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

~

.

The Blind Men and the Billiard-Ball

We are all familiar with the parable of the Blind Men and the Elephant:   although it does suggest something about the limits of human perception (and, by parabolic implication, cognition), it is essentially a testimony to the complexity of elephants.**

Had the blind men been, rather, fondling a billiard-ball, they would have agreed on its characteristics.  From whatever angle the blindman came, he would conclude:  Smooth!  Round!***

[** Likewise characteristic is the excellence of elephants.

For proof and examples, click here.]

[*** If wise, each blindman would testify only to the local properties of the surface, such as the Gaussian curvature.

Additionally, even a sighted man, given free run all over the object, must needs refrain from asseverating, that what he felt and saw is all there is to the geometry -- the perceptual two-sphere might rather have been a cross-section of an unperceived hypersphere, of which it forms  but a negligeable part.]

Less obviously, such blindmen would concur as to the nature of the Stone–Čech compactification of any given set.  For, from whatever angle this grand object were approached, the verdict would be identical:  Maximal!  Universal!

~

Contrast the subject of Algebraic Topology.

I just finished reading one elementary introduction to this subject, and leafing through two others entitled Algebraic Topology  which, though scarcely elementary, claim the reassuring subtitle of An Introduction or A First Course.

The first labors long in the messy, mucking-about-with-triangles world of simplexes, as a lead-in to simplicial homology.

The second spurns these “rigid gadgets”,  and hews to the purer path of singular homology.

The third abjures homology altogether.

[Appendix]  Weiteres zur Elefanten-Mechanik:

It is well known that slowly-moving things obey classical mechanics.

-- Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p. 269

Corollary:

Elephants obey classical mechanics.

Egyenletesen sürü számsorozatok

We have oft lamented, how … hard things are, particularly in math and physics.  Let alone actually coming up with any worthwhile contributions yourself, nor even simply “keeping up with the literature”, but merely :  taking in -- understanding one  classic work done forty years, fifty years, a century ago.

I have on my nightstand  volume one of John von Neumann’s Collected Works (so titled, in English, and published in 1961).   Now, most Americans, myself certainly included, think of von Neumann -- “Johnny” to those who knew him -- as a Princeton luminary equal to those of Gödel and Einstein, and who, even more than they, was significantly responsible for ushering American into the front ranks of modern math and science, and best known to the general public as the co-author of the Theory of Games and Economic Behavior (1944), whose subsequent significance has only grown.

Yet those more familiar with the man will also be aware that he was born in Budapest, where he attended a German-speaking high school, eventually pursuing post-graduate studies in Switzerland and teaching at the University of Berlin.  He moved to Princeton in 1930 (a wise move, as things turned out), and remained there to the end of his days.

Accordingly, the reader confronted with his Collected Works, will be prepared for many of the early papers to be in German.   For this reader, that presents no problem at all.  But in any event, the Pergamon Press has organized this collection, not by date, but by topic, the first volume containing  “Logic, Theory of Sets, and Quantum Mechanics” (spot the odd man out, but anyway).   For the technical reader, rather than the biographer, that is convenient and commendable.  (Cf. the philologian Hugo Schuchardt, in “Sachen und Wörter”, second paragraph:  “Stofflich geordnete Glossare  gewähren manche Aufklärung, die in alphabetische geordneten vermißt sind.”)   And you would expect that, thus organized, each volume would contain a mixture of articles in German and (later) English, with the exception of those treating of the theory of automata, or the game theory, which he never wrote about before coming to America, since he invented both of them here.

Yet the first volume contains no single article in English -- but does have one in Hungarian, whose title I have placed in the subject-field of this post,  pour épouvanter les érudits.

ON DEPTH AND BREADTH (überarbeitet)

            Someone (and since I forget who, it is as if the words had passed into proverb) once said: “A brilliant person is one of whom I think: I could have done such work, if only I were ten times smarter.  But in the case of a genius [here the reference was, if memory serves, to Feynman, and the epithet might actually have been not “genius” but “magician”], I can’t even imagine how it might be done.” 

            That is the depth dimension, often commented upon.

            Less obtrusively, because  by its nature  never to be noticed save over time,  is a baffling precellence of breadth.

