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.

.

Paradox (for Lent)

"Riddle this me, a thing that's three,
 and yet is only one."
"That is the blessed Trinity,
Father, Ghost, and Son."

"How then a man who dropped and died,
yet walks upon the mold?"
"That is the resurrected Christ,
of whom that tale is told."

"What then of him  who penned the prayer,
yet has forgotten faith?"
"Great God, that wretched man am I;
Help thou mine unbelief."


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[For a treatise on the Trinity by Murphy the penitent detective, click here.]

Lenten reflections

For some reason, I cannot offer these directly yet.
But you might check out the reflections of the penitent private detective:

Stripes

Wounds

Trinity

Orthoëpy of the Doxology

 

On the rare occasions when, as a child, I attended services -- with friends or relatives, and usually Lutheran -- they sang (and eventually I tried to sing with them), this simple song, a quatrain of tetrameters:

Praise God, from Whom all blessings flow;

Praise Him, all creatures here below;

Praise Him above, ye Heavenly Host;

Praise Father, Son, and Holy Ghost.

They called it “The Doxology”.  It was sung to a tune which, I later learned, is known familiarly as “the Old One Hundredth”.

Simply as written, it might almost be a quatrain from that other simple song (though that one is stress-timed, the syllable-count being variable):

Praise the Lord for the humble woodchuck,
praise the Lord for the mouse and mole,
praise the Lord for the backyard visits
that sustain the soul.

(That last line is a trimeter -- stress the “that”.)

But there is a subtlety, a depth, a twist to the Doxology, as we actually sang it.  After the straightforward naturalness of the first verse, an iambic tetrameter, the emphasis suddenly turns trochaic.   The way we sang it was:

PRAISE him ALL creaTURES here BE-low

The stress is now wrenchingly contrasemantic -- autonomous -- thrawn.

And likewise for the third verse -- though here, additionally, we slipped out of the familiar major key, to enter some mode more minor or indeed medieval:

PRAISE him A-bove YE hea-ven-LY host

before finally returning to the sunny uplands of iambic rhythm and major key.

[Note: The technical term for such displaced stress metri causâ  is "wrenched accent."]

~

For long, I did not know what this might mean.

Life took an ever more secular course, and I did not hear the song again until, in middle age, I at last was baptized, joining the Presbyterian church in Princeton.  There, we usually sang a sprightlier version,  with no sprung-rhythm, nor dark modes, and with some jolly “hallelujahs” thrown in.   But the old tones of the Old One Hundredth  still dwelt in memory.

The Doxology.   It was a numinous, a fearsome word, long and unEnglish and utterly unused in the circles in which I usually moved, and yet as normal and natural as “hot dog” or “baseball” to these believers.   It concretized the wall between the churched and the unchurched .  Occasionally, when staying with relatives, I would get a glimpse of what went on within those walls:  yet still it stood. between me without and them within.

And even within the everyday American Protestant context itself, it had a taste of strangeness.   The business of that thrawn stress in the middle verses  always gnawed at me, and left me alert for such things.  (I later wrote a paper called “Contra-semantic Sentence Stress”, taking off from the patter of stewardesses.)  I never analyzed the thing, vaguely assuming that, as does happen in verse, the choice of words was such as momentarily to throw the beat off-kilter -- many poets do this deftly, deliberately, just the right amount, to avoid any tedium or singsong quality.   But if you look at the verses as printed, you immediately see that is not so.  We could just as easily have sung them parallel to the first verse:

Praise HIM all CREAT-tures HERE be-LOW-hoh…..

But now here is the key -- and it is a key to Christianity as well, that faith that embraces paradox -- and parasemantics.   For, if we had sung it straight through in that simple way, refraining from any blue notes or celadon notes or burnt-umbrous notes as well, it truly would have been like the Humble Woodchuck -- a children’s song.  Nothing wrong with that.  A lot right with that.   But genuine Christianity is not by any means so simple.   It states that flat out, at the outset, with the startling, baffling, thrilling doctrine of the Trinity, which sets all reductivizing simplification at defiance.  And the doxology as we sang it, in that southern-Californian church, our Fords and Buicks shimmering out in the parking lot, to whisk us off afterwards for some post-worship hamburgers -- as we sang it, it gave us a taste of strangeness, a hint of darkness  absent from our workaday American lives, in that sunlit time before Kennedy was slain.

