Sunday, January 30, 2011

A “Memento” Joke


Our friend Snarla just sent in this:

The tachyon leaves.  The bartender says:  “We don’t serve your kind here.”  A tachyon walks into a bar.

This excellent joke (excellent on several levels) is given extra poignancy by the barroom scene in “Memento”, where the waitress, understandably doubting Leonard’s serial amnesia, tests him by slowing drooling -- in full view -- into the tankard she has just drawn for him. -- The next thing you see, he’s sitting at a little table, and gratefully accepts the tankard.   And she, with a pang and sudden pity, withdraws it to get a new one, saying “That one’s dusty.”

Saturday, January 29, 2011

Real as Rocks


Behold a stone.
A parable:
Practical Pete knows only that you can use this kind of rock to build sturdy houses.
Chemical Clem knows only that this kind of rock has a high iron content.
Geologist George can tell you it is metamorphic in origin.
Laboratory Larry can tell you its crystalline structure.

Relativists tend to stop there, and “celebrate” the tribal knowledge of whatever faction, each equally valid and independent of the rest -- even overt contradictions make no difference to them.   And we are with them part-way, in that we don’t particularly privilege the science-sounding assessments of Clem et ilk, over that of Practical Pete.  The thesis of Realism about Rocks is simply that, should Clem learn a bit about the building trades, he will come to agree with Pete on the suitability of this rock for dwellings.  And if Pete takes a lab course, he’ll wind up agreeing with Larry.  And so forth.  For the qualities lie, not in us, but in the rock itself.  But as for the politics of the thing (which we don’t stress, since the relativists are not intellectually interesting opponents): whoever had an integrated synthetic overview of all those properties (with perhaps a few more needed to explain the interconnections among the properties, e.g. the crystaline structure related to the lack of friability and hence suitability for sturdy building), has an unambiguously superior understanding of that rock than does someone who (like our friend the hedgehog) knows only One Big Thing.
            We would further assert -- and this is not definatory of Realism, but is more the result of observation and experience -- that none of these summations of views can exhaust all there is to know and say about that rock, precisely because it is so real -- discovered, not invented.  (You might almost say:  Begotten, not made…)

Compare:
Abel is a complete empiricist.  Various practical considerations (perhaps, estimating how much fencing material shall be needed for each new circular field) have led him to make ever-more-precise measurements on the ratio of the circumference to radius of a number of circles of different sizes.  (By good fortune, these occur naturally on his planet, in great abundance.)  All the measurements tally exactly to within experimental error.

Baker lives motionless in a lightless world, and knows nothing of shape or size.  But since (by way of compensation) his lifespan is a billion years, he has plenty of time to explore the wonders of infinite series.  And he was astonished to learn that four times the alternating sum-and-difference of the reciprocals of the odd natural numbers (one minues a third plus a fifth, etc.) turns out to be six times the sum of the natural numbers squared; and furthermore, equal to the product of  (well, consult your favorite book of math tables, for many more surprising examples.)

Charlie is a wave-function, living in a world of waves.  He quickly hits upon e to the i-pi-theta as key to his world.

Now Doug (just to make things interesting) has no particular intellectual pretentions or attainments at all:  but one day God takes him aside and reveals to him the infinite decimal expansion of pi.  Puzzled, Doug accepts the revelation.

Realism about pi is exactly parallel to realism about rocks.   The longer and more accurately Abel pursues his experiments, the more his measurements will converge on Doug’s value.   The farther out Baker calculates his infinites series, the more they will agree with what was revealed.  Deeper mathematical insights will connect the geometric and trigonometric and wave-theoretical and number-theoretical properties of pi.

*
            William James (The Principles of Psychology (1890), vol. II, p. 334) makes the same point about the nearly endless ways of describing a thing, taking as his example a piece of paper -- “a combustible, a writing surface, a thin thing, a hydrocarbonaceous thing…” -- concluding “the reality overflows these purposes at every pore”.  
All true; though the example, being an artefact rather than a natural object, proves less than it might, since the object in question was made for certain purposes -- our purposes -- by us;  and thus made (by no means begotten), its other qualities flow only so far.               
Having invented the coffee-cup to hold our coffee, or tweezers to tweeze, we do not subsequently discover that said cup is the perfect containment structure for magnetic monopoles, nor that tweezers are the precise shape required to receive meaning-of-life narrowcasts from the Orion nebula.   Whereas a rock might prove to have any number of further, unexpected properties: suitably ground, it might be medicinal; suitably sliced, it might reveal ancient fossiles; further analysis might show it to have reached our solar system from far away, preserving evidence of different, earlier  settings of the physical constants elsewhere in the cosmos.  And the wonderful properties of pi, likewise, may never be exhausted: because we found it, we didn’t make it;  He did.

