The Realist Vernacular

            What follows is neither proof nor argument, nor philosophy of any sort, but rather an exercise in sociolinguistics.  [And as such, a sort of sociological preparation for the thread announced here.]  The point is simply to exhibit a common way of talking among contemporary mathematicians, as well as some physicists and philosophers -- an easy style of conversation  that you would never imagine exists, if most of your acquaintance with science and its philosophy is mediated by figures like Daniel Dennett and Richard Dawkins.

            To cite evidence of theistic talk from the learned men of history, from antiquity through the nineteenth century, would be pointless, since the mode was well-nigh universal, at all times and in all realms.  Yet in our present day, most of the habitués of faculty clubs and coffee-houses  have managed to satisfy themselves, that all those who ever lived, in history, from Pythagorus  to the Einstein of “Der Herrgott würfelt nicht”, were, without exception,  imbeciles, and simply didn’t know what they were talking about, when they talked that way.  At last mankind has seen the light (or the darkness, rather);  we speak only of what is sensible and sniffable, like Donald Trump.  And who should be so rash as to cite the testimony of a mere Plato, or Galileo, or Cantor, or Gödel, against the brass certitude of so eminent a scholar as Sir Christopher Hitchens, Ph.D?

            Therefore I shall limit myself to recent citations.  A few examples from among many, snatched at random from  recent reading.

            Again, note:   I am not implying anything about the theology, per se, of the people here quoted, let alone suggesting that they never miss Mass.  Indeed a Dennett -- a roaring atheist -- can still indulge in such turns of phrases as "we can take advantage of the God's-eye perspective we have temporarily adopted" (the Devil can quote Scripture to his purpose).   So far, this is just corpus linguistics;  but more anon.

(1) Theistic language

 (a) mathematics

Arthur Koestler, The Act of Creation (1964):

Karl Friedrich Gauss described how he finally proved a theorem on which he had worked unsuccessfully for four years:  “At last, two days ago, I succeeded, not by dint of painful effort, but so to speak  by the grace of God.”

Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 219:

In one of the most cited discussions in his much-quoted book, Kuhn [1962] talks of scientific decision in terms of “conversion experience” and “faith”.

Richard de Millo et al., “Social Processes and Proofs”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 271:

The classical view does not require that an ordinary proof be accompanied by its formal counterpart; on the contrary, there are mathematically sound reasons for allowing the gods to formalize most of our arguments.

Similar language is often used in formulating the contemporary concept of a “hypertask”.  Cf. likewise

Shaughan Lavine, Understanding the Infinite (1994), p. 55:

            For Cantor … “countable” meant countable by God.

and similarly

Michael Potter, Set Theory and its Philosophy (2004), p. 250, re the plausibility of the Axiom of Choice:

… generalizing to the uncountable case  by appeal to the idea than an ideal being could achieve the choices required of him (or perhaps Him).

If we adopt some particular postulate system for abstract set theory, and agree that the criterion for accepting an intuitive set-theoretic argument  is that its analogue can be justified by the postuates, then we are  in efect  agreeing that the set of all intuitive sets  endowed with the membership relation  is a configuration satisfying the postulates.  We can have at best  intuitive reasons for believing this.  It is really an act of faith.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 153

Gregory Chaitin, “Gödel’s Theorem and Information”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 306:

If God tells one how many different programs of size less than N halt, this can be expressed as an N-bit base-two numeral, and from it  one could eventually deduce  which of these programs half  and which do not.  An alternative divine revelation would be knowing that program of size less than N which takes longest to halt.

Robin Wilson, Four Colors Suffice (2002), p. 214, recounting a mathematician’s reaction to the immensely long and repetitive, unsurveyable, computer-aided proof of the Four Color Conjecture:

God wouldn’t let the theorem be proved by a method as terrible as that!

 (b) physics

Hermann Weyl, Symmetry (1952):

Contingency is an essential feature of the world.   Clarke  in his controversy with Leibniz  admitted the latter’s principle of sufficient reason, but added that the sufficient reason often lies in the mere will of God.  I think, here Leibniz the rationalist is definitely wrong, and Clarke [the theist] on the right track.  But it would have been more sincere to deny the principle of sufficient reason altogether, instead of making God responsible for all that is unreason in the world.

Abdus Salam (1988):

God created just two dimensions -- one of space and one of time. … At a later epoch, there was a phase transition to four dimensions, plus six internal ones.

