“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.]

~

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

~

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).  

~

[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)

For Whom the Bell Tollens

Miscellaneous additions to our essay on the logic and rhetoric of modus tollens,

https://worldofdrjustice.blogspot.com/2015/12/modus-tollens-tollendus-est-iterum-re.html

(1) Psychological observations

People seldom intuit what is unpalatable to them.  [Moreover, one can] eliminate any undesirable indirect implication of their special insight  by means of an additional hilfs-intuition, liquidating the embassassing logical relation.

-- Ernest Gellner, The Devil in Modern Philosophy (1974), p. 95

In short, one ‘answers’ the sceptic  by striding across logical gaps  to conclusions inconsistent with premises that one does not contest.

-- John Watkins, Science and Scepticism (1984), p. 34

(2) The reductio ad absurdum /”self-mate” gambit:

The traditional argument for the primacy of acceleration-retardation  rests on the absurdity of denying it.

-- Stephen Jay Gould, Ontogeny and Phylogeny (1977), p. 216

Chomsky’s book Syntactic Structures, which is regarded by some as a foundation-stone for this kind of activity, has been described by no less an authority than Roman Jakobson  as an argumentum a contrario (Jakobson, 1959), showing the impossibility of the whole enterprise.

-- Hilary Putnam, “Some Issues in the Theory of Grammar”, in  Mind, Language, and Reality (1975), p. 85

Mysteries of Cricket

Cricket matches occasionally end in a tie, and often in a draw;  but that does not make it a futile game;  if no side ever won, the game would lose its point.

-- John Watkins, Science and Skepticism (1984), p. 279

Assignment:  Distinguish tie from draw.

https://www.youtube.com/watch?v=F6FdBZQw2zQ

I am at last resigned to the fact that I shall never understand String Theory;  and a grasp of cricket  must likewise remain  forever beyond my means.

Given that, Kipling’s dig about “the flannelled fools at the wicket” was quite unfair;  you practically have to be a barrister to make sense of the game.

~

More re the point of a given genre of game:

Within the vocabulary of chess, it makes no sense to say ‘That was the one and only move which would achieve checkmate, but was it the right move to make?’  … as someone might ask, whose purpose in playing chess  was to amuse a small child  rather than to win.

-- Alasdair MacIntyre, After Virtue (1981; ²1984), p.  125

(And what is the point, of this game of life? --  Perhaps, to amuse the gods.)

It would be interesting to imagine extragalactic sociologists or philosophers, provided with a complete, objective and uncommented, play-by-play record of every cricket game ever played, and every chess game ever played; we wonder whether, from these, they could deduce what the point was.  Cf. & contrast the card‘game’ that we children called ‘War’, in which again, the ‘point’ is to ‘win’,  only, each player’s moves being completely determined, the resemblance to any normal game is but feeble;  it’s really just a long-drawn-out, inefficient way  of staging a single coin-flip.

~

Back to cricket, with its arcane distinctions and welter of rules -- surely exceding those of tennis (let along ping-pong), basketball, soccer  and much else.    These come at a cost (we want to shout, Just play the game!), but indeed to such an extent, that what would, from a practical point of view, be counter-productive, must surely serve some sociopsychological purpose, delineating the cultural boundaries of the Tight Little Island.   Rather like the Japanese language, with its complex system of grammatical honorifics, which no-one can master who hasn’t grown up in the culture from birth.  Or Anglo-Catholic worship ritual: unlike in an evangelical church, where anyone can wander in off the street and fit right in, for an outsider to attempt to participate  is continually to find yourself sitting when (suddenly, inexplicably) everyone else is standing up,  or standing up when they are sitting down, or kneeling, or crossing themselves.

For an American analogue, we would suggest, not so much our National Pastime of baseball, which physically most resembles cricket, but American football, with its ever-expanding rulebook, its minute schedules of differing penalties for everything from twitching (on the offensive line -- “false motion” -- or the defensive -- “offsides”) to spiking the ball; and its many game-interrupting circumstances. -- What physically most resembles football is rugby;  but here the playing of the game -- the continuous unscripted motion, as in soccer or basketball -- quite trumps the courtly ceremony characteristic of football.

