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

Updates to Essays

Click on the title  for the essay in full.

To “The Umbrella Man”:

Arthur Koestler, on the attitude of himself and his companions in Cloudcuckooland, up to the shock of September 1939, when they finally realized that the time of postponements and appeasements had finally passed:

They had lived so long under the sign of the Umbrella,

that they found it difficult to believe

that the age of the Sword had come.

-- The Scum of the Earth (1941, 2^nd. edn. 1955)

[Anyone who is inclined to see a parallel with certain quite current events, is free to do so.]

To “De Amore”:

Marriage is a duel to the death, which no man of honour should decline.

-- G.K. Chesterton, Manalive (1912)

Man to Matrimony:  Are you satisfaktionsfähig ?

Matrimony:  Aye;  and there is little else that is, here below.

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]

To “Minimalism in Physics”:

On the extra profusion of different string theories, a mathematician remarks dryly,

It was hardly an idea calculated to appeal to a man with a taste for desert landscapes  … There are more than 10^500 versions of string theory  lounging indolently about.

-- David Berlinski,  The Deniable Darwin (2009), p. 532-3

Philosophical Scratch-Pad (bis)

Continuing our wildly successful “Scratch-Pad” feature,  we now inaugurate an on-deck circle for stray philosophy-related quotations,  pursuant to their eventual incorporation into the appropriate Host Post.

[Cf.  Causality]

In The Decline of the West [Der Untergang des Abendlandes],  Spengler had argued that the idea of destiny was the answer to the narrow and lifeless theory of causality  used by historians, and that ‘the destiny-logic of the world-becoming’ was to be preferred to ‘the causal logic of notion and law’.

-- Gordon Craig, Germany 1866-1945 (1978), p. 492

That is mistily/mystically worded;  to decipher, consider the metaphor of embryonic development  (largely pre-programmed -- ‘destined’) versus the vicissitudes of post-uterine life.  -- That observation is not necessarily to defend Oswald Spengler’s dark doctrine -- indeed, my initial purpose in quoting it  was mere mockery -- but simply to render it parsable to the contemporary mind.

And yet and yet -- as we watch history  before our wincing eyes unfolding, in all its irrationality, its mad rush to doom, we must wonder:  mayhap he was right.


[Cf. What is Truth?]

The objection is not only false, but very much the reverse of the facts.

-- G.K. Chesterton, All Things Considered (1908)

(This is more than wordplay -- it is the difference between contradictory and contrary.  Something can fail to be strictly true, based on a triviality.  Chesterton is here after bigger game.)

[Cf. Consilience]

The intellectual structure of micro-economic theory  is very similar to that of theories controlling the behavior of perfect gases in physics.

-- David Berlinski, The Deniable Darwin (2009), p. 129

(This is probably hogwash;  but still, it’ll wash your hog bright ‘n’ shiny.)

[Cf. Babylonian mathematics]

Egyptian engineers  working under the pharaohs  knew that the angles of a triangle  sum to more or less one hundred and eighty degrees.  The number appears as a free parameter in their theories, something given by experience and experiment.  The Greeks, on the other hand, could prove what the Egyptians could only calculate.

-- David Berlinski, The Deniable Darwin (2009), p. 268

A mathematical scratchpad (even further scratched)

[There simply isn’t time, at least before retirement, to integrate each thought-balloon as it bubbles up -- a proto-insight or pre-idea -- into the appropriate essayistic context in finished form.   Yet to leave these on the desktop equivalent of a desk drawer is to tempt the Reaper.   Therefore I shall place some of them here -- philosophical post-it notes;  mathematical Zettel.]

[Cf. De stultitia]

Our focus  in the essay of that name, is on the plight of those sorry souls (99.9999999 % of us) who fail to grasp what Grothendieck, or Witten, or whom-have-you, saw easily enough.

Distinct from, though related to, this, are questions of which (at the forefront of science) we are permitted a glimpse,  but which nobody understands.  As:

On cosmogenesis:

The whole vast imposing structure  organizes iteself  from absolutely  nothing.

This is not simply  difficult to grasp.   It  is    incomprehensible.

-- David Berlinski,  “Was There a Big Bang?” (1998), collected in :  The Deniable Darwin (2009), p. 229

And:

All this leaves us  where we so often find ourselves.  We are confronted with certain open questions.  We do not know the answers, but what is worse, we have no clear idea -- no idea whatsoever -- of how they might be answered. 

