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

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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

On Scope and Difficulty

Literature and philosophy are fortunate in object, deficient in methodology. You can create whatever you like, and it can be quite unexpected and wonderful; but there’s no generally accepted method to check that you aren’t going off the deep end – or the shallow. So we wind up with diffuse or frivolous productions (like much of fiction and metaphysics) or even noxious ones (nihilism, postmodernism, nouveau roman). Attempts to add some formal rigor, some “constraints”, may produce abortions (like sestinas, or leipogrammatic novels), or shrunken offspring of once-great ideals (the nigglings of the Linguistic Turn, vice the Big Bow-Wow issues of Aquinas or Plato).

Physics is fortunate in methodology, restricted in its object – the actual physical world. In principle this might have led to a rather thin, grey science, had the structure of the cosmos been much simpler than that of the mind. (If you study universes other than the actual one, you’re doing mathematical cosmogony, not physics.) And indeed there were times when physicists thought they were already close to a Theory of Everything and could soon shut up shop. In practice, the universe turns out to have even a minimum intricacy that beggars our comprehension and stretches our mathematics.

Psychology, like physics, aspired to be a science, and here its object was no real limitation to the ambitions of the human mind, since it was itself coextensive with the human mind. But methodological bad luck seems to have produced some nonsense – cf. Frederick Crews on psychoanalysis, passim.

Mathematics is supremely fortunate in methodology, and in one sense unrestricted in object: you create the space you explore. Initially much of the space was suggested by aspects of Nature; but already with the hypothetico-deductive method, math had in principle cut its ties. As MacLane puts it, “The axiomatic method is a declaration of independence for mathematics. … A scientific theory can be falsified by factual data, while a mathematical theory cannot be so falsified.” So the actual angles of your triangle of photonic geodesics do not total 180 degrees? No skin off Euclid; go talk with Riemann or Bolyai down the hall.

The result has been an exfoliation of astonishing depth and invention. What has been dreamt up and perfected is far wilder than anything a poet or novelist has ever thought of attempting. Yet as the field becomes fragmented, there may develop a lack of check. It was rumored already at the birth of Relativity that only a dozen men around the world could understand it; how much more so as we clamber ever higher on the shoulders of those who stand on the shoulders of those who stand (or totter) on the shoulders of giants. In principle you could have some subspecialty in which the going theory – the generally acknowledged state of the art – was understood by only one person, its originator – and perhaps not even by him, once the inspiration had passed.  (No mere hypothetical -- cf. the sad confession of Mordell: 
"I have a poor memory, and cannot remember many of my results or proofs, let alone prove them again."

The case is different in literature. There is the class of “poet’s poet”, but it is not thus homotopic to zero. Once the circle of admirers shrinks to just the poet and his mom, we conclude, “He’s a bad poet.” In the case of some theoretical development which, say, only Noam Elkies understands and can vouch for, the rest of us may be uncertain what to think.

There have been dead ends in mathematics, like the “unit fractions” which are the theme of the Rhind Papyrus. Perhaps there are likewise now what may prove to be some overactive meristems.

*

Difficulty is not itself a measure of depth. Some things are hard in a bad way: we might call it “complication without genuine complexity”. Example: tax law; etiquette at the court of the Pooh-Bah.

Learning fifty languages fluently is hard – doubtless no one’s ever done it – and also broadening (“quot linguae, tot homines”, though the relationship is really sublinear), but after the second or third foreign language, you’re not going appreciably deeper. By contrast, an extra layer of genuine depth is attained when a specifically restricted set is studied by the methods of Indo-Germanistik. Here the languages don’t just sit side by side, but form a parallax, are projected backwards and forwards again according to rules – diachronic rules qualitatively different from the rules of any given languages – of reconstruction.

A major thrust of Chomsky’s movement was to go deeper, much deeper – “deep structure” wasn’t the half of it. A number of American students, ahistorical by inclination in any case, fancied that his focus on ordinary sentences of modern American English freed them from the drudgery of philology, and from considering any work done more recently than last year; and were startled when he published Cartesian Linguistics, with its serious consideration of Port-Royal. And bemused to discover that he is the same N. Chomsky of computer-science Chomsky-hierarchy fame. But for this deep thinker, it did all hang together: the rationalist philosophy, the intense algebraic abstraction (arriving finally at rules for which we no longer have conscious linguistic intuition), and the rich deductive structure of the working theory itself. This latter indeed posed problems for the would-be Chomskyan. No doubt inevitably, given their penchant for depth rather than breadth, many of its luminaries were somewhat deficient in workaday polyglot knowledge; and in accord with their abstract and theoretical bent, were not above manhandling the odd awkward fact, and massaging ill-fitting data along the edges:  Galileo as against Tycho Brahe. Now, if the theory had been superficial, errors might easily have been blacked out, like typos. But tout se tenait. Erroneous data would contribute to a particular formulation of, say, c-command, which itself then participated in myriad new structures and deductions. The student would have to carry forward through all this a mental footnote of doubt, adding little fuzzy-logic indices of partial truth to each succeeding syllogism. It was all hard enough to follow as a true believer; as a skeptic, I found it exhausting.

Contemporary mathematics seems weirdly, wildly deeper than anything else we do. By the time you get (as I have not) to, say, cohomology of sheaves (or its further offspring, which I cannot even name), you have passed through several layers of qualitatively different, structurally rich mentation. The further this development continues, and the more recherché the mental capacities required, the fewer people will have even a potential neurological capacity to follow where it leads. Our ancestors having survived the sabretooth, there is no qualitative difference in athletic capacities over mankind; despite the gushing of scribblers over this or that “sports genius”, there is really no analog of a Chomsky or an Elkies – such a person would have to actually be able to fly. But these exotic exaptations of more ethereal cognitive capacities may give rise to truly incommunicable insights, like different species which, with the best will in the world – man and faithful dog, or chimp, or dolphin – fail to communicate to one another anything at all complex. Something of the sort apparently happened some time ago in the field of music.

~ ~ ~

Postscript on difficulty: quotations from an author of unusual scope.

Ian Hacking, "Historical Ontology" (2002), p. 32. He engagingly admits, "I do find it very hard to make sense of Descartes, even after reading commentaries, predecessors, and more arcane texts of the same period. The more I make consistent sense of him, the more he seems to me to inhabit an alien universe." Accordingly, those who fancy they understand him, are living in a fools' paradise:

Ian Hacking, "Historical Ontology" (2002), p. 56: "It is paradoxical that Descartes can speak directly to sophomores whose conception of the world seems to be distressingly achronic -- yet the better you know the text, the more you realize that only the most arduous hermeneutical scholarship can make much sense out of it at all."

Ian Hacking, "Historical Ontology" (2002), p. 220. More modesty: "Chunks of Wittgenstein's writings have appeared every year or so since his death in 1951. They have the effect of time-release capsules. This is salutary if you think, as I do, that many of us skim his words and forget."