“Refutation” inflation (re-updated)

[The present note is an exercise in logic and linguistic hygiene.  It is not political per se, and in particular is agnostic as to the facts and merits of the tangled case under discussion.]

[Original post-date 16 May 2017]
One of the top stories in today’s crowded news:

The family of slain Democratic National Committee staffer Seth Rich rejected Fox News reports that he had leaked work e-mails to WikiLeaks before he was fatally shot last year in the District.

The reports, which gained traction on social media, said an FBI forensics examination showed Rich transferred 44,053 DNC e-mails and 17,761 attachments to a now-deceased WikiLeaks director.

Rich’s parents, Joel and Mary Ann, said Tuesday through a spokesman that they do not believe their son gave any information to WikiLeaks.

https://www.washingtonpost.com/local/public-safety/family-of-slain-seth-rich-says-reports-he-fed-wikileaks-dnc-info-are-untrue/2017/05/16/9b32ef9c-3a46-11e7-8854-21f359183e8c_story.html?utm_term=.f0f43ca07a0f

That is admirably and neutrally stated.  However, some news sources are reporting the same facts with headlines like “Seth Rich Parents Refute New Claims On Wikileaks Contact”.   Therein lies a confusion.

To refute is (in its original, non-catachrestic sense) to disprove.  The allegations in question are perfectly precise and emprical, subject to either (partial) refutation or (partial) confirmation.

But the only party in a position to refute the allegations is someone who professionally and forensically examined the laptop in question.    Does it contain such material, or does it not?  The family is in no position to “refute” the allegation, however false it may be.  Indeed, on the Post account, they cannot really be said even to have denied the allegations;  they simply said they don’t believe them.  A perfectly rational stance; but not exactly a denial (for after all, how would they know -- if their son had been secretly betraying his employer, why would he inform his family?), and certainly not a refutation.

Increasingly, the less careful media uses refute where deny would be appropriate.  Part of this may be simple semantic weakness on the journalists’ part (to which many other technical terms, like impeach, are subject), but partly also to the fact that deny has accumulated invidious connotations, as though anyone who “denies” X  is himself shady in some way.   That is a legitimate worry;  other synonyms are available (the family discounted/pooh-poohed/scoffed at/… the allegations) which lack such connotations.  Better to use these than to induce a crucial ambiguity in the verb refute, in a way that renders it inapt for precise usage.

Part of the problem in the fluidity of use of refute  might be  not political, but cognitive and linguistic:  confusion with the paronym rebut.  A rebuttal is not quite as decisive as a refutation, but supposed to be more evidential and structured than a mere denial.  Mere denial is a weak defense indeed, available even to the ghosts of five-year-olds, as in the comic strip "The Family Circus".

A further semantic distinction:  re discussions within the first Nixon administration, of Kevin Phillips’ The Emerging Republican Majority:

Phillips had not been refuted by the West Wing,  but his thesis had been rejected.
--  Patrick Buchanan,  Nixon’s White House Wars (2017), p. 146

Note:  There are other ways of disposing of an allegation, other than outright refutation:  you may undermine, or infirm, or discredit it, in various ways.   Thus, if a witness presenting himself as Dr. Smith (M.D. Harvard) testifies that the deceased died of psoriasis, another doctor (or team thereof) might refute that testimony (on its own ground) by presenting evidence that the deceased had a huge malignant brain tumor but had never had a skin condition.  But anyone -- say, a lowly clerk at Harvard Medical School -- could discredit the testimony on entirely other grounds, by showing that Smith never attended Harvard Medical School, nor (with a bit of extra digging) ever so much as finished high school.  That would be devastating counter-evidence, but not a “refutation” in the technical sense.  (Logically, Smith might nonetheless have blundered upon the correct explanation of the demise.)

One can’t help suspecting that the media’s terminological laxity might be connected to an epistemological weakness:  presenting counter-allegations as evidentially telling (whether or not they are actually awarded the accolade “refutation”) although (consider the source) they are suspect or undermined at the outset, as coming from the accused's family, or attorney, or partner in crime.  Some of these are treated with great journalistic reverence, and actually pass into folklore  --  "he didn't do nuttin' " (spoken while the perp is actually in flagrante), or, post-hoc, “he didn’t have a gun” (though one was found in his possession, surrounded by spent cartridges), and where all else fails and guilt is  ... irrefutable, “he was hoping to go to college”, “he was starting to turn his life around”.