(Such considerations illuminate the celebrated "unreasonable effectiveness of mathematics" -- not just in the physical sciences, but in distant-lying parts of mathematics itself.  It's all from the same workshop.)

So, our motto:
Transcendental Numbers -- As Real as Rocks. ©

Friday, January 28, 2011

Scorecard



In these essays, I am not concerned to tussle in detail with any of the panoply of agnostic philosophies currently on offer in the cultural Mall of America, nor even particularly to distinguish among them.  Their various designating terms may even be used loosely, as the phrase requires -- “nattering nominalists”, “sodden solipsists”, "deviant deconstructionists", "prurient proctoscoptists", and so on.  The point is rather to explicate, and make more plausible, Cantorian Realism, with a nod in the direction of its nice fit with theism.   I capitalize the word Realism, not to exalt it, but to distinguish it from the everyday use of the word, which can shade over into almost an opposite meaning, as in Realpolitik (the capitalization there is an artefact of German).   I don’t capitalize nominalists because, well, they suck.
Nor, despite the keystone role of “visibilium omnium et invisibilium”, am I the least interested in any philosophical debates about Abstract vs. Concrete.   The integers could be made of green cheese, for all that matters.

So, here’s all you need to keep track of the action:

The Good Guys (in the golden jerseys):


A typical Realist, in native dress

The Bad Guys (in the purple jerseys):
nominalists, solipsists, subjectivists, perspectivalists, epiphenomenalists, eliminativists,  phenomenalists, behaviorists, occasionalists;
finitists, intuitionists, constructivists, formalists [these, all in the mathematical sense];
fatalists, predestinarians; instrumentalists [by which I mean, not strummers of the ukelele, but the theory-dissing attitude to science],
idealists [in the non-idealistic sense],
sociologismists [an awkward coinage, but we need to distinguish these from actual sociologists, who are  in principle  honest souls  like plumbers];
gnostics;  cultural constructionists, moral relativists, ‘internal’ ‘realists’, situation ethicists, irrationalists, tribalists, identity-politicians, deconstructionists, post-modernists, nihilists, diabolists, atavists, primitivists, Donald Trump, bestialists, methodological onanists, proctoscopic introspectionists, ……

A clowder of Nominalists


Guys we bar, though there is something to be said for them :
empiricists, verificationists, logical positivists; existentialists

Guys that maybe get a bad rap:
mysterians, mystics, quietists; coherentists

The score so far:
Good Guys: 700000.    Bad Guys:  0.

Tuesday, January 25, 2011

Further updates


[update to  this ]

I confessed that perceptual/imaginative disability with burning cheeks;  but am somewhat comforted to come across this testimony from a fellow-sufferer:

It is impossible to imagine a four-dimensional space.  I personally find it hard enough to visualize three-dimensional space.
-- Stephen Hawking, A Brief History of Time (1988; 2nd edn. 1996) p. 24

And now this, from William James, The Principles of Psychology (1890), vol. II, p. 275, re our apprehension of space:
It is a notion, if ever there was one;  and no intuition.  Most of us apprehend it in the barest symbolic abridgement;  and if  perchance  we ever do try to make it more adequate, we just add one image of sensible extension to another  until we are tired.  Most of us are obliged to turn round  and drop the thought of the space in front of us  when we think of that behind.

Alas, only too true!  Indeed (confiteor), I myself cannot simultaneously chew gum and conceive either.

[and to this]

P.A.M. Dirac, The Principles of Quantum Mechanics (4th edn. 1958), p. 311:

The Schrödinger picture is unsuited for dealing with quantum electrodynamics, because the vacuum fluctuations play such a dominant role in it. … They get bypassed when one uses the Heisenberg picture, and one is then able to concentrate on qualities that are of physical importance.


I’d Like to Add Just One Thing


[This is a continuation of a thread begun here.]


