Words of the Day: sublunary, postlapsarian, …

Re Melanesians:

“When a native says that he is a man, he means that he is a man and not a ghost; not that he is a man and not a beast.”

-- Robert Louis Stevenson, In the South Seas (1896)

~

Perhaps the first word we heard, ourselves still children, which, rather than simply denoting something new (like a previously unencountered animal), designated something old in a new way, and thus reframed us, was:   Earthling.   In terms of reference, the word is synonymous with people;  but its intension  (with an -s-, not -t-; a term of art among linguistic philosophers, referring to the way it picks its referent out) is different.  (Thus likewise morning star and evening star, both referring -- though from different lookouts -- to Venus.) 

We meet this first in science fiction, and it permanently expands the mind.   When the exploits of the spacemen are forgotten  along with other tales of the nursery, we yet retain the spaciousness of the new view.  “I went to the mall with three of my friends”;  “I went to the mall with three…. fellow Earthlings”:  we sense the wider world beyond our plankboard stage.

(We just came across a mirror-term to earthling:

Suppose that an archaeologist who had visited us from outer space  were trying to explain human history to his fellow spacelings.

-- Jared Diamond, The Rise and Fall of the Third Chimpanzee (1991), p. 172 )

Another take on the same referents (namely, our earthly selves) is human.  Though the word is of course by now quite common, it was not always so, being a scientific word, borrowed from Latin (humanus, from homo; if there be a further relation to humus, then we have an interesting parallel -- again ‘earthling’, but now in the sense of ‘sons of the soil’).   By this term we are distinguished from animals;  but as everyone learns that perfectly well from an infant age, we do not need this word to teach us the species perspective.

Rather otherwise is mortal -- again from Latin, from the word for ‘death’.  In this secularized age, the average reader might think of this as a kind of moldy Sunday-school word, but the original sense among the pagan Greeks and Romans  was as opposed, not to God, but to the immortals -- the gods.  This is the sense that survives in the phrase “What fools we mortals be” (in Shakespeare’s most robustly pagan play).
An unexpected limitation in this word immortal  is evidenced by the following splendid epigram:

The actual infinity of a Platonist  is as seen by a mathematician who is eternal;

The potential infinity of the Intuitionists  is as seen by a mathematican who is merely immortal.

This is brilliant.  We expect the distinction to be between an idealized immortal mathematician, with all the time in the world to count to infinity, and a mortal one, who feels time’s wingèd chariot hurrying near, though he might make do with Supertasks:  accordingly, we initially take eternal simply as a stylistic synonym for immortal, before the second strophe brings us up short.  The Cumaean sibyl was merely immortal -- and consequently longed for easeful death.   Whereas the Platonist beholds the world of Forms  sub specie aeternitatis.  

(The epigram above comes not from a treatise of theology, but from a book of set theory and its logic.  I am quoting from memory, and possibly amiss, since Google finds, not that, but mostly philosophical and theological works.

-- Ha!  wait, found it, nestling on my shelves.   It is from Understanding the Infinite, by Shaughan Lavine.  He states it thus:

The idealization of experience that yields the actual infinite of classical mathematics  is that of the Eternal Mathematician, while the one that yields the potential infinite of intuitionistic mathematics  is that of the Immortal Mathematician.

But I like my misquotation rather better, and so shall leave it.)

~

A particularly delightful word,  containing, as were it a microcosm, a whole philosophy, is the word sublunary:  meaning, in the old cosmology, all that is to be found beneath the Sphere of the Moon.   (The allusion is not to the Moon’s own sphericity, wonderful though that be,  but to the celestial shell whereon that glowing queen is constrained to move  in regal splendour.)  Which is simply to say:  our everyday world;  except that now, instead of moving about in it like a tadpole in a pond, taking it all for granted, our thoughts now float upwards, and our hearts do yearn.

A simple swain -- yet fain to peer, beyond the sphere

The sublunary world is, in other words, this life Here Below.  And this brings us to a curious etymological fact:  for in Arabic, there are two common words meaning ‘world’:   al-`âlam, which is neutral, like world;  and al-dunyâ, which adds the notion of a contrast:  though now, not to any physical supralunary realm, but to al-âxirah ‘the hereafter’.   Morphologically, dunyâ is a feminine elative adjective, meaning ‘lower, nearer’:  thus precisely encapsulating the notions expressed by sublunary and Here Below, but in a single morpheme.

