likely unfamiliar to you unless you have studied mathematics, and which motivated this whole post in the first place.   In mathematical writing, this use is extremely common -- but probably unnoticed as being of the technical, special terminology of that subject, whether by adepts or by laymen.  An example:

Semisimple Lie algebras can be factored in a unique way (up to rearrangement) as a direct sum of simple Lie algebras.

-- Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 232

(Here the “up to” clause modifies unique.) Which is to say:  Their factors are unique, but the order of these is not (just as 3 × 5 and 5 × 3  equally  equal  15.)

Commonest of all:  unique/uniquely “up to isomorphism”.

~

Suddenly the blood froze in my veins.

Here I thought I’d noticed something clever;  but it is increasingly difficult to say anything original these days, as Wikipedia’s omniscence takes in ever-larger swaths of the noösphere.    And indeed -- unlike the offline lexicographers, Wiki has (like Kilroy) already been there.

http://en.wikipedia.org/wiki/Up_to

That article notes that, ‘in informal contexts’, modulo X is, like up to X,  often used in this sense of ‘considering as equivalent  all entitites differing only in their X-value’.    The idioms are not quite equivalent.  First,  modulo -- and especially its abbreviation mod when followed by an integer -- originated strictly in one formal context, namely number theory.   Second, modulo in non-number-theoretic uses has leaked out into the general educated population;  as,

Chairman:  “And so, the ‘ayes’ have it unanimously.”

A handful of agitated spectators in the balcony:  “No! No!  Shame!”

Chairman (unperturbed):  “… modulo certain yelps from the peanut gallery.”

Here it means no more than “apart from”, “setting aside”. -- By contrast, up to is never used in this extended way.

As an additional excellence, there is a corresponding article in French:

http://fr.wikipedia.org/wiki/%C3%80_quelque_chose_pr%C3%A8s

But, in line with what we said earlier, that article is contentually different, since we are here dealing, not with an overt terminus technicus like homology/homologie, which are used the same way withing mathematics,  but a cryptonym cobbled-together out of tiny unassuming semantically-multivalent function-words.  Thus, the nearest French equivalent of “up to X”, namely “à X près”,  is synonymous in some uses and not-quite-so in others.

~

Re Karl Popper on philosophical theories:

He then, bar a hairsbreadth, equates these with scientific theories.

-- Margaret Masterman, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 72

Cf. “except on a set of measure zero”.

Pugnaciously vacuous definitions

At the beginning of his preface to the introductory exposition Philosophy of Logic (1970),  W.V.O. Quine quotes Tweedledee:

“Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t.  That’s logic.”

That was a bit of nineteenth century donnish waggery (though it does give rather the flavor of modal logics and possible-world semantics  in our own day).

Quine then drolly goes on:

If pressed to supplement Tweedledee’s ostensive definition of logic  with a discursive definition of the same subject, I would say that logic is the systematic study of the logical truths.

We pause for Homeric laughter.

Anyone schooled in the lore of Russell’s paradox and impredicative definitions, will recognize that the professor is having his fun.

He then has some more:

Pressed further, I would say that a sentence is logically true  if all sentences with its grammatical structure are true.  Pressed further still, I would say to read this book.

(Ah!  ‘Tis a plaint  we ourselves have often made:  Buy my books!)

[Note:  Contrast the  equally terse but quite unselfreferential definition of logic in

R. Goldblatt, Topoi , 2^nd edn. 1984:  “the study of the canons of deductive reasoning”.  Word for word, that’s very good.]

The jest is not without a subtle content, assuming as it tacitly does that there are such things as logical truths (a subject we see in a different light after Quine’s own extended attack on the analytic/synthetic distinction).   And it is quite in line with Quine’s celebrated existential bon mot:

    “To be is to be the value of a variable.”

To non-initiates, that will be mystifying;  to semi-initiates, cheeky; to familiar navigators of Quinespace, an ultra-compact allusion to his perennial concern with ontology in relation to quantification.

Put more pugnaciously:

Existence is -- what existential quantification expresses.

 -- W.V.O. Quine, “Existence and Quantification” 

[Further Quinefun  available here:
http://worldofdrjustice.blogspot.com/2015/01/quine-on-bunnies.html ]

~

The celebrated Cambridge Philosopher of Bland, G. E. Moore, wrote in his Principia Ethica:

If I am asked, “What is good?”, my answer is:  Good is good;  and that is the end of the matter.