~

There is generally a core of rules that characterize the essence of a given kind of game.  As for the rest, they contribute to its aura of spectacle,  but are not constitutive of the game.

The essense of football is simply that one guy runs with the ball toward a goal, and the other guys try to stop him.   Early football (and kids’ football even today) was largely just that.   Then they added the possibility of a forward pass:  that was a significant addition to the core, constitutive of an extension of the game (Football  v. 2.0), but one in which the earlier nucleus remained aufgehoben:  after all, there are plenty of running plays even in the NFL, and if you laid them all end-to-end, you’d have a simulacrum of original football (though to be sure, they are subtly different, since each play is instinct with the possibility that someone might throw the ball).

The constitutive rules of chess are the powers of the pieces.   That much is sine qua non -- you couldn’t very well specify the moves of all the pieces except the knight, letting that fellow to joust around as the mood might move him.   But now you’ve got a game, which could just as well exist without niggling side-rules like that of castling, or capture en passant (let alone purely ceremonial, non-structural rules like “j’adoube”).

Metrization, Mensuration, Measurement

We earlier treated the matter of a metric on a topological space, in a series of essays beginning here:

The Urysohn Metrization Theorem:  for real this time

Now, as lagniappe, we offer a pair of Metrization Epigrams, which the budding mathematician, stuck for an opener with that leotard-clad vision at the espresso bar, can use for a pick-up line:

 A discrete space is like the autistic atomism of the Tractatus, or Leibnizian monads.

An indiscrete space is (as one writer charmingly put it), “really quite crowded:  each point is an accumulation point of every other set.”  (One pictures the Jellyby children in Bleak House, ever tripping over one another’s legs.)

(Believe me, chicks go wild over such things.  Or at least, if you are like most gangly Adam's-apple-challenged graduate-students in math, it’s your last best shot.)

~

It is by no means only topological spaces that one might wish to subject to a metric: all kinds of things, really:    Which species lie how close to which others (and different metrics -- phenotypic, cladistic, etc. -- yield different results);  which languages are neighbors in linguistic space (again there is a phenotypic/cladistic distinction:  descent vs. Sprachbund); which people have a natural affinity with which other (seating-plans at dinner-parties; blind dates; etc.)  And more generally, what is the curvature tensor of the noösphere?

As (from a philosopher of science):

One sometimes wants to say that a theory has much more testable content than some other statement:  for instance, the Newtonian theory has much more testable content than ‘The moon orbits the earth’.  But our apparatus, as it stands, does not entitle us to say this.  We have no metric for testable content.

--John Watkins, Science and Skepticism (1984), p.  185

Compare Keynes’ critique of relative subjective probabilities.

Here a noted philologian on the notion as applied to languages:

Was nun die Sache selbst anlangt, so meine ich  daß immer Sprache und Sprache, mögen sie auch noch so weit auseinander liegen, in wissenschaftlichem Sinn  enger zusammengehören  als Sprache und Literature, seien es auch die  desselben Volkes.

-- Hugo Schuchardt, “Über die Lautgesetze”  (1885), in Leo Spitzer, ed., Hugo Schuchardt-Brevier (1921; 2^nd edn. 1928), p. 85

And here, indeed, he polemicizes against the nominalist treatment or ‘indiscrete toplogy” of diachronic linguistics:

Ist es denn nun nicht an sich ganz gleichgültig, ob rom. andare von adnare oder addare oder ambulare oder einem keltischen Verbalstamm herkommt;  ob in diesem Dialecte  l zu r  und in jenem  r zu l  wird usw.?   Welchen Sinn haben alle die Tausende etymologischer und morphologischer Korrespondenzen, die Tausende von Lautgesetzen, solange sie isoliert bleiben, solange sie nicht in höhere Ordnungen aufgelöst werden?

-- Hugo Schuchardt, “Über die Lautgesetze”  (1885), in Leo Spitzer, ed., Hugo Schuchardt-Brevier (1921; 2^nd edn. 1928), p. 84

~

This metaphor of ‘metrization’, outside the exact sciences, is very loose, as it is not strictly needed -- for taxonomic purposes, a more approximate neighborhood-system will suffice (a “Uniformity”) so to speak -- and still less is to be obtained.