(The term sublunary has joined its age-mates in the medieval museum;  yet we still retain a kindred metaphor, "everything under the sun".)

~

At last we arrive at the final word of our title, postlapsarian :  a word that contains multitudes.  It refers to life subsequent to the Disaster in the Garden -- the only life that any of us ordinary folks have ever known.

In Adam’s Fall

we sinned all.

Our earliest ancestors, rueing the day

*

For a portrait of Grace and Reprobation,

try this:

The Penitent and the Impenitent Thief

 *

Once again, it is a word that picks us out, every one of us (for even Eve and Adam were postlapsarian at the end), but in a new way, thus differing most starkly from the bland philosophical agnosticism of the coreferential but non-synonymous term human.   It bears within it a deep and stark perspective -- one which we are inclined to disregard, as we scramble for sales at the mall, or lounge back glassy-eyed before the goggle-box (the behavior, however, belying the complacency).  

Yet in a better age, it was borne well in mind;  as when Condillac, in the preface to his L’art de penser (1780), though that work is seen in retrospect as having paved the way to atheism, yet was careful to remark,  that his analyses apply only to the postlapsarian soul:  much as a myrmecologist  (one more modest than Edward Wilson), should preface his monograph with the caveat that the generalizations made therein  might not apply to the world outside the termitary.   Our favorite Neothomist speaks of

… la précaution qu’il prend,  au début de son Art de penser, de rappeler qu’il va décrire l’âme telle qu’elle est à présent, après le péché originel.  Avant le péché, elle avait des idées  antérieures à l’usage qu’elle fait des sens,

«mais les choses ont changé  depuis sa désobéissance».

-- Etienne Gilson, Linguistique et philosophie (1969), p. 28

 Condillac, perhaps having second thoughts about what he unleashed 

We are, thus, all of us, earthlings, mortals, sublunary and postlapsarian.   Each of these words picks us out from among all else in the Creation, yet each from a different angle, in a way that enlarges our humanity.

~

Linguistic footnote:  We have chosen these philosophically rich words for the fun of it; but the basic phenomena here under discussion  occur more widely.  Thus cordate and nephrophoric (in their somewhat specialized use among philosophers) ‘having a heart’ and ‘having kidneys”:  non-synonymous but co-referential.

Or, to take an example quite similar to that of sublunary:

The expression dry land evokes the ocean in a way that land itself does not,  and by this very fact seems filled with sea breezes -- or rather, tempests, since the particular light in which the land is thereby set, is as a place of safety, reached at last after perilous voyages.  "Earthling"-style designation of the denizens of this default environment, from the salty perspective of the mariner: landlubbers.
Likewise terra firma:  it’s a place where you can finally find your foothold, after your return -- O Earthling -- from your voyage to outer space.

~

[update July 2020]

 

Mensch is a rather friendly-sounding German for any human being;  in Yiddish-English, a mensch is a solid, kind-hearted guy.   In the following, the word is contrasted invidiously with ‘policemen’, who thus are delimited outside the circle of humanity:

“ein Mensch schwer und vier Polizisten leicht verletzt. „ https://www.welt.de/vermischtes/article211457365/Stuttgart-Festnahmen-wegen-Auseinandersetzungen-vier-Polizisten-verletzt.html#Comments

Reader comments:

“ein Mensch schwer und vier Polizisten leicht verletzt. „ So klingt linker Sprachgebrauch. Macht ihr Journalisten schon unterbewusst. Als ob Polizisten keine Menschen sind.

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Ein Mensch und 4 Polizisten- habt ihr das aus Versehen aus einem linksextremen Kampfblatt übernommen??

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Contra Constructivism

“Only an atheist

 need fear infinity.”

Note:   In this place,

 http://worldofdrjustice.blogspot.com/2011/11/ask-dr-massey.html

an unafraid theologian presents an Orthodox account of the doctrine of absolution and the remission of sins,  conscientiously noting  along the way  a possible mathematical objection to the Church’s handling of infinity in this connection, which he is not personally qualified to resolve.

