Risen indeed

 

An Egyptian-American Muslim colleague of mine  received an e-mail from his Egyptian-American Coptic friend, addressed to a roster of friends and acquaintances both Christian and Muslim.   The message read, in its entirety:

Have a blessed Easter.

Risen indeed.

Now, my colleague occasionally has difficulties with nuances of English, even after a couple of decades of residence in the U.S.   But he did understand that this terse and telegraphic phrase asserted the Resurrection of Jesus.

Though Muslim to the core, he is quite accepting of the other Abrahamic faiths (indeed, has more than once attended services with Christian friends), and he was happy to receive this blessing from a friend.   He then asked me, in confidence (since he knows that I am candid about such things), whether all Christian denominations believed in the Resurrection.

            “Um … yes,” I replied.   (The “Um” part was to stall for time while the taken-aback-sensation passed.  In a similar fashion, some years ago, a young woman colleague, university-educated and soon to be well traveled, asked me out of the blue, “Do Jews believe that Jesus is the Messiah?”   -- “Um … no,” was my reply.)

His own reply was a surprise.  “Then we believe the same thing!” he exclaimed, beaming.   For the Koran (he went on) proclaimed that God “lifted” Jesus up to Paradise (rafa`a).   I expressed satisfaction at this congruence, and privately desired that the message might spread to outfits like ISIL and Boko Haram, who might, please, accordingly  knock it off.

He went on to inform me that (according to Muslims) Jesus of Nazareth (upon whom be peace) had not actually been crucified.   I had heard this motif before in connection with the Ahmadiyya (a pure simulacrum had been crucified in his place); but my colleague added a neat twist, reminiscent of the trope wherein the executioner hoists with his own petard:   God had caused the very soldier leading Jesus to Golgotha  to suddenly take on the mild carpenter’s appearance.  

“That’s not in the Koran,” I frowned.  -- “No,” he conceded.

“Hadith?”  (There are tens of thousands of these, some sounder than others, you never known what might be found in some of the doubtful ones.)  --  He considered.  “In the culture,” he finally said.

Which is to say:  Folklore, like “December 25”.   (None but the most ignorant Christian will take offense if you state that we really have no idea what day Jesus was born on.)

Well -- He is born, and He is risen.  Peace on Earth, good will to men.

A vision common to Christianity and Islam

(Nur mit ein bißchen anderen Worten)

~

[Philological footnote]  The Arabic, rafa`, is literally ‘raised’.   Thus, it is not quite equivalent to resurrexit, in what might elsewhere be a small  but is here a crucial difference:  rafa` is comptabile with Jesus having been raised to Paradise while still alive, like Elijah.  There is nothing in the Koran saying that he was risen from the dead.

~

With our linguistic caps on, let us look more closely at the wording of that greeting.   To a Muslim with an uncertain grasp of English, it might seem no more than you might find on a Hallmark card.   It is thus the perfect two-tier or encapsulated expression:   no more than a smooth rock, or black box, to the uninitiated;  but a treasure-chest  which springs open at the click, to initiates.

The standard American greetings that those of my generation grew up with  were “Happy Easter” and “Merry Christmas”.   But as those holdays became increasingly commercialized, some observent Christians have come increasingly to say “Blessed Easter” or “Easter Blessings”, to make it plain that they are actually referring to the holy-day and not simply a time to hunt eggs or find discounts at the Mall (or as a counterpunch to the latterday degradation of even “Merry Christmas” to “Season’s Greetings”).  You would not say it to someone you knew to be an atheist;  though you might say it to a kindly stranger.

 

“Risen indeed”, by contrast, is not used  even by the generality of Christians, let alone to those of other faiths or none.  It specifically echoes the antiphonal style of worship, priest and congregation:      

    

   “The Lord be with you” 

   Rx: “And also with you.”   

   “Christ is risen.” 

   Rx: “He is risen indeed.”    

 

And by leaving off the first two words of the latter formula, pronoun and auxiliary verb, the writer made it clear that he was echoing the ritual greeting, which you may hear in Orthodox, Catholic and Episcopal services, but not among churches whose liturgy or worship-style is not antiphonal.

Among the Eastern Orthodox, this formula is known as the Paschal troparion:   Χριστὸς ἀνέστη.