As, a pair of British linguists comments:

One recent attempt by French researchers  has given us the term dialectometry, which describes a formula for indexing the dialect ‘distance’ of any two speakers in a survey.  So far, the utility of the index has not been demonstrated.

--J.K. Chambers & Peter Trudgill, Dialectology (1980), p. 112

This, in the synchronic arena, is reminiscent of the glottochronology of Morris Swadesh, who attempted a sort of carbon-dating of linguistic evolution, based on an assumed universal rate of lexical decay, in the absence of direct evidence.

[Update 17 January 2016]  Another cautionary tale about the fetishization of metrics:

Two of our most vital industries, health care and education, have become increasingly subjected to metrics and measurements. Of course, we need to hold professionals accountable. But the focus on numbers has gone too far. We’re hitting the targets, but missing the point.

http://www.nytimes.com/2016/01/17/opinion/sunday/how-measurement-fails-doctors-and-teachers.html?ref=international

Philologisches:

Whether, in that opening paragraph, the author wrote “metrics and measurements” intending to refer to two distinct though related concepts, or whether it was just an idle bit of synonymic accumulation like “bequeath and bestow” for those who might be unfamiliar with the somewhat technical word metric, is there unclear.  But it does raise a linguistic point.

The word metric, and its close kin meter, metrical, metrization, derive from Greek.

Mensuration and commensurable  go back to Latin mensura.

Measure comes ultimately from that Latin word as well, but via the phonetics of medieval French.

The same Indo-European root  is said to lie at the base of all of them.

English, an etymological patchwork, has some tendency to layer its vocabulary by origin, Greek roots being reserved for the most technical, followed by Latin,  with the Saxon vocabulary as jack of all work.  In this, it contrasts with German:  to Graeco-English oxygen, hydrogen, nitrogen  correspond homely-sounding Germanic compounds Sauerstoff (‘sour-stuff’), Wasserstoff, Stickstoff.   And Freud’s German originals for the English ego and id, were nothing but nominalizations of the ordinary pronouns, das Ich & das Es.

Roughly such tiering is at work in our Wortfeld of ‘measure’. 

Measure sounds reasonably English (though partly just because it chimes with pleasure and leisure, which are likewise French words in disguise), and is in everyday use for all purposes (though it also has technical specializations, as in mathematical measure theory).

Latinate mensuration (little used) is scarcely more than a stuffy synonym for ‘measuring’;  commensurate has everyday though businesslike uses (“a salary commensurate with the job responsibilities”); while commensurable is mostly technical, whether in its mathematical sense, or its more recent philosophic sense (“commensurable discourses”).

Metric is kind of a green-eyeshade/clipboard sort of word at best.  It becomes fully technical in mathematical uses like metric space and semi-metric,  finally soaring off into the intellectual empyrean with metrizable.

~

Still wearing our lexicographer’s hat, here is an attestation for a word with which I had previously been unfamiliar, used by a philosopher of science.  After a rather confusing Gedankenexperiment judging the verifiability of a physical geometric hypothesis, involving all sorts of skulduggery with measuring rods, and “tampering with the semantic anchorage of the word congruent”  (that’ll get you two weeks in the clinky  without the option), and in which Albert Einstein (from beyond the grave) plays a role like that of Fantomas, battling the equally post-mortem shade of Pierre Duhem,  our professor writes:

The required resort to the introduction of a spatial dependence of the thermal coefficients  might well not be open to Einstein.  Hence, in order to retain Euclideanism, it would then be necessary to remetrize the space.  ….  Einstein’s geometric articulation of that thesis  does not leave room for saving it by resorting to a remetrization in the sense of making the length of the rod vary with position or orientation  even after it has been corrected for idiosyncratic distortions.  But why saddle the Duhemian thesis as such  with a restriction peculiar to Einstein’s particular version of it?  And thus why not allow Duhem to save his thesis by countenancing those alterations in the congruence definition which are remetrizations?