[Update 17 August 2017]  Bringing it back to Wikileaks:

Assange once told me that he did not “accept” the allegation that Russia had provided him e-mails through a third party,  which of course was different from saying that the allegation was untrue.

-- Raffi Khatchadourian, in 21 Aug 2017 The New Yorker]

But nor did he make any move to refute or cast doubt on the allegation:

I asked if he was even able to know the chain of custody of his election material before it came to him.  He declined to answer.

[Side-note:  for that phrase of forensic science chain of custody,  cf. the term of hadith stemmatology,  isnâd.]

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A particularly piquant use of the term “refutation” occurs in the mathematical polylogue by Imre Lakatos, Proofs and Refutations (1976).  The title impishly echoes that of Karl Popper’s better-known Conjectures and Refutations (1962).  But whereas that title reflected the expection rough-and-tumble of normal science, Lakotos’ phrase produces a double-take:  if a “proof” gets “refuted”, it wasn’t really a proof to begin with, but only a purported proof.  But Lakatos is not referring to those (relatively rare) instances of purported proofs that turned out to be fatally flawed, and left no progeny in mathematics.  Rather, he considers mathematical demonstrations that were all right so far as they went, but which contained hidden assumptions.  These being unearthed in a “refutation”, the original proof, or something much like it, gets deepened, until further unsuspected subtleties become revealed.    He offers a dialectic analysis of the process of mathematizing.   The result does not demote mathematical truth to a mere just-so story, as among nihilists and relativists.  It rather offers a more epistemologically modest picture of the mathematical enterprise (the fallible human excavation of a transcendental reality, a Platonist would say), in which the notions of “proof” and “refutation” both get toned down a bit, and the process becomes a bit more like developing a software package, finding and fixing bugs along the way.  The result is real progress.

For a more technical discussion of refutation and its semantic field, try

http://worldofdrjustice.blogspot.com/2016/01/refutation-vs-disconfirmation.html

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The flip side of the coin, by which the media use artificially strengthened language when presenting the allegations of the victim class and their attorneys, is artificial down-grading when the allegations come from the authorities.   As, a headline from a moment ago:

South San Francisco police officers on Wednesday morning shot and killed a man who they say was allegedly armed with a shotgun.

Either “they say” or “allegedly” would be an adequate and appropriate editorial distancing from the official police statement.   Together, they are at best redundant, or, if taken literally, false:  the police did not state “He was allegedly armed with a shotgun”;  such a statement would be in place if, say, the police had not actually seen the shotgun, but a bystander reported -- alleged -- such possession of a shotgun (which had  been abstracted from the crime scene by the time the police arrived).  

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[Addendum] Further vocabulary.

In the following sophisticated example, the verb nullify is used, not in the sense ‘refute’ exactly, but ‘to render null and void’

-- not to prove the falsity of a statement, but to disable its presuppositions:

I turn now to the objection that, even if probability-scepticism does not nullify the concept of truth, it does nullify the idea that science should aspire after truth.

-- John Watkins, Science and Scepticism (1984), p. 162

Indeed, Lakotos’ impish use of refute is roughly the same idea.

~

A bonus from Classical Antiquity:

Pericles, also, was a hearer of Zeno, the Eleatic, who treated of natural philosophy  in the same manner as Parmenides did, but had also perfected himself in an art of his own, for refuting and silencing  opponents in argument;  as Timon of Phlius describes it --

Also the two-edged tongue  of mighty Zeno, who,

say what one would, could argue it untrue.

-- Plutarch’s Lives (Dryden’s translation)

Manichean Mathematics

In overwrought politics and religion, it is commonplace to demonize some of one’s foes.  Thus, for Christians, the Antichrist;  for Muslims, the comparable Dajjâl.   On the cover of the current Time, Hilary Clinton appears with a pair of horns.  So it goes.

Although disputes exist (both personal and professional) within the ranks of mathematicians, you really don’t find that sort of thing, so far as I know.   About as far as it goes  is witty dismissals of some subdiscipline;  as, Cantor’s innovations in the transfinite, which some dismissed as “more theology than mathematics” -- theology, though, not diabolism.  Or Category Theory as “the higher macramé” (as wits had it back when I was in college.  Since then, the Category approach has knit -- or rather knotted, to continue the metaphor -- some quite interesting structures, that seem to hold.)