The English Wikipedia has a particlarly nice article on this topic, as does the Russian version:

       http://en.wikipedia.org/wiki/Paschal_greeting

Christus resurrexit! Resurrexit vere!

[Update]  The whole offering-and-response style recalls, actually, Arabic.  There are dozens or scores, if not hundreds, of formulaic greetings and blessings, each with a prescribed response.   This scarcely exists in English, apart from “Thank you” -- “You’re welcome”.  (Arabic:  shukran. -- `afwan.)

Thus:  To al-salaam `aleykum, you reply Wa-`alaykum al-salaam.  (“Hello.” -- “Hello back.”)

To SabaaH al-xayr (“Good morning”), SabaaH al-nuur (literally “Morning of light”;  with many further variants.)

To na`iiman! (a salutation to someone who has evidently just bathed, or shaved, or had a haircut):  Allah yin`im `ayleek.

And so on.

There is thus, both a theological aspect to the responsory mode of worship, and a cultural/aesthetic one:  both typical of the Mideast.

[Further update]  Apart from the context of worship, such ritualized exchanges are typical of traditional societies, and indeed help constitute their social cement.   Contemporary American is sadly lacking in these:  a lonely survival is “Thank you” -- “You’re welcome” -- and even this may go the way of the dodo  as the custom of so much as thanking  fades, along with that relic of our mother’s day, the Thank-You Note.

(As, https://www.washingtonpost.com/advice/2025/08/26/carolyn-hax-cash-gifts-thank-you-notes/

Nieces prefer getting cash gifts and not sending thank-you notes.   Sending cash feels like paying bills  to this aunt, and nieces’ lack of response to gifts  makes it even worse.)

 

But the forefathers of our ancestors in the northern isles, were replete with these.   A fossil of one survives in the mysterious word wassail, now known to most speakers only in some old Christmas carols.  In origin, it is a salutation; grammatically, an imperative.  And it came with its own antiphon:

 

Host:  Wassail!  [cognate with Old Norse ves heill ‘be hale’, as in “hale and hearty” and “to your health”]

Guest:  Drink-hail!

 

Ah, the days, those were the days;  those were the days  indeed.

 

Equinox: Measure for Measure

A-dance as had their grandsires never sinned,

the giddy throng now treads the village green.

“Farewell forever to the Winter’s frost --

Let Summer be our air  forevermore.

Tear down the tower-clock of seasons’ whirl,

and let us frolic in oblivion,

our ever-noonday joys  unmarked, untimed,

and unreliev’d by memory of dark.”

But as our nature, from the Maker’s forge,

is blent of virtue, yet with vice annealed,

so shows the iron-bound justice of the Judge

in equal measure  of the Light, with Night.

Modus Ponens Paradox

Dixit Algorizmi:

* All Elephants are Mortal.

* Babar is an Elephant.

* Yet -- Babar is Eternal.

How do we account for this discrepancy?

For discussion, see:

https://worldofdrjustice.blogspot.com/2015/12/modus-tollens-tollendus-est-iterum-re.html

For additional crispy thought-food, click on any of the Labels below. 

For those who could care a fig for philosophy(**), and are interested only in the mighty Elephant King, click here.

(**) Technically known as the ficus philosophiae.

Ontological discontinuity (at the nether end)

As was known to all until perhaps the 18^th century (when certain mechanists would strive to deny it, as do certain homunculi  even in our own day), there is an ontological discontinuity between Man and Beast; a matter touched upon here.  But what is the case, within the order of the Beasts;  or indeed, between the least of these, and the mere madness of atoms?

We have  from time to time  penned odd little poems, or perplexing parables,  which in fact do seek to glimpse that phase-change, from random atoms  to the least of God’s creatures -- the infinum of the scala naturae.  For a representative sample, try this:

https://worldofdrjustice.blogspot.com/2012/10/gabriels-pep-talk-to-bugs.html

For the full roster of such things, here:

https://worldofdrjustice.blogspot.com/search/label/bugs

Faith and Ceremony

G. K. Chesterton, originally Protestant, then a convert to Catholicism, defends his church against Iconoclasts;  his remarks do equally well in defense of Eastern Orthodoxy.

