Minimalism vs. Nihilism

I’m currently reading an intermittently entertaining book by Jim Holt, with the simultaneously catchy and off-putting title  Why Does the World Exist?   (2012).    I actually gave my mother a copy for Christmas when it came out:  not having yet read it myself, but counting on that ever-companionable writer to make big ideas plain to the public, the notion being that she would read it, and then we would discuss it in our weekly phone-calls (we live on opposite seaboards:  she in a nursing-home on the west coast,  I in a sheltered workshop on the east).   Unfortunately, Alzheimer’s intervened, and we can no longer discuss anything substantive;  her days of wine and metaphysics are behind her.  Though, now, as I finally get around to reading it myself, it would seem that, in this particular case, she hasn’t missed much.  The problem is, his clear-headed addressal of that perennial topic of sophomore seminars -- Why is there Something, rather than Nothing? --  would have made an excellent laconic magazine-article, but Holt somehow wound up with a book contract for this ultimately (indeed, quickly) sterile subject, and has to pad things out.  So he takes various nugatory arguments by e.g. “metaphysical nihilists” more seriously than they warrant, and duly deflates them, but takes several pages doing so.

(Mr. Holt’s genteel approach to such gentlemen  is to humor them -- though, to his credit, ultimately to refute them.  Ours is rather to rip their throats out;  cf.  our diatribe contra Eliminative Materialism.)

The exercise calls to mind James Surowiecki’s canny column in this week’s New Yorker, “Punditonomics”, where he notes the (obvious) economic incentive to spew out short-shelflife twaddle if all you want is eyeballs.   A lot of that has been evoked by the disappearance of Malaysia Air Flight 370 (I actually work a few feet away from some Bureau integrees who are involved with that case, and poll them every so often.  BLUF:  UFOs are not involved.)   As Surowiecki reports,

The peak (or nadir) of the speculative frenzy  came when CNN anchor Don Lemon wondered aloud whether the plane might have been swallowed up by a mini black hole.

(Ceci renoue avec notre thème du néant.)  When a retired general who now shills as a CNN commentator, failed to cough out an equally (yet distinct!) OTT hypothesis, suggesting we wait and look at the facts,  the Host replied, revealingly,

“You know how cable news works, don’t you?  We got time to fill here!”

And thus, likewise, Mr Holt has some pages to fill, that were better left (in the spirit of Nothingness) elegantly blank.

On pp. 59-62, Holt gamely attempts a riff on nothingness as, “as Leibniz was the first to point out, the simplest of all possible realities,”  adding solicitiously, in case the reader has instinctually at this point  razzed in disgust, “Simplicity is greatly prized in science.”  (So, pipe down, you-all in the back row.)  In mathematics, by contrast, the technical term for this (the null group, or anything else) is:  trivial.

In the course of this, though, he does raise a somewhat interesting question:  “If our world turns out to have an infinite census of objects, why should it be, say, aleph-2 rather than aleph-29?  Only the Null World escapes this kind of arbitrariness.”  (The same observation applies to a finite ontology as well, of course.)  And the answer to that, we would suggest, as we have argued in a series of essays (Theologia Mathematica), is to take the invisible world seriously, quite on a par (as Gödel argued) with the visible (and  arguably  moreso).  Such infinities as exist, exist, and those that don’t, don’t, and there you have it.  (Of course, should there turn out to be some unexpected supremum like aleph-29, we will look for a principled understanding of that surprising fact.)   This line of thought has been applied more seriously to the problem of the cardinality of the continuum:  aleph-1 seems the only non-arbitrary value.

~

So what do you do when the puns on Nothingness run out?   If you’re a good author, like Jim Holt, you don’t simply repeat yourself, or inflate a chapter like a bicycle-tire:  You take the show on the road.   And so we are treated to a travelogue of several European and American capitals, meeting various Colorful Characters along the way.   It’s like a thematic tour, only instead of the theme being Chocolate or Wines,  it’s Big Ideas;  and instead of Mansions of the Rich and Famous,  it’s Cramped Apartments of the Brainy and Loquacious.

And thus we are whisked to Pittsburgh, where we make the acquaintance of the legendary Adolf Grünbaum, a Morris-Zapp-like personality, still robustly vigorous in advanced age.   And in addition to various choice philosophical obiter dicta, we are regaled with a memorable tale of the nighttime drive to the restaurant atop Mount Washington, the voluble crusty atheist at the wheel.  (B.L.U.F. : Never drive with the guy.)