Fortunately, I was able to reply in complete reassurance on this point:  the Church doctrine is fully consonant with set-theory in its post-Cantorian understanding.

For more:

http://worldofdrjustice.blogspot.com/search/label/infinity

Constructivist Angelology

But yet when considered, may help us to enlarge our thoughts  towards greater perfections of it  in superior ranks of spirits. … The several degrees of angels  may probably have larger views.

-- John Locke, An Essay Concerning Human Understanding (1690)

Man’s understanding, though allied to the angelical, operates differently.  The angels understand intuitively, man by the painful use of the discursive reason.

-- E. Tillyard, The Elizabethan World Picture (1942)

It is presumably not obvious to the chimpanzee (or, if this be setting his smarts too low, to the humble woodchuck) that for all m, n in Z, m + n = n + m.  Nevertheless, in his daily scurryings and burrowings, he will repeatedly meet up with particular instantiations of this modest truth.

            For the woodchuck (at any event the southern northeastern lesser striped variety) builds a number of nests and other temporary dwellings, each of which has the framework of a variously triangulated  polyhedron, built tinkertoy-fashion from a fixed number of sticks.  Now, gathering them one by one would take too long, nor can the tidy woodchuck stand to have any sticks left over.  So when constructing his summer dwelling -- an icosahedron, which needs thirty sticks (did I get that right? My calculating powers are not much beyond those of a woodchuck) -- he normally harvests a jubjub bush, which has twenty-two sticks of exactly the right specs and which blooms in the spring, then rounds it out with the eight-sticked glubglub bush, which sprouts slightly later. 

But then one year, the blooming of the jubjub was delayed, and the woodchucks despaired.  All but one, the enterprising Willie, who went doggedly (or groundhoggishly) ahead  and harvested the available glubglub, supplementing this  when the jubjub arrived slightly later.  This remarkable exploit was recorded in the annals: for 22 then 8, one may substitute 8 then 22.

            It was subsequently found that a mubmub bush (18 sticks) followed by a nubnub bush (12) would do just as well – und zwar, in either order!  This fact too was recorded.

            The years went by, then the centuries, and the millennia, and the annals grew to seven times seventy stout volumes, densely filled with such arcana as: a cube-for-cubs may be constructed of a lublub (7) plus a rubrub (5), and this in either order; and so on for billions of examples.  All this was considered a branch of botany, a purely empirical science.

            By this means, the woodchucks arrived at an analogue of Babylonian mathematics.

Interlude:   A physicist depicts the arithmetical state-of-play in a papyrus from Egyptian/Babylonian times:

It records the resolution of a great number of fractions  into a sum of aliquot parts,  the original numerator always being 2:  as, for instance,

2/97 = 1/56 + 1/679 + 1/776

But no rules are given for effecting such resolutions, and the whole treatise seems to be a mere compendium of results obtained by repeated trials.

-- James Jeans, The Growth of Physical Science (1947 [posthum.]; 2^nd edn. 1951), p. 11

            Until one day one Wisedome Woodchuck, a distant descendant of Willie, figured the whole thing out, and in a remarkable demonstration of only eighty pages (rather hard to follow, but sound), showed that m + n = n + m  was a perfectly general fact, replacing the seven-times-seventy volumes at a stroke, and freeing up his brethren for yet further architectural innovations, which previously had been shunned, as their particulars were not yet in the book.  The annals were placed in a museum, which the elder woodchucks might still visit, marveling at favorite exhibits (as who could forget that remarkable winter, when 5,878 + 519 turned out to be equal to 519 + 5,878?  A tour de force!). Meanwhile generations of young woodchucks (the pride and despair of their parents, who could not follow them into Canaan, with their aging brains) studied Wisedome’s proof, breaking their little heads against it.

           

Meanwhile in Metropolis… The humans, learning of this, politely saluted Wisedome’s modest accomplishment, and experienced a pang of sympathy for woodchuck-kind; yet felt no inclination to visit their Museum of Particular Results: for which they felt, indeed, a kind of horror.  And even the general result, while true, is somehow to us not truly interesting. In any case we are all too busy wrestling with the Riemann Hypothesis, to have time to look back.