   
  This thought arises as I finish Ian Hacking’s The Taming of Chance.  Hacking is unfailingly intelligent, but not deep beyond one’s reach.  He has a modest, easy style.  He’s not a bibliographic bully, forever quoting Kant or Toqueville or whoever you haven’t read, a propos of everything and not always with necessity (“Will the Red Sox prevail next season?  Only time, as Aquinas would have it, will tell.”).  He does not throw up a Potemkin village of supposed sources, a bibliography stuffed with Lukacs and Plato, longer and denser than the slender article it supports, topped off with epigraphs from Wallace Stevens and Valéry. Yet with everything of him I come across, he reveals a new side of him.  A polymath;  forsooth, a polytope.

            The first thing of his I read – and warmly recommend – was The Emergence of Probability. Here he seemed one of those philosophical mathematicians-manqués of the stripe of Putnam or Quine, who write with wit and erudition about technical subjects, but (one eventually finds) pretty much plow the same furrow in their various works.  Only, in later works he reveals himself as a closet Foucault fan (to this reader, as far from Kolmogorov as, I don’t know, a Red Sox fan).   And he’s written complete books on purely psychological subjects.  In The Taming of Chance, he returns to the subject of Emergence from a (crucially) slightly different angle, so that the book covers much of the same ground  but with a very different feel and specificity.   It doesn’t even have a bibliography, you have to dig it out of the footnotes.  Yet see what the spade unearths! Page after page of primary sources, like one William Turnbull’s Treatise on the Strength, Flexure, and Stiffness of Cast Iron Beams (London, 1831), or of secondary ones from obscure nooks, like the Tübinger Zeitschrift für Staatswissenschaft 4 (1863).  He cites Bernoulli, fair enough – but wait, not (or rather, not only) the celebrated Jakob of the Ars Conjectandi (which --though I haven’t read it and probably never will-- is a recognized landmark in the field),  but Johann – yet wait again, not that Johann, that familiar Johann “Jock” Bernouilli whose works you probably have on your night-table, but the later one; and not his mathematical publications neither, but rather his  no doubt in its day rightly celebrated  Reisen durch Brandenburg, Pommern, Preussen, Curland, Russland und Pohlen, published in four volumes over the course of that busy year of 1779-80 (in memory still green), and of which Hacking’s unobtrusive morsel is quarried from volume two.

            This is not what is meant by being merely, or even hugely, “well-read”.  Well-read means you’ve read all the classics throughout history in every major language in every field, and absolutely every scrap of anything at all in your own particular specialty (published or in preprint – or in papyrus --  ancient or modern), plus the odd condiment from among contemporary novels, Doonesbury,  or the folks at Port-Royal.  That’s hard enough-- I read all the time; always have; and am still just hitting the high points (and forgetting much).  Yet given 400 hours in a week instead of forty, and a couple of lifetimes tacked end to end, plus total recall, one can imagine doing it.  But the “Compte rendu des travaux de la Société phrénologique pendant le cours de l’année 1839”, which Hacking uses to such telling effect, is not on anybody’s reading list.  The ergodic reader ranging over all space  would perish in the heat death of the universe before striking upon this. 

            In fact, the whole situation puts me in mind of what Lszlew Brnzowlski so memorably said in his (partially encrypted) diary entry for 3 April 1648 (U. Tbilisi MS #9784-K) --…  No it doesn’t.  Haven’t read it.  Never heard of it.

Postscript.

            

Hacking himself can strike a stance of amazement at (allegedly) recondite knowledge.  In a chapter on philology (in Historical Ontology, p. 141), citing Foucault’s discussion of Schegel, Bopp, and Grimm, he exclaims: “Who on earth was Bopp?”  (One of the most celebrated names in the history of philology, that’s who.)  This is no doubt a pose, to let the reader catch his breath, and to chummily pretend that we’re all just struggling along here together, lads -- the way someone expounding physics for a lay audience will strive to make it all “more accessible” by sprinkling-in little apology-markers (“what physicists call the ‘spin’ of a particle known as the ‘electron’”;  “the so-called ‘Higgs boson’”;  “something called a ‘photon’”).  But shortly thereafter, Hacking himself calmly goes on to dust off such truly obscure proto-philologists as, um, “Chladenius”...

And if all that is making you feel stupid (as it certainly should),  here’s something to make you feel even stupider:

        De Stultitiâ

*

One naturally does not notice, upon first or second acquaintance, any lack of breadth  in any favorite author of your own.  You go to him for some few things, whatever they might be, and are delighted to find them again.  We do not reproach the humble hamburger, for failing to be a quail;  nor the quarterback, for lacking the qualities of a third-baseman; nor Dickens, for his obstinate indifference to the more recent developments of the higher calculus; nor indeed the Gospels, for lacking tomorrow’s weather forecast. Yet if some author comes to be our guide, our lens upon the variety of life, we do  in time  notice any astigmatism, or narrowness of the field of view.

