Riemann, Chomsky, Fido

‘Tis of great use to the sailor  to know the length of his line, though he cannot with it fathom all the depths of the ocean.  (Locke, Essay, I.i.6)

It was with the warmest empathy that we read the other day  of the collie with a 200-item Wort- und Spielzeug-schatz -- treasury of words and playthings.
[Note:  This essay was originally written in response to the report of a German dog-prodigy, in 2007.  For the latest such story, see http://www.nytimes.com/2011/01/18/science/18dog.html?hpw ]

As a semanticist, he is the perfect embodiment of what is traditionally known as the Fido-‘Fido’ theory:  this word refers to this object, and the way it gets its reference is – Fido grabs it in his mouth!  A saving detail, somewhat distinguishing him from the complexity of a cash register – or rather (since a cash register, in addition to popping up a digit when you press a key, can also calculate) from a mere inert pegboard with 200 labeled pictures – was his action when presented with an unfamiliar word: he fetched an unfamiliar toy!  Thus, to describe him, we need 201 lines of (utterly non-recursive) code:  if, then, else. (Game, set, match.)

 [For an alternate view of Fido and the Essence of Language, Cf.  Dr. Max Müller’s Bau-Wau Theorie, by Dr. Christoph Gottlieb Voigtmann (Leipzig, 1865).]

            Fido can fetch, but he doesn’t exfoliate.  The structure of his language is the structure of his toybox – a heap of unrelated items. If he could also speak – play with the words as he plays with the toys – magic might happen.  We gaze back with saddened understanding into the empty depths of his eager brown eyes.

(Per Wittgenstein, though, if he could, we’d be disappointed:  “Wenn der Löwe sprechen könnte, wir könnten ihn nicht verstehen.”)

            Empathetic, since we are ourselves in much the same predicament, faced with any subject for which we lack an inborn knack. In language we are all born-geniuses; in mathematics (most of us), born-morons. How poignant to hear the French, who tout their language as logic itself, stumble through the simple act of counting:

“….fifty, sixty…uhh..sixty-ten (soixante-dix)…mmm….four-twenties (quatre-vingts)….four-twenty-ten (quatre-vingt-dix -- I kid you not; and as to the spelling, sic)…”

or say they’ll meet you “today in eight” (aujourd’hui en huit; meaning: seven days from today).

            When  young, the mind at its most resilient, I put my shoulder to the boulder, majoring in math.  Over time  I have found it a labor of Sisyphus.  When I’m not actively pushing it, it rolls back downhill.  And even when giving it my all, I can only fetch things as they are pointed out.  I have never learned to chatter in math.

            By contrast, in learning new words, new languages, new styles, I really am standing on the shoulders of giants, feet firmly rooted in the innate.  No need to constantly “keep up” one’s Spanish.  It’s there.  When an author stretches your syntax – Nabokov or Proust – it gets digested, metabolized, incorporated into the mental flesh.

            At one level, mathematics is a language; and for a time, may give us the feeling of a similar mastery.  The notation is so powerful, it lets us deal with complex sentences with deceptive ease.  But I have found that the various gimmicks and shortcuts become a substitute for thought: I might scramble up to the next terrace, but then I had to pull the ladder up behind me, because the strength of understanding is only as long as that ladder.

            Thus: You learn, with understanding, why a certain integrand can be transformed into another, that lends itself more readily to integration by known rules.  But this understanding then becomes encapsulated, a black box.   Like the law of cosines or any other once-derived, once-felt formula, it becomes formulaic.  Whereas:  from a structure like “Bobby was scolded by his teacher” we get to “the man who is widely believed to have been credited with this discovery” intuitively, without cutting the ties.

            There are other things we’re really really good at, like real-time visual analysis.  The more you learn about what is involved, the more miraculous it seems.  When you realize the fragmented, ambiguous nature of the input, and the coordination required on our part, it is amazing that we can thread our way across a room without tripping over some tensor.

 ***

            One night, math studies years behind me, in the dark without the crutch of a textbook or chalkboard in front of my nose, I tried to recall what I once knew, and for the most part could not.  Floating facts and stray derivations, like isolated lines of remembered verse.  Then with the resolute despair of Descartes at his stove, I tried to discover: all right, what do I know, that isn’t mere memorization?

            It wasn’t much.  I could visualize, intuit, that 2 x 3 = 6, by picturing the boxcar pattern on a die.  This even yielded the commutative truth: 6 = 3 x 2 (just tilt your head to the side).  Fifteen was harder, but was accessible via a triplet of quincunxes, each quincunx mentally held in place with the fingers of one hand.  And already the commutation required a different pattern, one big quincunx, each spot a little triangle.

            Invention gave out by twenty-one.  One can picture a triad of “dotted boxcars”, but what is 7 + 7 + 7 (let alone 3 + 3 + 3 + 3 + 3 + 3 + 3)?  One can dully, dutifully count (which is merely mechanical, bearing no relation to the gut grokking of a quincunx as five), or recall “3 x 7 = 21” from the times-table – a mere boilerplate formula like “a stitch in time saves nine”.

            So, I got as far as boxcars, but shall never catch up to the taxicab, where Ramanujan spotted “1729” as the face of an old friend.

            Shapes in space are also hard.   (Again Locke, Essay II xix.13:  "In a man who speaks of a chiliadron, or a body of a thousand sides, the idea of the figure may be very confused, though that of the number may be very distinct.")
Once, dipping into a bit of knot theory, and finding that I soon had to haul up the ladder to proceed, I went back to square one, and tried to understand a simple knot in the same direct way that we understand a softball or a football.  I made a wire model of a trefoil, turned it every whichway, closed my eyes and followed it with my fingers.  Falling asleep, I would imagine myself (like Mr. Tompkins) on a roller-coaster ride  shaped just like that, sensing when another part of the track would pass above or below, feeling the centrifugal tug.  I can barely do it, strain though I may  -- I throw on the light in a panic and stare at the wire.

            Again Locke (Essay II.x.4):

The memory is very weak:  ideas in the mind quickly fade, and often vanish quite out of the understanding, leaving no more footsteps, or remaining characters of themselves, than shadows do  flying over fields of corn…

            Years of effort have sadly ratified the epigram of Novalis: "Zur Mathematik gelangt Man nur durch eine Theophanie."

            De profundis, ideo, clamo: Riemann – eleison!  Poincaré – eleison!

            Solâ gratiâ.