Meanwhile in Elysium, where throne the angels sensu strictior, the lowest order of angelic beings sensu lato, a mock compliment is paid to Andrew Wiles, who finally figured out that little Fermat puzzle, with which the angel-kind  are wont to amuse the nursery.  Not that the angels arrived earlier at his proof, nor any refinement thereof.  They simply scoop up a few infinities of integers with their fractal fingers, twist them this way and that—and see, it doesn’t fit!  Simple.

            Moreover, all facts about all structures of ordinal type omega, whether or not deducible by any finite axiomatization, are equally transparent to the angels. They just look.

            So, is Elysium the mathematical Paradise?  Not quite…

            In a remarkably lucid and accessible article*, which should be packed into every pupil’s lunchbox by a considerate mom, Gödel observes that our continuing failure to resolve Cantor’s continuum problem, left over from the previous century, is quite an embarrassment.  It means that we are unable to wrap our minds around the very simplest multiplication problem possible, beyond the finite ones that these days can scarcely stump a woodchuck. Namely, two times two (times two, times two – keep going).  He writes:

            “It is easily proved that the power of the continuum is equal to 2^(aleph-nought). So the continuum problem turns out to be a question from the ‘multiplication table’ of cardinal numbers: namely, the problem of evaluating a certain infinite product (in fact the simplest non-trivial one that can be formed).  There is, however, not one infinite product (of factors > 1) for which so much as an upper bound for its value can be assigned. […] It is not even known whether or not m < n implies 2^m < 2^n.” 

            We are  so to speak  staring helplessly  at a pile of sticks.

            Nor does the subsequent Cantor+Cohen demonstration of the independence of the continuum hypothesis  from a particular system of axioms for set theory   set the matter aside. Gödel had already anticipated Cohen’s result, and wrote:

A proof of the undecidability of Cantor’s conjecture from the accepted axioms of set theory (in contradistinction, e.g., to the proof of the transcendency of pi) would by no means solve the problem.  For if the meanings of the primitive terms of set theory … are accepted as sound, it follows that the set-theoretical concepts and theorems describe some well-determined reality, in which Cantor’s conjecture must either be true or false.

            Indeed Gödel suspects that the Cantor conjecture is actually, factually false: which means that somewhere, among the actual literal real numbers, there is hiding a set of cardinality intermediate between aleph-nought and its power set, with definite members which the angels could name.  Not, however, the lowest order thereof; this lies beyond them.  But at the next step up, the archangels hang these sets from mobiles over their infants’ cribs.  In fact a woodchuck may somewhere inadvertantly have used one of these sets for nesting materials, and even now lies sleeping on it – a night of troubled dreams.

            So much for a simple pancake-stack of omega-many deuces – the limit of the lower-angels’ ken.  What about the square root of omega-to-the-omega; or cross sections of fibre bundles on toroidal cap-omega-cross-theta space? For each level of angels, there will be something beyond them that they just don’t get.

*

There are two poles of the range of approaches to the problem of infinities.  One is that of the badger-like Brouwer, who simply sweeps the chessmen to the floor, folds up the board and goes home.  (An only somewhat more amenable figure, says Gödel, is Weyl, who allows as how there might be something to board games, but suggests we play checkers – or Chutes ‘n Ladders – rather than chess.)  The other pole says:  Infinities are tricky, but they all exist, and are present to the Infinite Mind. Gödel himself uses that term, e.g. noting that Ramsey’s admission of formulae of (countably) infinite length  might be constructivistic for an infinite mind  but not for our own.  Gödel does not, however, seem to feel much need for any desperate appeal to such a mind, in the course of an ordinary day, since he -- like Badger’s amiable friend the Water-Rat-- is a thoroughgoing Realist, and comfortable as such in his own skin.  For him the assumption of infinite classes “is quite as legitimate as the assumption of physical bodies, and there is quite as much reason to believe in their existence.”  The outwardly gloomy Austrian  is really the jolly Dr. Johnson of set theory.