So it was with some surprise that I ran across this passage:

Euclid has been the evil genius … for the history of mathematics.

-- Imre Lakatos, Proofs and Refutations (1976), p. 140

(Having just the other day watched the old Fritz Lang film, “The Testament of Dr Mabuse”, this phrase delivered a particular chill.) 

In the course of his book, Lakatos does permit himself some unkind or dismissive digs at certain mathematicians, historians of the field, or styles of mathematizing, but this formulation seemed extreme.  Yet as the author’s footnote immediately reveals, it is taken straight from so sober a source as R. B. Braithwaite, who (in 1953) called Euclid the “evil genius of philosophy of science”  -- though, to be sure, the “good genius of mathematics" itself.   (One pictures twin miniature Euclids, one red with a tail, one white with wings, perched on Donald Duck’s shoulders and whispering conflicting counsels.)

~

This mild and forgettable aside  yet brought back memories of Berkeley days, when some of my friends were refugees from the group then known as the National Caucus of Labor Committees, a sect around the guru Lynn Marcus (later known as Lyndon Larouche).   That formation made diabolization a positive organizing principle of their style of thought.   What made it all so fascinating to follow, was trying to guess whom they would seize on to demonize: Zbigniew Brzezinski, all right, but -- the Queen of England?  (And that, lastingly.  Never was a doily of an old lady more improbably cast.)   And then things got really interesting when the dichotomization was extended to figures from intellectual history.   As, Plato they liked (yay, Plato);  but he appeared ever in a duality, contrasting with “the evil Aristotle”.

The NCLC even extended this perspective to mathematics, their particular hero being Riemann (again, hoorah hoorah).   Whom they chose as the anti-Riemann  I forget, but anyhow it was impressive that a group that was vying for converts in an arena crowded with such aggressively ignorant rival leftist sects (for the NCLC was formally of the Left at that time)  as the Weathermen, various Maoist groups, the "Symbionese" 'Liberation' '''Army''', the Black Panthers and so forth,  had even heard of such figures as Riemann, and deemed them of world-historical importance.

(Note:  It is by way of arcane allusion to that episode, that I titled my ongoing feuilleton of dark doings among the Illuminati, the Riemann Conspiracy.  Something like three people will get the joke.)

Is .... this .... the face of Evil ??

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Climbing Mount Ineffable

[A Lenten meditation]

Darwinians use a nicely heuristic image for evolution as blind climbings of a fitness-landscape.  Richard Dawkins sharpens the metaphor with the title of his (excellent) book, Climbing Mount Improbable : thus recognizing that, though he is a staunch proponent of Natural Selection, evolving something as nearly perfect as a Penguin  is not a slam-dunk.

The pinnacle of Natural Selection  so far

(Indeed, penguins represent a classic case of Irreducible Complexity:  remove one single feather, and the creature is not nearly so cute.)

Our own essays have recurred to a mountaineering metaphor, in support of Platonism (the Realism position in math).  Namely:  Team A sets out to conquer Mount A from its forbidding southern face;  Team B sets out to conquer Mount B  from its frigid north one.  They meet at the summit, to their mutual surprise.   A equaled B, all along!  This attests to the reality of the mountain, prior to and independent of all human endeavor.  (For ‘mountain’ read:  the truths of mathematics, arrived at independently  by various researchers  using quite disparate methods.)

Now, Imre Lakatos  is neither (statically) Realist nor Nominalist -- he is a Dialectician, and juggles both views.   In the course of a quite intricate examination of the evolution of the Euler characteristic  (don’t imagine you really understand the following sentence unless you have worked through that monograph), he remarks in a footnote:

As far as naïve classification is concerned, nominalists are close to the truth when claiming that the only thing that polyhedra have in common  is their name.  But after a few centuries of proofs and refutations, as the theory of polyhedra develops, and theoretical classification replaces naïve classification,  the balance changes in favour of the realist.

-- Imre Lakatos, Proofs and Refutations (1976), p. 92  (**)

Presumably what motivated this formulation, was the experience of beginning with a hopeful conjecture that soon is drowned in a welter of disparate counterexamples; yet with time and hard analysis, we do progressively manage to sort things out -- as though our fumblings were being guided by something real, though unseen.

