Sunday, April 7, 2013

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 old 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

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 , 2nd 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.


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:

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 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.

Für psychologisch tiefgreifende Krimis,
in pikanter amerikanischer Mundart,
und christlich gesinnt,
klicken Sie bitte hier:

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 it's 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:


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 was 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.)

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:

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

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!
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:

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:

Vacuums, Voids, and the Latest Ideas about the Origins of the Universe
By John D. Barrow

The Biography of a Dangerous Idea
By Charles Seife

A Natural History of Zero
By Robert Kaplan

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