The fundamental theorem of enumeration, independently discovered by several anonymous cave dwellers, states that the number of elements in a set  is the sum over all elements of that set  of the constant function 1.
-- Doron Zeilberger, “Enumerative and Algebraic Combinatorics”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 67

The very simplest thing one can do with the natural numbers, beyong simply admiring them, is to add two of them together.  Do we understand how to do this?


[Update:   My mistake.
The very simplest thing you can do is, given one of them, take its successor.  Addition is a binary relation; whereas

Counting-one-more is a unary operation in the set of numbers.
-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 88]

            I don’t mean practically:  that we occasionally get our sums wrong, no more shows that we can’t add – let alone that there is any paradox at the heart of addition – than an occasional stutter or solecism  shows that we can’t talk – always provided that, in each case, we recognize our error when it is pointed out. (“You’re right; I meant to say ‘357’/’palimpsest’.")  I mean, conceptually:  Do we truly understand what addition is, beyond our comforting successes up to this point, in stacking one smallish number on top of another?

            Time for a parable.  Farmer John pulls into his long driveway, rolls to a stop, turns off the ignition, and announces that he knows how to drive; been driving for up’erds o’ forty years, in fact.  To all appearances, we must agree.  But then we learn that he has never driven over 5 mph, never used reverse gear  nor indeed any gear beyond the first, never driven on a highway  nor indeed any route but that half-mile stretch between his driveway and the barn, and thus never had to use turn signals (or headlights, or windshield wipers, etc.), or the brakes.  We might then say that he doesn’t quite know how to drive.   He then retorts:  But this is driving, this is what I mean by the word, just this and nothing more; and I do it perfectly. 
            At this point one could make the usual observations about language-games, you say tomahto I say tomayto, all that, but I wish to point to something quite different: to something not linguistical at all, nor a “matter of semantics”, nor of convention: something out there, and quite real:  the automobile itself.  A top-of-the-line Mercedes, as it happens (seems rather wasted on our farmer friend).  John does what he does with it, and may call that what he likes; but we may note as an objective fact  -- true, as it were, across all possible worlds – that he has not exhausted the capabilities of the actual machine.  Can you say that you have eaten, if you have chewed, but not swallowed?

            In his Wittgenstein, Kripke tries to outline a skeptical problem about addition, by defining a new operation, quus, which corresponds to plus sometimes and not others, and getting all into a lather about that.  The move is much like Nelson Goodman’s celebrated/execrated blue vs. grue, a nice enough puzzle that was funny the first time.  Essentially, quus is plus when the summands are small enough, and collapses thereafter.  It’s the sort of artificial move that can give skepticism a bad name. (Nor am I here to clear that name.  Doubt the existence of your own head, if you must; but go do it behind the barn.)
            Now in fact, real arithmetical puzzles do exist, even in the matter of addition, without the invention of any artificial operations.  (I shall now sit down and sup with the devil of skepticism; but observe this long spoon.)   If you go on long enough, addition does totter – and almost falls; yet at that point where the nominalist bellows, “Ist gerichtet!”, a sweeter and yet mightier voice calls: “Ist gerettet!”; as we shall hear.

            Meanwhile back in parable country, Farmer Jim has been introduced to a horseless carriage for the first time. He admires its sleek lines, its metallic glint, its rumble when the engine is turned on.  He gets in, rolls forward one foot, and gets out.  “Nice,” he says, “very nice.  But I can go farther on my horse.”