            For some time, Orwell was my cherished author, seemingly spanning great territories: as novelist, memoirist, and critic – nay, ever as a fighter at the front --: noted in particular for his splended essay attacking insularity, “Inside the Whale”.   It was only a chance remark of his, apologizing for not having read more than several dozen of the (delightful, but) trifling novels of P.G. Wodehouse, that it struck me, how much had escaped his notice while he was so occupied:  scilicet, virtually every world-shaking intellectual development, be it in physics, mathematics, biology, philosophy,  from the mid-19^th century   on.

            Tolkien created a world, Middle Earth,  commended (in Webster’s Encyclopedia of Literature) as “one of the more detailed of all fantasy worlds”, yet which always struck me as narrow to the point of suffocation.  GKC  creates an  in some ways  similar world, but which has airholes into the infinite, which let us breathe. C.S. Lewis, in his childhood, together with his brother, created such a puppet-theatre world, reminiscences of which survive in his adult fiction; but he bursts its walls decisively  in all his essays.  His world is wider; retaining (in his own metaphor) the furniture of the nursery, side by side with the cold hard sunlight of the new world.

            It was in the course of reacquainting myself, with the saga of Greek antiquity, and Roman valor, that I happened to notice  the absence of any echo thereof, in any of the aforementioned authors. (The mindworld of Lewis embraces Northern mythology, but nothing earlier.)

*

Other such instances.

Re Otto Rank, Ernest Jones speaks of

… his truly vast erudition;  it was quite mysterious  how he found the time to read all that he did.  One of the compliments I treasure in my life  was when he asked me wherever I had found all that material in one of my non-medical essays;  that the omniscient Rank should be impressed, signified much.

-- Ernest Jones, Freud: Years of Maturity (1955), p. 160

Richard Fortey, Earth (2004), p. 314:

            One of the chief proponents of the [Snowball Earth] theory is Paul Hoffman at Harvard, one of those American academics who seem to have twice their fair share of energy. If expertise is defined as knowing  more and more  about less and less,  I am at a loss to describe what it is to know more and more  about more and more – but that is the Hoffmann condition.

*

In a work, written in English, and  ostensibly directed at engineers and physicists, subsequent to a discussion of infinitesimal deformation and path systems, we read:

We shall not go further into this approach here.  It is done quite simply and naturally in a classic paper by J. Radon.  Indeed, since this paper is one of the clearest and most elegant in the entire history of the calculus of variations, we prefer to suggest to the reader  that he consult it directly.

-- Robert Hermann, Differential Geometry and the Calculus of Variations (1968), p. 257

An admirable suggestion!  One hopes, however, that our pocket-protector-sporting anglophone engineer has been keeping up his German, and has ready access to an enormous research library, since it was published in that language (“Zum Problem von Lagrange”) back in 1928, in a relatively obscure series (Hamburg. Math. Einzelschriften, Leipzig).

*

[Update Jan 2012]  Freeman Dyson on his depth stage and his breadth stage:
http://moreintelligentlife.com/content/ideas/charles-nevin/60-year-job-freeman-dyson

~            ~            ~

For a different essay covering distinct though similar ground, try this:
http://worldofdrjustice.blogspot.com/2010/12/on-scope-and-difficulty.html

For depth itself, as deep as it gets (which is in mathematics),  this:

http://worldofdrjustice.blogspot.com/2014/01/a-dive-to-depths-expanded.html

A Dive to the Depths (expanded)

A phrase you will often meet in higher mathematics, and almost nowhere else, is:

“a deep result”

O loveliest of monostichs, thou !

The very notion of what ‘deep’ means, in such a context, is itself deep;  indeed, too deep for me, at present.   This, owing to a crippling condition of mathematical oligophrenia.  --  which, however, I pray that time and diligence might partly palliate.  (For a glimpse into the terrible sufferings of mathematical oligophreniacs, click here, if you dare.)  Yet I am putting up this skeletal promissory-note of a post, so that there will be a space to scribble insights as they wake me in the night.