Similar remarks, I would cautiously submit (under correction), might apply to the multimillennial evolution of theology, in the Historical Church.

Apart from the dogmatic mouthpiece ‘Episilon’ in his classic dialogue/sotie, Lakatos suggests that we have not reaching the summit of any mathematical mountain, and perhaps never will.  With that I concur wholeheartedly;  only admonishing, that the summit is there.  Nor are we likely to get much insight into the internal workings of the Godhead, this side the eschaton;  but those workings are there as well. 

And so we strategically retreat  to the more modest metaphor of the base-camp.  We never quite reach the summit, but with luck and elbow-grease, we might climb high enough that we can detect the smoke from the campfires of the North Face team.

(The theological analogue here is one of the favorite themes of C.S. Lewis, in The Abolition of Man and other works:  the anticipation of Christian insights in other traditions.)

The mathematical upshot of all this, goes back to a repeated theme of these essays:  the distinction between our (human, contingent, fallible) mathematicizing, and the (antecedently existent, transcending) mathematical truth.   (For anyone who might dispute that, answer this:  Did the Universe even exist, prior to Newton?)

The theological upshot -- Well, longstoryshort:  Don’t go cutting off heads, merely because you imagine you perceive some straw in your neighbor’s eye.

(**)  More pointedly: 

… the problem of finding out where God drew the boundary dividing Eulerian from non-Eulerian polyhedra.  But there is no reason to believe that the term ‘Eulerian’ occurred in God’s blueprint of the universe at all.

-- -- Imre Lakatos, Proofs and Refutations (1976), p. 68

Amen.  But He has a blueprint, of which our notion of “polyhedra” is a primitive glimpse.

Similar remarks might apply to the Trinity.

~

Another parallel:

Consider a theological scholar  working on an apparent inconsistency between two Biblical passages.  Theological doctrine assures him that the Bible, properly understood, contains no inconsistencies.  His task is to provide a gloss that offers a convincing reconciliation of the two passages.  Such work seems essentially analogous to ‘normal’ scientific research as depicted by Kuhn;  and there are grounds for supposing that he would not repudiate the analogy.

--  John Watkins, “Against ‘Normal Science’”, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 33

Internal, External, Universal

[Today’s theologico-mathematical analogy may be stretched, far-fetched;  but ‘tis the Lord’s day, a time meet for meditation  more at large.

For more extensive reflections, focusing on Realism in both domains, consult the essay series that begins here.]

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Instead of defining the properties of a collection by reference to its members -- its internal  structure -- one can proceed by reference to its external relationships with other collections.

-- R. Goldblatt, Topoi , 2^nd edn. 1984

I am reminded of the Christian critique of narcissistic individualism, so telling for our own day, when it has become a very plague, both sapping the individual character, and corrupting the polity as it forms an algal bloom as identity politics.  This view was made more acute, and very contemporary, by C.S.Lewis in The Four Loves and elsewhere, with its metaphor that health lies neither in religious solipsism (the “inner light”, which he decries) nor in that solipsism-à-deux of “looking into each other’s eyes”, but rather in mutually apprehending some external thing, of which we each see aspects, though along different sight-lines.

There are traditional notions of something large and out-there, above us and beyond us;  but these are vague and unstructured, and have perhaps grown stale through overfamiliarity (though we have never understood them well enough to have leave to dismiss them out of hand).   So let us turn to consider a mathematical notion of something containing -- something larger than what you started with, yet perfectly contained within itself:  not spreading over us like a fog, but rounding us out.  The technical name for this is comforting, downright cozy:  compactification.  (The Water Rat of Wind in the Willows  pictures his snug and tidy den.)

Compactness has turned out to be one of the most central notions of topology, a field which itself is about as central as you can get.  For details, see Wikipedia (that paradisal repository of all that is known, or could ever be known);  but the takeaway is, that it is a quite vaunting generalization of the idea of finiteness.  Such spaces are nice to work with.

Thus for instance:  consider the open interval (0,1).  It is not too intimidating (apart from its harbored continuum), but it is irksomely incomplete, in that a well-regulated sequence of points -- ½,¼, 1/8 … -- can march off towards nullity,  yet nullity they find not, nor unity neither  should they march the other way.  We can complete this space, and simultaneously compactify it, in an obvious way:  just add the points zero and one at either end, to get the closed interval [0,1].  Now all is well.