For see -- it does not bid just us to praise.  All creatures.  In particular and in primo:  the ducks, whose chorus of simple unstinting praise sails up to Heaven from every field and pond.   And even as we sing, our praise is paralleled, by a higher choir, unseen but singing too, aloud and full-throated, upon heights upon clouds upon Eternity and upon Time  -- World Without End.

~   Praise   Him   ~

~

A story that will haunt your dreams

Murphy and the Magic Pawnshop

 Also available for your Nook

~

[Appendix]

Other examples of effectively used contramorphological ictus:

“Is she dreaming?” -- “Yes -  I - thínk - so.”

“Is she pretty?” -- “Yes - ev-ér - so.”

 --  “Jennifer Juniper” (Donovan)
[Somehow, this last line wouldn't quite work with normal lexical stress.]

https://www.youtube.com/watch?v=M1cESSh-SqU

(And indeed the name itself, as pronounced in the refrain:  Jennifér Junipér.

 This seemingly arbitrary ictus is redeemed in the final, French-language verse,
 French being given to breath-group end-stress.)

~

A further prosodic wrinkle  is that the doxology is often performed while drawing out the final three or four syllables of each line:

Praise God, from Whom all blesss - sings - flooowww;

Praise Him, all creatures   heere -  beee - looowwww;

Praise Him above, ye Heaven - lyy -  Hooost;

Praise Father, Son, and  Ho - lyy - Ghooost.

Compare this Proustian observation:

Le marchand d’habits psalmodiait: « Habits, marchand d’habits,  ha… bits »,  avec la même pause entre les deux dernières syllabes d’habits  que s’il eût entonné en plain-chant: « Per omnia saecula saeculo… rum » …

-- Marcel Proust, La Prisonnière (1919; p. 118 of the Pléiade edition)

New Year’s Resolutions

Here’s what’s on-deck for 2012 !!!

(1)  To contemplate the blessed doctrine of the Trinity.

(2)  To, um, ….

(3) To ….    ?????

(x)  ???????????????????????????????????????????????????????????

~ ~ ~

My mind’s as yet too weak to compass this.  But the pen may yet mechanically record, such glimpses as I come across in the writings of others.

~

Father Schall, S.J., in The Order of Things (2007), calls the Trinity “the inner life of the Godhead.”

~

Roger Scruton, in whom the light of logic burns  brightly alive, draws an intriguing conclusion:

God, says the Christian, has three natures, and we come to him by three separate paths:  when we worship him as transcendental law-giver;  when we encounter him incarnate;  and when the Holy Spirit moves through us in its work of concord.  It follows that there are three modes of rebellion against God:  the repudiation of law;  the assault on the sanctity of the human person;  and the desecration of the work of the spirit.

Modern Philosophy (1994), p. 474

What I know for sure about the Trinity

[This space intentionally --  though reluctantly --  left blank …]

(For an alternate perspective, from a Catholic detective, click here.)

On Symbols

We spoke earlier of the relative adequacy, as a “symbol”, of the traditional OT picture of Jehovah,  to the uncountably infinite trans-reality of the Godhead Him-/It-/Blorg-/self.  This symbol is, literally, infinitely inadequate.  And yet adequate to our understanding, depending upon how high we have climbed on the ladder;  and crucially more adequate than certain alternatives, proposed by various paganisms and modern heresies.  Our purpose here is broadly  to defend the very idea of using symbols, and indeed the intellectual integrity of an admittedly threadbare image, simply by pointing out that such reduction to the palpable  is no mental vice unique to religion, but may be found even within -- nay, not science merely, to show that would be child's-play:  but even within that most abstract and impalpable exercise in mental extension, mathematics.