But perhaps that is where we should be left:  in the dark, tortured by confusing hints, … and a sense that, dear God, we really do not yet understand.

-- David Berlinski,  “God, Man, and Physics” collected in :  The Deniable Darwin (2009), p. 270

~

The hardest part of a subject is the beginning.  Once a certain stage is passed, we gain confidence  and feel that, if need be, we could carry on by ourselves.

-- John Synge & Byron Griffith,  Principles of Mechanics (1942, 1959), p. 506

Alas, that has not been my experience at all.

Any technical subject is like a whirligig, which rotates faster and faster until the centrifugal force throws you off.   It’s like the Peter Principle, everyone eventually reaching his own personal level of incompetence;  only, in math and in physics, these levels stack indefinitely towards heaven, so that a few of us can ascend quite a ways, before we are finally out of our element.

[Cf.  Any Ideas? ] 

Recent years have seen striking developments in the conceptual organization of mathematics.  There developments use certain new concepts  such as “module”, “category”, and “morphism”  which are algebraic in character.

-- Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3^rd edn. 1999), p. vii

The reason they speak here of new “concepts” rather than additional structures  is that the notions referred to do not exist merely within algebra, but serve to organize other mathematical fields as well.

~

In an exterior view of the finished product, we see structure mathematics as largely logical or deductive:  P entails Q.

But from the interior standpoint of the practicing mathematician (and here, though we refer to the ‘actio’ sense of mathematicizing as opposed to the actum or product, the interest is not psychological but ideational), a key verb is rather motivate:  P motivates Q.   An illustration of this special vocabulary:  “The desire to extend Fourier L² to L^p spaces  motivates the Riesz interpolation theorem.”

~

More vocabulary from the conceptual domain:  thrust, as in the following passage

The Heisenberg uncertainty principle:  The mathematical thrust of the principle can be formulated in terms of a relation between a function and its Fourier transform.  The basic underlying law, formulated in its vaguest and most general form [i.e., its most intuitive formulation], states that a function and its Fourier transform cannot both be essentially localized.

-- Elias Stein & Rami Shakarchi, Fourier Analysis (2003), p. 158

~   ~   ~

[Cf.  On Depth]

When I made my original discovery of radiation from black holes, it seemed a miracle that a rather messy calculation should lead to emission that was exactly thermal.  However, joint work with Jim Hartle and Gary Gibbons  uncovered the deep reason.

-- Stephen Hawking, in: Stephen Hawking & Roger Penrose, The Nature of Space and Time (1996), p. 44

For the mathematician, contrasting with messy  are simple and elegant -- yet in the following, even these don’t get you to the yonder side, where Depth dwells:

Having derived the equation for the vibrating string, we now explain two methods to solve it:

(1) using traveling waves;

(2) using the superposition of standing waves.

While the first approach is very simple and elegant, it does not give full insight into the problem.

-- Elias Stein & Rami Shakarchi, Fourier Analysis (2003), p.  8

String theory is sometimes described as a theory that was invented backwards … People had pieces of it quite well worked out  without understanding the deep meaning of their results. … Math is funny that way.  Formulas can sometimes be manipulated, checked, and extended  witnout being deeply understood.

--Steven Gubser, The Little Book of String Theory (2010), p. 2

[Cf. Consilence in Mathematics]

Horizontal consilience:

… the structure theorem for finitely generated groups -- a fine illustration of conceptual unification.

-- Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3^rd edn. 1999), p. vi

Mathematics is a coherent, interlocking whole, and advances in one area  often lead to advances elsewhere.

-- Ian Stewart,  How to Cut a Cake (2006), p. 89

*

Commercial Break

A private detective  confronts the uncanny;

an ecclesiastical mystery:

Murphy Calls In a Specialist

*

[Cf. Language and Math]

Expressing himself in the language of fluxions and fluents, Newton managed to conceal his insights in a notation that was miraculously maladroit.  Not so Leibniz.  The language of mathematics and mathematics itself  are mutually sustaining.

-- David Berlinski, Newton’s Gift (2000), p. 57

Note:  The first clause of that observation does not actually relate to the point about notation (as opposed to vocabulary), and is silly in itself.  The concepts were new, so obviously any term for these would either be an out-and-out neologism, or a semantic hijacking of an extant word.   There is nothing lexically more rebarbative about fluent and fluxion than about derivative, differential, infinitessimal. 