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 91, alludes to Roger Penrose’s paraphrase of cosmic censorship:  “God abhors a naked singularity.”   The (jocularly) theistic language is especially odd and noteworthy, since the epigram here being echoed -- “Nature abhors a vacuum” (Latin:  horror vacui) does not use it.

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 126f.:

These laws [of physics] may have originally been decreed by God, but it appears that he has since left the universe to evolve according to them  and does not now intervene…

That, then (for that author) leaves only the explanation of the settings of the parameters for initial conditions:

One possible answer is to say that God chose the initial configuration of the universe for reasons that we cannot hope to understand.  This would certainly have been within the power of an omnipotent being, but if he had started it off in such an incomprehensible way, why did he choose to let it evolve according to laws that we could understand?

Simon Blackburn, Think (1999), p. 69, in a section called “A Scientific Model”:

Classical physics identifies the temperature of a gas with the mean kinetic energies of the molecules that compose it.  So in making hot gases, God has only one thing to fix…

Blackburn is a philosopher, not a physicist; as commonly in that community, he is writing in an “as if” mode;  but still, the choice of words is noteworthy.

John Gribbin & Martin Rees, quoted in Edward Harrison, Cosmology (2^nd edn. 2000), p. 489:  Absent the strong anthropic principle,

If there is a unique ‘theory of everything’, then … we would have to accept it as genuinely coincidental, or even providential, that the constants determined by high-energy physics happen to lie in the narrowly restricted range that allows complexity and consciousness to evolve…

(“Providential”… a lovely word…)

Paul Davies, The Goldilocks Enigma (2006), p. 3:

It appeared to Hoyle as if a superintellect had been ‘monkeying’ with the laws of physics. … Like the porridge in the tale of Goldilocks … the universe seems to be ‘just right’ for life.

Paul Davies, The Goldilocks Enigma (2006), p. 236:

Most theoretical physicists are Platonists in the way they conceptualize the laws of physics as precise mathematical relationships  possessing a real, independent existence…

Shing-Tung Yau, The Shape of Inner Space (2010), p. 101:

As Robert Greene puts it, “you’re trying to find the one metric given you by God.”

Michael Atiyah re string theory, quoted in Shing-Tung Yau, The Shape of Inner Space (2010), p. 292:

They’re onto something, obviously.  Whether that something is what God’s created for the universe  remains to be seen.  But if He didn’t do it for the universe, it must have been for something.

This statement is reminiscent of the Principle of Plenitude; and recalls Einstein’s celebrated quip, anent the possible failure of an experiment to confirm his prediction: "Da täte mir halt der liebe Gott leid; die Theorie stimmt doch."

Less seriously, but using a religious metaphor: J. L. Synge, Relativity:  The General Theory (1960), p.  ix:

It is to support Minkowski’s way of looking at relativity that I find myself pursuing the hard path of a missionary.

More serious is the following.  The author is discussing the vexed question of the Collapse of the Wave-Packet upon observation (related to:  If a tree falls in a forest, and no-one’s around, does it make a sound?  -- here rewritten in Oxford terms, replacing the forest by a quad), and quotes the old limerick:

Dear Sir, Your astonishment’s odd;

I am always about in the Quad.

And that’s why the tree

Will continue to be,

Since observed by Yours faithfully, God.

He comments (J. C. Polkinghorne, The Quantum World (1984), p. 67):

Divine reduction of wavepackets would be an overkill, since it would operate everywhere and always, forcing the electron each time to go through a definite slit.  The point about measurement is that it only occurs spasmodically.

Note the predicted empirical consequences of God in the Quad!   Then, between square brackets, the author adds:

This observation is in accord with the classic theological understanding of creation, which sees God as the ground and support of all that is (in our terms, the guarantor of the Schrödinger equation), but not as an object among objects (no collapser of wavepackets).

This aside is in fact central:  by the time he wrote this, Polkinghorne had quit his endowed Chair in physics to become a village vicar.

(c ) analytical philosophy

Michael Dummett, Truth and other enigmas (1978), p. 15, discussing character as (it may be, untested) hidden propensities:

If B still wishes to maintain the necessity of ‘Either Jones was brave or he was not’, he will have to old  either that there must be some fact of the sort to which we usually appeal … or else that there is some fact of an extraordinary kind, perhaps known only to God.

