Magister magistri

Eugene Wigner, himself no dummyhead (or, to use the phrase that Pauli grudgingly conceded to Einstein, “not so stupid), a Nobel Prize physicist, said, concerning the mathematician/physicist/you-name-it John von Neumann,

“Whenever I talked with the sharpest intellect whom I have known -- with von Neumann -- I always had the impression  that only hé was fully awake, that I was halfway in a dream.”

(quoted in George Dyson, Darwin among the Machines (1997), p. 77.)

For an overview of the abjectly humble outlook  among many of those who would generally be considered quite gifted, see

  http://worldofdrjustice.blogspot.com/2013/03/de-stultitia-updated.html

Narcissism versus Arrogance (and Hubris)

Back to the correlation to narcissism, of various scientific disciplines.  (This is the continuation of an essay begun here.)

It is beyond contest, that (say) chemists and civil engineers have a reputation for very low doses of Drama, compared with (say) physicists.   But what are the traits that gave physicists such a rep?

It is seldom really narcissism, for that is a matter of individual, personal psychology.  If there is self-regard, that among physicists is more that of the guild -- a corporative trait, which might strike outsiders as arrogant. 

Ann Finkbeiner, in her fine history of the elite advisory group known as the Jasons (who were overwhelmingly physicists), gives an anecdote from an oceanographer (p. 138), recalling an uncharacteristic foray into oceanography by the Jasons:

I did resent this “the ocean is a wonderful summer playground for smart physicists who can do it in their spare time”.  You may know this thing called physics arrogance.  It’s real.

Indeed, Finkbeiner reveals (p. xvii) that at one point “I wanted to call this book The Arrogance of Physicists.”

Now, arrogance can co-exist with narcissism, and flavor it;   but they by no means need be co-present.  Thus, the eponym, Narcissus himself, was not arrogant in the least:  he was dreamily, solipsistically folded-in on himself.

Consider the humble confession of quantum wizard Wolfgang Pauli in 1925:

At the moment, physics is again terribly confused.  In any case, it is difficult for me, and I wish I had been a movie comedian or something of the sort, and had never heard of physics.

Yet many anecdotes are told about Pauli’s arrogance -- within physics : e.g. “What Professor Einstein says” (he concedes) “is not so stupid”.  Or the damning phrase “Not even wrong” (recently the title of a book attacking the very integrity of String Theory).   Or consider this joke (quoted by Ann Finkelbein in The Jasons, 2006):

A physicist gets to heaven  and God asks him if there’s anything he’d like to know, and the physicist says, “Yes, please.  Why is the Fine Structure Constant 1/137?”  God gives him the explanation, and the physicist says, “No, that’s wrong.”

Ironic note:   Any such explanation from God  would in fact have been wrong -- since (as we now know) the value of the Fine Structure Constant, though close to 1/137, is not exactly that, and thus is not a ration of (small) integers at all:  it’s just one of those boring physical parameters that drone on and on, and could just as well have been somewhat different.  The conjecture that it might actually be a pure integer ratio  caused a certain amount of numerological excitement back in the day, but that turned out to be a false alarm.

~

We have frequently written about the related but distinct notions of depth  versus difficulty (in math and science).   This comparison/contrast of narcissism versus arrogance (the former being key to our American culture, the latter to our politics) may likewise prove fruitful.

Stepping out a further circle of semantic nuance, we find hubris.  Like narcissism and arrogance, this can be an individual characteristic, as it was among the Greeks who named the concept.   But it can also characterize the goals and self-conception of an entire discipline. And beginning at least in the twentieth century, if not earlier, physics could well be described as hubristic.

[TBC]

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.

~

Psychological

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".

~

Mathematical

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:
http://worldofdrjustice.blogspot.com/2011/12/adventures-in-algebraic-geometry.html ]

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:
http://www.linguasacrapublishing.com/justice.html

Frontiers of Pataphysics

Of all the laws of physics, the Pauli Exclusion Principle seems the most like a fiat.  Two innocent fermions would like to snuggle together in the same quantum state, just like their buddies the bosons do (technically, it’s called “bundling”), when in pops Professor Pauli and says (for reasons best known to himself):  “NEINNN!  Es ist streng verboten !!”