Consider, then, one of the simplest of mathematical objects:   the torus.  You have all seen this described as the “surface of a donut”.  Set aside the humbly sensuous coffee-dunking aspect of this:  the very essence of the image, be it of inner-tube or anchor-ring (the latter metaphor  was favored by our ancesters), is already a concession to our infirmity:  specifically, to our own creaturely incarnation in three spatial dimensions (this particular Sitz im Leben being of no mathematical significance whatsoever).  It is a symbol -- and quite an adequate one -- of the embedded torus:  but not (and this is crucial) of the torus as mathematicians now understand it, as conceived within itself:  but rather as embedded (with some bending) into the cramped space in which we dwell.   Beholding the donut, you think:  How curvaceous, surely more curvy than the surface of a tennis ball.  And yet, in its essence, it is not curved at all: Its intrinsic geometry is completely flat; and this “inner flatness” manifests itself  even in the embedded object, whose Euler characteristic turns out to be zero (as against 2 for the sphere).  (For a magisterial exposition of all this, see Jeffrey Weeks,  The Shape of Space.)   A more adequate symbol would be simply a rectangle with opposite edges identified -- or better yet, the toroidal covering-space which carpets R x  R with infinite replications of this patch-sample.  (Already how distantly we have left behind the donut !)   This symbol is still not the torus itself, as it thrones in Platonic heaven;  but it is a symbol which, unlike that of the donut, is wonderfully productive of new ideas.  As: Identify one pair of opposite edges as before, in parallel; but the other pair, anti-parallel, “with a twist”.  The result is a Klein bottle, not embeddable in three-space at all.  (Though four will suffice.)   Or, as:  Instead of parallel-identification of opposite edges of a square, perform an orientation-matching identification of opposite faces of a cube (thus resulting in a universe that is finite though Euclidean -- something they told me was impossible, when I was but a little boy in short-pants).   Or indeed (and now the usual geometrical torus of the nursery, seems as pale a reflection of the basic idea, as does that java-sodden donut of the former):  Consider any product (finite or infinite, countable or not) of R-mod-1 with itself -- a torus let loose from the leading-strings.
Similar such exercises result in surfaces we could never previously have imagined, as the very best scientists did not, but which, looked at correctly, become almost intuitive.  One of them might be the shape of the very space we live in.

It is, I suspect (speaking under correction), in part for such intellectual   reasons, that the Church Fathers, cognizant of the very partial truth of the God-the-Father Yahweh image, attempted to fill the picture out, with the doctrine of a Trinity (a blessed doctrine, it may be:  but arrived at, not handed to us on a platter in the Bible).  The symbol is not fruitful if you press it into fetishistic ends, snorfling up triads wherever they may be found; but it is an improvement on what went before.

Parsing “Trinitarian Minimalism”

            “Make everything as simple as possible,   but no simpler.”

                   --  attributed to Einstein

As we exulted earlier, your favorite hangout for Cantorian Realism  is also wearing the yellow jersey in the global sweepstakes for googling that soon-to-be-famous phrase,

~ “Trinitarian Minimalism” ~

Now, one other site does come up for this collocation.  It contains the following delightful passage:

If the hidden identity of divine being is revealed within the revelatory content

of the Gospel story, then perhaps there is yet scope for a via media between the Scylla of trinitarian minimalism on the one hand  and the Charybdis of trinitarian maximalism on the other. One possible solution is to unpack further the content of the trinitarian theologoumenon by means of exploring the person and work of the Spirit.

(Sidenote on theologoumenon.  It is not every day that Dr. J. learns a new word;  this one’s a doozer.  Merriam-Webster defines it thus: “a theological statement or concept in the area of individual opinion rather than of authoritative doctrine”. -- Actually, my friend Jennifer taught me another one earlier this morning, equally a keeper:  hooptie. -- It’s all good.)

Of linguistic interest here is the fact that this syntagm of adjective-plus-noun  is, semantically, actually parsed in opposite ways  in either case.

(1)  The way I’ve been using it, I start from the broad notion of minimalism (a traditional term of the arts), extend it suggestively to mathematics and the hard sciences (largely in the sense of parsimony, but with retention of the echoes of other uses of the term);  then, alarmed at the possibilities of ‘minimalist excess’ (to coin a phrase) -- Nominalism, solipsism, atheism, "The more the universe seems comprehensible, the more it seems pointless" --   I attempt, in the spirit of Einstein, to buttress it from below, and --  fancifully, metaphorically -- apply the modifier “Trinitarian”.