First off -- the term "deep" does not mean simply ‘difficult’; indeed, though such results lie in the depths, and are not to be had for the asking, once you have somehow managed to fish one up, it may seem clarity itself.  Nor does merely being difficult make anything deep.  Any humongous brute-force calculation falls into that category;  for a more-substantive example, consider the Four-Color Hypothesis, which people suspected should be deep, but the proof that changed "Hypothesis" to "Theorem"  is a combination of clever tricks and elbow-grease.  The response of the mathematical community was disappointment:  "So, it turns out it wasn't an interesting conjecture after all."  (Of course, it may yet prove to be "interesting" in our cognitive human sense; that awaits a proof of an entirely different kind.)

We may go further, and put forward that an overarching purpose of mathematical research is to reveal something previously difficult  as now simple, when seen in the right way.  Again and again this has happened in history, beginning with the replacement of finger-counting by symbols, and of clunky symbols like Roman numerals by decimals.  For a more recent example:

Although Beurling’s own proof [characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to prove;  it depends on hardly anything more than the geometry of Hilbert space.  The profitable point of view  is not sequential but functional.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

Nor is there any simple similarity or opposition between being “deep”  and using what are now called “elementary” methods in number theory (as opposed to analytic methods), e.g. Alte Selberg’s proof of the Prime Number Theorem by elementary methods, compared with earlier analytic proofs by Hadamard and others.  Typically, proofs that restrict themselves to “elementary” methods are harder than those that permit themselves a more capacious toolkit;  but whether they ever, or generally, gain depth via this austere discipline, I have no idea.

In the meantime, some related posts outside of a mathematical context  are these:

            On Depth and Breadth

            On Scope and Difficulty

As appetizers, try the following hors-d’œuvres platter -- to follow which, however, we have as yet prepared no meal (as with our early essays on the Realist vernacular, this is more by way of linguistic warm-up):

The connection between linear transformations  and bilinear functionals  goes quite a bit deeper …

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 38

The Riesz representation theorem depends on some of the deeper parts of the theory of measure and integration.

--George Simmons, Introduction to Topology and Modern Analysis (1963), p.

The analogy between cyclotomic fields  and fields formed from the points of finite order on elliptic curves   is very deep.

-- Neil Koblitz, 1993

The notion of Kan extensions is the deeper form of the basic constructions of adjoints.  We end with the observation that all concepts of category theory are Kan extensions.

-- Saunders MacLane, Categories for the Working Mathematician (1971; 2^nd ed. 1998), p. vii

The continued-fraction representation of real numbers is deeper than the decimal expansion.

-- Roger Penrose, 2004

There are deep ties between enumerative geometry and Ramanujan’s tau function.

Contrast:

The useful fact about products of projections  lies near the surface.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p.  47

In the same work, while acknowledging the possibility of a lattice-theoretic formulation of the subspaces of Hilbert space, he dismisses this possibility as “trite” (p. 22).

*     *     *

~ Commercial break ~

Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

*     *     *

Much commoner than “trite” is trivial, which is virtually a terminus technicus of mathematical practice.  Let one quote stand for all:

One of the useful conclusions we can draw from Theorem 2 [to the effect that the norm of a Hermitian operator equals the supremum of its eigenvalues] is that the spectrum of a Hermitian operator  is not empty.  This is not a trivial conclusion.  We shall obtain the corresponding fact for normal operators  only after the application of a lot more relatively deep analysis.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 55

Psychosociological note:   Attempt to imagine the effect on our pumped-up math-major  freshman or sophomore brains, hearing dismissed as “trivial” (with a wave of the hand) propositions which, but a year before, we ourselves would not have begun to understand, and which indeed most of our countrymen will go peacefully to their graves without understanding.  (For more on the hubris involved, click here.)

Also note:  What counts as “trivial” is relative to where you stand.  Thus, in the very next sentence, Halmos adds:  “We hereby report that the spectrum of an arbitrary operator is also not empty;  since we shall have no occasion to make use of this fact, we shall not enter into its proof.”  The proof, one gathers, is more difficult still.  But when once you have mounted, and stand upon that summit, the fact that Hermitian operators in particular have eigenvalues, is trivial indeed.

Leave it to mathematics to recruit even the notion of triviality into some highly non-trivial constructions.   E.g.

A topological space over X is called a locally trivial fibration if every x in X has a neighborhood over which Y is trivial.

-- Klaus Jänich,  Topology (1980; Eng. trans. 1984), p. 129

And:

Sard’s Theorem … is … a highly non-trivial  theorem  which is elementary in the sense that it uses only the notion of a differentiable map.