[Afternote:   I have just begun reading the Collected Works of Adolf Grünbaum, the first volume of which  came out in 2013.  Evidently the wily old philosopher was delighted by the rather Zorba-the-Greek portrait he gets in Jim Holt’s book, for the dust jacket quotes Why Does the World Exist to the effect that “In the philosophical world, Grünbaum is a man of immense stature.  He is arguably the greatest living philosopher of science.”  On page one of the Introduction, the editor goes on to quote Holt at greater length along those lines.]

“…Mmmyess… Immmmense, that is quite the word ….”

Then before you know it -- like James Bond hopping from metropolis to metropolis -- we are in Paris, this time without the excuse of any actual philosopher to interview, the slender connective being the Café de Flore, where Sartre used to squat, and doodle about le néant (Why is there quelque-chose rather than rien du tout?, that is to say).   We never get any philosophy here, but the author does offer a vivid tableau of the nightlife of fashionable Eurotrash:

At a table in the back  I spotted Karl Lagerfeld, with his characteristic ponytail, dark glasses, and high white collar, in hushed conversation with one of his muses, who was wearing what looked like black lipstick.  Other than that, the place was pretty much empty: le Néant.

But then there was a noisy burst of activity.  A woman of a certain age … breezed through the front door, accompanied by a pair of what appeared to be Cuban gigolos  dressed in shell suits.  Giggling and grinding their teeth, this trio sat down with us  and began to jabber away.  The woman’s face was a sallow mask of leathery jollity, and she talked in a low croak  that put me in mind of Jeanne Moreau. … It seemed a good time to leave.

(Should you ask:  Why are there gigolos, rather than nothing -- the question is unanswerable.)

[Update 10 May 2014]  Jim Holt takes on Derek Parfit; wins by forfeit:
http://worldofdrjustice.blogspot.com/2014/05/have-theory-will-travel.html

A Dive to the Depths (expanded)

A phrase you will often meet in higher mathematics, and almost nowhere else, is:

“a deep result”

O loveliest of monostichs, thou !

The very notion of what ‘deep’ means, in such a context, is itself deep;  indeed, too deep for me, at present.   This, owing to a crippling condition of mathematical oligophrenia.  --  which, however, I pray that time and diligence might partly palliate.  (For a glimpse into the terrible sufferings of mathematical oligophreniacs, click here, if you dare.)  Yet I am putting up this skeletal promissory-note of a post, so that there will be a space to scribble insights as they wake me in the night.

First off -- the term does not mean simply ‘difficult’; indeed, though such results lie in the depths, and are not to be had for the asking, once you have somehow managed to fish one up, it may seem clarity itself.  Nor does merely being difficult make anything deep.  Any humongous brute-force calculation falls into that category;  for a more-substantive example, consider the Four-Color Hypothesis, which people suspected should be deep, but the proof that changed "Hypothesis" to "Theorem"  is a combination of clever tricks and elbow-grease.  The response of the mathematical community was disappointment:  "So, it turns out it wasn't an interesting conjecture after all."  (Of course, it may yet prove to be "interesting" in our cognitive human sense; that awaits a proof of an entirely different kind.)

We may go further, and put forward that an overarching purpose of mathematical research is to reveal something previously difficult  as now simple, when seen in the right way.  Again and again this has happened in history, beginning with the replacement of finger-counting by symbols, and of clunky symbols like Roman numerals by decimals.  For a more recent example:

Although Beurling’s own proof [characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to prove;  it depends on hardly anything more than the geometry of Hilbert space.  The profitable point of view  is not sequential but functional.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

Nor is there any simple similarity or opposition between being “deep”  and using what are now called “elementary” methods in number theory (as opposed to analytic methods), e.g. Alte Selberg’s proof of the Prime Number Theorem by elementary methods, compared with earlier analytic proofs by Hadamard and others.  Typically, proofs that restrict themselves to “elementary” methods are harder than those that permit themselves a more capacious toolkit;  but whether they ever, or generally, gain depth via this austere discipline, I have no idea.

In the meantime, some related posts outside of a mathematical context  are these:

            On Depth and Breadth

            On Scope and Difficulty

As appetizers, try the following hors-d’œuvres platter -- to follow which, however, we have as yet prepared no meal (as with our early essays on the Realist vernacular, this is more by way of linguistic warm-up):

The connection between linear transformations  and bilinear functionals  goes quite a bit deeper …

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 38

The Riesz representation theorem depends on some of the deeper parts of the theory of measure and integration.