*

[Cf. What is Mathematics.]

Betrand Russell, in The Principles of Mathematics (1903):

Mathematics is the class of all propositions of the form ‘p implies q’ …

The appended dribble of dots replace additional uninteresting clauses, which rob the sally of its epigrammatic pithiness, while yet failing to throw any light upon the subject.   It is a definition for people with no interest in the dark loamy richness of actual math as such;  and worthy of the author whose massive Principia Mathematica could as well have been titled Why Math Isn’t Interesting After All.

The characterization becomes even less interesting when you reflect that the expression “p implies q”, in logician’s lingo, is mere ‘material implication’ -- what would be better dubbed immaterial implication, since it involves no notion of causation or even logical entailment (and is thus immaterial to any actual problem), but is neither more nor less than another way of saying “either not-p, or q”.

[cf. Axiomatics in its Element]

Two citations illustrating the insight that axiomatizations, though perhaps logically prior, are pragmatically post-hoc:

The order of nature, and the order of logical dependence, are not the same as the order of our discoveries.

-- Morris Cohen & Ernest Nagel,  An Introduction to Logic and Scientific Method (1934)

Not all axiom systems are formal systems, and formalization need not lead to axiomatization.

The axiomatic method is an orderly way of summarizing experience.

-- Hao Wang, Popular Lectures in Mathematical Logic  (1981), p. 11

[Cf. Oligophrenia mathematica]

I am not a mathematician, but a math groupie;  not even a math wannabe (as I once was, back in Math 55), but a math wannedabe.

And in fact, considered coldly, I did not then  even rise to the level of a wannabe, but only a meta-wannabe, a wannawannabe:  someone who wished that his dearest wish was for mathematics, but who, truth to tell, was more interested in history and literature.

[Cf. Minimalism in Mathematics]

On Ramanujan’s notebooks:

There were thousands of theorems, corollaries, and examples.  For page after page, they stretched on, rarely watered down by proof or explanation, almost aphoristic in their compression, all their mathematical truths  boiled down to a line or two.

-- Robert Kanigel, The Man who Knew Infinity, p. 204

The reasons for this were twofold.  Ramanujan himself was not particularly aphoristic.   But he had never absorbed the modern notion of proof, which would take up so much more space;  and as a poor man in India, he suffered from a shortage of paper.

*

Für psychologisch tiefgreifende Krimis,

in pikanter amerikanischer Mundart,

und christlich gesinnt,

klicken Sie bitte hier:

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

*

[Sui generis]

On Ramanujan, who grew up in India, and in mathematics  was largely self-taught:

He was like a species that had branched off from the main evolutionary line  and, like an Australian echidna or a Galapagos tortoise, had come to occupy a biological niche all his own.

-- Robert Kanigel, The Man who Knew Infinity (1991), p. 61

The unusual career of Ramanujan  is one of the most celebrated biographies in the history of mathematics.   His achievements in the face of relative intellectual adversity as a child of modest means in the rural subcontinent,  are indeed inspiring, and warm the hearts of those in quest of Diversity -- whence, for those who can decipher the trobar clus of modern peri-academic patois, the book’s subtitle,  “A Life of the Genius Ramanujan”.   (Genius he indisputably was;  but the word these days is mainly used to celebrate anyone other than straight white males -- a “genius at basketball” or whatever.)

Yet the larger lesson is not how divergent Ramanujan was, but how much in the mainstream of things:  He did not found a new field of mathematics, he worked within number theory.   And this fact in turn reminds us of two characterizations of math as a whole, on which we have often dwelt:

            (a)  It is not something we invent out of whole cloth, it is something we discover.   This must channel our discoveries, just as the facts of the actual universe  discipline physics.

            (b)  Mathematics has already, for at least two hundred years, been uniquely rich conceptually  among human endeavors.   In a landscape embracing Cantorian set theory, algebraic geometry, and topos theory, it is next to impossible to come up with something unprecedentedly deep and strange;  in any case, Ramanujan did not.   To return to the metaphor:  We certainly treasure our quirky friend the echidna;  but only to someone whose zoological experience extended no further than a European barnyard, would he seem all that aberrant.   In a world of social insects, benthic hypothermophiles, and communal slime-molds, the echidna seems like just one more furry friend.

