Monday, January 6, 2014

Difficulty is Hard

[Note:  Rather in the spirit of those of our essays which we have labeled “faux-naïf”, the title of this one might be called “pseudo-stupid”.  
Compare a formulation we likewise favor, “Infinity is big.”   That epigram is double-edged.  First, it mimics the naïve astonishment that the novice feels, not only upon being introduced to the idea of infinity, but even large-but-finite things like a googolplex.  (As a child, I marveled over that one, much as I marveled over the brontosaurus, and for the same reasons.)  But beyond that, it alludes to the fact that infinity is much bigger than you can imagine when you first meet it as “1,2,3, …. keep going forever”.  And this, in two qualitatively different ways:  (a)  The whole “Hilbert’s hotel” Marx-Brothers-stateroom routines you can play with countable infinity (well described by Rudy Rucker in Infinity and the Mind).  (b) That countable infinity, for all its capaciousness, is merely the smallest infinity; beyond it lies the uncountable infinity which denumerates the real numbers.   That one you can still kind of get a handle on;  but then in turn, infinitely many much larger infinities  rise beyond.

Too, the epigram is tricky to turn around into ‘Finitude is small’.  For, although anything finite is immeasurably smaller than infinity -- infinitessimally so -- so too is any given finite quantity, not immeasurably small to be sure (the ratio can be measured exactly, and differs for different quatities, unlike the case when comparing it with infinity), still unimaginably small (in psychologically evident sense which could be more rigorously defined) with respect to some other finite quantity, which therefore is unimaginably larger than it is.  (Think Graham's Number, or some iterated Ackermann function thereof.) There is, indeed, a lot of elbow room in the land of the finite.  To get a handle on it at all, you stop talking about individual quantities altogether, and instead investigate rates of growth of various kinds of function.  Some have been discovered which increase with a dizzying rapidity, next to which the proverbial “exponential growth” is like watching paint dry.

The concept of “difficulty” is not nearly so dizzying as that;  still, here as well there are at least two levels.  (1)  That felt by the ordinary layman, “Gee, this stuff is hard.”  (2)  A sharper and deeper sensation felt by many of those who have devoted a lifetime of study and practice to math and the sciences:  “Some of this stuff is difficult in ways I never even knew existed."

And, rounding out the paradox hidden in the apparent tautology,  the apparent converse is false:  for ease does not come easy, but only with much practice, and a certain gift.]

In the post linked to immediately below, we examined essayistically  the peculiar difficulty of mathematics -- not merely the well-known fact that a majority of schoolchildren find that algebra hurts their head, but that everyone, all the way to the top of the professional pinnacle, eventually butts up against something that baffles them, and weighs on their brain:

            De Stultitiâ

In the following, we surveyed less drastic analogues of the phenomenon, in such fields as linguistics and physics:
            On Scope and Difficulty

In the following series of essays, we examined the (difficult) question of intellectual depth, comparing and contrasting that with the (mostly psychological, not particularly deep) notion of difficulty:

           On Depth

Now (in the spirit of that last essay-series), we pass  to views internal to the field;  and this in two perspectives:

(1) Psychological:  simply a scattering of quotations, illustrative of the groans and misereres, of those who have attempted to scale this cognitive Olympus.

(2)  Mathematical:  Hints at ways in which certain areas or aspects of mathematics can be qualitatively “difficult”, quite apart from any intellectual limitations of its practictioners.



Otto Hahn, My Life (1968), p. 90: "I remember Professor Rubens once asking me: `How do you manage to distinguish between all these names and remember all their chemical properties into the bargain?  It's all so frightfully complicated!'"

Imre Lakotos' catty footnote in Lakotos & Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 114: "Neurath's [1935] shows that he never grasped Popper's simple argument."

Ronald Clark, Einstein: the Life and Times (1971), p. 333: Wolfgang Pauli, quite sure of his own brilliance, nonetheless wrote to a friend in the 1920's: "Physics is very muddled again at the moment; it is much too hard for me anyway, and I wish I were a movie comedian or something like that  and had never heard anything about physics."

Freeman Dyson, Disturbing the Universe (1979), p. 54: at Cornell, "Hans [Bethe] was using the old cookbook quantum mechanics that Dick [Feynman] couldn't understand.  Dick was using his own private quantum mechanics that nobody else could understand."

Mark Kac, Enigmas of Chance (1985), p. 112: "I had a look at some of Wiener's work on Brownian motion  but found it extremely difficult to follow."
& p. 115:  Kac contributed to the invariance principle, which is "now textbook stuff".  Yet "a recent book on the subject  is outside my comprehension."  [Note that this does not mean, "contains much material that was new to me", but rather:  "Even after working my way through the book, I cannot understand it.  God willing the next generation will be able to."]

Richard Rhodes, reviewing Abraham Pais' biography of Niels Bohr in NYTimes Book Review, 26 I 92: "It's sometimes heavy going, and I was reminded along the way of Luis Alvarez telling me that when he read Mr. Pais's biography of Einstein  he'd skipped the hard parts.  If a Nobel laureate could skip the hard parts, so can we all."

John Langlands, in his first of a series of IAS lectures (fall 99), said he'd wanted to be a physicist, but physics was "too difficult", so he had to settle for being a humble mathematics professor at the Institute for Advanced Studies.