(2)  Our theologian, by contrast, starts from Trinitarianism, in its usual non-metaphorical sense;  then, seeking to distinguish among stances within this basic viewpoint, avails himself of some nice-sounding nouns lying ready to hand, minimalism and maximalism.   These do not have prior theological standing, and the resultant phrase (which, if I do say so myself, has quite a ring to it) is quite non-standard -- indeed, the page linked to above is the only such use in all of googlespace.

The meanings are thus quite distinct, though not actually at odds.

Now -- what adds a piquant semantic sauce to all this, is that the theological idea of the Trinity was not given to us explicitly in Scripture, but evolved over the centuries, in something like the scenario described in (1) !   It is presumably this fact to which our author alludes, when he applies to the doctrine the modest and cautious term theologoumenon.

The early Church began with monotheism, which it counterposed to the paganisms of the Greco-Roman world:  the which, rather than already-monotheistic Jewry, supplied the bulk of new converts.  But with time and contemplation, it became apparent that there was a danger (if you wish to call it that) that the whole enterprise might bleach out, in the direction pointed by Arianism, and exemplified today by Unitarianism:  a vague feel-good theism without bite. Revelation was sought, and the result is the Credo as we say it today.

Adventures in Minimalism

In my original post on the matter, in an essay of brilliant condensation, I said all that needs to be said on the fascinating topic of Minimalism.   However, it’s a rainy day, and there’s nothing good on TV.  So I figure I’ll while away an hour or so saying some more stuff about some other stuff.

~ ~ ~

Arthur Koestler, The Act of Creation (1964), p. 339, quotes Hemingway’s advice to budding writers:

     “The more bloody good stuff you cut out,
      the more bloody good your novel will be.”

~ ~ ~

The Iliad of Minimalism:  a mock-heroic epic

  We weigh our words

  in careful prose.

  Be one found wanting,

  away it g---.

[* An earlier version of this epic poem ending in “goes”.  But it quickly became apparent  that this was to insult the intelligence of the reader, since the word is implicit in the rhyme-scheme.]

~ ~ ~

Minimalism:  Mode d’emploi

WARNING  :   Minimalism can be deadly if misused.

To avoid the swamps of Buddhism, solipsism, atheism, and nihilism,

adhere strictly  -- as in a gale you would grab the mast -- to the blessed doctrine of the Trinity.

(Hmm…  Cantorian Realism…. Trinitarian Minimalism… I’m liking it … possibilities there … )

Credo

[Theologia Mathematica, ch. 2.   Continues this.]

        

There is a phrase within a clause within a prayer, of which I am particularly fond, and upon which I ponder ceaselessly:  “…the Father almighty, Maker of Heaven and Earth,

and of all things visible and invisible…”

… visibilium omnium et invisibilium.  Doubtless each line of the credo might profitably exfoliate into a tome,

http://en.wikipedia.org/wiki/Mass_%28music%29#III._Credo

http://www.youtube.com/watch?v=fjtxKpHSXzg&feature=related

but let us now, here, pause for a moment at this one.

           

            The phrase is no mere afterthought.  There are in fact a whale of a lot of invisibles out there, and it’s not just ghosts, or disembodied ectoplasm, or ethereal unstructured mush.  Nor do I mean “dark matter” or “dark energy”, though apparently there are gobs & gobs of that as well: more than of ordinary matter, which now rattles about in the cosmos like spare change.  (Or, to coin a phrase, like angry candy.) No, dark-whatever, doubtless all jolly stuff, and quaint in its way, but no more inherently fascinating than, say, rabbits.  It’s merely the library-paste and plasticine that happened to be lying about when God got around to making this particular universe on some particular day, possibly with leftovers from some earlier practice project; and now it hangs about, drifting moodily hither and yon, like so much unemployed blancmange.  Being invisible doesn’t make it significant or interesting.  Let us have no fetish about the invisible.  If for some reason the credo had said, “… and of all things probable and improbable,” and if the improbable had somehow been mostly ignored, yet contained most of what was of interest in the universe, then we’d be talking about the improbable; or the fantastical; or the ironical.  The entities  I shall be getting at here  are not significant because they’re invisible; I’d be even willing to concede that they’re significant despite being invisible, that visibility would be one further and delightful perfection, one which we may someday hope to glimpse.  In any event, what is meant here is the mathematical scaffolding, on which the sun and the moon and the quarks hang  like so much laundry.  That is, the plan of the thing, so much more permanent and pervasive than the things themselves.  I mean the symphonic score  from which our ephemeral melodies derive.

