Emmy Noether zum Geburtstag

Alles Gute zum Geburtstag, Emmy !

Today's Google-doodle commemorates Emmy Noether.   Noether really is a significant creative figure in the history of mathematics.

Unlike Ada Lovelace, she wasn't anyone's famulus, but forged her own way independently.   And, despite the ferocious abstraction of her research, she was a nourishing and accessible person to her circle of mathematicians.  As one leading algebraist wrote in his memoirs:

Emmy Noether jouait le rôle de mère poule protectrice.

-- André Weil, Souvenirs d’apprentissage (1991)

~

As an undergraduate math major at Harvard, I had barely heard of Noether;  and this, for three good reasons, none of which had to do with Noether herself.

(1)  Undergrads generally need to learn the subject (i.e., what’s on the test) and not worry about its history.

(2) The exact sciences generally hew to “Whig history”:  whatever the present state of the field, is all you really need to know.  The pioneers were blunderers.

(3) For math specifically, the Platonic outlook  singularly minimizes the role of the individual researcher.   All mathematical truths are already stored-up in Platonic heaven;  whoso proves this theorem or that, has simply managed to claw loose a pre-existing nugget from the hoard.

The closest I came to having heard of her, was that I’d heard of (but not studied) “Noetherian rings” (an adjective I still do not know how to pronounce).  These figured prominently in the introductory algebra text by Van der Waerden, one of her pupils.

Now:   Noetherian rings (along with the eponymous modules) are all very well in their way;  but they must take their place in a menagerie of such constructions. 

Thus:  If you stick with it  long enough,  you will learn such gems as this:

In a short exact sequence of R-modules, A and C noetherian imply B noetherian, and conversely.

--  Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3^rd edn. 1999), p. 380

Hmm!  Mmyess!  Good to know!  Must make a note of that!

But that is not why she’s famous.

~ ~ ~

Emmy Noether

Separated at birth ??

Emma Goldman

 ~ ~ ~

The discovery (or invention) of Noetherian rings, was internal to algebra.  What inscribed her name on the scrolls of History in letters of gold, was the linking of algebra to something seemingly outside itself:  specifically, the linking of each algebraic symmetry with a conservation law -- which puts her work at the center of modern physics.   In this it bears comparison with the previous Erlangen Program led by Felix Klein, which characterized the plethora of newly recognized geometries (prior to Gauss & Lobachevsky, there was but one) by their symmetry-groups.

Technically stated:

Noether’s principle:  To any continuous one-parameter group of symmetries of the Lagrangian,  there corresponds a conservation law for the associated Euler-Lagrange PDE.

--  Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 479

or (seen from the standpoint of QFT):

Noether’s theorem:  If a Lagrangian has a continuous symmetry, then there exists a current associated with that symmetry  that is conserved when the equations of motion are satisfied.

-- Matthew Schwartz, Quantum Field Theory and the Standard Model (2014), p. 34

Noether’s results antedate post-quantum particle physics, but their reach remains.  As, “If there is a gauge invariance, expect to find a conserved charge,” (Roger Penrose).  Penrose however goes on to note, that the principle is not unlimited:  it does not apply to derive the conservation of the energy-momentum in General Relativity (The Road to Reality (2004), p. 489-90).

On “The Nature of Mathematical Knowledge” (enlarged)

That question is about as interesting as the nature of our knowledge of elephants.  We are interested in the zoology of elephants, not in the specificities of classroom biology lessons, or the economics of zoos.  We are uninterested in each blind man’s subjective and partial report upon the individual organs of these splendid creatures.

Mathematical knowledge, like pachydermal knowledge, is imperfect knowledge of something real that exists independently of us. By contrast, just which images we manage to form of these objects  are very much dependent upon ourselves – and to that extent, of interest only to unemployed social workers.

As our former math teacher put it:

Mathematics has a real content which transcends the inadequacies of our efforts to formalize it.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. v.

Or, from a philosopher:

For most mathematicians most of the time, having a feel for what is evident is important, but it is also enough: There is no further need for a theory of what that feel is.