But there exists a less obvious kind of compactification, involving the addition of but one point (we pause, that you might wonder:  Yet how can this thing be?).  In turns out to be deeper, in that such a one-point compactification (via Alexandroff extension) is available for any locally compact Hausdorff space.  In the simple case of our open interval, conceptually you add a point at one end and bend the segment around to meet it.  The result is a little ring:  like all round things, it is ever so perfect and pleasing.

And our pleasure at this maneuver  is more than aesthetic, for the move applies as well to the entire real line R.  This space is complete in the standard Cauchy-sequence sense, yet it too is “incomplete” in a way, namely, in the sense that an infinite sequence might have no convergent subsequence (R is not 'sequentially compact', as they say in the trade):  the series (such as 1,2,3, …) may march off forever towards infinity, but “infinity isn’t there”.  We can both ‘complete’ and compactify it  by adding a “point at infinity”, replacing the standard metric with a bounded one (the resulting space being homeomorphic to what we started with), and then “round it around” to a ring-shape as before.

You see where we’re going with this.

Ah, but do you.  For mathematics has latterly progressed in ways considerably more intricate than simply sharpening our intuitions of infinity, so that, when we say that “God is infinite”, we can have something much more incisive in mind than simply “way bigger than an elephant”, with which our grandsires had to make do.  For geometry has been algebrized: beginning with Descartes, but zooming off in unexpected new directions with algebraic topology.

We have seen that there are varying ways of compactifying a given space.  In the context of Universal Algebra, a question arises:  For any given space, is there one way that is, in some sense, universal or canonical -- the “Mother of all compactifications” (to speak with Saddam Hussein)?  Indeed there is:  it is known as the Stone–Čech compactification. The result is universal in that any continuous map whatever, from our original space to a compact Hausdorff space, can be factored through the Stone–Čech compactification.  (Thus, the closure of (0,1) into [0,1] does not rate as Stone–Čech, since e.g. sin (1/x), defined on the open interval, does not extend to the closed.) -- Whoever can grasp this, will never consort with Nominalists again.

We have considered this matter in a particular area of point-set topology, but the notion of universality, as made precise by this notion of lifting a given map to procede through the universal, is quite general -- hair-raisingly general, in fact.  In general, “a morphism [is said to be] universal  [iff]  any other morphism into a system with this property  factors uniquely through the universal morphism.” (Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3^rd edn. 1999), p. 129.)

~   ~   ~

So much for the math.  And now for our dominical metaphor, offered in all humility.

We are, according to Scripture, but now also in a sense which might possibly someday be made relatively precise, made in (or better:  from) the image of our Maker.  Only, not visually (that were absurd, and gives rise to all the idolatries), nor yet (abstractly, or spiritually) isomorphically,  but rather: homomorphic images, of various types and sizes.  (Bonus:  homomorphic now becomes a graeco-latin pun.)  Whatever can apply to us, can apply to and through Him, in a manner made familiar by Category Theory.

And by what seems a kind of anticipation of the functorial view, the Historical Church chose precisely universality as its defining epithet:  catholicus.

(Yet who are these, streaming across the blasted landscape in despair, the wretched remnants of their mockeries  strapped to their backs?  Why, ‘tis the very tribe of atheists, quite put to flight!)

Within Set Theory, there is a notion reminiscent of all this:  the Reflection Principle.  It is very counterintuitive -- but then, so is life.

~

Appended Epigram

That God is simply the sum of All that Is, is mere pantheism.  We shall posit rather, that He is its Stone–Čech compactification. 

(Here we tread, not on dangerous, but on spongy ground, the sort that led into the swamp of the ‘God particle’.

Various defenses spring to mind, but I have a feeling that they are self-serving.  Taceamus igitur.)

Similar to our image of the lower thing being the homomorphic image of the higher:

The highest things often have “footprints”, as the medievals put it, among the lower things.

-- James Schall, S.J., The Order of Things (2007), p. 22

~

(All right, now we do something very wrong.  But my character, sapped by whoring after epigrams -- e’en as the bard  was slain by a pun --  cannot resist.

