Scenes from Office Life

Today -- March 14 -- our office celebrated International Pi Day(***), by bringing in a whole bunch of pies.  (March 14 --  3   1   4 -- get it?   Not funny, just fun.)  Apple, cherry, banana cream, pecan -- you name it.   And amazingly, for every single one of those pies, the ratio of its circumference to its diameter was -- you’ll never guess -- more or less 3.14 !!   Thus proving the theorem.

(***)   Actually Interplanetary Pi Day, since the value of that transcendental constant  is invariant throughout the universe.

[For further glimpses of our office life, click here.

For more mathy funnies,  here.]

Existence and Use

 Now let us return yet again, with our crude wooden mallet, and hammer anew on that empirical numerical square peg  which we are attempting to pound into a (so we hold) round hole, as round as a cow – the Platonic realm -- or as it might be, to embed it smoothly into Euclidean n-space.

I am not saying that numbers are ideas.  Or if they are that, they are that in addition, where they keep such scruffy company as the Idea of a Unicorn and the Idea of a Ham Sandwich. They are, rather, facts as hard as baseballs, packing quite as much wallop. (Also, admittedly, we are interested in the Pitcher …)

In pointing out the “reality” of numbers, we’re really doing much what we do when demonstrating the reality of a hammer, without venturing so much as a murmur as to what either of these things are “in themselves”.  Philosophy pretty much gave up on the ding an sich a long time ago, and the new quantum perspectives makes even ordinary objects seem more remote than ever.  The reality of a hammer, as against that of my imaginary rabbit friend,  consists in its experiential predictability: I feel its weight, you do too, we both agree it’s heavier than a walnut; it looks a certain way from this angle, another way from that, and these match up in a natural way, both visually and tactually, and so forth.  (Whereas you, with your confounded skepticism, scare away my imaginary rabbit-friend, and thus can never experience his charming and comforting presence.) The fact that we can also state that the hammer is right there, as opposed to another place, is nice: it’s one more fact about the hammer, but it’s not the essence of being real. In the case of a quantum particle, we cannot state that it is precisely there, though it’s somewhere around here for sure; but it does other things predictably enough  that we count it as part of the furniture of the universe, and not just some fluke.  It is likewise difficult to say “where” the Big Bang was when it popped, nor “where” our universe is now. The latter questions might make sense and they might not; we might be comfortably embedded in a multiverse (itself endowed with a metric), just a stone’s throw from Cosmos 87; or we might simply be, with nothing else to be relative to and hence no “where”.  But we are here, wherever “here” is; we are real. 

            Now, let us hammer away at this metaphor of math as a useful tool, and thus real.  (The point feels Wittgenstinian, though in general I am little in sympathy with his approach to math.)

            So.  A rock isn’t for anything.  Hence, unpoliced by pragmatics, rocks shade off into pebbles and sand and dust and molecules at one end, into boulders and cliffs and mountains and tectonic plates (or the entire Moon) at the other, with no clear boundaries.  (Unlike the boundaries among mathematical objects – numbers but others as well – which are sharp as a guillotine.)  A hammer can’t shade very far without ceasing to serve its defining function – it needs to serve to hammer things, to earn its keep – and thus ceasing to be a true hammer.

            Mathematical objects have this tool-like usefulness, and in that respect are more like hammers than like rocks.  And this does tend rather to tether them to this world, thus to make them more familiar, more acceptable to the Chamber of Commerce: for these are practical men, and like to know that a thing has a use, and hasn’t just been dreamt up in some pooftah parlor-game on a rainy afternoon.  Integers are useful for counting, Riemann manifolds for relativistic physics, and so forth.  But: cavete, gentlemen of the C-of-C:  they are not defined by these (human) uses. A toaster is so defined, but Mathematica thrones though we perish.  These objects exist prior to and independent of any such simian or homosapiensan applications – rather like rocks, the bare unappreciated rocks, which exist, in their original and ultimate solidity, prior to their being hewn into ashlars.

            One thing I am not claiming  is any special inevitability of this or that application of mathematical objects;  on this special subject I am comfortable with the nominalist position (the “hocus-pocus” position, vice “God’s truth”).  It is all too familiar, that this or that feature of physics, or of anything else, may be modeled (approached) by this particular mathematical formalism or that. Indeed, to make it all painfully plain:  Take the simplest application of the simplest mathematical object: the use of the natural numbers for enumeration.  The association is psychologically so tight, that some people doubtless imagine that N is defined by that application, and was even invented for that applicaton (by Homo sapiens or perhaps Neanderthal  underpaid mathematicians), the way a toaster was invented to toast bread, and is defined by this end.  If bread no longer existed, the toaster would, in a sense, cease to be. 