Initially, the Heisenberg Uncertainty Principle feels something like this:  like the children in the Märchen who are warned never to go too near a certain spot in the forest, or a certain room in the castle (or a certain tree in the Garden, if memory serves), physicists are warned not to inquire too nicely into both the location and the momentum of a particle, nor any other pair of conjugate quantities.   But it turns out that the principle need not be stipulated, but can be derived, in various ways.  Thus Feynman, in QED (p. 56), claims that the principle falls out of his favorite technique of adding-up little arrows  -- “There is no need for an uncertainty principle!”  And Stein & Skakarchi (Fourier Analysis, p. 160) show that it follows from that fact that, if a given function is ‘bunched’, then its Fourier transform cannot be:  in which case the mysterious Principle turns out to be a simple truth of mathematics, and not a peculiarity of physics.

*

My mind was brought back to these reflections while reviewing my distressingly slender résumé so far:  which, despite my world-celebrated discovery of the Higgs Boson (documented here), would scarcely suffice for a Nobel Prize in Physics, or even a Mitch ‘n’ Gladys Memorial Prize in Some Kind of Science.   Accordingly, we feel the need to beef it up a bit, with the following finding, arrived at entirely independently of Pauli, when my brother and I were respectively five and seven years old:

The Metapenguin Exclusion Principle ®

My brother and I developed this in what we may call (in retrospect, though at the time we did not call it anything at all) the “Voice Game”:  which consisted in this.

~

There were precisely three available voice-registers:  Normal, Low, and Squeaky.   Only one of us could be in any given register at one time.  What you actually said, in this register, was up to you;  in practice, we didn’t say much beyond “*I* have the lo-o-ow voice.”  -- “And I-yee have the squeeeaky voice!”

In principle  we would transition at random among the registers, in accordance with the statistics of Weak Decay.  But there was a symmetry-breaking consideration: Naturally, any kid would want the Squeaky voice.  I remember one time when my brother was occupying it, and I tried to entice him out of it by saying, in a hearty TV-commercial voice, as though it were the best thing in the world:  “*I* have thee Norrrmal voice.”   Hoping thereby to entice him to a Lyman transition from Squeaky to Low, his hope being thus to entice me to a Balmer transition up to Squeaky, whereupon he could grab Normal -- only to find that it wasn’t as much fun as it was cracked up to be.

Such was the Voice Game.

~

From this, the fermionic version associated with Pauli  follows easily as a special case.  We leave this as an exercise for the reader.

Truth Decay

The empirical character of a very successful theory  always grows stale, after a time.  We may then feel (as Poincaré did with respect to Newton’s theory) that the theory is nothing but a set of implicit definitions or conventions …

-- Karl Popper, “Truth, Rationality, and the Growth of Knowledge”, repr. in Conjectures and Refutations: the Growth of Scientific Knowledge (1962; page refs. to the Harper paperback reprint), p. 240

I’m currently reading The Shape of Inner Space, by Shing-Tung Yau -- the Yau of Calabi-Yau, which lies at the heart of string theory.    Unlike Smolin’s The Trouble with Physics, let alone Woit's Not Even Wrong, the author is not out to debunk the theory in any way, especially as he was one of its mathematical progenitors:  nor to puff it, like Brian Greene, since  unlike those contentious authors, Yau is not a physicist by trade, but a pure geometer.  But towards the end of the book, he is led to exclaim: 

Given that much of string theory now hinges on compactifications of Calabi-Yau manifolds, which have these moduli with their associated massless scalar fields  and particles that don’t appear to exist, is string theory itself doomed?

And he quotes physicist Burton Richter against those quasi-nihilistic latitudinarians who have (as a recourse of despair) embraced “the landscape” (short version:  Anything Goes):

To them the reductionist voyage that has taken physics so far has come to an end.  Since that is what they believe, I can’t understand why they don’t take up something else -- macramé, for example.

There have, throughout history, been repeated instances of premature prophecying of “the End of Physics”:  but  there   the idea was that we were close to having solved everything;  never, that physics might one day become permanently stuck.