-- Shaughan Lavine, Understanding the Infinite (1994)

*

Actually, it should not be deduced from the above, that I am somehow dissing our big friends the elephants.  A Theory of Elephants is considerably more promising than that idle grail of the physicists, a Theory of Everything.

The outstanding problem in the Theory of Elephants is the ontological status of BABAR – THE KING !!!!!

His Majesty ... The King !!!

An agnostic or non-Realist version, wilfully po-faced:

Our deductivist proposes clean answers to philosophical questions.  What is mathematics about?  Nothing … What is mathematical knowledge?  It is knowledge of what follows from what.  Mathematical knowledge is logical knowledge.

-- Stewart Shapiro, Thinking about Mathematics (2000), p. 150

*

*

There is a systematic ambiguity (roughly that of actio versus actum) to the term mathematics:

(I) The praxis of mathematizing.  This is a human pastime, comparable to needlework or basketball.

(II) The truths of mathematics.  Or, more or less synonymously: The real (though invisible) world.  These, in themselves, bear no dependency upon human practice or to any species whatever;  they existed before we were born.

The above may count as a polemical reformulation of roughly the dichotomy in the title of Hao Wang’s fine essay, “The Theory and Practice of Mathematics”.

*

In 1950, Raymond Wilder gave an address, “The Cultural Basis of Mathematics”, reprinted in  various places, and later wrote a whole book on the subject, Mathematics as a Cultural System (1981).   Bien-pensant commentors treat these with grave respect;  but the notion is practically nonsense.  For, if we take mathematics in sense (II) – the only sense of interest to us here – that is like saying “The cultural basis of elephants”:  there is none.  There is a cultural basis of circus stunts, of mouse- and peanut-myths, of Dumbo, but not of elephants themselves.  Their basis is their own four feet.

Why should the culture of mathematics  (necessarily, sensu I) – that is, the foibles of mathematicians – retain our attention?  The purely “human side” of mathematicians  is generally less interesting than that of country music stars.  A lot of mathematicians are pretty Asperger’s, frankly.
(For a poignant illustration of this, read The Genius in My Basement.)

There is, we grant, a certain interest in the sociology of mathematics, or in biographies of the great mathematicians. Intellectually, it is on a level with gossip about the off-court antics of basketball stars.  Fun, but of no mathematical (or basketball) interest.   It’s just a way for the mind to chew gum while it’s too exhausted to do anything substantial.  To get real, do math (or play basketball).

Not to come down too hard on the small geeky community that does follow the doings of math and physics whizzes; I number myself among them.   It would even be neat  if, instead of collecting baseball cards, people collected mathematician cards (“Trajea two Steven Smales for a John Milnor!”) .  -- By “people”, I here mean “eight-year olds”.

*

More interesting is the purported “reduction of mathematics to logic”.  It is not initially clear, however,  to what extent this program, if successful on its own terms, would enlighten us as to mathematics-sensu-(II), as opposed to the sense-(I) territory of our own mathematical formulations and formalizations (these being, after all, largely for mere convenience).  It might be more along the lines of the demonstration of the equivalence of the Heisenberg-style matrix-mechanics formulation with that of the Schroedinger-style wave formulation, of quantum mechanics. That feat didn't tell us all that much about the actual phenomena of physics,  apart from the fact that the world is a many-splendored thing, and can be described -- blind-man-fondling-elephant-fashion -- in a variety of ways.  It’s more like deciding whether today’s symposium shall be conducted in English or in French.

*

That said --

We argued here that the axiomatic method is cognitively post-hoc, and that  only in cases where (as with the Euclidean axioms) their positing is transparently motivated by our experience of the sensible world, is a top-down, axiomatic presentation  pedagogically sound.   Thus similarly in physics:

In lecture after lecture, and essay after essay,  Einstein began, not with an introduction to the subject at hand, but with an overview of how he’d arrived at that subject, or of how scientists in general  arrive at subjects in general. … For Einstein himself, the results of science had become incomprehensible without an understanding of the processes that led to them.