--George Simmons, Introduction to Topology and Modern Analysis (1963), p.

The analogy between cyclotomic fields  and fields formed from the points of finite order on elliptic curves   is very deep.

-- Neil Koblitz, 1993

The notion of Kan extensions is the deeper form of the basic constructions of adjoints.  We end with the observation that all concepts of category theory are Kan extensions.

-- Saunders MacLane, Categories for the Working Mathematician (1971; 2^nd ed. 1998), p. vii

The continued-fraction representation of real numbers is deeper than the decimal expansion.

-- Roger Penrose, 2004

There are deep ties between enumerative geometry and Ramanujan’s tau function.

Contrast:

The useful fact about products of projections  lies near the surface.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p.  47

In the same work, while acknowledging the possibility of a lattice-theoretic formulation of the subspaces of Hilbert space, he dismisses this possibility as “trite” (p. 22).

*     *     *

~ Commercial break ~

Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

*     *     *

Much commoner than “trite” is trivial, which is virtually a terminus technicus of mathematical practice.  Let one quote stand for all:

One of the useful conclusions we can draw from Theorem 2 [to the effect that the norm of a Hermitian operator equals the supremum of its eigenvalues] is that the spectrum of a Hermitian operator  is not empty.  This is not a trivial conclusion.  We shall obtain the corresponding fact for normal operators  only after the application of a lot more relatively deep analysis.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 55

Psychosociological note:   Attempt to imagine the effect on our pumped-up math-major  freshman or sophomore brains, hearing dismissed as “trivial” (with a wave of the hand) propositions which, but a year before, we ourselves would not have begun to understand, and which indeed most of our countrymen will go peacefully to their graves without understanding.  (For more on the hubris involved, click here.)

Also note:  What counts as “trivial” is relative to where you stand.  Thus, in the very next sentence, Halmos adds:  “We hereby report that the spectrum of an arbitrary operator is also not empty;  since we shall have no occasion to make use of this fact, we shall not enter into its proof.”  The proof, one gathers, is more difficult still.  But when once you have mounted, and stand upon that summit, the fact that Hermitian operators in particular have eigenvalues, is trivial indeed.

Leave it to mathematics to recruit even the notion of triviality into some highly non-trivial constructions.   E.g.

A topological space over X is called a locally trivial fibration if every x in X has a neighborhood over which Y is trivial.

-- Klaus Jänich,  Topology (1980; Eng. trans. 1984), p. 129

And:

Sard’s Theorem … is … a highly non-trivial  theorem  which is elementary in the sense that it uses only the notion of a differentiable map.

-- Shlomo Sternberg, Lectures on Differential Geometry (1964), p.

The terms deep and elementary (here in the everday sense, and not the special number-theoretic meaning mentioned above) are not antonyms, but they do contrast:

The spectral theorem implies that every normal operator has a large supply of invariant subspaces;  this is classical  and can be considered elementary by now.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

This dependence of “depth-perception” upon experience accumulated with the passage of time, does not refute the notion of relative mathematical depth as ill-defined.   What is intuitive though difficult to put into words  is a notion of “deeper than” rather than of absolute depth.  To the giant, neither the pond nor the puddle appears deep;  but the pond is deeper  for all that.

~

Re the classification of simple algebras (“simple”, to be sure, in a certain technical sense, meaning roughly: incredibly complex and difficult):

The tools employed  are not deep; they are just, so to speak, linear algebra  raised to the nth power.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 122

~

Further attestations.

A pioneer of Hilbert space theory, expounding the then-contemporary state-of-the-art for a nonspecialist mathematical audience, particularly as regards dilations and extensions of operators:

There do not seem to be any conspicuous and challenging yes-or-no questions that serve to indicate the direction in which the search for new results might begin,  but I have faith.  There is depth in the subject;  the trouble is that the surface has not been explored enough  to show where the deepest parts lie.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

Writes a premier algebraic-topologist:

There are geometric problems which require the use of the multiplicative structure of the topological invariants.  Such problems are deeper than those which can be solved by considering the additive structure alone.

Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 105

Re Lie algebras and a closed subgroup H of the full linear group:

It is furthermore true (and this is deeper) that these one-parameter subgroups  fill a neighborhood of the identity in H, and consequently generate H if H is connected.  …The converse part of the correspondence  involves a subtlety of the type that makes the study of Lie groups a quite sophisticated topic.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 118

Re the Hodge conjecture:

It arises deep within the subject, at a high level of abstraction;  and the only way to reach it  is by way of those layers of increasing abstraction.