*

It was with these passages ringing in my mind, that I picked up Jonah Lehrer’s essay in the current issue of The New Yorker:  “The Truth Wears Off”.   I won’t summarize it -- it is brilliantly written, go read it -- but merely comment that the widespread scientific predicament there depicted -- in physics and biology and psychology and beyond -- seems even worse that the slightly softened interpretation that the author tries to present as a possible explanation or compromise.   Neither “happenstance” nor unconscious bias  can explain the range of what is supposedly going on.  E.g. the researcher who repeatedly failed to replicate his own results, not for want of trying, should  if anyone  have been prey to such bias.

[Cultural comment, to be developed if anyone’s interested:  the scientific standing of Rhine’s experiments in ESP.
-- Indeed, this just in:
http://www.nytimes.com/2011/01/06/science/06esp.html?ref=global-home&pagewanted=print  ]

*

As Pauli wrote to Kronig in 1925:
"At the moment, physics is again terribly confused.  In any case, it is difficult for me, and I wish I had been a movie comedian or something of the sort, and had never heard of physics."
Or cosmologist Edward Harrison in 1975, saying that endless expansion "would make the whole universe meaningless.  If that were true, I would quit, and spend my life raising roses." 

*

There is a thread, a thought, that links these observations;  but  once again -- Wovon man nicht sprechen kann, darüber muss man schweigen.

*

The Shape of Inner Space is something of an intellectual autobiography, in addition to an overview of Calabi-Yau.  And quite an engagingly modest one, as these things go.   By the end of it, you have a comfortable familiarity with the author, and are full of friendly feelings.

Immediately after finishing it, I happened to re-skim George Szpiro’s equally well-written volume, Poincaré’s Prize , and was startled to see that the sinister puppet-master depicted in the chapter “The Gang of Four, Plus Two”, is none other than Shing-Tung Yau.  When I originally read the book, the name meant nothing to me; but now… it was like rounding the corner, and encountering your own cousin brandishing a knife.

Now… in the grander scheme of things, so what;  we all have our faults.  But this account of academic rivalries  gibed with my surprise at searching Shape’s index for the name of Woit or Smolin, and not finding them.  Whatever their merits or demerits, these are popular recent book-length treatments of string-theory, and one might expect that Yau would reply to them, if only to rake them over the coals.  But only towards the end of his book  does he mention theirs, in briefest passing:  and then, without uttering their names (as one might be forced to allude to Mein Kampf, but draw the line at penning the name of its author).

Yau does, however, offer a kind of reply, as he demonstrates, pretty convincingly, that String Theory, the beneficiary of much math, has in turn created, and inspired, solid math in its own right, which will survive independently of whether the particular universe of our own sojourn  happens to embody the physics.   Indeed, the fact that Ed Witten received a Fields Medal, rather than a Nobel Prize for Physics, pretty much demonstrates that.

*

Addendum:

Pauli was not alone in his lament.   Schrödinger to Bohr, 1926:

If we are going to stick to this damn quantum-jumping, then I regret that I ever had anything to do with quantum theory.

And Einstein in 1924, re the idea of renouncing strict causality:

I would rather be a cobbler, or even an employee in a gambling house, than a physicist.

This is rather a rarified kind of job-dissatisfaction.  It is as though a carpenter, towards the end of a long and successful career, were to cry out, “Had I known that wood is mere cellulose, rather than a habitation for dryads,  I would have preferred to have been anything -- a plumber or a physicist -- rather than this!”

But the phenomenon does exist.  The author in which I found those two quotes -- J. C. Polkinghorne, The Quantum World (1984), p. 53 -- after a distinguished career as a particle physicist, resigned his chair, and became a vicar in a country parish in Kent.  It was from this Father-Brown-like living that he published his popular little volume with Princeton University Press.

Similar instances could be cited, notably Alexander Grothendieck.  (Motif: “Goodbye to All That.”)

 

[Update Feb 2014]  A statistical reason for “truth decay”, with examples:

Most scientists would look at a P value of 0.01 and “say that there was just a 1% chance” of the result being a false alarm. “But they would be wrong.”

Rather than being convenient shorthand for significance, the P value is a specific measure developed to test whether results touted as evidence for an effect are likely to be observed if the effect is not real. It says nothing about the likelihood of the effect in the first place.

http://www.nature.com/news/scientific-method-statistical-errors-1.14700