-- “The Falsifiability of Theories”, in: Adolf Grünbaum, Collected Works, vol. I (2013), p. 72-3

As indicated, I couldn’t really follow the dialectical taffy-pull in that conterfactual-strewn discussion, but simply cite the passage as though on one of those “citation slips” we used to rely on at Merriam-Webster.

~

I just now happened upon a passage which we quoted earlier in another context (here):  a use, by a mathematician (or if you prefer, a logician) of the in-itself-not-expressly-mathematical term measurement,  not in a technical mathematical sense such as measure zero or measure theory,  but sliding mathwards towards concepts very far from any plain man’s conception of “measurement” (as: wholly non-numerical, non-quantitative   fundamental group):

Mathematics is, as it has always been, largely the science of measurement.  But “measurement” must here be understood as referring to more than the meter stick.  The genus of a topological figure  measures one of its aspects;  objects of genus zero  are in a sense simpler than those of higher genus. 

There are many dimensions of measurement ….:  characteristic, transcendence degree, cardinality, fundamental group … Occasionally we are so successful in the science of measurement  that we can completely characterize an object … by giving, as it were, its latitude and longitude:  its measurements in the relevant dimensions.

-- Herbert Enderton, “Elements of Recursion Theory”, in:  Jon Barwise, ed. Handbook of Mathematical Logic (1977), p. 554

I originally cited that as a not-especially-successful attempt (in its first sentence) at a one-line characterization of What Mathematics Is.    Measurement, in the usual sense, is common to a great many studious activities, from chemistry to engineering to dressmaking.   He only manages, in what follows, to make that characterization  more or less work, by moving the goalposts  and re-defining measurement in his own Pickwickian sense.
 

~

A philosophically alert historian of physics  calls attention to a linguistico-philosophical subtlety in the word measurement as it is used in quantum theory:

In all cases, an observation is accompanied by a measurement.  The converse is not true, however, for we may quite well perform a measurement and yet fail to observe the result.  [ndlr:  That much is true but trifling, but then he goes on to make his point.]  Now in considering the disturbance generated by an observation, we must make clear that the disturbance is caused by the physical measurement, and not by the cognitive act whereby the result of the measurement is comprehended by the percipient. … The observation is rendered possible by the collision of the photon with the particle, and hence it is this collision which constitutes the measurement.

-- A. D’Abro, The Rise of the New Physics (1939), vol. II, p. 667

Here he is not making an actio/actum distinction in the term measurement (though one exists; for the actum, “Her measurements are 36, 28, 36”), for we are still dealing with actio here:  but he is at pains to remove the connotation of a (human) action -- a human act, which one foggy school of thought has deemed central and essential  to all of quantum physics.  Observation, D’Abro is saying, is a human action;  measurement is whatever triggers the collapse of the wave-function.

~

This whole question of measurement  is, for the man meditating over brandy, frankly pretty annoying.    We want to know the scheme of things -- if equations be at the base of it, well and good, the more general the better.  (As:  Hamiltonian dynamics;  Set Theory; Topology.)  Beholding Saint Peter’s or the Taj Mahal, we wish to savor the whole, and perhaps to penetrate to the aesthetic and formal ideas behind them;  but we do not wish to know the length or this or that member in centimenters, nor how much the materials cost, etc. Such matters are distinctly hypo-ouranian.

In latterday musings upon quantum theory, measurement has been lifted to a role rather like that of (in earlier days) action, or conservation of energy -- or rather, like that of the Demiurge, bringing entities (or the values of their parameters) into existence.  Yet in practice, they don’t always even tell you much about what you are trying to measure.

Measurements on the force of attraction between two electric charges  will not  in general  verify Coulomb’s law.  We observe that the force depends  in some peculiar way  upon the position of external charges, which suggests to us that the measured effects  are not entirely due to the system in question, namely, the two test charges.

-- Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p. 524

A (non)Definition of Depth

We’ve posted a number of reflections about the idea of “depth” in (especially) mathematics and related science (for the complete list of these, click here: http://worldofdrjustice.blogspot.com/search/label/depth ), without ever really defining the term.  And this, for a reason:

Two interrelated ideas that have been widely assumed to be unanalysable  are those of one scientific theory being deeper and more unified than another.