Richard Panek, The Invisible Century (2004), p. 153-4

C'est exact;  and the farther physics wanders from our human experience, and the father math develops beyond anything the world has seen before, the more necessary such a psycho-cognitive ladder does become.

*

Something like the dichotomy outlined above  must have been behind André Weil’s tart remark, in “History of Mathematics” (reprinted in Collected Works v. III as (1978b)):

Some universities have established chairs for “the history and philosophy of mathematics”;  it is hard for me to imagine  what those two have in common.

For:  the one is situated and contingent, the other timeless and beyond place.

*

Footnote:   These remarks about mathematics  apply  mutatis mutandis  to the Deity.  Deliberately confusing the distinction between truth and praxis, Karen Armstrong wrote a book -- a minor best-seller -- with the impudent title A History of God.  (At least she put A, not The; probably saved herself an extra millennium in Purgatory right there.)

*

Lakatos’ classic dialectical-dialogue Proofs and Refutations (you see the Hegel-style paradox already in the title), though focussing on the (as he persuasively argues, in the course of a detailed case-study spanning many decades) micro-level mess of actual mathematical progress, is yet Realist at its core:  the subtitle is “The Logic of Mathematical Discovery”, not “The Sociology of  ‘Mathematical’ Invention”.   We quoted him in another context  thus:

As far as naïve classification is concerned, nominalists are close to the truth when claiming that the only thing that polyhedra have in common  is their name.  But after a few centuries of proofs and refutations, as the theory of polyhedra develops, and theoretical classification replaces naïve classification,  the balance changes in favour of the realist.

-- Imre Lakatos, Proofs and Refutations (1976), p. 92

In an appendix to the main work, he offers a Hegelian formulation, one which (by the time the reader has progressed this far) has a certain paradoxical piquancy:

Mathematics, this product of human activity, ‘alienates itself’ [in the sense of Hegel and Marx] from the human activity, which has been producing it.  It becomes a living, growing organism, that acquires a certain autonomy [emphasis in original] from the activity that produced  it.  … The activity of human mathematicians, as it appears in history, is only a fumbling realisation of the wonderful dialectic of mathematical ideas.
-- Imre Lakatos, Proofs and Refutations (1976), p. 146

Plato, in his Paradise, smiles.

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 …

*

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*

~

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:

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*

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.)

Too Much of Nothing

[Note:  This title is an allusion to a Bob Dylan song.  Peter Paul and Mary did a wonderful cover of this, which achieved some airtime, some forty-odd years ago.  But I cannot find it on the Web.  The one older (B&W) video you do find   is not the album version, but a performance that is much more up-tempo, sunnier, and, well, shallower.  The haunting version that still echoes in my mind had strange harmonies and blue-notes, and funky harmonica.  As during the verse:

It’s ALL been done be-fo-ore,

It’s all  been  written  in  a bo-oo-k…

When there's too much of noth-innng

No-bod-y should loo-oo-ook!

The melody of the second line is just a monotone, but the way they rendered it, with tensely close harmony, so that the male and the female voice become indistinguishable -- another voice peels off from the crown and weaves its own magic -- more plaintive, more revival-like, than the versions available now. -- Unless, of course, the song has actually fermented in memory, grown richer over the years as I replayed it in my head, ripening and deepening like a fine red wine…

Anyone who can point me to this version  will earn my gratitude.]

~     ~

  

Gorgias presented his nihilist arguments in On Non-Existence; however, the original text is no longer extant.

-- Wikipedia, re the early Greek sophist

Etienne Gilson reports (La philosophie au Moyen Age, p. 196) re the work of one Frédégise (d. 834),

Epistola de nihilo et tenebris, où il soutient que le néant et les ténèbres  sont quelque chose, et non pas seulement l’absence de quelque chose.