An early post against ultra-Darwinism  mentioned -- purely in passing -- the Urysohn Metrization Theorem;  after which, to my embarrassment, this site received a number of serious enquiries after that worthy result.   Actually  it was kind of cool.  And so, to accommodate surfers who are mathematically advanced but lousy spellers, we add these:

Stone-Cech

Stone-Čeck

Stone-Ček

Stone-Czech

Stone-check

Stone- tchèque

Stone-Tscheck
pStone-pČech  [the p is silent ...])

~ ~ ~

All that is rather by way of somewhat remedying the obvious insufficiences of St Anselm’s Ontological Argument, while yet retaining sympathy with his project.

The images/metaphors  of the Scala Naturae, and the Ladder of Abstraction, both point ever-upwards, as if to some final lodestar or ultimate Utmost, without  of course  proving the existence of any such thing.  There is also something empirically amiss, in that both visions are linear -- and reality is generally not like that.    More to the point would be Partially Ordered Sets -- and that gets us straight to the door of Zorn’s lemma:

Suppose a partially ordered set P has the property that every totally ordered subset has an upper bound in P. Then the set P contains at least one maximal element.

Now, that Maximal Element -- remind you of Anyone?

Stairway to Paradise

This is a more robust analogy than that of the long extension-ladder, but it probably won’t buy us anything of theological import.   Note in particular that the various upper bounds referred to must lie in P:   P is already complete.   Whereas a simile for the Godhead would more likely be along the lines of Inaccessible Cardinals, or Proper Classes,  ever beyond iterative reach.

C.S. Lewis drops a remarkable aside, in the final paragraph of his essay “The Language of Religion”:

I sometimes wonder whether the Ontological Argument did not itself arise as a partially unsuccessful translation of an experience without concepts or words.

-- Christian Reflections (1967), p. 141

(Nota bene:  There are intellectual as well as emotional such experiences, as in mathematical insight -- at least, without words.  Brouwer once characterized mathematics as “an essentially languageless activity of the mind”.

More here.)

Lewis’s essay, incidentally, is  gem, developing at length  an idea he has often sketched, concerning the evolving adequacy of language to non-everyday puzzles like theology and math.  In that spirit, we have offered a couple of vizualizable new analogies to play around with:  Universal Compactification, and Partially Ordered Sets.

Lewis’s linguistic point is continuous with his opposition to intellectual “Whig history”.   Thus, if our ancestors spoke of God as though He had a white beard, and depicted him this way in art, it is not because they were morons;  indeed, such a depiction did not, at the time, constitute an asserted denial of the thesis that God is incorporeal:  for that later thesis simply lies (intellectually and chronologically) beyond the original level of discussion.

(In similar fashion, if I state that “the red vehicle was stationary at the time of the collision", that is not meant to deny the thesis that the earth rotates on its axis, and moreover revolves around the sun.)

Exactly the same point can be made with respect to the praxis of mathematics.  (I mean its ever-evolving practice by actual mathematicians, rather than the arguably  timeless, transcendental truths of Mathematics itself, as it resides in the mind of the Creator.)

Thus, Wikipedia (re Imre Lakatos):

Lakatos re-examines the history of the calculus, with special regard to Augustin-Louis Cauchy and the concept of uniform convergence, in the light of non-standard analysis. Lakatos is concerned that historians of mathematics should not judge the evolution of mathematics in terms of currently fashionable theories. As an illustration, he examines Cauchy's proof that the sum of a series of continuous functions is itself continuous. Lakatos is critical of those who would see Cauchy's proof, with its failure to make explicit a suitable convergence hypothesis, merely as an inadequate approach to Weierstrassian analysis. Lakatos sees in such an approach a failure to realize that Cauchy's concept of the continuum differed from currently dominant views.

Lakatos’ dialectical insights are worked out at length in the multisided dialogue (a ‘polygonal’ conversation, as it were), Proofs and Refutations.

[Update April 2017]  I had rather hoped to have added a “Footnote to CSL” with that shtick about creatures as homomorphic images (of various cuts and complexity) of their Creator, a more flexible metaphor than Lewis’ example of the faces of a cube.  But upon re-reading his essay “Transposition”, I learn that Transposition is his term for much the same thing -- he even uses the term algebraic in that connection.  The whole idea is worked-out exquisitely in that place.

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. ]

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For further musings from this pen,

check here:

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

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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.)

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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”.

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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!

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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