~

~  Posthumous Endorsement ~

"If I were alive today, and in the mood for a mystery,

this is what I'd be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(My name is Ramanujan, and I approved this message.)

~         ~

~

.

CHAOS, COINCIDENCE, CORRELATION, CONSPIRACY

Daily life is a tissue of loose ends, shot through with snarls and dropped stitches. That society nevertheless does not unravel, seems due to the fact that most people, though in detail unpredictable, are essentially similar and basically sane; due additionally to conservative/equilibrating structures evolved over the centuries; and perhaps to some further, anti-entropic factor, still obscure  but whose presence we may suspect (guardian angels would be the simplest explanation; something equally speculative though less intuitive, concerning negentropy or self-organizing systems, could also be alluded to).  Things pretty much jog along and usually eventually work out; but the details of any particular transaction  often may not bear looking into.

            One gets a glimpse of this  whenever, by some chance, a particular circumstance must be looked into further – something went a lot or a little bit wrong,  that, for some reason, this time, needs to be corrected or explained  and not just brushed aside as usual.  One often finds, upon turning over a rock long left undisturbed, a welter of wriggling vermin: errors previously unsuspected, data missing or misfiled, notes contradicting what one had previously believed (while their author has since left the company, or died prematurely).  Normally, none of this is sinister, but simply shows, on the human level, an analog (not an effect;  a metaphor, merely) of the underlying quantum uncertainty and atomic chaos  which nevertheless  ultimately manages to cohere in this coffee cup or that paperweight  sitting placidly, bien sage, on your desk.  If, however, the original affair is shady, the unexplained details which investigation unearths  lie in its shadow.  Let one instance stand for all: the celebrated “18-minute gap” in that tape, during Watergate.  Personally I find it utterly plausible that the gap resulted from the same gremlins that hide our spectacles, lose our umbrellas, and eat óne out of each pair of laundered socks.  But in context, it did look suspicious.

            An instinct not to let go of such suspicions  is buttressed by cases in which investigation of some misdeed,  severe enough to catch the atte.ntion of the media or the D.A.,  reveals a pot-pourri of previously unreported malfeasance, as in the Watergate affair.  Novelists and cineasts take the trend a satisfying step further with variations on the “Blow-Up” motif, in which a chance following-up of some minor discrepancy, out of mere curiosity, leads to a Vast Conspiracy. 

            The problem becomes acute when the stakes are high, the affair complex, and the discrepancies numerous (as they inevitably will be in any far-reaching affair, even absent any sinister underpinnings).  As, the Kennedy Assassination, or 9/11.

            We are currently witnessing a whole industry, tying our current President and his associates – both laterally (the present Administration) and longitudinally (the Bush family, going back to Prescott) – to all manner of skullduggery.  At the far fringe are the conspiracy theorists (stronger in Europe than here) who initially maintained, and some still maintain, that 9/11 was actually planned and carried out by a U.S. cabal, the airplanes being remotely controlled from the ground. A tad less extreme is the allegation, popular among some less reflective Arabs, that Israel did it.  Another tad, the position that,  while the op was indeed al-Qaeda’s, the Administration knew (sub-position 1, more florid) that that attack was due on that day, or (subposition 2, more reality-based, pointing to the August 6 memo) knew in a general way that such a thing might happen, and deliberately did nothing, so that they could use it as an excuse to proceed with their nefarious plans.  Certainly there have been agents provocateurs in history; but I tend to disbelieve in any gross such allegation concerning a present-day American administration, since every potential conspirator must be aware that, the way things stand in America today, no conspiracy broader than two twins in a cave  can stay secret for long – people write books and go on talk shows  to expose much milder grievances. 

            More centrist – if only with respect to such as these – is the new movie, “Fahrenheit 9/11”.  To his credit, Michael Moore does not even allude to such darker fantasies – indeed he indirectly exonerates the Administration, or at least Bush himself, of complicity,  if only through imputation of cluelessness. Bush’s blank look while the WTC attack vied for his attention with the Pet Goat, is itself a piece of evidence against all but a watered-down version of the last of the above-listed positions.  (Moore himself voiceovers a plausible thought-balloon in keeping with the last one, along the lines of: “Dang, wasn’t there some memo about something like this? Maybe I should have spent more time at the office and less on the golf course.”)  In one of his books, Moore hints at something more conspiratorial, but there fingers the Saudis (he really doesn’t like the Saudis), not the Illuminati or the Carlyle Group. And in the film, he does raise a number of other serious suspicions, none – again to his credit – original with himself, but dramatizations of charges explored at length by serious journalists in books. 