So far, a trifling with words, but:

Il serait absurde de dire: nihil désigne une chose, si l’on admettait  en même temps  que nihil  signifie le néant.  Or, c’est précisément ce que nie Frédégise.  Le nihil auquel il pense  est celui dont Dieu a créé le monde, ex nihilo, c’est-à-dire  une sorte de matière commune et indifférenciée…

That final clause might strike us as a quibble, a bait-and-switch; but it is similar to the once-trending view of cosmogony known as Steady State (though here it was particles coming into being out of nothing, here and there on an ongoing basis);  the contemporary "Universe for Free" idea is a modern version.   Further, the affordances of later mathematics  would allow him to stick with his guns, without smuggling in any ontology via the back door.

*

Travaillant au noir,

le détective  se trouve aux prises

avec le Saint-Esprit

C’est quoi, le péché irrémissible ???

*

Thus consider the empty set -- something with which, once the New Math struck, every schoolchild has been familiar.   Concretely, it is nothing;  but abstractly, it is something, and this, by the same necessary motion of the mind, which considers Laurel and Hardy a pair, over and above the separate existence of Mr. Laurel and Mr. Hardy.  -- So far so vacuous, perhaps; but at least innocent of any theological special pleading.

Yet later mathematicians  contrived to construct the whole of the natural numbers -- and with that, the scaffolding of all that is  or (as it might be) may be -- out of this same empty-set.   One approach (Zermelo) identifies zero with the same; one, with the set containing the empty set;  two, with the set containing that.  Another approach (von Neumann), with more craft, took the empty set, and the set containing (nothing but) the empty set (quite a different thing altogether) -- well, the details are unimportant.  And lest you imagine that such an exercise were the idle fiddling of someone with too much time on their hands, know that John von Neumann was one of the hardest of hard heads; a pioneer of computer science; a mainstay of the Princeton intellectual-social scene; and possessed of clearances  of which you or I can only dream.

The empty set is thus Nothingnes, Reified.  So far -- despite von Neumann’s pulling all of set theory out of an empty hat -- you may be unimpressed.  After all, the use of this term can often be (as Quine likes to put it) “paraphrased away”:  e.g.  “A ∩ B = ∅” means neither more nor less than “A and B have no elements in common”.  But “experience has shown that the admission of ∅ as an actual object  enhances and simplifies the theory” (R. Goldblatt, Topoi , 2^nd edn. 1984).

~

For all its humble nothingness, Nothing has an awful lot of distinct names in mathematics.

The empty set itself is known also as the void set.  The set on which a function is zero is its zero set, and the complement of this is its cozero set.
The nullity of an operator is the dimension of its nullspace (or, in the case of a homomorphism of groups, the rank of its kernel).  More exotically:  the Nullstellensatz is a foundation of algebraic geometry.
In an algebraic structure, an element is termed nilpotent if it can be raised to a power to give zero.  This doesn’t mean that the element is ‘powerless’, as the etymology might suggest.  Thus, in the cyclic group of order p^2 (p a prime), p itself is nilpotent since its square is zero;  but it maps every other nonzero element to something nonzero.

Further:

critical point: one where the derivative of the defining function is zero.

zero set: the set of all points mapped to zero by a function

singular (matrix): one whose derivative is zero

closed (differential form): one whose differential is zero

annihilator (of a subspace): the set of all functionals that map the subspace to zero

Cf. also the notion of a “set of measure zero”, which is not quite as zero-y as it sounds, since the ones we are interested in are generally infinite.

~

Etymological footnote.

In his memoirs, Souvenirs d’apprentissage (1991), André Weil, who has much larger accomplishments to his credit, takes a certain satisfaction in having introduced a new symbol for the empty set, which stuck:

Ø

He explains that he took this from the Norwegian alphabet, “et j’étais seul  dans Bourbaki  à le connaître”.

In his droll conspectus of mathematical history, David Berlinski, Infinite Ascent (2005) adds:

To the empty set is reserved the symbol Ø, the figure now in use in daily life to signify access denied -- a symbolic spillover, I suppose, from its original suggestion of a canceled eye.

Note:  A more profound comment  than might initially appear, given the Mediterranean context of the "Evil Eye" ...