            Properly apprehended, it is a structure of – crystalline palaces, transparent and thus largely invisible to the untutored gaze, save as the light of insight  glances off them at an angle, and so catches the inner eye.  These ideal edifices are as hard and as chiseled and as real, as our own makeshift dungeons of stone: nay, more real, for these intricate perfections are the prototypes, whereof our own poor earthbound shantytown is but the fallen, partial, semi-crumbled, quasi-scrambled, half-forgotten misremembered afterimage.  They are, it is true, invisible: but in part (in increasing part) -- not unimaginable.  Through intense and lifelong study, we may – by luck, or grace, or mental sweat – eventually acquire a glimpse of their upper ramparts, from which turrets rise, from whence pennants flutter – flutter in a plenitude, an infinitude of dimensions, one upon the other like palace halls; so that our own most swirling ballet or crashing waves  are but as the slogging of an ant  trapped between the narrow glass walls of the ant-farm.

            These diaphanous entities, being (as we shall argue) a part of the Creation, display a different side of God, from what we customarily encounter.  Or rather, as it may be, many different sides: the mystery of the Trinity becomes the mystery of the Infinity.  For we must not think of “Math” as just some subject in school, or as a section at the bookstore, beyond “Gardening” and next to “Pets”.  For one thing, there is just so much of it,  acres of math like fields of wheat, with more unfolding with each passing day, and much which, when first met with, seems qualitatively, drastically diverse:  not like different species, say a wolf and a fox, but like different phyla – a microbe and a mastodon.  And even as science has discovered some of the commonalities between mastodons and microbes, in the process deepening our appreciation of each, so too does the steady, then accelerated, and finally springing advances of our collective understanding – as it might be, the Mathematical Overmind – deepen and widen and heighten and… beyonden  our sense of the unitary structure of All There Is.  The whole enterprise is so fantastic, with such unity-in-diversity (again, compare the Trinity) that whole new fields have evolved at a metalevel, just to keep tabs on it all:  Set Theory and Proof Theory, to police our reasonings, and Category Theory, to provide display cases for all the genera of the menagerie, in the museum of the mind. 

            At the bookstore-cum-giftshop in Hilbert’s celebrated Hotel, you will find aisles for:  History; Fiction (including Astrology and Economics); Physics ‘n’ Chemistry; Biology; Number Theory; Point-set Topology; Algebraic Topology; Algebraic K-Theory; Topological K-Theory; Real Analysis; Complex Analysis; The Riemann Hypothesis; Sheaf Theory; Topos Theory; The Poincaré Conjecture.;  and Miscellaneous (i.e., gardening, computers, self-help, sports, celebrities, stamp-collecting, and all the rest).  There is no separate section for Theology, since that overlaps all of them.

            Now, none of this is exactly new: it is paleo/retro-NeoPlatonism. A retread, you may say, and twice-refried.  Yet there is now much more concrete substance to the view, than was available to Plato or Plotinus. They might imagine the cube and the icosahedron, and Kepler might attempt to stuff the planetary orbits into such homely and visible (risible) boxes, but they never encountered a Riemann manifold – or rather, they did, because we live in one, but they couldn’t see it  so they couldn’t imagine it, any more than they imagined fibre bundles or E8. 

            The Atheist, viewing a world charged with the grandeur of God -- which is difficult to ignore, since it will flame out, like shining from (to coin a figure) shook foil -- sniffs and dismisses it as tinsel: seeing the shook foil but not the Shaker.  So too the Nominalist, beholding or rather failing to behold  the serried ranks of theorems rising like seraphim beyond sight, regards these as a mere medley of contingent things, simply frothed out of someone’s brain, and which might, like a limerick or a pop-tune, have frothed out into something quite different.  (This is if anything the more charitable of contemporary dismissals, vice the dismissal of math and science as being merely the Eurocentric patriarchal dogma of the racist sexist agist ablist ruling class…)

*

            The nominalist viewpoint was given epigrammatic utterance (1886) by Kronecker, thus:

“Die ganzen Zahlen hat der liebe Gott gemacht; alles andere ist Menschenwerk.”