            It has been correctly observed that the basic motivation behind some of the most persistent of conspiracy theorists  is not mere irrationalism, but a perhaps excessive penchant for rationality – demanding to make sense of things that, well, don’t.  The distinction is hard to spot, since any empirically competent, level-headed investigator  tends to get swallowed up in a crowd of crazies.

            Another possible determinant, perhaps less often observed: in ascribing a pattern to the welter of events, as deriving from a plot, the theorizer may be acquiescing in a demand for personalization.  For, the confusing flux of the world  often lacks palpable causes.  Surely it can’t just all be random – one suspects deep dark forces at work. A plot puts a face on it.  But usually this is an explanatory short-circuit. For indeed there are forces, as deep and dark as those of hydrodynamics, but they are not even rational, let alone personal.

Empirical question: Does a tendancy towards florid conspiracy-mongering  correlate negatively with religious belief?  It is an observation  due perhaps to many, but prominently expressed in many a delightful story of Father Brown, that superstition tends to flow into the vacuum left by an absence of faith.  Perhaps belief in a heavenly Father – however distant, however inscrutable – leaves one less likely to imagine that the world’s workings may be ascribed to some shave-pate villain in a bunker somewhere, stroking a cat.

Against this, however:  the observed fixation of Book-of-Revelation-style evangelicals  on the Antichrist, working through the New World Order.  They do, so they say, believe in God;  a lot of good it’s done them.

            There is a healthy kernel to the epistemological attitude which, in its pathological exfoliation, becomes conspiracy-mindedness: a simple stance of “Wait a minute.”  For as Locke observed, and as subsequent scientific and historical discoveries  retrospectively further buttress: that “some (and those the most) taking things upon trust, misemploy their power of assent, by lazily enslaving their minds, to the dictates and dominion of others, in doctrines, which it is their duty  carefully to examine.”  (Essay I.iv.22)

            In an unhealthy development, skepticism becomes  not a key to inquiry, but a pose of imperviousness in the face of counter-evidence.  Michael Rutschky, in a recent contribution to the Sueddeutsche Zeitung, described the situation in Germany, where “Ich bleibe skeptisch” has become a slogan – not, like that of the fabled Missouran (“You’ll have to show me”) inviting actual demonstration, but issuing a warning that, whatever evidence you may bring, it will be dismissed as smoke and mirrors. This sclerotic ‘tough-mindedness’ in the face of empricism  is often coupled with a soft side for Verschwörungstheorie.  At an extreme, this leads to political infantilization.  That the phenomenon seems more to be met with in Europe (at least, comparing their educated reading public with that of America, and setting aside the seething cauldron of the Web sites and talk shows) may cohere  with an objective semi-infantilization or anyhow sidelining  of Europe itself, in the age of the unilateral hyperpower.

*

The detective versus an unnamed evil thing:

A tale of madness, and of the uncanny --

Murphy Calls-in a Specialist

Available for Kindle or Nook

*

[Footnote May 2013: 

The above was written ca. 2005;  since that time, the fragile cortices of many of our countrymen have cracked, while paranoid fantasies sprout like weeds.  Yet the circumstances tend to reinforce the conjecture above, that hands far from the tillers of power  are freed for idle mischief.   When in office, Republicans are crafty and cynical; when out, they lose their moorings.]

[Sidenote]  There is a quite respectable practice of trying-to-see-hidden-patterns everywhere:  it's called Science.  For which see:

http://worldofdrjustice.blogspot.com/2011/10/consilience.html

That practice too  has its pathologies, which perhaps someday time will allow us to notice in this place.

We find both kinds of nisus united  in the figure of Newton.  On the one hand, his scientific Principia side; on the other, his esoteric researches (alchemical and theological), which bulked equally large in his life.  Re the latter, David Berlinski remarks (Newton’s Gift (2000), p. 68):  “Secretive natures are quite prepared to believe that the history of human affairs  is largely the history of a great many secrets."

Souvenirs d’apprentissage (bis)

I am currently reading the memoirs of André Weil, doyen of algebraic geometry, Souvenirs d’apprentissage.  Herewith a couple of notes.