 ~

In later Chomskyan linguistics, of the Government-and-Binding variety, "nothing" came to play a very large role indeed.  It came in different flavors -- trace and PRO -- and boasted the Empty Category Principle or ECP  as one of the pillars of the theory.

Yet such Nothings are as rich  as any Something:

So-called ‘empty’ categories  are not devoid of properties;  they are specified for syntactic features.  The term ‘empty’ refers [merely] to the fact that these categories are not associated with phonetic content.

-- Liliane Haegeman, Introduction to Government and Binding Theory (1991),  p. 288

Cf., indeed, Whorf’s celebrated treatment of the phrase/concept “empty gasoline cans”.

~

~  Posthumous Endorsement ~

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

this is what I would be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(I am Benjamin Lee Whorf, and I approved this message.)

~         ~

~

Epigrams re le néant:

What we understand about God is not nothing; it may even be infinitely much;  but it is a set of measure zero in the larger space of what is true.

-- Dr J, Tischreden

A literary forerunner of the playful paradox “too much of nothing”:

We used to sing together -- in my case very tunelessly. I had inherited a plentiful lack of musical genius from my Mother, who had neither ear nor voice.

-- Edmund Gosse, Father and Son (1907)

Tiens !  Ceci vient à propos:

“The Power of Nothing”, by the aptly-named Michael Specter, in the current issue of The New Yorker.

*

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*

I am currently reading a book by Russell Standish, called Theory of Nothing -- basically a way of referring "by duality" to a Theory of Everything.
Rather wittier  is this epigram from a detractor of string theory (as it reaches its Landscape dead-end):  the theory has "gone from being a Theory of Everything, to a Theory of Anything."  I.e., where predictive value is lost, because Anything Goes.

[Update 21 Feb 2012]  And now this:
http://www.nytimes.com/2012/02/21/science/space/cosmologists-try-to-explain-a-universe-springing-from-nothing.html?src=me&ref=general

~

In his substantial general survey of modern philosophy, Roger Scruton is obliged to take notice of the existentialists and phenomenologists.   Appropriately, even delightfully, he places these in a chapter titled “The Devil”.

Heidegger … famously argued that there is something that is true of Nothing, namely that it ‘noths’ (Das Nichts nichtet.)

-- Roger Scruton, Modern Philosophy (1994), p. 458

(Heidegger fiddles with words  like an imbecile playing with his faeces.)

Then on to Sartre and L’Etre et le Néant  (a book I discovered in high school and seized eagerly, having heard that it was hip and rebellious and edgy and kewl -- and was baffled to find that it sux):

Nothingness, he tells us, lies ‘coiled in the heart of Being, like a worm’.  I can encounter Nothing at any juncture:  for example, when I look for someone in a café where I expect to meet him, and he is not there.  The world is suddenly colured by his absence;  and this negative fact has a peculiar reality   all of its own.

But however strange this experience, it is surely not an archetype of evil.  Entering Les Deux Magots to find that Sartre is not there  is one of life’s blessings.

-- id.

~

There is a real sense in which “too much of nothing” applies to basic physics.

Most macroscopic objects behave in ways that do not tip us off to their quantum nature (save as what, for classical physics, was the paradox of the stability of atoms, was ultimately solved or at least frozen by the notion of quantization as applied to electron orbits).  But when we get rid of such objects -- when we attempt to get rid of everything, all matter whatever, and scrape down to the bare vacuum … the striven-for Nothing rebels, and that in the most violent way imaginable.  As a result of the Heisenberg Uncertainty Principle -- here, not as an epistemological limitation on ourselves, as (given its title) it is often taken to be, but as a literally creative faculty at the heart of Nature -- the classical featureless vacuum becomes instead the Quantum Foam:  a riot of virtual particles, and topologically  perhaps some nightmare analog of Alexander’s horned sphere.

One stroke of Alexander’s sword

will *not* undo this knot !