(“Ganzen Zahlen” means integers, but he may have meant only the non-negative integers, or “natural numbers”.)

Kronecker, rueing the day that ever he denied the transcendence of higher mathematics

(Randbemerkung:  A delightful swipe at Kronecker, well below the belt, and delivered with punch by a pugnacious Platonist, can be savored here:  The Continuum. )

Stephen Kleene translates, “God made the integers, all the rest is the work of man”, and glosses:

We cannot expect that the cognizance of the natural number sequence  can be reduced to that of anything essentially more primitive than itself.

(Introduction to Metamathematics, p. 19.)

The natural numbers – non-negative, non-zero whole numbers -- were, indeed, the only numbers recognized by the Pythagoreans.  One reads somewhere of a Pythagorean  casting himself into the sea in despair, upon encountering the proof of the non-rationality of the square root of 2; in fact, such self-drowning might just as well have been prompted by the sight of half an apple, since once you accept fractions, you are heading straight for E8.

            For in admitting the integers as being in no wise contingent – the work, indeed, of the Necessary Being – while desiring to dismiss the rest (“Menschenwerk” sounds even more like a kindergartner’s art project than the more Biblical “the work of man”), Kronecker has left the castle of mathematical agnosticism unguarded, by leaving open its postern gate.  Suppose we were – setting aside centuries of other riches – to begin by restricting ourselves to the laws of the natural numbers.  We would notice (as Euclid noticed) the primes, and require, for their adequate handling, great heaps of Number Theory.  Now, this already is no small thing.  You could fill a succession of lifetimes with nothing but Number Theory.  New discoveries emerge daily – some of them with such grave implications that they are actually classified, and at compartmented levels well beyond Top Secret, which does well enough for the design of an airplane or the movement of troops.  But the wealth is not just quantitative.  For to adequately handle nothing more than the primes, you need all of Number Theory, including even Analytic Number Theory, which brings in the continuum and the charmingly designated “imaginary” numbers (now much easier to imagine, though they remain as invisible as the number “23”), including indeed the Riemann Hypothesis, which already brings us to the frontier of knowledge with its outstanding unsolved problems.  Soon the whole of mathematics would come tumbling in through the unguarded door. 

[Note:  It’s never that simple.  I am aware that Kronecker himself was even more nominalistic that the famous quotation might suggest, as he did not accept the integers as a finished totality.  He went to some effort to derive results in a way that makes no use of such a totality.  Sort of neat if you can pull it off, like building a castle entirely out of toothpicks.  But if the idea of an actual infinity is problematic, that of the integers somehow running out of breath is even moreso.  I shall accept the natural numbers as given, and shall, for polemical purposes,  portray, as our foil, a sort of idealized Kronecker (who may even now be repenting of his nominalism, in some warm place) – the Kronecker of the quote – as accepting them as well.]

[As to the actual forked-radish of that name,  Joseph Dauben remarks (Georg Cantor, p. 66):

No-one could have been more opposed to Cantor’s ideas, nor have done more to damage his early career, than Leopold Kronecker.

Really, were it not for his key concession that the natural numbers are God-given, Kronecker might well be the villain of the piece.  As George Szpiro wrote of him (in Poincare’s Prize):  “Kronecker would not accept anything if it had not been invented by himself.”   Actually that can’t be quite accurate, since on that account he would not accept the integers… But anyhow, a perfect summary of the solipsist/nominalist epistemology.]




[The essay continues here.]

Topological Theology

A friend wrote in:

>Don't Muslims believe in the same "god" that Christians and Jews do?  

Sober orthodoxy of each camp would doubtless concede that the others “believe in” the “same” God (whatever it means to “believe in” something ultimately unknowable  this side the grave, and whatever “same” means  when comparing finitely described infinites); adding simply, that they worship Him amiss.

In the matter of the actual ontology of the Godhead, only Christianity is vulnerable here, owing to the doctrine of the Trinity.  Those who have not plumbed the subtlety of this doctrine  may be forgiven for imagining we worship three of Him.  And indeed, when jihadists fulminate against “polytheists”, they mean us, and on this account.