-- Shlomo Sternberg, Lectures on Differential Geometry (1964)

The terms deep and elementary (here in the everday sense, and not the special number-theoretic meaning mentioned above) are not antonyms, but they do contrast:

The spectral theorem implies that every normal operator has a large supply of invariant subspaces;  this is classical  and can be considered elementary by now.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

This dependence of “depth-perception” upon experience accumulated with the passage of time, does not refute the notion of relative mathematical depth as ill-defined.   What is intuitive though difficult to put into words  is a notion of “deeper than” rather than of absolute depth.  To the giant, neither the pond nor the puddle appears deep;  but the pond is deeper  for all that.

~

Re the classification of simple algebras (“simple”, to be sure, in a certain technical sense, meaning roughly: incredibly complex and difficult):

The tools employed  are not deep; they are just, so to speak, linear algebra  raised to the nth power.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 122

~

Further attestations.

A pioneer of Hilbert space theory, expounding the then-contemporary state-of-the-art for a nonspecialist mathematical audience, particularly as regards dilations and extensions of operators:

There do not seem to be any conspicuous and challenging yes-or-no questions that serve to indicate the direction in which the search for new results might begin,  but I have faith.  There is depth in the subject;  the trouble is that the surface has not been explored enough  to show where the deepest parts lie.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

Writes a premier algebraic-topologist:

There are geometric problems which require the use of the multiplicative structure of the topological invariants.  Such problems are deeper than those which can be solved by considering the additive structure alone.

Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 105

Re Lie algebras and a closed subgroup H of the full linear group:

It is furthermore true (and this is deeper) that these one-parameter subgroups  fill a neighborhood of the identity in H, and consequently generate H if H is connected.  …The converse part of the correspondence  involves a subtlety of the type that makes the study of Lie groups a quite sophisticated topic.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 118

Re the Hodge conjecture:

It arises deep within the subject, at a high level of abstraction;  and the only way to reach it  is by way of those layers of increasing abstraction.

-- Keith Devlin, The Millennium Problems (2002), p. 9

This is a truly adult depiction of depth: as lying deep within the layers of the enigmatic onion.  It is not a case where you can just swallow some peyote and see it all in a flash.  
(For more, compare:  The Ladder of Abstraction.)

Writes a philosopher:

The axioms are not logical truths ... Their truth is established by intuitions which lie too deep for proof, since all proof depends on them.

-- Roger Scruton, Modern Philosophy (1994), p. 392

An example from outside the field of mathematics -- though it is a mathematician who is writing this:

Just how a protein manages to organize itself in space, using only the sequence of its own amino acids, remains a mystery, perhaps the deepest in computational biology.

-- David Berlinski, “What Brings a World into Being” (2001), collected in:  The Deniable Darwin (2009), p. 243

~

Related vocabulary:

Related to the concept of depth (which focusses on the root of things) is that of richness (regarding the blossoms that bloom from this root).   Hadamard adopts this metaphor explicitly:

Application’s constant relation to theory  is the same as that of the leaf to the tree:  one supports the other, but the former feeds the latter.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 125

Further examples:

The algebra of composition of maps  resembles the algebra of multiplication of numbers,  but its interpretation  is much richer.

-- Lawvere & Schanuel, Conceptual Mathematics (1997), p. 11

… a recent, and surprising, theoretical advance by Winkler.  He shows that, for many … algebraically closed fields, the “free Skolemization” has a model companion.

-- Angus Macintyre, “Model Completeness”, in: Jon Barwise, ed. Handbook of Mathematical Logic (1977), p. 164

The concept of thickness [in graph theory] is the deep mathematical idea that underlies the recreational puzzle of Earth/Moon maps.
-- Ian Stewart,  How to Cut a Cake (2006), p. 126

~      ~      ~

Having convinced ourselves, by the examples of mathematics, that there is more to this assessment-word deep than an emotional or impressionistic grunt,  we look to some cases outside of mathematics where an idea has been similarly assessed.

Some discoveries provide answers to questions.   Others are so deep  that they cast questions in a new light,  showing that previous mysteries  were misperceived.

-- Brian Greene, Fabric of the Cosmos

We are not at home in the world, and this homelessness is a deep truth about our condition.

-- Roger Scruton, Modern Philosophy (1994), p. 464

T.S. Eliot affirms that what is past and what is present, even what might have been, indicate a present purpose.  This is a metaphysical point of great depth.