-- Keith Devlin, The Millennium Problems (2002), p. 9

This is a truly adult depiction of depth: as lying deep within the layers of the enigmatic onion.  It is not a case where you can just swallow some peyote and see it all in a flash.  
(For more, compare:  The Ladder of Abstraction.)

Writes a philosopher:

The axioms are not logical truths ... Their truth is established by intuitions which lie too deep for proof, since all proof depends on them.

-- Roger Scruton, Modern Philosophy (1994), p. 392

An example from outside the field of mathematics -- though it is a mathematician who is writing this:

Just how a protein manages to organize itself in space, using only the sequence of its own amino acids, remains a mystery, perhaps the deepest in computational biology.

-- David Berlinski, “What Brings a World into Being” (2001), collected in:  The Deniable Darwin (2009), p. 243

~

Related vocabulary:

Related to the concept of depth (which focusses on the root of things) is that of richness (regarding the blossoms that bloom from this root).   Hadamard adopts this metaphor explicitly:

Application’s constant relation to theory  is the same as that of the leaf to the tree:  one supports the other, but the former feeds the latter.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 125

Further examples:

The algebra of composition of maps  resembles the algebra of multiplication of numbers,  but its interpretation  is much richer.

-- Lawvere & Schanuel, Conceptual Mathematics (1997), p. 11

… a recent, and surprising, theoretical advance by Winkler.  He shows that, for many … algebraically closed fields, the “free Skolemization” has a model companion.

-- Angus Macintyre, “Model Completeness”, in: Jon Barwise, ed. Handbook of Mathematical Logic (1977), p. 164

The concept of thickness [in graph theory] is the deep mathematical idea that underlies the recreational puzzle of Earth/Moon maps.
-- Ian Stewart,  How to Cut a Cake (2006), p. 126

~      ~      ~

Having convinced ourselves, by the examples of mathematics, that there is more to this assessment-word deep than an emotional or impressionistic grunt,  we look to some cases outside of mathematics where an idea has been similarly assessed.

Some discoveries provide answers to questions.   Others are so deep  that they cast questions in a new light,  showing that previous mysteries  were misperceived.

-- Brian Greene, Fabric of the Cosmos

We are not at home in the world, and this homelessness is a deep truth about our condition.

-- Roger Scruton, Modern Philosophy (1994), p. 464

T.S. Eliot affirms that what is past and what is present, even what might have been, indicate a present purpose.  This is a metaphysical point of great depth.

-- James Schall, S.J., The Order of Things (2007), p. 69

And, more prosaically, but no less tellingly for all that:

Although running Bain Capital required a lot more brains and savvy than playing roulette does -- a lot more brains and savvy than most of us could even pretend to possess -- the job was not conceptually deep.  Romney did not develop a model of the world from the business of private equity. … “He’s not a very notional leader,” [said] Romney’s campaign spokesman …

-- Louis Menand, “Money Pol”, The New Yorker (19 III 2012)

~      ~      ~

This is quite aside  from the path of mathematics, but -- it may be, that such depth is displayed in quite distantly allied regions:  all tracing back to Him, perhaps by some functorial construction.  In that spirit, this:

http://www.youtube.com/watch?v=s5xDEyuXaOY&feature=fvwrel

The final anguish  of the Asian bride  suggests the depth  of the Riemann Hypothesis.

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]

~     ~     ~

Somewhat less far off the path …  Deep is indeed the term of art  in mathematics, antonymic to trivial.   Now compare, from another discipline, the word profound, in reference to Newton’s perplexing, little-known  philosophical-speculative opus:

That it is exclusively mystical  I do not believe -- that there is a mystical element  seems certain.  I hope that  one day  some profound student -- no one less will suffice -- will study this mass of papers.

-- E. Andrade, quoted in James Newman, ed. World of Mathematics (1956), p. 273

~ ~ ~

Above, we saw the distinction deep vs. difficult.  Here now even the latter concept is bilayered:

Although Beurling’s own proof [characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to prove;  it depends on hardly anything more than the geometry of Hilbert space.  The profitable point of view  is not sequential but functional.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

In infinite dimensions, it is tedious but not difficult  to construct spaces  strictly  but not uniformly  convex.