Update 25 III 2012]   And on that very topic… a book review in this morning’s NYTimes:  A Universe from Nothing, by Lawrence Krauss.  No new information or argument in this book, apparently;  it all comes down to what the meaning of Nothing is.

The book offers the sort of thing we have repeatedly polemicized against -- and with a “Nyaah nyahh I told you so” Afterword by Richard Dawkins, no less -- but as the reviewer, David Albert, does a fine job himself of polemicizing (“Let me put those niceties aside  and try to be quick, crude, and concrete” -- You go, guy, gloves off!) your correspondent can maintain his decorous Sunday Lenten silence, and merely link.

(The online site unaccountably buries the article, so this is something of a public service.)

http://www.nytimes.com/2012/03/25/books/review/a-universe-from-nothing-by-lawrence-m-krauss.html?_r=1&sq=&st=cse&%2334;david%20albert=&scp=1&%2334;%20krauss=&pagewanted=print

The subtitle of the book under review is “Why there is Something Rather than Nothing”.

Our own remarks on the matter  can be consulted  here:

http://worldofdrjustice.blogspot.com/2011/01/why-is-there-something-rather-than.html

[Update 7 April 2012]  Krauss provides a summary of his views here:

http://www.latimes.com/news/opinion/commentary/la-oe-krauss-cosmology-design-universe-20120401,0,4136597.story

Arts & Letters Daily provides a typically bone-headed teaser in its link to the article:

"Physics has undermined logic. Even nothingness is not what it seemed. The universe is devoid of meaning. That’s not such a bad thing."

*     *     *

~ 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.

*     *     *

[Update  25 April 2012]  They’re at it  again…

ALDaily, shilling for nihilism, links to an interview with Krauss in the Atlantic , with this come-on:

More and more, physics is encroaching on philosophy. No surprise that philosophers feel threatened. They should, says Lawrence Krauss. Science progresses, and philosophy doesn’t.

As for the actual article … you can’t make this stuff up.  Their roving reporter just happened to hook up with the distinguished nihilist as he was coming from … a memorial service for professional atheist Christopher Hitchens.

If I were to write that, you’d think it was unfair satire.

[Update 10 June 2012] Jim Holt on l'affaire Krauss:

Physicists, Stop the Churlishness

A KERFUFFLE has broken out between philosophy and physics. It began earlier this spring when a philosopher (David Albert) gave a sharply negative review in this paper to a book by a physicist (Lawrence Krauss) that purported to solve, by purely scientific means, the mystery of the universe’s existence. The physicist responded to the review by calling the philosopher who wrote it “moronic” and arguing that philosophy, unlike physics, makes no progress and is rather boring, if not totally useless.

 [Update 30 September 2012]   Crashaw, “On Hope”:

Dear Hope! Earth’s dowry, and Heaven’s debt,

the entity of things that are not yet.

Subtlest, but surest being!  Thou by whom

our Nothing hath a definition.

(Note for scansion:  definition is here quinquesyllabic.  -- As is, indeed, the word quinquesyllabic itself.)

 ~

More on the vicissitudes of Nothing in mathematics.

In  the introduction to Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 14, we are presented with the following syllogism:

* Nothing is better than lifelong happiness.

* But a cheese sandwich is better than nothing.

* Therefore, a cheese sandwich is better than lifelong happiness.

Nobody buys that;  but it’s hard to put your finger on where the reasoning goes off the rails.   The author rewords the first proposition as

For all x, lifelong happiness is at least as good as x.

Here the seeming noun nothing was covertly a quantifier.   Whereas  “the second sentence cannot be rewritten in these terms  because the word nothing is not playing the role of a quantifier.  Its nearest mathematical equivalent is something like the empty set.”