-- John Watkins, Science and Skepticism (1984), p. xiii

The idea of theoretical depth has considerable importance in Popper’s philosophy of science.  “If at all possible, we are after deep theories.”  But he was pessimistic about the possibility of any sharp characterisation of the idea.

-- John Watkins, Science and Skepticism (1984), p. 188

And that, not necessarily for any ‘deep’ reason -- not correlating intimately with the intricacies of physics, say -- but much as it is hard to characterize sharply such multifaceted (or blobby) concepts as beauty or game.    And here, I must sympathize with the archetypal philistine of a hundred New Yorker cartoons,  genially conceding, “I don’t know much about art, but I know what I like.”   A mathematician or physicist may not be able to define depth in a way that would satisfy the notoriously finicky tribe of philosophers:  but he knows it when he sees it.  And smiles.

So, sorry, no necessary-and-sufficient conditions, nor even a rough-and-ready dictionary definition;  but anyhow, an epigram:

Whewell’s requirment that a deep hypothesis, one that gets hold of nature’s ‘alphabet’ as he put it, must enable us ‘to explain … cases of a kind  different from those which we contemplated in the formation of our hypothesis.

-- John Watkins, Science and Skepticism (1984), p. 190

(A similar metaphor, more popular since Whewell’s day:  a good theory must “cut Nature at the joints”.)

~

Hmm, now I’ve piqued my own curiosity.  How does a practicing lexicographer go about characterizing the term deep in the sense(s) of interest here?

Okay, for starter’s, a British one -- Collins English Dictionary (1979):

deep:  … (6) difficult to understand or penetrate; abstruse

(7)  learned or intellectually demanding: a deep discussion

Sense (6) is of no interest.  We went to some effort in this post (“A Dive to theDepths”) to distinguish depth from difficulty.  (For the list of posts re difficulty: http://worldofdrjustice.blogspot.com/search/label/difficulty .)   Sense (6) would be used by a lazy man, giving up -- “Too deep for me.”   Here he is not even using the word with its native literal resonance -- he could quite as well speak with poor Mr Tulliver, who often confessed that the world was “too many” for him.   Indeed, sense (6) might deserve one of those non-semantic/non-grammatical, sociolinguistic labels like “ -- Not in polite use”:  here,  “-- Not used in this sense by serious thinkers.”

Actually, by a twist of pragmatics, the phrase does get used by serious thinkers -- but typically as an ironic put-down.  Thus:

The towering 19^th-century mathematician Hamilton  labored hard on his “Law of Hodographic Isochronism”,

but when he sent it to John Herschel (whom Hankins calls “the best-known and best-regarded British scientist of his time” [p. 134]), he got the reply:  “You are fairly got out of my depth”.  (This was Herschel’s regular response when he did not have the time or inclination to follow Hamilton’s long analytical excursions.)

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

And again, roughly a century later:  After presenting a rather absurd and convoluted, goalpost-moving series of proposals from Lakatos and Morrall:

I am out of my depth with a claim of this kind.

-- John Watkins, Science and Skepticism (1984), p. 334

Here he is being disengenuously self-deprecating;  and his reply is all the more biting.

~

A more general thought, though, on depth versus difficulty.   I almost wrote that the latter was “much less interesting” -- though really, that depends on your day-job.  If you teach primary school, you need not (ex cathedra) worry your head one bit about depth in our sense;  whereas you must ever be alert to the perils of difficulty.  To epigrammatize into a dichotomy:  Depth (again, in our privileged sense) inheres in the subject itself;  Difficulty is relative to the practical limitations of some species (be it human, chimp, or the poor fly stuck in the fly-bottle) when grappling with the problems in that subject.   Since the philosophy of this blog is Platonist, we have little interest in the latter (no intellectual interest;  though some emotional interest, maudlin or morbid, as here).