-- Prof. Lewis, University of Alberta, course in Functional Analysis, 1982

The proof that Reidemesiter moves  and planar isotopy  suffice to get us from any one projection of a knot  to any other projection of that knot  is not particularly difficult;  however, it is technically involved.

-- Colin Adams, The Knot Book (1994), p. 15

And here the original distinction is made even sharper:  though they are not interchangeable, depth tracks with generality, which in turn tracks (on the plane of praxis -- of proof) with simplicity:

Re the Denjoy-Young-Saks Theorem on the derived numbers of functions:

As we would expect  in view of the great generality of the final statement of the theorem,  the proof due to Saks is of extreme simplicity.

-- F. Riez & B. Sz.-Nagy, Leçons d’analyse fonctionelle [references to the English translation, Functional Analysis, 1955], p. 17

~

Here a leading mathematician laments the shallowness of his understanding of something he himself proved (regarding representations of a Kac-Moody algebra, as it happens):

My proof of this result was technically quite involved.  I was able to explain how the Langlands dual group appeared, but even now, more than twenty years later, I still find mysterious why it appears.  I solved the problem, but it was ultimately unsatisfying to feel that something just appeared out of thin air.

-- Edward Frenkel, Love & Math (2013), p. 181

This is setting oneself high standards indeed.   Shakespeare probably did not lie awake o’ nights fretting how the devil he ever came to write Hamlet;  Mozart did not find the bread of pleasure at having written the Sonata in A  turning to ashes at the thought that it might have been dictated to his unconscious  by an angel.

~

A near-synonym of the math-word deep, but shorn of all irrelevant aesthetic echo, is:  “highly nontrivial”.   The term is decidedly commendatory, though to a layman it might sound like faint praise, as were one to dub one’s lady-love “seriously unugly”.  The expression may be extensionally impeccable, but ‘twould never pass in a sonnet.

 Further:

In set theory, a forcing extension in Cohen’s sense  is reminiscent of algebraic extensions of a field, but

… the forcing method is far more complex, both conceptually and technically, involving set-theoretic, combinatorial, topological, logical, and metamathematical aspects.

-- Joan Bagaria “Set Theory”,  in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 625

Although he does not use the term "deep" here, that expanded characterization  "complex, both conceptually and technically", especially the "conceptually" part, points in that direction.

~

One motive for Frege’s choice  was again generality:

Does not the ground of arithmetic lie deeper than that of all empirical knowledge, deeper even than that of geometry?

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 184

… [Cantor’s] remarks on functions of several variables (where the provability of theorems  was deepening the level of rigour in analysis)

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 223

 

.

Help Dr Justice Get His Yacht ! !

An Emotional Appeal

Can you imagine life without a yacht ?  I can’t.   Just picking up signals here from such outstanding/upstanding leaders as Mitt Romney and Donald Trump, for whom the 99% are like dirt beneath the toenails.  For them, only the 1% count -- and that’s actually something of a euphemism, the real club being limited to a handful of guys who Skullandkeyed at Yale  and prepped at Choate.  And the only way anyone can be any good (on this view) is for everyone to have a yacht that is bigger than anyone else's.  (There may be a subtle paradox in this scheme;  our logicians are working on it.)

So, how is poor Dr Justice ever to get his yacht?     )* sob *(

We ran this past our Economic Logic department, and they came back with a nifty solution:

 => Simply become very very rich !

Ahh… you ask (a stickler for details);  but how is that to come about.

Obviously, not by working; that’s for chumps; here’s why:

http://www.whatmittpays.com/

So how?

=> Simple.  Just sell millions upon millions of books.

Ahhhh, you add, with a knowing look:  And how do you go about that?

Again, simple.  

The best way is to run in the Republican Presidential primary.  Your wild & wacky behavior will ensure enormous publicity and consequent sales, long after your joke campaign is over.

But what about the rest of us?

Ahhh….  Simplicity itself.

=>  Just ensure that millions of people watch this video:

http://www.youtube.com/watch?v=FwzOXzZWUik

No-one who watches that video can help buying  a copy -- two, ten copies -- of that rollicking funnovel, Murphy on the Mount.

AHHHhhhhhhhhhh, you say (sly one), and how do you propose to manage that  ??

We-ll … That’s where you come in -- all of you -- my faithful fans -- standing there upstandingly in the farms and fields, gazing out to the horizon -- squall comin’ in …. better get them cows backina barn...

=>  Simply click on that link and “Like” the video !  YouTube tots those stats up, and, before you know it, you’ve got a yacht.

A clever method.  Patent pending.