That is:  If you have to ask, I can’t tell you.  (Even so, that pseudo-definition is preferable to the nihilist/reductionist dismissal by the Eliminative Materialists.)

Indeed, I shall now venture a definition  quite in Moore’s spirit:

Truth is … what is true (and known to God as such), quite apart from human-knowability, let alone provability (whether by finitistic, intuitionistic, or classical means).

~

From a college textbook:

A measure space consists of a set X equipped with two fundamental objects:

(1) a σ-algebra M of “measurable” sets, which is a non-empty collection of subsets of X closed under complements and countable unions and intersections.

(2) A measure μ: M -> [0, ∞] with the following defining property …

-- Elias Stein & Rami Shakarchi, Real Analysis (2005), p. 263

The bolded terms are there defined; but the term measurable sets is not -- the clause that follows reminds us of the definition of a σ-algebra instead.  The authors acknowledge their sleight-of-hand (effectively remedied by material elsewhere in the book) by placing the offending word in quotation marks. -- Note that this typographic care exemplifies the semantic Akribie of math writers, which we have elsewhere praised.

Not all math authors are as onomastically aware as those.  As:

Old-fashioned text-books  tend to start off with mystifying definition of these terms:  Euclid’s own definition,  “A point is that which has no part”, is a good example.  After a perfunctory discussion of these, the author clears his throat, begins a new chapter, and gets going with some concrete examples:  the definitions are mercifully forgotten.

-- Stephen Toulmin, The Philosophy of Science (1953), p. 72

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To define -- literally, ‘delimit’ -- is by way of penning-in the definiendum with antecedently familiar landmarks.   To say that none such exist, is to refuse definition;  as in this classic hymn:

There is nothin' like a dame,
Nothin' -- in -  the -  world,
There is nothin' you can name
That is anything like a dame!

-- “South Pacific”

By this point  we have passed definitely from the realm of Oxbridge drollery to that of Joe on the Boat.  And here, indeed, we discover a whole subculture of definitional legerdemain, noticed in the delightful Dictionary of American Slang, by Wentworth & Flexner (1967).   In the Supplement we find this entry:

blivit   (n.)

   Anything unnecessary, confused, or annoying.  Lit. defined as “10 pounds of shit in a 5-pound bag.”  Orig. W.W. II Army use.  The word is seldom heard except when the speaker uses it in order to define it;  hence the word is actually a joke.

As a former harmless-drudge chez Merriam-Webster, I salute that as a gem of the lexicographic art.

A classic development of the pugnaciously uncooperative definition -- Lexicography with an Attitude -- is The Devil’s Dictionary, by Ambrose Bierce.  (Sample:  “fork: an instrument for putting pieces of dead animals into the mouth.”)  Similarly impish was Hobbes’ definition of paradox:  “an opinion not yet generally received”.

A Pugnaciously Vacuous definition of the meaning of life  can be viewed here:

http://worldofdrjustice.blogspot.com/2012/03/ask-dr-science.html

~

Quite other than such conscious humor, are cases like this:

Hamilton follows the Kantian notion of time closely in his “Essay on Algebra as the Science of Pure Time”.  Since the inner sense of time is more general than the outer sense of space, Hamilton concludes that algebra is a more general and fundamental branch of mathematics than geometry.

-- Thomas Hankins, Sir William Rowan Hamilton (1980), p. 268

As the discoverer of quaternions, Hamilton has as much right as anyone to deliver himself of after-dinner remarks (a genre of public speaking to which he was particularly devoted) about the nature of algebra;  but this one is horse-hockey.

 

(Psychohistorical note:  Hamilton was for a time utterly immersed in Kant;  this characterization of algebra as the “Science of Pure Time” stems from psycho-philosophical exuberance, rather than algebraical expertise. 

A curious tentative parallel might be made with Hamilton’s countryman G.K. Chesteron, who likewise was given to flights of literary exuberance;  both were in marital situations requiring a great deal of self-sacrifice,  which they met with infinite patience;  and both were given to a kind of idealism  which some might diagnose as compensatory.

Okay, beyond our pay-grade.  Yet as Silvan Schweber affirms in his perceptive review [Isis, 1982] of this exemplary socioscientific biography:  “Hankins has eschewed giving psychoanalytical interpretations, [but] to anyone interested in the psychodynamics of creativity, William Rowan Hamilton presents a fascinating case study.”)