The first thing you notice is that he is a very graceful writer when he so chooses.  You do not see this in his mathematical writings, which are straightforward and businesslike, when not (very occasionally) interrupted  by some dyspeptic outburst (we quoted one of them here).   And as he stresses the importance of hewing to original languages whenever possible (he himself was an admirable polyglot), we shall so hew here.

~

Weil prefaces his book with a tribute to his late wife:

Notre mariage a été de ceux qui font mentir La Rochefoucauld.  Fulsere vere candidi mihi soles…

For an elucidation of that Latin tag, I naturally turned to that fons sapientiae, Dr. Massey, who replied by return of post:

"Bright suns truly shone for me"

It's from Catullus Carmina 8, in which the poet is depressed after being dumped by his lover Lesbia.

~

Since Weil was born in 1906 (and ripened early), and the memoirs were not published until 1991, we sometimes get the benefit of the long view.   Alluding to the current Lake-Wobegone system of American puericulture, he remarks:

N’est-il pas étrange que l’émulation se trouve honnie à présent  comme ressort pédagogique, alors que l’esprit de compétition, dans presque tout les domaines, n’a peut-être jamais été si âpre qu’il l’est aujourd’hui ?

He also speaks somewhat dismissively of “the New Math” fad in schools, counterposing the value of a traditional grammatical education for training the mind:

Est-ce pure coïncidence  que l’Inde, avec Pânini, ait inventé la grammaire  avant d’inventer la numération décimale  et les nombres négatifs,  et que  par la suite  grammaire et algèbre aient pris  toutes deux  dans la civilisation médiévale de langue arabe  l’essor que l’on sait ?  Naguère on a cru préparer les petits enfants à l’étude des mathématiques  en les forçant à parler d’ensembles, de bijections, de nombres cardinaux  et de l’ensemble vide.  Peut-être n’y étais-je pas moins bien préparé par l’étude de l’analyse grammatical …

Later in the volume (p. 120) we read of another glancing yet important brush between linguistics and pure mathematics :  the ushering of structure, rather than number, to center stage:

Quant au choix du mot de structure, mes souvenirs sont en défaut;  mais à cette époque  il était déjà entré … dans le vocabulaire des linguistes, et je conservais des contacts avec ce milieu, et tout particulièrement avec Emile Benveniste …

*

Si cela vous parle,

savourez la série noire

en argot authentique d’Amérique :

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

*

~

André Weil was the brother of the better-known Simone Weil.  In the preface, he excuses himself for alluding to her but little, pleading that he has already said what he has to say, to her biographer.  But, recounting a stroll with some monks at Santo Domingo de Silos, he writes:

De leur conversation, au cours de la promenade rituelle dans le cloître, il m’est resté une phrase.  Comme il était question d’un saint au comportement quelque peu excentrique, l’un d’eux fit observer doucement, «Mais le christianisme est une folie» («el cristianismo es una locura»); ce propos, parfaitement orthodoxe, m’est souvent revenu à l’esprit au sujet de la vie de ma sœur.

~

Weil’s 1938 cri de cœur, “Science française”, which begins "J'en ai assez!", and which was refused publication at the time, is reprinted in the Œuvres scientifiques,  as was its eventual post-war airing.  But since neither version names names, nor does Wikipedia mention the incidents in question s.v. Jean Perrin,  here is a tidbit (p. 126):

L’une des cliques en question, et sans doute la plus puissante, avait  à sa tête  le physicien Jean Perrin, prix Nobel, … inventeur du C.N.R.S.  Non content des moyens importants dont il disposait déjà, il imagina de créer toute une hiérarchie de médailles  assorties de récompenses pécuniaires,  depuis la grande médaille d’or  jusqu’aux médaillettes … Il n’était pas difficile de soupçonner que la devise en serait:
«Nul n’aura de l’esprit   que nous  et nos amis.»

*

Pour d’autres friandises

de la confiserie 

du docteur Justice,

consultez:

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

*

I am reminded of a comment one writer made  on the much-ballyhoo’d creation of some new honor or other (it may have been the MacArthur):  “another lap in the meritocratic rat-race”.

~

All told, the book is mathematically disappointing.  I don’t mean that it should have been stuffed with equations.   But we do hope for some insight into mathematical ideation, such as is furnished by the memoirs of Hardy or of Hadamard.   For one thing, the field he helped to found, algebraic geometry, has the reputation of being one of the most ferociously abstract of all human endeavors.   It must be very different working in that field, or in topos theory, from solving the four-color problem or classifying finite simple groups.  But of this we get not an inkling. 