            Only now there’s a problem, of a sort which did not confront the schoolmen, who never counted on the uncountable:  the Infinite Mind is all very well, but -- Which infinity did you have in mind?

            Who comprehends *everything*? God does, by definition. Yet He cannot be simply the crown on a tower of constructively ascending intelligences.  He is like an “inaccessible cardinal” – and not the first.  Nor perhaps ‘the last’, if there is no last.  Whatever He might be, there is Cantor in the wings, grinning, waiting to perform a Power Set on God, yielding – what?  -- Nothing one can begin to commence to pretend that we can approach with our sadly finite understanding.

            All of which suggests, if nothing else does,  that God is something more and other than an alternately wrathful and affectionate granddad  with a perfectly enormous white beard – however much longer that beard might be, than the stubble which disfigures your chin or mine.  Who one day, apparently from sheer idleness, as one might choose chocolate, chose the Jews.  Who later, some say, cast a Jove-like eye  on a certain Palestinian virgin.  And who at present is very angry indeed with the Democrats (or the Ravens, or whomever).  Yet what He in fact might be, we cannot even begin to imagine anyone’s beginning to conceive.  (Cf. the suggestion of 1 Kings 8:27  that the heavens themselves have heavens (and so on up); and that the whole omega-tower of them  cannot encompass God.)

*     *     *

~ Commercial break ~

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We now return you to your regularly scheduled essay.

*     *     *

            We actually wind up with a sort of hamstringing of the Ontological Argument. Notoriously its conclusion does not really follow from its premise;  but now even its premise limps: “Since we can imagine a Perfect Being…”  But that’s just it, we can’t!  Not even little infinite bits of one! Yet paradoxically (and God reportedly loves paradox – at least Chesterton does, His publicity agent on Earth), this seeming stomping on the prostrate corpse of the offspring of Anselm, this despairing cry that somehow even Infinity does not suffice, so far from opening the agora  to legions of snickering atheists chanting “Toleja so!”, points somehow upward, -- outward,   -- onward ….  Praise Him!

Postscript:

John Locke himself, normally regarded as the Poster Boy for Empiricism, of I'm-from-Missouri common-sensicality, yet delivers himself of this (Essay, III.vi.12):

That there should be more species of intelligent creatures above us, than there are of sensible and material below us, is probable to me from hence:  that in all the visible corporeal world, we see no chasms, or gaps.

That is to say:  The gap between ourselves, and God, must somehow be filled, according to the Principle of Plenitude.

And again (IV.iii.23):

He that will consider the infinite power … of the Creator of all things, will find reason to think, it was not all laid out upon so inconsiderable, mean, and impotent a creature, as he will find man to be;  who  in all probability, is one of the lowest of all intellectual beings …

Angels of all sorts are naturally beyond our discovery, and all those intelligences, whereof ‘tis likely there are more orders than of corporeal substances, are things, whereof our natural faculties give us no certain account at all.

Since theism is far from central to Locke’s Essay, it is curious to see the emphasis on this scala naturae idea.

--------------

*”What is Cantor’s Continuum Problem?”, repr. Benacerraf & Putnam, eds., Philosophy of Mathematics.

~

Postscript:  For the possibility that the structure of certain mathematical truths relating to an infinite domain  might resist any but a case-by-case “Babylonian” approach, cf. the quotation from Michael Dummett towards the end of this post:

http://worldofdrjustice.blogspot.com/2012/01/what-is-truth.html

Compare further (re ascending ranks of abstraction and generality):

"The Ladder of Abstraction"

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Twinned Eternities

In the poem portion of Nabokov's Pale Fire (1962), we read:

Outstare the stars. Infinite foretime and infinite aftertime.

The basic lexicon of Classical Arabic, as it happens (with its wealth of words), distinguishes these two, with a brief, monomorphemic word for each:

أبد  [abad] prospective eternity (with a terminus post quem but no end)

أزل [azal] retrospective eternity (with a terminus ante quem but no beginning)

Reflecting separately on these  raises interesting philosophical questions, e.g., regarding the indestructibility of the soul.  But as a merely intellectual matter, the picture will be clear to anyone who has got as far as geometry or calculus -- "half-infinite" directed intervals (such as the non-negative reals).