Dummett’s use of the term, to characterise the viewpoint of a hypothetical philosopher inclined to Realism, is so to speak opaque, not representing his own view, which is rather (id, p. 150) that “only a philosophically quite naïve person would adopt  realist view of statements about character”.  Only such, or Saint Peter.

(2) Realist language

Arthur Koestler, The Act of Creation (1964):

Gauss is reported to have said: “I have had my solutions for a long time, but I do not yet know how I am to arrive at them.”

Klaus Jänich,  Topology (1980; Eng. trans. 1984), p. 35:

When topological groups are found in nature [emphasis added], they  are generally not given abstractly as a set G with a composition law and a topology, but concretely, as a group of transformations…

This is not really anymore outrageously realist than saying “When a set of six objects is found in nature…”   (say, the familiar six-pack; although the one at my side is already down to five, and it’s not even noon).

And again, p. 157:

Covering spaces very often “occur in nature”:  that is, one comes across them spontaneously, while studying entirely different problems.

Shaughan Lavine, Understanding the Infinite (1994), p. 160: 

We seem to have nontrivial intuitions concerning the infinite, going far beyond simple things like Extensionality, Pairing, or even Power Set.

Shing-Tung Yau, The Shape of Inner Space (2010), p. ix:

The strength of this discipline [i.e., mathematics] lies not simply in its ability to explain physical reality…, because to a mathematician, mathematics is reality.

If all you know of math is elementary arithmetic, this statement may lack punch.   But in the upper reaches of set theory, topology and so on, it embraces a world of miracles and of monsters.

~ ~ ~

It is important to note, that this is the way Realists talk en famille.  The talk is casual, often not literal, yet is meant in some serious sense.  It is not to be compared with the tawdry pseudo-theological, pseudo-mystical sort of gibberish that popular authors (and even some serious ones, yielding no doubt to the Satanic promptings of the marketing department) use to gin up their pap for the masses -- like “God particle” for the freaking Higgs boson.

It may be objected (it will be objected;  it has been objected) that such expressions, in a modern mouth, are a mere façon de parler.  To which we reply (with Whorf), that a façon de parler tends to cohere with a façon de penser. 

Nor is it a refutation of the point here made, to adduce agnostic pseudepigrapha from any of the gentlemen here quoted.   We are each a walking contradiction;  we contain multitudes.   C.S. Lewis himself  confessed that he tended to be a cranky agnostic at dawn, but a theist later, when he’d had tea and was more himself.

Once again:   The point here is not to assert that so-and-so among our near contemporaries  is or is not a believer.  Indeed it will strengthen my eventual case, if many of them are not in fact believers, yet find themselves attracted, or guided, or driven, to Realist or Theistic language, whether for convenience, or (in the case of Cantor, Gödel, and Einstein) something deeper.

So, a very modest initial move, a sort of pawn to king’s four.  (Only later -- much later -- shall we see if we can capture Satan’s Queen.)   So far we hold simply, that occasional use of Realist or Theist language, in serious discourse, is not  in and of itself  diagnostic for Trisomy 21.

Adventures in Algebraic Geometry

The closest I ever came -- and that  unwittingly -- to a brush with algebraic geometry,   was in freshman calculus.  (This was back in the ‘sixties -- before your time.)  The instructor was Robin Hartshorne, a young and winsome elf of a man.  He was an engaging lecturer, teaching from  or at least in parallel to  a beguiling text (Spivak’s Calculus, so utterly different in spirit from the dry Thomas treatise that had repelled me in high school to the point of dropping the class);  moreover he was -- now that I think back on it -- the first deeply intelligent teacher I’d ever had (though there were to be others -- notably Gleason):  up through high school, there had been no hint that such creatures even existed.

Still, he wore his learning lightly, like his tweeds.  The class was fun.  Particularly endearing -- though also startling, at the time -- was one day in the second semester, when he was presenting the topic of definite integrals -- painful but necessary, rather like a rectal exam.   He was chalking away, when all at once he seized up, staring in bemusement at the blackboard;  then turned to us with a sheepish grin.

“It’s been a long time since I’ve done one of these,” he said.