~

Another subcategory -- already bordered on by Hamilton’s epigram for the definition of algebra -- is formed by definitions which, while not vacuous, we might label Pugnaciously Perverse.   The poet Coleridge was (unfortunately) Hamilton’s philosophical mentor, even as regards what Science ought to be; and he defined that subject  thus:

“any chain of truths which are either absolutely certain, or necessarily true for the human mind, from the laws and constitution of the mind itself.  In neither case is our conviction derived, or capable of receiving any addition, from outward experience, or empirical data.”

-- Thomas Hankins, Sir William Rowan Hamilton (1980), p. 268

(That first sentence does oddly prefigure the sort of prose  churned out by the truckload by the epigones of Donald Davidson.  -- Note too the anticipation of Post-Modernism.)

~

Another non-cooperative move in the orismological game, is pooh-poohing the very definiendum -- denying that there is anything coherent to define.  One Christian writer (C.S. Lewis or Hilaire Belloc, I forget which) once said testily, that the notion of “the Renaissance” was a will-o’-the-wisp, used  by secular writers to mean “whatever I like that happened in the fifteenth and sixteenth centuries.”

~

Composing a truly vacuous definition  is harder than you’d think.

Thus, consider the Euclidean definition of a point (“that which has no part”) which our philosopher friend scoffed at;  and let us phrase it even more egregiously:

point:  a point-like figure

That actually has cognitive content.  It means:  To visualize what is meant when geometers refer to a punctum (as opposed to a linea, etc.), think of something like a pencil-point.  Do not think of something like a dance-floor, or the cosmos, or an elephant.

And:

set:  a set of elements

Here the definition is so far from vacuous that someone could reasonably object that it is actually false, since it excludes the null-set.   (In this ‘definition’, the stress, so to speak, is on “elements”; “set” could be replaced by “bunch” or “passle” or “bucketload”.)

For indeed:

There is no direct circularity  if we presuppose sets in our study of sets (or induction in our study of induction), since the first occurrence of the word is in the metalanguage, the second in the object language.

-- Michael Potter, Set Theory and its Philosophy (2004) , p. 9

In further defense of impredicativity:

Impredicative definitions are necessary for ordinary mathematics, as they are unproblematic if one adopts a realist attitude about the objects defined -- realist in just the sense that the objects exist in advance of the definitions, that they are picked out by the definitions, not created by them.  That imposes a substantial constraint on any acceptable philosophy of mathematics.

-- Shaughan Lavine, Understanding the Infinite (1994), p. 107

~

The grandfather of all tautological definitions is the one given by Yahweh in Exodus: 

~ ~ ~  I  Am   That  I  Am  ~ ~ ~

Yet at the same time, that is the best definition that could be given, since, in a common view of the Abrahamic religions, any limitative predication would be false.   (That view led in particular to the via negativa,  which insisted on the denial of all suggested predicates of the One.)

~

Another style of coyness with definition  is illustrated in the following, immediately after the authors have introduced Maxwell’s Equations  for E and H in free space:

At first we do not attempt to give physical meaning to these symbols.  We merely say:  let us assume that there exist physical quantities represented by symbols having the indicated properties, and see what these equations say about the quantities.  From the last two equations [to the effect that E and H -- whatever they might be -- have divergence zero], it is clear that E and H are solenoidal …

-- Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p. 303

After a few pages of discussion, the authors summarize:

In a very real sense, therefore, these equations may be said to constitute a definition of E and H.

-- Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p. 306

Thus, back where we began.  The dangled definition in intuitive terms, is ultimately withheld:   No dessert until you finish your broccoli;  then -- Your broccoli was your dessert.

They subsequently reinforce the apophatic stance:

Physical meaning of E and H:  … The only real importance of the quantities is involved in the fact that they satisfy the field equations. … It seems most logical to go the whole way and treat E and H as defined by the field equations in all cases.  The commoner definitions can then be looked upon as mere picturizations. -- Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p. 311

E & H:  They Are That They Are.

~

Lakatos offers his translation of an epigram from Poincaré:

    Mathematics is the art of giving the same name to different things.

Now, that is not vacuous (i.e., vacuously true, or anyhow only infinitessimally informative), since it is egregiously false;  but it is a witticism, not a blooper, since we all know that Poincaré was perhaps the leading mathematician of his time -- he has some cards up his sleeve, which he will slip out when it suits him.

For a series of essays on the art of definition, with especial reference to math,
try these:
http://worldofdrjustice.blogspot.com/search/label/definition

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 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. 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:
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:

http://www.linguasacrapublishing.com/justice.html

*

~

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, 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 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(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 ∂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, 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:

The Ladder of Abstraction

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