In Laughter in the Dark (English version 1938), Nabokov plays with the twinned hemi-eternities, in a passage full of satire:

“Haven’t I met your sister once?” queried Dorianna in her lovely bass voice.

“My sister is in Heaven,” answered Rex gravely.

“Oh, I’m sorry,” said Dorianna.

“Never was born,” he added.

Much is rustling beneath the sheen of this. First, Rex, po-faced, lays a logical snare for Dorianna, knowing that she would automatically assume that the now-absent soul is residing in the after-eternity (Arabic abad), which is all that most people ever think about, or even conceive to exist. An extra twist is provided by Dorianna’s automatic, flustered response at the unexpected reply – the conveyed sense is meant to be, “I’m sorry for your loss,” but simply stringing bits of syntax together yields, “I’m sorry that your sister is in Heaven.” (**) Rex then, indirectly pointing out the ambiguity in his initial statement, and implicitly rebuking Dorianna for failure to perceive ambiguity (again, in whatever sphere, most people don’t), reveals that this sororal Arlésienne is in fact at present domiciled in the ante-eternity (Arabic azal).

That passage, incidentally, rests as well on the paradox alluded to last time, that of the indestructibility of the soul. Intuitively, we think of a soul as the sort of thing (being brewed, after all, from the aethers of the Eternal One) that doesn’t simply wink out of existence. But by the same token, it shouldn’t be the sort of thing that simply winks in – in which case it must, as in the Platonic conception, be “stored up” somewhere (Rex specifies Paradise as the ‘green room’ or on-deck circle, though probably more from a faux-polite convenance than from any committed Platonism).

That observation does not exhaust the (depthless!) well of paradox subtending such insight. Follow it out, and you’ll encounter a retrospective version of the quantum-mechanical “many-worlds” scenario.

(**) P.G. Wodehouse plays on the same effect. In his world, earls are crotchety and idiosyncratic, and butlers are perforce imperturbable in the face of all.

            “Dash it, Beach, this egg is undercooked!”

            “Yes, sir.”

            “What – is the cook out sick again?”

            “Yes, sir.”

            “Well, I’ll be damned!”

            “Yes, sir.”

(Picture an infinitesimal lengthening of the glide initial: “Yyyes, sir.” The joke is, the double-meaning will be completely lost on the earl.)

~

It was the contention of “The Stokes Conjecture” (a chapter in my book, The Semantics of Form in Arabic)  that possession of such a neatly twinned pair of monomorphemes as azal and abad, as opposed to such roundabout (and nonce) phrasal confections as “infinite aftertime”, should track with a greater tendency of speakers of the language so outfitted, to be clear on the subject, for it to be cognitively relatively accessible, and thus to accrete around it  further developments both morphological (derivata) and semantic (metaphors).   Whether this be the case for Arabic letters and Islamic theology, I leave to my learnèd readers to make out.

~

The casual reference above, to the isomorphism of azal and abad, as constituents of infinite Time, to the half-rays  as subsets of the real line, conceals a textural disparity, in point of richness.   Not, as you might imagine, in favor of the human/experiential conception of Eternity, but the mathematical/intellectual science of the Real Line.

We really don’t know what to do with Eternity -- and literally, wouldn’t know what to do in it.    Like the silence of the infinite spatial reaches that so dismayed Pascal, we stand aghast at the prospect of doing anything “forever”, be it strumming harps or standing around on clouds swapping New Yorker captions.   The prospect of an infinite afterlife, for which we are supposed to yearn, is strictly baffling.

(Note that there is nothing heretical in that observation of human psychology;  notably, C.S. Lewis was converted to Christianity  before any sort of belief in or appreciation for  an infinite afterlife  was granted him.)
(For a mathematician's take on how the afterlife shall be spent, try this.
For an equine perspective,  this.)
(I riff upon the bafflement in the azal case, here.)