From that perspective, sense (7) is also disappointing.   A subject itself (such as algebraic geometry or M-theory) cannot be called “learned” (i.e., learnèd):  that epithet might only be applied to whoever is gassing on about it.  “Intellectually demanding” could be applied either to a subject or a particular discussion or presentation thereof.  And that quality might be due to anything from the intellectual limitations of the audience (“The concept of evidence is too intellectually demanding for Trump voters”)  -- thus, back to sense (6) -- to an (overly) condensed presentation on the part of the lecturer, to actual depth inherent to the subject (and which would still be apparent to an angel, who understood the subject perfectly well).   Thus, neither sense goes far towards elucidating what mathematicians mean when they refer to a “deep result”.

~

Curious now whether my old alma-mater Merriam-Webster  did any better, I looked it up in their Collegiate Dictionary (Eleventh Edition),  I found something quite different.

First, their treatment of the geospatial, ‘literal’ sense of the term, from which all others ultimately derive, is unexpectedly rich and reticulated (I almost wrote:  “deep”), containing sub-subsenses like

deep  1 b (1) : extending well inward from an outer surface <a ~ gash>

(That is the sort of distinction you come up with when you are working from a generous deskful of carefully chosen citation-slips, rather than copying other dictionaries  or pulling the definition out of your butt.)

But then things sort of fall apart.  The sense “difficult” is not treated as a top-level numbered sense, as in the Collins, but as a subsense of a sense not defined save as the sum of its (rather disparate) subsenses:

deep  3 a : difficult to penetrate or comprehend : recondite < ~ mathematical problems>

3 b : mysterious, obscure <a ~ dark secret >

3 c : grave in nature or effect <in  ~est disgrace>

3 d :  of penetrating intellect : wise <a ~ deep thinker >

along with several more lying well off our axis of interest.   And oddly, despite all the careful hair-splitting, nothing really corresponding to Collins’ (7).

Thus, we still have come no further towards our goal.

~

The subject of depth, unlike that of mathematics, or Christianity (or oahspe),  tends not to attract disquisitions of the “What  is ….?”  sort.   We tried our hand at one for math (here), basically coming up with little more than a florilegium of blind-men-and-the-elephant stabs at it, for the overly general definiendum mathematics itself;  more fruitful was the task of characterizing topics within mathematics, like affine connection or topology, since here (at least for the former example) the definer’s intention is more in the nature of targeted enlightenment  than an after-dinner speech :  the result was a nice bouquet of epigrams.

~

Back to Depth vs Difficulty.    

(1) A deep remark or insight  is associated, not with presenting difficulties (as in Collins sense (6) ), but -- quite the contrary -- with resolving them.

A humble but poignant case  has been recounted here (Induction/Recursion), where a problem that had seemed difficult (to New Jersey third-graders, back in the complacent days before the impact of Sputnik  had filtered down to elementary school) -- that of multiplying multi-digit numbers -- suddenly became transparent, under the impact of an insight which (relative to what we had learned so far, most of it from the Mickey Mouse Club) might qualify as (qualifiedly) deep. 

(2) Above, we made something akin to an actio/actum distinction between difficulty and depth (human activity vs. the subject itself);   yet now we may make an additional distinction, on the same -- human -- side of the Platonic/psychological divide.  You might call it horizontal/vertical,  syntagmatic/paradigmatic :  judging words (concepts) by the company they keep.

~

Let us recur to that subsense in Webster’s Colleagiate,  “3 b : mysterious, obscure”, and consider the idea of  deep as it appears in company with that of being hidden:

The mind is in a sad state, when Sleep, the all-involving, cannot confine her spectres within the dim region of her sway, but suffers them to break forth, affighting this actual life, with secrets that perchance belong to a deeper one.

-- Nathaniel Hawthorne, “The Birthmark”

(I.e., a deeper, hidden something, that somehow itself  amounts to a “life”.)

And from a mathematical physicist:

It is indeed true that we can prove, from this kind of Euclidean argument,  that squares, made up of right angles, actually do exist.  But there is a deep issue hiding here.

-- Roger Penrose,  The Road to Reality (2004), p. 28

Namely (tying in with cosmology):

His fourth postulate asserts the equality of all right angles.  … In effect, the fourth postulate is asserting the isotropy and homogeneity of space.

-- Roger Penrose,  The Road to Reality (2004), p. 29

~

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