Though of course these shows and pageants  meant immeasurably much to me, and profoundly affected my emotions,  yet I do not base my belief on such emotions, still less on such pageants or shows.   I was myself received into the Catholic Church  in a small tin shed, painted brick-red, which stood among the sculleries and outhouses of a Railway Hotel.

-- G.K. Chesterton, The Resurrection of Rome (1930)

Substitutionary atonement

It is a repeated motif of the thriller-series “24”, that Jack Bauer offers himself up as a sacrifice -- in Season Six, for one specific hostage.   We marveled at that them here:

http://worldofdrjustice.blogspot.com/2012/07/24-season-vi-executive-summary.html

Against the background of all the other whiz-bang implausibilities of that show, this one was not the most far-fetched, but it still seemed a stretch.

And yet now we have a real-life authentic hero of law enforcement, who did exactly that during yesterday's terror attack in France, and died for his pains:

Arnaud Beltrame

https://www.valeursactuelles.com/societe/arnaud-beltrame-les-francais-ont-recu-un-modele-94219

The  Christian doctrine with which such acts are blessed, is called “Substitutionary atonement”.  

Jack Bauer received no thanks for his gambit; les gendarmes de France are likewise best appreciated by the bobos-gauchos  when letting themselves be stoned or shot at, rather than firing back.   A few of them have been getting somewhat annoyed:

    http://worldofdrjustice.blogspot.com/2016/10/entre-bleus.html

A side-bet for “Pascal’s Wager”

For the most part, Pascal’s Wager gets little respect these days, whether among believers or agnostics.  But it turns out you can take Pascal’s logic quite seriously, and wind up … backing a horse of another color entirely.

Such are the Yazidis:  devil-worshippers by repute;  but not, it appears, Satanists.  Let us explain.

Historically, Satanists do not form any single actual church, apart from occasional aberrations:

http://www.chicagotribune.com/news/nationworld/ct-satanic-temple-monument-minnesota-20170506-story.html
Rather, they are individuals of a particular bent, who, like Faust or Aleister Crowley, out of diabolical Superbia, dabble in the dark arts.  They generally (like Faust or Crowley) formed part of some ambient mainstream faith, before defecting in service of the Prince of Lies.

As for what is really going on with the Yazidis, it is difficult to know, since they themselves -- a hermetically closed sect -- are mum on the subject, and their neighbors, being all of them detractors, cannot well be relied upon.  But roughly (and since we are dealing here with theological logic, and not with Mideast anthropology, that rough cut will do): 

(Not history, but rational reconstruction):
At some point around the dawn of the sect, its elders reasoned thus:

(1)  God the Father, maker of heaven and earth, is notoriously forgiving.  -- Here they reason with Heinrich Heine, whose reported last-words, trifling with God on his death-bed, were:

Bien sûr, il me pardonnera; c'est son métier.

(2)  Satan, on the other hand, has an evil reputation:  vicious and vindictive.  You don’t want to get on his Enemies List.

(3)  Ergo:  Placate Satan (in words, at least) in this life;  and hope for forgiveness in the next.

Such a prudentially propitiatory policy  may be compared with paying protection-money to Al Capone.  Doesn’t mean you like the guy;  it’s just an expense of doing business.

In summary:  Yazidis, in the old phrase, “hold a candle to the Devil”, but they do not actually (interiorly, in their heart of heartgs) worship hIM.

Se non è vero, è ben trovato;  think of it as a Gedankenexperiment.

The pros and cons of Pascal’s Wager  are reviewed here:

http://worldofdrjustice.blogspot.com/2017/03/in-defense-of-pascals-wager-expanded.html

Doctor Justice names the penguins

[Prefatory remarks:
In this space, we intend, favente deo, to begin a quest of world-historical import:  the Naming of the Penguins.   The project is to appear in installments.  To prepare yourselves, read (or re-read),  L’Ile des pingouins, by Anatole France.]

(1) We read in Scripture, how that Adam did name the beasts:

And out of the ground the Lord God formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof.
And Adam gave names to all cattle, and to the fowl of the air, and to every beast of the field ..

Further research reveals, however, that our Original Parent did not name each beast individually, but only by kind:   Behold, these are the lions, these the lambs;  these the hawks and kites and crows … 

Thus, much work remains to be done.  And as a linear descendant of Adam (on my mother’s side -- father’s too, in fact), I feel it is incumbent to me to take up this cross.