The Real Line, by contrast, is … infinitely diverting.   To begin with, even in a low-focus broad view (abstracting from the  so to speak  “quantum foam” of the infinitessimally inspected continuum), it harbors a great many other infinities within itself.  Thus, that infinite ray or half-line,  [0, ∞ ), is topologically equivalent to a mere half-open interval,  [0, 1):  in both cases, you can cover the space with a countably infinite sequence of disjoint intervals, no finite subset of which can do the job.  Thus, for the ray: [0,1), [1,2), [2,3) …. etc;  for the interval, [0, ½) [½,¾), [¾, 7/8) …

Further, the infinitude lies not only in thus stretching out forever, but in drilling down.   Any interval (even one that is closed, and thus compact) harbors infinitely many intervals as subsets;  and any such patch contains an uncountably infinite collection of points.  (We recall Dyson’s title, “Infinite in All Directions”, but with ‘directions’ differing qualitatively rather than simply like a compass needle.)

And those are just the appetizers.  The continuum is a regular zoo of exotic creatures -- the Cantor set, the Borel sets, projective sets, and a host of others studied in the discipline of “Descriptive Set Theory”.

~

As for those specifically twinned infinities, staring at each other from opposite sides of a mirror, they have a counterpart in modern physics, which takes reversibility of Charge, Parity, and Time (separately for some processes, in combination for others) as a kind of credo, like Liberté, Egalité, Fraternité.   Traditionally, time-reversed solutions were usually dismissed out of hand as unphysical;  other, more venturesome theorists, embraced them, telling fables of antiparticles as simply particles moving backwards in time, among other scenarios that chill the blood.

[ShoutOut:  Many thanks to Djinn ibn Sayârah  for help in formatting this for posting here.]

[Afternote]  My friend the Arabian theologian writes in :

Mark Twain wrote, regarding death:

“I do not fear death. I had been dead for billions and billions of years before I was born, and had not suffered the slightest inconvenience from it.”

That's kinda funny for as far as it goes, but no one who has experienced consciousness can be seriously flippant about it being snuffed out.

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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

Thoughts 'n' Things

  Rudy Rucker, Infinity and the Mind, p. 38:

(**) Just as a rock is already in the Universe, whether or not someone is handling it,  an idea is already in the Mindscape, whether or not someone is thinking it.

This is itself a pleasant thought, recalling the ditty about God-in-the-quad; but in actual fact – I don’t think so.

(So you see—I am not an uncritical Platonist.  Platonic heaven must be so gerrymandered, as to exclude such things as cheese doodles and Sponge Bob Squarepants.)

The actual universe has (for example) -- whatever geometry it has:  regardless of whether there are rational creatures capable of understanding it, let alone deriving it.  Likewise the landscape of math.  But particular formulations of physics, and perhaps even of math – matrix mechanics v. wave mechanics, Cauchy analysis vs. non-standard analysis – do not exist in complete independence from their proponents.  They are, one might say, propositions, not objects.  The objects (or patterns, or whatever they are)  exist  even in the absence of  a person to spout propositions about them; but the propositions require a proposer.  – Nothing specially abstract here; the same thing is true of rocks.  This rock exists independently of any finite mind, but: “There lies a rock” and “Behold that rock!” and “What a rock that is!” must come out of some actual someone’s mind or mouth.

            The unbridledly idealistic view in (**) conjures up a skyscape of untethered thought-balloons.  It is pleasant to contemplate, in a comic-strip sort of way, but not to be taken too seriously.  For one thing, unlike the situation with mathematical truths, where anyone at any place or time might discover them, there is no way for a rational creature in another galaxy or dimension to reach out and grab one of those thought-balloons by the tail;  he is required to blow his own bubbles.  Whereas the structures of mathematics are like fixed landmarks, which one encounters again and again, from different approaches.  For instance:  Yang-Mills gauge theories, discovered by the physics expedition; and connections on fibre-bundles, discovered by the math team; and lo, they meet in the middle.  Likewise group-theory.  Different body-parts of this have been grabbed onto by matrix theory, algebra (symmetries of solutions to equations), geometry (the Erlangen program), particle physics (glad you could get here; meet Sophus Lie), and in time it becomes clear that it’s all part of the same elephant.  Whether they come from physics, or mathematics, or computer science, two such explorers may not realise that they have come upon the same mountain, till they have circled around it a bit and compared notes.  And this happens repeatedly.  We may summarize in an epigram:  The mindscape of mathematics is a multidimensional torus:  whatever direction you set off in, you eventually wind up back at Hilbert’s Hotel.