~

Now -- if the anecdote stopped there, it would be just one more ultimately stupid instance of Genius Porn :   the populace cooing contentedly when told that Einstein flunked grade-school math,  or that Gauss was late to learn to speak, or that Erdös tried to cut a grapefruit with a butter-knife  (which indeed he did, though it was a craftily calculated move).   These falsely flatter our vanity.  They are the opposite of a much better genre of joke, which you need a bit of math to actually understand, such as the one about von Neumann and the summation of infinite series.

For the incident, trivial until viewed in the light of later developments, did  there and then  plant the seed of doubt and wonder, as I sat theretofore clueless in the second row.   His being momentarily at a loss  struck me  at the time  as quite surprising, almost inexplicable:  as though a test-pilot, stepping from his aircraft into his roadster, were to stare at the ignition and say, “Remind me how to start one of these.”

Had I continued with a chemistry major as originally planned (well, originally-originally an English major, until I realized, with chill horror, the error of my ways), and thus retreated or perhaps advanced  depending on how you look at it, into ever-more-technical intricacies  and mechanical practicalities, the significance of that incident would never have become apparent.  But as it was, Hartshorne’s class (and Spivak’s sparkle) were instrumental in turning me towards a concentration in pure mathematics -- which was the only kind of mathematics they really taught at Harvard (golden memories of that climate of abstraction here), and eventually towards having a go at Berkeley towards a Ph.D.   Accordingly I was to be introduced to as-yet-unsuspected levels of intellectual depth, such that each, compared with the one before (I speak loosely;  they are basically incomparable), is as the definite integral to 2 + 2:  rising like the serried ranks of angels.  And though I myself never progressed beyond the level of the cherubim, it was sufficient to glimpse that empyrean wherein, indeed, you might forget the particular monkey-tricks used to solve thorny individual definite integrals (basically you just memorize these, storing them for reference like tools in a toolkit, unless you’re Euler or von Neumann, in which case you re-derive them instantly from scratch, or simply perform a brute-force numerical calculation in your head).

As each glowing level is added, the one below  becomes obsolete ...

Moreover, the tired old cart-horse of the calculus was, it turns out, very far from the centers of Hartshorne’s research interests, which are almost unimaginably abstract.   In that pokey little classroom in Massachusetts, he was really only on loan to us from Sagittarius, having once studied with Grothendieck, a confirmed extraterrestrial.  It was bruited about that he had something to do with something called projective geometry;  but only much later was I to learn that he is one of the pioneers of …

sheaf theory

… a topic so ferociously abstract, that even to define what sheaves are  is utterly beyond me.  (Wikipedia doesn’t even try, observing that “their correct definition is rather technical”.)  Nay, wert thou to gaze upon this theory naked -- not even to speak, not even to breathe of its further generalizations in topos theory -- ‘twould make thine eyes, like stars, to start from their spheres, and thine each particular hair -- nay more, thou wouldst  in sooth  explode in flames,  as did dame Semele,  when  all unheeding she beheld,   unveiled,  great Zeus in all his lightning !
 

~

~  Posthumous Endorsement ~

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

this is what I'd be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(Je m'appelle Evariste Galois, and I approved this message.)

~         ~

~

~

As so often when some movement of math has been seen streaking off westwards  out into the void, presumably never to been seen again by mortal man, it reappears shining in the east, reborn in some applicable form -- thus suggesting, you will notice, that the global topology of the noösphere is toroidal.  In the present case, algebraic topology has come to be crucial in such applications as: string theory (via its prior discovery of Calabi-Yau manifolds), coding theory,  cryptography and steganography -- which means that the juicy bits are probably highly classified.

And this raises the interesting paradox, which Epimenides would have relished, whether someone like Hartshorne is Cleared for the contents of his own head.

Robin Hartshorne as seen in a recent spectral image.  Since the old days, he seems to have sprouted quite a bundle of fibres over his base-space.

 

~

I recently happened upon an essay by the usually abstruse Samuel Eilenberg (of Eilenberg-and-Steenrod notoriety), written for a collection sponsored by the Office of Naval Research.  In deference, perhaps, to the needs of our seamen, Professor Eilenberg permits himself some observations and analogies   that lie within the reach of the common folk.  Thus:

The analogy between sheaves and covering spaces  is very close.

-- Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 110

Bingo !  Covering-spaces I get.  Of course, I don’t actually see the analogy, but at least it’s a comfort, knowing that there is one.

[Update, Thanksgiving 2014]   Edward Frenkel, noticing my perplexity, kindly sent in this explanation:

Coverings and sheaves are related. And it's not just an analogy. A covering space is an example of a sheaf (the simplest example): It is a sheaf of finite sets (provided that the covering is finite).