~

In a single review from March 25, 2001, the New York Times Book Review treats together  the following three titles:

THE BOOK OF NOTHING

Vacuums, Voids, and the Latest Ideas about the Origins of the Universe

By John D. Barrow

ZERO

The Biography of a Dangerous Idea

By Charles Seife

THE NOTHING THAT IS

A Natural History of Zero

By Robert Kaplan

Uniform Spaces

[The following does not rise even to the level of an essay-in-progress;  more like a thought-in-progress, or even (saving your presence) a difficult bowel-movement.   But the hordes of typist-elves in the cavernous warehouses of WDJ  have yet to present anything brought to perfection this morning, and I wished not to disappoint the milling crowds that swarm this site each weekend, bringing the whole family, Sister Sue and Fido too, gawking at the glittering thoughtfronts -- the polemics, the poems, the darling little monostichs (these we can all afford) -- while shaking their heads sadly at the Trinitarian Minimalism and Cantorian Realism (out of our price-range) -- all  save one diminutive child towards the back of the bunch, eyes riveted on the prize, instinct with penetrating understanding…]

We saw here the dialectic of mathematical invention (not trying to be too Hegelian here -- think of it as an ensouled pendulum) whereby, beginning with the everyday world we live in -- I almost wrote ‘space’, but that would be to get ahead of our tale -- we abstract from the clutter of minute-to-minute experience, and conceive of it all happening within a space.   We then formalize that space with the Euclidean axioms.   We then familiarize ourselves with this new mind-environment, solving tricky problems and whatnot for a couple of thousand years, then -- since we have long effectively been working in the World of the Unseen -- very lightly generalize to Euclidean spaces of any finite dimension  -- a bit of a stretch biologically, but where, mathematically, everything works pretty much as before.

Meanwhile independently, mathematical analysis had proceeded apace, not necessarily concerned with the geometrical substrate as such, but piling up its own increasingly intricate problematics.   Then by an ideational leap which is of the essence of mathematics, and into which simply listening to lectures and slogging through the problem-sets at the end of the chapters, gives you no insight at all (executive summary:  Mathematicians are like gods), a clutch of bold spirits, bearing in mind certain delicate problems such as infinite sequences of functions and their convergence, generalized the stage on which such pageants play out, from the Euclidean to the general topological.   (The history has here been brutally telescoped.)  Something of the sort was in any case needed to save the Euclidean picture itself, since infinite-dimensional spaces were now required (even by physics),  and the finite-dimensional structures would not generalize in any straightforward way.

General topological spaces being a wildly assorted bag, various restrictions are put on them, for one purpose or another, to allow deduction and calculation.  One of these is metrizability, which we examined in the essay on Urysohn.   That has the advantage of preserving much of our hard-won familiarity with the Euclidean metric, while allowing a vast array of new metrics for particular purposes. (For example:  the by-now-familiar Lorentz metric of Einsteinian spacetime.  Once mind-boggling, yet now -- in this vaster context -- almost cuddly.)  These in turn can be slightly re-generalized, by considering pseudometrics; or further regimented, with the concept of a norm, which in turn may be relaxed into a seminorm;  and so it goes.

~

A quite different and likewise fruitful generalization of metric spaces  is the notion of a Uniform Space, introduced by algebraic geometer André Weil, in “Sur les espaces à structure uniforme et sur la topologie générale” (reprinted in volume I of his Collected Papers as [1937]).   He broaches it with a bang:

La notion de distance  est utilisée dans de nombreux travaux de topologie, [mais] l’on s’explique mal qu’elle soit venue à jouer un pareil rôle  dans une branche des mathématiques  où elle n’est, à proprement parler, qu’une intruse…

On voit apparaître ici  cette hypothèse du dénombrable (dite aussi, on ne sait pourquoi, de séparabilité),  malfaisant parasite qui infeste tant de livres … dont il affaiblit la portée  tout en nuisant à une claire compréhension des phénomènes.  … La conscience d’un mathématicien, s’il en possède [!], doit répugner à faire intervenir une hypothèse superflue …

Strong words !   The notion of metric, he claims, is not simply too restrictive, but is the wrong sort of notion for topology -- a cuckoo’s-egg in the nest.   And indeed, minus the polemics, James Dugundji makes the same point (Topology, p. 200):

A metric … can be regarded a providing a measure of nearness that is applicable throughout the space  … This notion of uniform smallness is not a topological concept :  equivalent metrics specify different sets as being equally small.