It turns out that Shing-Tung Yau likes this montane metaphor as well.  Cf. The Shape of Inner Space (2010), p. 103:

A mathematical proof is a bit like climbing a mountain.

And he nicely outlines the Yang-Mills case (p. 290):

The physicist Chen Ning Yang was similarly astonished to find that the Yang-Mills equations, which describe the forces between particles, are rooted in gauge theories in physics  that bear striking resemblances to ideas in bundle theory, which mathematicians began developing three decades earlier, as Yang put it, “without reference to the physical world”.  When he asked the geometer S. S. Chern how it was possible that “mathematicians dream up these concepts out of nowhere,” Chern protested, “No, no.  These concepts were not dreamed up.  They were natural and real.”

            Contrast the case with “thoughts”.  Supposititious entities of the mindscape, even some popular thought-balloon, tethered to a billion different heads, need never be rediscoverable by another explorer, nor acknowledged as real should he simply be grabbed by the lapel by one of the thinkers, and treated to an exposition of same.  For example, the notion held dear by countless generations of schoolboys around the globe, of the uniquely funny nature of flatulence, will never appear among the gravely ellipsoidal thought-balloons of the solons of Fdrmrphlandia; even “funny”, for them, is not well-defined, and not particularly worth defining.

Now, probably Rucker meant to restrict the realm of “ideas” to just some of them.  Not, “Wouldn’t it be fun to dip Suzy’s pigtail into the inkwell!”, but things like “The square of the hypotenuse is equal to the sum of the squares on the other two sides.”  Fine; but careful, here.  The Pythagorean theorem has  as its basis  a fact about Euclidean geometry, in every possible world; just as Fermat’s Last Theorem expresses (in a possibly somewhat contingent and imperfect way) a fact about the natural numbers.   But a fact is not the same thing as an idea.  As a matter of fact, there is a coffee stain on this shirt; but “the idea of this coffee-stained shirt” is no strut or girder of God’s architectonics.  An idea concerning a fact of mathematics, in a finite mind,  may bear – must bear -- but an imperfect relation to the fact itself (‘fact’ here used broadly: it may refer to a wildly transfinite complexus of relations, some of them perhaps perceptible only to angels).   Most people’s ideas of mathematical truths bear as much relation to the truths themselves  as does a crayon scribble to the Sistine Chapel  which it might (based merely upon memory of a fleeting ill-lit glimpse) attempt to depict.  To posit that all truths of mathematics exist as Ideas in God’s mind, is logically allowable, but really adds nothing, and is in any case unknowable. To identify these truths with the neuronal states of the pitiful meat-wads sloshing around in our half-cracked crania, is to add nothing at all, but is rather to detract.

[Appendix]  Karl Kraus apparently entertained a notion of independent or pre-existent thoughts.  He speaks of someone being

von der Präformiertheit der Gedanken  überzeugt, und davon daß der schöpferische Mensch  nur ein erwähltes Gefäß ist; und davon, daß die Gedanken und die Gedichte da waren  vor den Dichtern und Denkern.

-- “Heine und die Folgen”, reprinted in J. Franzen, The Kraus Project, p. 88

The whole ‘meme’ idea (itself a meme) is similar -- not that the various Chiclet-thoughtlets were truly Platonically pre-existing, but that, once hatched, they lead a promiscuous existence, wandering into people’s minds  like pollen into our air-passages.

~

Footnotes from the 19^th century:

Dedekind … allowed his philosophy of mind  much reign, with a ‘proof’ that “there are infinite systems”;  for he gave  as evidence “the totality S of all things, which may be objects of my thought”, since  as well as any of its elements s,  it contained also “the thought s’ that can be the object of my thought …This ‘proof’ did not gain a good reception.”

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 105

For Frege,

In contrast to subjective ‘ideas’ (Vorstellungen), ‘thought’ was intended in an objective sense, rather like state of affairs, sharable among thinkers  and indeed independent of anyone thinking then.

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 190