Above all,  Weil was long a core member of one of the most sociologically remarkable mathematical activities of all time:  the Bourbaki group, which labored collectively and published anonymously.   What was that like ?  

Apart from the pranks and in-jokes  characteristic of any working group, we are not given a glimpse.

So, frustrated at the reticence of one of its founding members,  I have no recourse but to quote the following satirical evaluation:

Named after a French general of widely admired stupidity, the Bourbaki was founded in the 1930’s  .. A committee was formed  and pedagogical improvements discussed.

This is the myth.  In all of French history, no mathematician of standing has ever concerned himself with the welfare of his students.  The Bourbaki was founded to amused the members of the Bourbaki.

To a man, these mathematicians believed that their first order of business was to correct, and, if possible, eliminate, the work of other mathematicians.

-- David Berlinski, Infinite Ascent (2005)

~

En fin de compte … Our attempt to experience mathematics more richly  by reading the memoirs of its practitioners, is like that of the gum-snapping beautician devouring the latest tabloids for the off-screen escapades of her favorite stars.  In both cases, we court a simulacrum of what is ultimately inaccessible.  Indeed, the beautician is  if anything  launched upon a more reasonable quest.   If you think that Lindsay Lohan is an interesting person, then her antics in the National Equirer should be satisfying, since that shallow cipher is little more than the sum of her antics.   Whereas the well-written, travelogue-y accounts of André Weil  gave no sense of what it is like to have his sort of towering mathematical mind -- they might have been written by anyone.  (Quine’s memoir, The Time of my Life, was disappointing in the same way.)

~

For another not-so-close encounter with algebraic geometry, via the man and not the math, click here:

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

~

André Weil’s teacher Jaques Hadamard, a major figure of number theory and cryptography, is best known to the lay public for his booklet Psychology of Invention in the Mathematical Field;  my father, no mathematician, but a typical subscriber to Eisenhower-era Scientific American, had it on his shelves, where I made its acquaintance in high school.  Spurred yet disappointed by Weil’s memoir, I ordered what I presumed to be the French original of this book, via InterLibrary Loan;  and in due course  it arrived at our local library.

Mais encore -- quelle déception !  For much the same reasons as Weil, Hadamard had fled (in 1940) to the United States, and indeed specifically to Princeton.  And it was there that he wrote that memoir, in English, which Princeton University Press brought out in 1945.

France just doesn’t know how to hold onto its mathematicians -- as Weil was already complaining in 1938.   And it was at Princeton that I made the acquaintance of the likewise-exiled French mathematician who earned a Fields medal for proving the Weil Conjectures, Pierre Deligne (we were fellow parents at the Princeton Friends School, and met to plan-out Math Day).   A distinguished intellectual genealogy, all very baronial -- but abroad.

~

For more from this pen, including a soon-to-be-released new title:

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

~ Afterword ~

I have from time to time -- fitfully, fretfully -- pecked away at some of the works of that French collectivity Bourbaki, without profit or enjoyment.   There is, then, a certain wry comfort in this assessment by their celebrated countryman René Thom:

No new theorem of any importance came out of the immese effort at systematization of Nicolas Bourbaki -- which in itself is not a true formalization, because Bourbaki uses a nonformalized metalanguage.

-- René Thom, “’Modern’ Mathematics (1971), repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 73

[Update]  The slim pickings of mathematical morsels in Weil’s memoir  are slightly supplemented by an anecdote in a book I just finished reading, the highly entertaining Genius in my Basement (2011), by Alex Masters.   (The genius alluded to  is not Weil  but group-theorist Simon Norton;  more elsewhere anon.)  The anthropologist Lévi-Srauss, he relates, baffled at what structure underlay Australian marriage taboos 

… went around New York … banging on the doors of mathematicians.  The first was dismissive:  “Mathematics has four operations, and marriage is not one of them.”  But the second was the young and brilliant André Weil… “When in doubt,” cried Mr. Weil, “look for the group!” and he bustled Lévi-Strauss off the street  into his study.  Within a few days, Weil had solved the problem.

(“Group” in the sense of Group Theory, of course;  though indeed sibs and clans can be relevant.)