Gigerenzer et al, The Empire of Chance (1989), p. 97: Ronald Fischer's writings are "not always transparent  to even the most hermeneutic reader".

John Conway, 17 XI 1999: "I studied Quantum Mechanics with Dirac. Quantum Mechanics is hard to understand, even when you can answer the questions on the exams.  And I couldn't answer the questions on the exams anymore."  [Yet another mathematical genius who found physics "too hard".]

David Berlinski, The Advent of the Algorithm (2000), p. 157: "Gödel lectured on his own results … the mathematicians (and philosophers) at Princeton for the most part could not and did not understand a word of what he said…"  [Note:  Here, nevertheless, the audience was mathematically the most sophisticated in the world.]

I.M.Yaglom, Felix Klein and Sophus Lie (1988), p. 54:
“Lobschevsky’s colleagues  failed to understand his work.  Since they did not want to write negative reviews, they simply ‘lost’ the text.”

I.M.Yaglom, Felix Klein and Sophus Lie (1988), p. 55:
“This symbolic language, using a minimum of words, made it very difficult for Bolyai’s contemporaries to read his great work.”

I.M.Yaglom, Felix Klein and Sophus Lie (1988), p. 57:
“His review was extremely negative.  Bunyakovsky failed to understand Lobachevsky’s ideas.”

I.M.Yaglom, Felix Klein and Sophus Lie (1988), p. 61:
“The audience listened attentively to Riemann’s lecture “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen”, but did not understand it.”

I.M.Yaglom, Felix Klein and Sophus Lie (1988), p. 78:
“Readers were unprepared for Grassmann’s approach and for his idiosyncratic style… Grassmann’s first book was ignored by mathematicians.”

I.M.Yaglom, Felix Klein and Sophus Lie (1988), p. 148:
The letter that Gaulois wrote on the eve of his death was published, “but, obviously, the item was not understood by anyone at the time  and was ignored.”

I.M.Yaglom, Felix Klein and Sophus Lie (1988), p. 153:
“Typically, the officers who proposed the problem  refused at first to consider Monge’s solution, being certain that his mathematical training was insufficient for solving it.”

I.M.Yaglom, Felix Klein and Sophus Lie (1988), p. 177:
“Further explanations proving the mathematical validity of all of Klein’s constructions  were not convincing: he who does not wish to see, will not see.”

Hamilton's intellectual biographer calls that mathematician's  Lectures on Quaternions "hundreds  of all but impenetrable pages".


First, certain subfields within mathematics are considered inherently substantially more difficult than others, at least for new entrants:

A Vertex Operator Algebra is an infinite-dimensional, Z+-graded vector space with infinitely many products.  It is not an easy definition, and there are no easy examples.
-- Terry Gannon, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 539

In reference to a certain operation on elliptic curves:

This construction can be regarded as the very beginning of Hodge theory, a powerful branch of algebraic geometry  with a reputation for extreme difficulty.
-- Jordan Ellenberg, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 191
[As for garden-variety algebraic-geometry, that is formidable enough: ]

Second, certain familiar problems, now considered elementary, turn out to be very difficult to solve in real generality and with proper rigor.    Thus, one of the first problems you meet in freshman physics is that of the Vibrating String.  Later, after mastering calculus and advanced calculus, you move on to Real Analysis -- and meet the thing again. Browsing through the standard textbok of F. Riesz & B. Sz.-Nagy, Leçons d’analyse fonctionelle [translated as Functional Analysis, 1955], I was surprised to find, well towards the end of the book, a chapter “Applications to the Vibrating String Problem”.
Similarly, one author remarks that only in recent times have certain classic problems in physics been settled rigorously, using the full arsenal of topology -- but that topologists are given scant credit, since the physicists imagined they had settled these matters long ago (though their proofs were fallacious).

Or cf. Charles Fefferman, who, in his article on the Navier-Stokes equation, places front and center  its status as a surprisingly tough nut to crack:

The Euler and Navier-Stokes equations describe the motion of an idealized fluid.  They are important in science and engineering, yet they are very poorly understood.  They present a major challenge to mathematics. … Although the Euler equation is 250 years old, and the Navier-Stokes equation well over 100 years old, there is no consensus as to whether Navier-Stokes or Euler solutions exist for all time, or whether instead they “break down” at a finite time. 
in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 193-4

All this is of more than academic interest, since Navier-Stokes rules hydrodynamics which governs the oceans and the atmosphere, and hence determines whether we shall all go blithely on or whether shall one day disappear in a polar vortex or the like.  (As I write [7 January 2014], the temperature has been hovering around zero Fahrenheit, but with a high of 72 forecast for Saturday -- four days from now.  It feels as though we may have entered a region of unstable vorticity.)

(Thus spooked, I read on, and on p. 196 encountered this:

In the Euler equation … solutions can behave very strangely.  A two-dimensional fluid that is initially at rest, and subject to no outside forces, can suddenly start moving …

For the past few days, I’ve been reading a novel by Stephen King, and passages like that cause the hairs on the back of the neck to bristle like quills upon the proverbial porpentine.)


For more from this pen, try this:

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