Where to begin?   Well, more numerous than all the beasts of the field, are the Penguins, of pure repute:

Appellavitque Adam nominibus suis cuncta animantia

Accordingly, all the penguins of Antarctica have lined up single-file, and are passing patiently beneath my hand, as I sain each one, and give each one its name before God.

“Fluffy; Chubby; Tumtums; Blackie; Whitey; Roly-poly; Poly-Roly …”

As of press-time, Dr Justice has individually named eighty thousand penguins; just five million more to go.

(2)  [Update, 28 April 2017]

Five hundred thousand and counting:

“….  Fishsnitcher, Egghuddler, Iceberg Bertie, Lollybop,

 Antarctic Archie, Austrobird,  Gus the Glacier Guy, Snowmelt…”

Te baptizo, Nitide !

[TBC, DV….]

Another Lewisian non-epigram

“There is no morality in Heaven.”

-- CSL, Letters to Malcolm

[baffling out of context, sensible within]

The Trinity (triune unity -- re-renewed)

[In observance of Maundy Thursday,  a re-post  from yesteryear.]

I spend a fair amount of time in the company of Muslims these days;  indeed, at present, by an accident of the seating-chart, I probably spend more time in close propinquity with Muslims, than with Christians, Hindus, Jainists, Jews, and Zoroastrians combined.   (Lotta LDS, though.  Plus all that could change with the next re-org, as my next podmates might be Zoroastrians.)   [Update March 2017:  And now, in fact, our branch chief is a Zoroastrian.] There is an effort of good-will on both sides;  my Sunni neighbor points eagerly to passages in the Koran, where good things are promised to ‘believers’ (mu’miniin) rather than specifically ‘Muslims’ (muslimiin).  A kind-hearted man, he hopes to be with me in Paradise, and not to gaze down on me roasting in Hell.  (`Uqbaalak, ya shaykh.)

Now, we Christians know implicitly, that the doctrine of the Trinity is no polytheism:

that Father, Son, and Holy Ghost, are not at all like Apollo, Zeus, and Hera, say, but more like le père Dupont, at once father, and Frenchman, and fireman.   But to explain this to our Muslim friends, is difficult.   I often have to fall back on saying:  the Trinity is a Christian mystery; it may be true or false (in some transcendental sense of those categories), but it is no descendent, direct or indirect, of the sort of pagan polytheism which stuffed the Kaaba with idols, and peopled the trees with dryads, and Olympus with squabbling gods.  For us as for you, God Himself -- Allah -- Yahweh -- e’en He -- is indeed One.

Yet this unity by no means necessitates or logically entails, that there could be no parts at variance within the Godhead, even to dissension.  (Of course, their absence might be a truth of the Church, and thus beyond dispute;  I am speaking here only of logic, the only subject in which I have been ordained.)  How indeed could we ever know otherwise (save by some enigmatic Revelation)?  After all:  We are made in His image, and we ourselves are a bundle of contradictions;  and He contains us, as a proper part.  (Additionally, that Eloi, eloi passage would seem to point in that direction.)

~

C. S. Lewis saw rightly when he compared the notion of the Trinity (purely as regards its intellectual coherence, rather than anything theological) with the ‘separate’ faces of a cube -- which latter is, however, nothing more nor less than the sum of all of them.  (Indeed, if you were to go with a sort of ‘projective cube’, with antipodal faces identified, our new Cube (topologically a three-torus) would even consist of exactly three parts.)

The historian of physics D’Abro offers a similar parable, along the lines of Abbot’s Flatland (A. D’Abro, The Rise of the New Physics (1939), vol. II, p. 653), his quarry being however, not the Trinity, but the (in some ways similar) “wave-particle duality”:  or, as we might term it, the wave-particle identity.  He imagines our various perceptions of something we believe to be one entity, but which sometimes seems a triangle, and at others, a circle:  eventually we realize that it is a cone, seen now this way, now that.  And, regarding the separateness/unity of electricity and magnetism:

The theory of relativity brought about the fusion of the two aspects, no longer by utilizing the background of 3-dimensional space, but by introducing the more refined background of 4-dimensional space-time.   The underlying entity, the partial aspects of which are electric and magnetic, were found to be the 4-dimensional electromagnetic tensor  situated in space-time.