            Not so the natural numbers.  Fie, lest ye imagine, that they require our little home-improvement projects for their existence.  And as a (particularly smarting)  proof, consider this:  They are not even needed for enumeration.  (Yet mark:  Though seemingly to their discredit, this rather redounds to their greater glory.)  For:

            One could quite well enumerate things, and even perform simple arithmetic, without the slightest notion of integers, or indeed of counting.  You do it thus:

            First sculpt a monkey, and set it next the temple. Then (using your duplicator-zapper, left behind by those visitors from the Companion of Sirius), duplicate the statue and add another one just like it.  Put the resulting statue-grouping a bit behind the (unitary) first.  Now duplicate that, and add a monkey. Put this a bit behind the last; and so on.

            Now, if you want to figure out (say) how many children there are in your family, have them line up next to the rows of monkeys, beginning with the one in front, and have each child hold hands with one monkey.  Keep walkng back along the ranks until each child is paired with exactly one monkey. 

            You can perform addition and subtraction (and, more laboriously, multiplication) by similar purely physical means, having no more arithmétic notion of what you’re up to than do the Pirana.  To be sure:  Various arithmetical relations are implicit in this system, for instance “greater-than” means you have to walk farther along, away from the temple, till  you find a match.  But these relations need never, in practice, become conceptually explicit.  Like the Pirana, you can get by with no names for numbers as such, certainly no systematic numbering.  First, operation of the system, being purely physical, can be performed without words or symbols.  But suppose you do want names for each individual sculpture-row (as for: “Meet you at the (117)”.) Simply color the nearest group (the unit group) yellow, the next one purple, the next one green, the next one (say) black-and-white stripes, and then the next one pink polka-dots on an argent field; mark the next one simply with a little white flag; the next, with a happy-face; and “so” forth. No rhyme and no reason; none needed.  It covers the (finite) maximum group of statuary you have thus far needed to resort to.  Should the sets ever fail you, simply zap-dupe the hindermost and add a monkey, dubbing the new addition however you like.

            The fable is not idle.  Some of the skepticism about the “reality” – that is, the transcendental uniformity – of mathematical objects, may stem, I suspect, from a more justified skepicism about the God’s-truth view of applications of mathematical objects.  Thus, “Tensors are real because the were precisely what Einstein required for his field equations.”  Morris Kline (among others) saw more clearly: it is possible (or, in his view, actually likely) that tensors are not precisely what Einstein needed, and that something will supercede them.  Tensors had already been discovered (or, as the fools say: invented), and  as it turned out, they served him well enough.  Einstein notoriously lacked the mathematical tools ready to hand, when he came up with his intuitive take on gravitation.  He wandered down the hall to the math department, or maybe met some guy in a bar, and the guy says, Yo, Al, try this, always worked for me; and Al says, Good enough.  Sort of like going next door to borrow a hammer only they don’t have one but they have a paperweight and it will do.

Let’s take it a step further:  We built up the integers above, concretely, Peano-fashion, via the successor-function.  But having reached that height, we may throw away the ladder.  Notice that the hidden isometry between the size of the integer and the number of monkeys in the statuary that represents it, is logically unnecessary.  So, to save raw materials, we replace each statuary group by a simple stele, surmounted by the arbitrary symbol that was chosen to represent the group.  Counting then will amount to reciting a memorized list, “… Tyrolean hat, bowling ball, chartreuse lozenge, woodchuck rampant, dot, ….”  This is essentially what we settled down to with: “January, February, March….” (originally the list was etymologically more numerical, before those designations were ousted in favor of various godlings and tyrants.)

This, by way of a reply to Quine, who states, in his reply to Charles Parsons (Hahn & Schilpp, ed., p.401):

I prefer to say, with Benacerraf, simply that there are no natural numbers, and there is no need of them, since whatever purposes we might have used them for  can be served by any progression.

 

True enough – witness our monkey-statues.  After all, anything an actual person will ever have to count is finite -- a stock of two googolplex counters should suffice nicely.  And if one is trying to answer the (probably ill-posed and unanswerable) question, “What is an integer, really?”, then the observation is perhaps telling.  We ourselves have no interest in the quiditas, the “inner threeness” of the number three. A monkey trio, or a tricorner hat, will do just fine. But the “purposes we might have used them for” are by no means as meager as that dismissive phrase might sound – the way one might use a radio as a paperweight, or a statuette as a hammer.  You can’t settle the Riemann Hypothesis with nothing but poker-chips.