Namely, given a covering p: C --> X of a manifold X, and given an open subset U of X, the set of sections of of the corresponding sheaf over U is just p^{-1}(U) [the preimage of U in C under p]. Note that the stalk of this sheaf over a point x of X is just the fiber over x [the set of points in C, which project down onto x under p; p^{-1}(x)].

The covering gives us a way to "glue" these fibers together (indeed, set-theoretically, C is the union of these fibers -- but it's more than that, because C is a manifold, just like X; so C is not a "disjoint" union of these fibers, they are really "glued" together in a particular way).

The simplest covering space is the trivial one: a union of N copies of X, each mapping identically to X under p.

Here is a non-trivial example: let X be a circle. Now take the Moebius strip in which this circle is the circle "in the middle." Take the "edge" of the strip -- this will be your C. Notice that for each point in your original circle X, there are two points in C. But C is NOT the union of two circles (which would be the trivial double covering). In fact, C is just ONE circle, covering another circle (our X) in a non-trivial fashion.

A general sheaf is very similar. The difference is that the fibers could be infinite, or they could be vector spaces, etc. But the idea is the same.

~

We earlier discussed  the memoir Souvenirs d’Apprentissage, by a pioneer of algebraic geometry, André Weil.   Avid for more, we got hold of a copy of the memoir Random Curves (2008), by a contemporary algebraic geometer,  Neal Koblitz, well known to anyone with an interest in Elliptic Curve Cryptography. (It came in via InterLibrary Loan -- interestingly, the lending institution turned out to be the U.S. Naval Academy in Annapolis.    Compare the publishing venue of the Eilenberg article referenced immediately above.  We salute the broad interests of our midshipmen!)

Both authors have a wide range of interests and experience outside of mathematics;  both engaged in extensive foreign travel;  and it is of this that they principally write:  the reader will enjoy these accounts for their own sake, but we set down the volumes with a twinge of disappointment, that we are no closer to insight about algebraic geometry than we were before.  However, one biographical detail did strikingly stand out.  Both authors took a principled stand against unjust wars:  and this, not simply by penning valiant Letters to the Editor from the safety of their studies, but by a brave and almost reckless defiance while actually serving in the armies of their respective countries:  actions that could easily have led to their injury or even death, and which did actually lead to their imprisonment (and, in Koblitz’s case, to a severe beating).  But what is truly remarkable is that, in both cases, they used their time in the slammer far more profitably, mathematically, than most of us  use ours  even in the best of circumstances.   It was there that Weil did his seminal work, and there that Koblitz returned in concentrated form to the practice of algebra which he had largely abandoned during two years of political turmoil.

Intrigued, I wrote to the latter author, inquiring whether, from the standpoint of Kolmogorov-style measure-theoretical probability theory, we may validly generalize from this sample of two (2);  and he was kind enough to reply:

I love generalizations based on small samples.  I'm sure a lot of algebraic geometry was nursed at the Indiantown Gap army stockade!

Thus encouraged, I here make bold to speculate about the martial philosophy of Robin Hartshorne.  I know nothing of his politics, but truly cannot imagine the man wielding an M-16.   Or a flyswatter, for that matter.   Were a mayfly to venture into his office, Hartshorne would no doubt observe the pattern of its flight (musing all the while on the brevity of this earthly life) and calculate whether that trajectory describes an elliptic curve. -- Which, come to think of it, it just well might.   Many mathematical treasures remain to be unearthed in the field of biology!  (For a few of these, see the fine book by Ian Stewart, Life’s Other Secret.)

~

A rather huffy response to modern algebraic geometry, which it is a pleasure to reproduce  mainly because I do not understand the subject, and which suggests (like the characterization of Category Theory back when I was in college, as “the higher macramé”) that (as with these new-fangled things called “computers”) I am perhaps not missing much:

Attempts to extend the geometry of second-order surfaces  and the algebra of quadratic forms  to objects of higher degrees  quickly leads to  the detritus of algebraic geometry, with its discouraging hierarchy of complicated degeneracies, and answers that can be computed only theoretically.

-- Vladimir I. Arnold, Lectures on Partial Differential Equations (Russian edition 1997; English translation 2004), Preface.

Harumph!  Hear hear!

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.