-- A. D’Abro, The Rise of the New Physics (1939), vol. II, p. 653

The Trinity, we may confide, whatever in its unknowable essence it may be, is at the very least as complex as a  tensor.

~

Pascal (Pensées, 1670 [posthum]), has a very odd passage, asserting the alienating nature of God’s complexity -- or perhaps not -plexity, but monolithicness :

S’il y a un Dieu, il est infiniment incompréhensible, puisque, n’ayant ni partie ni bornes, il n’a nul rapport à nous.  Nous sommes donc incapables de connaître  ni ce qu’il est, ni s’il est.

Ni s'il est ! -- A useful first step towards an antidote  might be to drop that assertion about God's lacking any parts.

.

In Defense of “Pascal’s Wager” (expanded)

I have always particularly disliked Pascal’s Wager.   First, on logical grounds, since, like Anselm’s Ontological Argument, it can just as easily lead to the Great Pumpkin or the Great Penguin, as to God as conceived by Christians (of whatever stripe).  Second, on psycho-religious.   Romeo does not pitch upon Juliet by a process of totting up her good points, and subtracting her bad (suitably weighted  statistically or otherwise), comparing the result with the similar figure for the other wenches in Verona, meanwhile having his accountant double-check for accuracy, and then proceeding (with degree X of Bayesian confidence) to the balcony of the lucky winner.   Nor do such cold considerations yield any sense of the Christian feelings and beliefs about God.

The tart-tongued psychologist-philosopher William James  felt a similar aversion.  In his essay “The Will to Believe” (1896), he writes:

You probably feel that, when religious faith expresses itself thus, in the language of the gaming-table, it is put to its last trumps.  … If we were ourselves in the place of the Deity, we should probably take particular pleasure in cutting off believers of this pattern  from their infinite reward.

"Va-banque on black !"

Nevertheless, I am uncomfortably aware of having embraced something somewhat akin -- a leap of faith across a crevasse of uncertainty, borne on the wings of reason -- in subscribing to the Nicene Creed.

Credo ut intelligam

But just now, perusing the thoughtful and chatty treatise of a (non-religious) philosopher, this:

Very few people today are impressed by Pascal’s wagers.  But decision theorists have long been using the arguments he invented … Pascal’s logic is sound. The trouble is that his starting points  no longer apply for most of us.  They are no longer “live possibilities”.

-- Ian Hacking, An Introduction to Probability and Inductive Logic (2001), p. 123

Bingo!   That crack about the Great Pumpkin  misses the mark, because that portly vegetable only comes into perspective when seen from afar, from some philosopher’s distant star.   For anyone else (with the possible exception of Seneca the younger -- vide “The Pumpkinification of Claudius”), that rotund worthy is not a Live Possibility.

Our purpose here is not to argue for the intellectual respectability of adhering to the Christian faith (or to that of Islam), but to pursue the (psycho)logical point.   For, you,  respected and right-thinking reader, have almost certainly (if you are reading these words) made a similar bargain yourself:  in embracing the Religion of Science.

Not knocking science, nor denying its partly empirical character:  ever since my seventeenth year, my almost every effort has been bent in pursuit of scientific understanding.   But such is not easily to be had, even for those who majored in scientific subjects at top-flight colleges  and never slacked or looked back.   Most of what we accept, at least outside our own field of research, we largely accept on faith.  And indeed, even within that field -- QFT or String Theory, say -- there are doctrines and assumptions you just swallow  and then push forward, not really questioning these, trying to come up with something publishable that hews to the Narrative, at least until you get tenure.   "Allez en avant -- la foi vous viendra." (That celebrated epigram was actually not uttered by a prior or a bishop, but by d'Alembert, anent the mysteries of the calculus.)   To reject science sweepingly is intellectually disasterous;  but embracing it involves certain … wagers.

(For more on the subject, try this:  Veracity and Verifiability.)