The uses to which numbers can be put, and the results we shall obtain with them, inhere in the structure of the set of integers itself.  It is universal, it is absolute, it is culture-free.  Numbers are Necessary, however you choose concretely to characterize them.  We may never get our hands on more than the trunk of this elephant, plus a couple of legs, but the elephant’s there, in his mighty totality.

Babylonians again

If you ever took an astronomy class and paid attention, you know that there are two different ways of defining a month:  sidereal month and synodic month.  Frankly, these tax my retention.  But the real picture is even more complicated.

Shlomo Sternberg, Celestial Mechanics (1969), p. 26:

The return [of the moon] to maximum velocity  takes slightly longer than a sidereal month, and slightly less than a synodic month.  The period of return to maximum velocity is known as the anomalistic period or anomalistic month.

Thus far, a curiosity of astronomy;  one which, one might venture at hazard, was discovered by some persistent fellow with a telescope, prior to the discoveries of quasars or the Red Shift, but probably not all that long before.

But no.   The facts were known in the second century B.C.E. -- and this, in dizzying detail:

Hipparchus gives the relation that 251 synodic periods  are almost exactly equal to 269 anomalistic periods.

Now, at this point, things become seriously weird:

What is most striking is that we find this ratio 251/269 used in a theoretical way in Babylonian tables.

            This startlingly early anticipation does not mean that the ancient Babylonians had been informed of the facts by visiting Space Aliens, as the New Agers would no doubt have it (if they have heard of this story at all).  But it does suggest a preternaturally diligent interest in astronomical niceties.

            And indeed, preternatural is here the operative word.  For as Sternberg several times mentions, the desire for extraordinarily precise knowledge of the celestial facts  stemmed, among the ancient Hebrews, from necessities of religious observance.   Islam likewise has such requirements, a fact perhaps not unconnected with another blazingly insightful observation of the late 9^th/early 10^th century (p. 20):

The motion of the apogee (beyond the effect due to precession) was apparently first discovered by the Arabian astronomer al-Battani.

These, then, are instances in which religion was the midwife of science.  Other instances are familiar from the field of linguistics, where the preservation and explication of sacred Scripture (the Vedas, the Bible, the Koran) was the motivation.

Babylonian mathematics

In The Character of Physical Law (1965), the physicist Feynman presents an informal dichotomy between the “Greek” and the “Babylonian” approach to science.  The Greek: axiomatic, systematic.  The Babylonian: ad-hoc bricolage.  I satirized the latter in the parable of the humble woodchuck (“Constructivist Angelology”).

In this morning’s reading, I happened upon this:

Shlomo Sternberg, Celestial Mechanics (1969), p. 1:

In most popular accounts, the contributions of the Babylonians …  are consistently underestimated.  The Babylonians, in addition to having accumulated impressive observational data, based their tables and predictions on a form of Fourier analysis (using what are now known as “spline functions” instead of trigonometrical series).

!!! -- I stand corrected.

Riemann, Chomsky, Fido

‘Tis of great use to the sailor  to know the length of his line, though he cannot with it fathom all the depths of the ocean.  (Locke, Essay, I.i.6)

It was with the warmest empathy that we read the other day  of the collie with a 200-item Wort- und Spielzeug-schatz -- treasury of words and playthings.
[Note:  This essay was originally written in response to the report of a German dog-prodigy, in 2007.  For the latest such story, see http://www.nytimes.com/2011/01/18/science/18dog.html?hpw ]

As a semanticist, he is the perfect embodiment of what is traditionally known as the Fido-‘Fido’ theory:  this word refers to this object, and the way it gets its reference is – Fido grabs it in his mouth!  A saving detail, somewhat distinguishing him from the complexity of a cash register – or rather (since a cash register, in addition to popping up a digit when you press a key, can also calculate) from a mere inert pegboard with 200 labeled pictures – was his action when presented with an unfamiliar word: he fetched an unfamiliar toy!  Thus, to describe him, we need 201 lines of (utterly non-recursive) code:  if, then, else. (Game, set, match.)

 [For an alternate view of Fido and the Essence of Language, Cf.  Dr. Max Müller’s Bau-Wau Theorie, by Dr. Christoph Gottlieb Voigtmann (Leipzig, 1865).]

            Fido can fetch, but he doesn’t exfoliate.  The structure of his language is the structure of his toybox – a heap of unrelated items. If he could also speak – play with the words as he plays with the toys – magic might happen.  We gaze back with saddened understanding into the empty depths of his eager brown eyes.

(Per Wittgenstein, though, if he could, we’d be disappointed:  “Wenn der Löwe sprechen könnte, wir könnten ihn nicht verstehen.”)