… Notice that, even in metric spaces, a continuous map may be uniformly continuous if one pair of metrics is used, but not uniformly continuous when another pair of equivalent metrics is used;  uniform continuity is therefore  not a topological concept.

(“Equivalent” metrics in the sense that they generate the same roster of open sets, which define the topology.)

Contrast a different -- and very fruitful -- restriction on general topological spaces, that of being compact Hausdorff.  This notion is strictly topological in spirit.

Footnote:   For another instance of Gallic arithmophobia, cf. the remarks of Weil’s countryman  Jean Dieudonné, in Foundations of Modern Analysis (1960), p. 141:

The fundamental idea of Calculus [is] the “local” approximation of functions by linear functions.  In the classical teaching of Calculus, this idea is immediately obscured  by the accidental fact that, on a one-dimensional vector space, there is a one-to-one correspondence between linear forms and numbers, and therefore the derivative at a point is defined [horresco referens !] as  number instead of a linear form.

In defense of Sir Isaac Newton, it must be observed, that our worthy ancestor was  quite understandably  interested in how fast something was going, at each moment:  to answer which question, he needed to invent the differential calculus.  Dieudonné, from the vantage point of centuries of progress, is looking ahead to function-spaces and dense subsets of special functions and like that.

~

The passages immediately above  evoke, unbidden, an untoward echo  characteristic of their times (the Thirties; the Sixties):  “unAmerican” and (failure to adhere to) “Chairman Mao’s Correct Line”.   But “topological” is not an all-or-nothing concept;  and we return to sanity  with jolly John Kelly (General Topology), in the chapter titled “Uniform Spaces”:

We deduce from a topological premise (that the space is compact) a non-topological conclusion (that a function is uniformly continuous).  This chapter is devoted to a study of quasi-topological results of this sort.

Even more telling is the remark by George Simmons, author of the superbly pedagogical Introduction to Topology and Modern Analysis (1963):

Some writers deal with the theory of metric  spaces as if it were merely a fragment of the general theory of topological spaces.  This practice is no doubt logically correct, but it seems to me to violate the natural relations between these topics, in which metric spaces motivate the more general theory.

Thus, it is scarcely fair, or psychologically realistic, to denounce the notion of metric as an “intruder” in topology, as Weil does.  Similarly:  you shouldn’t start off with categories and functors  before learning about  ordinary numbers and sets, even if categories prove ultimately more foundational.

That said, there does come a point where actual everyday examples impel one to consider such things as convergence and compactness  in a setting more general than a metric space.  As: pointwise convergence, which is a perfectly familiar non-exotic sort of convergence, but which cannot be seen as convergence with respect to a metric.

~     ~     ~

We have thus seen uniform space as a gentle generalization of metric spaces.  Since the point of the latter is often concerned largely with matters of limits and convergence, all we really need to know is what it means to get “closer and closer”;  we don’t need to put a number on how close, each step of the way.   This aspect was highlighted by André Weil, when he debuted the idea of uniform spaces, as a kind of intellectual hygiene.   But in practice,  quite as important to the introducer of uniform spaces is their natural application to topological groups, which come ready-made with a structure amenable to notions of nearness.

But there is more.   John Kelley, in his General Topology (1955), who devotes an entire chapter to uniform spaces, writes:

It should be emphasized that this is by no means the only framework in which uniformity can be studied.  It is possible to study a set X  together with a distinguished family of pseudo-metrics for X, or to distinguish a collection of covers of X where are to be uniform covers (roughly in the sense of the Lebesgue covering lemma).  One may also consider “metrics” with values in a structure less restricted than that of the real numbers.  All of these notions are essentially equivalent.

Such a situation illustrates a recurring intellectual theme of this series of essays (with both philosophical and mathematical applications), tagged as “Rome by different roads”.   There is a section on this notion in our essay Consilience in mathematics (indeed, in one sense, the entire notion of consilience in general  is related to this idea).