Bonus:  For a relatively friendly look at Anselmian ontology, try this.
For a previously-unpublished tractate of the Saint:
http://worldofdrjustice.blogspot.com/2015/01/saint-anselms-proof-of-perfection-of.html

[Update May 2017]  Late-breaking developments on the logico-philosophical front:
http://worldofdrjustice.blogspot.com/2017/05/a-side-bet-for-pascals-wager.html

The Urysohn Metrization Theorem: a biochemical derivation

(I)  Various physicists, working with cesium-filled chambers and the like, have reported results that, at least as reported in the press, seem supraluminal.   Their technique is admirable; their results, at least as entertaining as that of the Euler disk; but they by no means dethrone the position of the speed of light in a vacuum, c, which still reigns at the center of physics.  Their surprising lab results may be compared with the paradoxical distinctions of group-velocity vs. phase velocity, in wave phenomena, or unphysical illusions such as in the thought-experiment of flipping a flashlight left to right-- millions of years later, on the arc of heaven as it were, an illumination mimics that trajectory with distances multiplied by a zillion, so that the 'signal' moves supraluminally, though nothing is actually moving left to right. A curiosity.  Einstein in heaven is not tearing his wild white hair.

(II) Various neuroscientists, sticking electrodes into hapless experimental subjects, detected electrical blips in the brain, as many as seven seconds before the fellow (responding to the prompting, “Coffee tea or milk?”) at last says, “Tea please.”

Neuroscientists -- an excitable lot -- froth over this.  “We feel we choose, but we don’t” enthuses one from the University College, London (whom I shall not name, so as not to defame; quotation from 1 Sept 2011 Nature).  Furthermore (adds one of his henchmen, in the same issue):

It’s possible that what are now correlations  could at some point become causal connections between brain mechanisms and behaviours.  If that were the case, then it would threaten free will, on any definition by any philosopher.

[“… any … any…” : sic; sic.]

The only reason such illation might have any color of probability, is that the neuroscientists (whose characters were formed in the laboratory, pithing frogs) regularly choose actions too trivial to reflect our real humanity.  In the case above -- I lied about the Stewardess Trilemma, really all the experiment was about was pushing some damned button, a button which moreover had no effect upon anything.  Boring beyond belief. No doubt the experimental subjects were in a coma throughout the experiment, or Dreaming of Babylon;  meanwhile the burblings of the subcortical tissues continued, much like the similar borborygmus of the bowels;  until eventually a finger twitched:  Ah what the heck, push now.

Not to knock it -- such results might even scale up.  Thus, to take the creature dear to the Churchlands, the sea-slug:  quite possibly its “choice” to slime-right or slime-left, at any given instant, owes its promptings largely to meteorological and other influences, with any accompanying mentation bearing but the faintest resemblance to the decision to enter the priesthood, or to see if cohomological algebra can yield insights into Yang-Mills theory.  But it doesn’t scale up to us -- at least, not to us at our best.  Prior, say, to coming up with the Urysohn Metrization conjecture, and then coming up with its proof, there was plenty of unconscious ruminating, and well more than seven seconds’ worth.  No mathematician denies it; all proclaim it.  (Poincaré stepping onto the bus.)   But that doesn’t mean that the truth of the Urysohn Metrization Theorem, or even the validity or coherence of our proofs thereof, are caused by, or bear any structural resemblance to, the sort of subcerebral singultus that so fascinates the guys in the lab.  While button-pushing may be close enough to a reflex action, that you could imagine it might just fall out of biochemistry, without invoking mind or will at all, that really doesn't work to explain such mental palaces as the UMT.

After all:  Free will, like the Cosmos itself, is a gift from our Maker, not just something we picked up at WalMart.  Hence  there is no a-priori reason why it should be intuitively understandable, or even analytically explicable (without millennia of hard work, and even then may lie beyond our poor powers), any more than is the Cosmos.

Ostracean Epigram

“If you’ve never had a religious experience, it’s folly to believe in God.
 You might as well believe in the excellence of oysters,

 when you can’t eat them without being sick.”

(Quotation from Aldous Huxley’s novel of 1928, Point Counter Point, p. 13.)

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 ~

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

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"

.

Scala naturæ (downward direction)

Jesus preached to the people;

then Francis, a man, catechized the birds of the air

(as Saint Maël, we read, preached to the penguins).

Shall a penguin, then,

take pulpit to the ocean-folk,

filling them with such dim visions as might fit,

proportional to their subaqueous understanding?