            Empathetic, since we are ourselves in much the same predicament, faced with any subject for which we lack an inborn knack. In language we are all born-geniuses; in mathematics (most of us), born-morons. How poignant to hear the French, who tout their language as logic itself, stumble through the simple act of counting:

“….fifty, sixty…uhh..sixty-ten (soixante-dix)…mmm….four-twenties (quatre-vingts)….four-twenty-ten (quatre-vingt-dix -- I kid you not; and as to the spelling, sic)…”

or say they’ll meet you “today in eight” (aujourd’hui en huit; meaning: seven days from today).

            When  young, the mind at its most resilient, I put my shoulder to the boulder, majoring in math.  Over time  I have found it a labor of Sisyphus.  When I’m not actively pushing it, it rolls back downhill.  And even when giving it my all, I can only fetch things as they are pointed out.  I have never learned to chatter in math.

            By contrast, in learning new words, new languages, new styles, I really am standing on the shoulders of giants, feet firmly rooted in the innate.  No need to constantly “keep up” one’s Spanish.  It’s there.  When an author stretches your syntax – Nabokov or Proust – it gets digested, metabolized, incorporated into the mental flesh.

            At one level, mathematics is a language; and for a time, may give us the feeling of a similar mastery.  The notation is so powerful, it lets us deal with complex sentences with deceptive ease.  But I have found that the various gimmicks and shortcuts become a substitute for thought: I might scramble up to the next terrace, but then I had to pull the ladder up behind me, because the strength of understanding is only as long as that ladder.

            Thus: You learn, with understanding, why a certain integrand can be transformed into another, that lends itself more readily to integration by known rules.  But this understanding then becomes encapsulated, a black box.   Like the law of cosines or any other once-derived, once-felt formula, it becomes formulaic.  Whereas:  from a structure like “Bobby was scolded by his teacher” we get to “the man who is widely believed to have been credited with this discovery” intuitively, without cutting the ties.

            There are other things we’re really really good at, like real-time visual analysis.  The more you learn about what is involved, the more miraculous it seems.  When you realize the fragmented, ambiguous nature of the input, and the coordination required on our part, it is amazing that we can thread our way across a room without tripping over some tensor.

 ***

            One night, math studies years behind me, in the dark without the crutch of a textbook or chalkboard in front of my nose, I tried to recall what I once knew, and for the most part could not.  Floating facts and stray derivations, like isolated lines of remembered verse.  Then with the resolute despair of Descartes at his stove, I tried to discover: all right, what do I know, that isn’t mere memorization?

            It wasn’t much.  I could visualize, intuit, that 2 x 3 = 6, by picturing the boxcar pattern on a die.  This even yielded the commutative truth: 6 = 3 x 2 (just tilt your head to the side).  Fifteen was harder, but was accessible via a triplet of quincunxes, each quincunx mentally held in place with the fingers of one hand.  And already the commutation required a different pattern, one big quincunx, each spot a little triangle.

            Invention gave out by twenty-one.  One can picture a triad of “dotted boxcars”, but what is 7 + 7 + 7 (let alone 3 + 3 + 3 + 3 + 3 + 3 + 3)?  One can dully, dutifully count (which is merely mechanical, bearing no relation to the gut grokking of a quincunx as five), or recall “3 x 7 = 21” from the times-table – a mere boilerplate formula like “a stitch in time saves nine”.

            So, I got as far as boxcars, but shall never catch up to the taxicab, where Ramanujan spotted “1729” as the face of an old friend.

            Shapes in space are also hard.   (Again Locke, Essay II xix.13:  "In a man who speaks of a chiliadron, or a body of a thousand sides, the idea of the figure may be very confused, though that of the number may be very distinct.")
Once, dipping into a bit of knot theory, and finding that I soon had to haul up the ladder to proceed, I went back to square one, and tried to understand a simple knot in the same direct way that we understand a softball or a football.  I made a wire model of a trefoil, turned it every whichway, closed my eyes and followed it with my fingers.  Falling asleep, I would imagine myself (like Mr. Tompkins) on a roller-coaster ride  shaped just like that, sensing when another part of the track would pass above or below, feeling the centrifugal tug.  I can barely do it, strain though I may  -- I throw on the light in a panic and stare at the wire.

            Again Locke (Essay II.x.4):

The memory is very weak:  ideas in the mind quickly fade, and often vanish quite out of the understanding, leaving no more footsteps, or remaining characters of themselves, than shadows do  flying over fields of corn…

            Years of effort have sadly ratified the epigram of Novalis: "Zur Mathematik gelangt Man nur durch eine Theophanie."

            De profundis, ideo, clamo: Riemann – eleison!  Poincaré – eleison!

            Solâ gratiâ.

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.

            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 questions 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 Hungarian  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"

.