As for Islam, it really wreaths God in no new doctrines.  G.K. Chesterton quipped that Muslims were really Christians who had clung to the Arian heresy.

>And furthermore, doesn't the Jewish faith also teach that

>Jesus was a prophet instead of the son of God?

Jews do not recognize Jesus as a prophet; for his reception by the Sanhedrin, see Matthew 26:65-66.

Muslims do. You will see his name followed by “PBUH” (Peace Be Upon Him), just like that of Muhammad. Muslims even uphold the doctrine of the virgin birth (though not of divine paternity).  Jesus’ standing is quite high – Jesus and Muhammad are supposed to hold some sort of joint prayer breakfast at the Masjid al-Aqsa, at the end of time.

> If so, why is it then so critical to reject Islam and okay to tolerate Jews?

What’s going on  is that these religions have been constructed to be “backwards compatible”. Christians embrace the Old Testament, and add to it the New.  Muslims in principle recognize both OT and NT (though actual practice is gingerly), calling Jews and Christians “people of the Book”; and add to these, the Qur’an.  In this respect, Islam is in the same position “topologically” as the Church of the Latter-Day Saints, for they too embrace OT and NT but say: Behold, new Scripture.

In other words, the nodes of the Judeo-Christian family tree may be defined by the most recently acknowledged holy book in each case.  Graphically:

                                     OT [Judaism]

                                      |

                                      |

                                     NT [Christianity]

                                      |

                                    /  \

            Koran[Islam]        Book of Mormon[LDS]

[Lord knows what this’ll look like by the time it’s got through the mailer; it’s supposed to be shaped like an inverted “Y”.]

           

In this unmetrical-geometric perspective, Muslims & Mormons have equal claim (should they wish to make it) to be some sort of post-Christian Christians or Reformed Christians or Turbo Christians, just as Christians preserved-while-transforming [Hegelians have a word for this: “sublating” (aufheben)] much of the richness of Judaism, but defined themselves by their new scriptural contribution.  They (the M&M’s) even coincide, oddly, in having historically embraced something that utterly sticks in a Christian craw:  polygamy.

So much for the (theo-, topo-)logical view.  Culturally/metrically, the reality is of course quite different.  Most Americans (forgetful of Galatians 1:8) probably think of Mormons as being some sort of Christians, “you know, like the Jehovah’s Witnesses, they come to your door”.   Whereas Muslims are seen as decidedly Other – to the extent that, our correspondent suggests, many Americans don’t realize that “God” and “Allah”, like “Dios” and “Dieu”, are synonyms.

*

[Postscript: The New Yorker 22 Aug 2005 reported Billy Graham’s son Franklin  as having declared that Islam was “wicked, violent, and not of the same God.”]

For the views of our colleague Murphy on all this, see
Murphy on the “Allah”/”God” question
(in the "from the Cracker Barrel" feature, in the gutter along the right).

[Update 3 November 2014]  Murphy wrote that “Allah/God” riff, by way of encouraging Christian tolerance towards Muslims.  But these days, the bulk of the intolerance is on the other side, even with respect to this linguistic issue:

KOTA KINABALU, Malaysia — As the students knelt in a circle at a Christian kindergarten near the shores of the South China Sea, a 6-year-old girl in pigtails read out a chapter from a children’s Bible: “Sepuluh hukum dari Allah” – God’s Ten Commandments.

Technically, she broke the law.

According to a series of government orders and rulings by Malaysia’s Islamic councils, the word for God in the Malay language – “Allah” – is reserved for Muslims. Malay-language bibles are banned everywhere except inside churches. State regulations ban a list of words, including Allah, in any non-Muslim context.

Malaysia, with its collage of ethnic groups and religions, has a long history of tensions over issues ranging from dietary differences to the economic preferences enshrined in Malaysian law for the Malay Muslim majority.

Zainah Anwar, the founder of Sisters in Islam, a women’s rights group, describes a “headlong descent into a puritanical, extremist, intolerant brand of Islam in this country.”

“Malaysia’s moderate Islam is only touted for Western consumption.”

http://www.nytimes.com/2014/11/04/world/asia/in-malaysia-allah-is-reserved-for-muslims-only.html?_r=0