-- James Schall, S.J., The Order of Things (2007), p. 69

And, more prosaically, but no less tellingly for all that:

Although running Bain Capital required a lot more brains and savvy than playing roulette does -- a lot more brains and savvy than most of us could even pretend to possess -- the job was not conceptually deep.  Romney did not develop a model of the world from the business of private equity. … “He’s not a very notional leader,” [said] Romney’s campaign spokesman …

-- Louis Menand, “Money Pol”, The New Yorker (19 III 2012)

~      ~      ~

This is quite aside  from the path of mathematics, but -- it may be, that such depth is displayed in quite distantly allied regions:  all tracing back to Him, perhaps by some functorial construction.  In that spirit, this:

http://www.youtube.com/watch?v=s5xDEyuXaOY&feature=fvwrel

The final anguish  of the Asian bride  suggests the depth  of the Riemann Hypothesis.

The enigma of a woman’s heart,

finally espied  by a Private Eye,

for less than the price  of a Valentine …

This Rose

[Kindle]  [Nook]

~     ~     ~

Somewhat less far off the path …  Deep is indeed the term of art  in mathematics, antonymic to trivial.   Now compare, from another discipline, the word profound, in reference to Newton’s perplexing, little-known  philosophical-speculative opus:

That it is exclusively mystical  I do not believe -- that there is a mystical element  seems certain.  I hope that  one day  some profound student -- no one less will suffice -- will study this mass of papers.

-- E. Andrade, quoted in James Newman, ed. World of Mathematics (1956), p. 273

~ ~ ~

Above, we saw the distinction deep vs. difficult.  Here now even the latter concept is bilayered:

Although Beurling’s own proof [characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to prove;  it depends on hardly anything more than the geometry of Hilbert space.  The profitable point of view  is not sequential but functional.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

In infinite dimensions, it is tedious but not difficult  to construct spaces  strictly  but not uniformly  convex.

-- Prof. Lewis, University of Alberta, course in Functional Analysis, 1982

The proof that Reidemesiter moves  and planar isotopy  suffice to get us from any one projection of a knot  to any other projection of that knot  is not particularly difficult;  however, it is technically involved.

-- Colin Adams, The Knot Book (1994), p. 15

And here the original distinction is made even sharper:  though they are not interchangeable, depth tracks with generality, which in turn tracks (on the plane of praxis -- of proof) with simplicity:

Re the Denjoy-Young-Saks Theorem on the derived numbers of functions:

As we would expect  in view of the great generality of the final statement of the theorem,  the proof due to Saks is of extreme simplicity.

-- F. Riez & B. Sz.-Nagy, Leçons d’analyse fonctionelle [references to the English translation, Functional Analysis, 1955], p. 17

~

Here a leading mathematician laments the shallowness of his understanding of something he himself proved (regarding representations of a Kac-Moody algebra, as it happens):

My proof of this result was technically quite involved.  I was able to explain how the Langlands dual group appeared, but even now, more than twenty years later, I still find mysterious why it appears.  I solved the problem, but it was ultimately unsatisfying to feel that something just appeared out of thin air.

-- Edward Frenkel, Love & Math (2013), p. 181

This is setting oneself high standards indeed.   Shakespeare probably did not lie awake o’ nights fretting how the devil he ever came to write Hamlet;  Mozart did not find the bread of pleasure at having written the Sonata in A  turning to ashes at the thought that it might have been dictated to his unconscious  by an angel.

~

A near-synonym of the math-word deep, but shorn of all irrelevant aesthetic echo, is:  “highly nontrivial”.   The term is decidedly commendatory, though to a layman it might sound like faint praise, as were one to dub one’s lady-love “seriously unugly”.  The expression may be extensionally impeccable, but ‘twould never pass in a sonnet.

 Further:

In set theory, a forcing extension in Cohen’s sense  is reminiscent of algebraic extensions of a field, but

… the forcing method is far more complex, both conceptually and technically, involving set-theoretic, combinatorial, topological, logical, and metamathematical aspects.

-- Joan Bagaria “Set Theory”,  in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 625

Although he does not use the term "deep" here, that expanded characterization  "complex, both conceptually and technically", especially the "conceptually" part, points in that direction.

~

One motive for Frege’s choice  was again generality:

Does not the ground of arithmetic lie deeper than that of all empirical knowledge, deeper even than that of geometry?

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 184

… [Cantor’s] remarks on functions of several variables (where the provability of theorems  was deepening the level of rigour in analysis)

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 223

 

.