            Likewise with addition.  Although the core and essence of addition is indeed simply that of tacking one number onto another, it is of the essence of math, as of language, that the operation is recursive: having done it  you can do it again, with the output of the first addition  an input to the second one;  and so forth, for a while; then stop.
            Now  we could in fact stop here, with no further concepts or developments, and have a perfectly coherent, and quite useful, operation of addition.   Had we not been created but a little lower than the angels, we probably would.  Every sum, let us say of a, b,c, d, and e, is to be performed thus:
            (((( a+b) + c) + d) + e)
This model for addition we may dub that of the Downs Syndrome Grocery Clerk (a familiar figure).  The items to be tallied  come along a conveyor belt, seriatim, and are rung up  one by one  until the items run out.  The details of the arithmetic have been exported to the cash register, just as the details of definite integrals are often exported to computers or math tables.  Let's not have any Searlian "Chinese Room" nonsense now: So long as the clerk punches in the integer written on each item as it comes, he is indeed adding; he has the entire system under his belt, as far as it goes.
            Consider now a more advanced grocery clerk.  After some glitch on the conveyor belt, two items (let’s keep it simple) arrive together.  Which shall he ring up first?  At this point, the Downs Syndrome model breaks down; we need something a little stronger.  Well, infinitely stronger, in fact, but let’s not emphasize that point just yet.  We add the proviso that addition of natural numbers is commutative: take the addends in whatever order you like.  Moreover, this clerk – who, let us say, has no cash register, but must do the sums in his head or on paper --sometimes saves himself some trouble, thus:  Presented with
            apple (25 cents) plus banana (30 cents) plus ten oranges @15 cents each
he does not add each orange successively to the previous sum of the apple and banana, but adds their own total sum to what proceeded:  .25 + .30 + 1.50    To justify this, we require that addition be associative:  for instance,  (a + b) + c = a + (b + c). 
            Put these two operations together, then, given that there is no upper limit on the (finite) number of things that can be added, you have now added either one mighty fire-breathing rule involving advanced quantification over infinite sets and strings, or else an infinite number of finite rule-schemata, each of which is an instance of the taboo’d fire-breather.  Either way, you’ve made a huge step, and you’re still just a grocery clerk.

            This strengthened system of addition is adequate to all the needs of the grocery.  But now we step out to the playing fields, where Achilles is racing a tortoise (who was given an advance lead).  A philosopher who (here with some reason) doubts his own head, points out that Achilles can never catch up with the tortoise.  We point out that he can – indeed look, he just has – but to do so we had to add up an infinite series,
            1 + 1/2 + 1/4  +1/8 + ….
Now we are facing yet another sort of infinity: not any actual infinite number (we shall still shun that, for now), nor infinitely many rule-schemata describing finitary processes, but a procedure with infinitely many terms.   So, are we cool with that?  We’d better be;  because look:  Achilles won.

            Now the finitist pounces.  “Does your grocery store allow rebates?”, he asks, innocently enough.
            “Why, yes.”
            “A-ha!  Then you must allow negative numbers in your sums.”
            “Well, yes, that can be done.  Our cash register is actually programmed for that.”
            “Good.  For now I’ve got you.” And he shows us the infinite sum
                1 – 1 + 1 – 1 + 1 – 1 + 1 ….
            Now, as it stands, that expression is ambiguous – though no more so than “1 + 2 + 3…”.  We allow ourselves to make do with expressions like the latter, because we agreed that you may group the terms however you like; it makes no difference.  Only now it does:
            (1 – 1) + (1 – 1) + …
yields partial sums  0, 0, 0, … and so converges to 0;
            1  (-1  + 1) (-1 + 1)
yields partial sums 1,1,1,… and so converges to 1; whereas
            ((…((1) – 1) +1) -1) ……………..
yields partial sums 1, 0, 1, 0, and so doesn’t converge at all.
            And worse is to come.  In steps the concierge of the Hilbert Hotel, and reassigns the guests to new rooms:  each guest in a room of even number k, is moved to room 2k.  Now the sum looks like this:
            1 + 1 -1  + 1 + 1 – 1 + 1 + 1 – 1 ….
which, suitably grouped by the threes of the minimal ecurring pattern, yields
            1 + 1 + 1 + ….
which diverges to infinity.
            So much (our finitist cries in triumph) for your easy accomodation of infinities – it has led you right over a cliff!  Be ye content therefore with finite sums, with finite everything.  Let Achilles  forever lag  behind that tortoise, in this finite life; abjure for aye the everlasting; and worship ye the finite godling, Mbumbo, lord of all the dumbos, creator of all things visible and that’s it.

            At this point, we really are properly chastened; we do not know what to reply.  But let us look back, to earlier testaments, and see if they provide guidance.
            Often in history, mathematicians have shrunk back, with something like horror, upon encountering something ontologically unprecedented.  So it was with the irrationals, the imaginaries, the non-Euclidean geometries, the infinitessimals (here the shrinking was much delayed, and the unshrinking rather recent), and much else.  And had they experienced a permanent failure of will – or of trust in the creations (or as it may be, lineaments) of our infinite Father – and never returned to the subject, the initial shock might have attained, in time, the force of a precept, an Awful Warning.  Yea, it is related, that in the distant past, a sailor proclaimed the irrationals, and was cast into the sea.  Yea and another, in a farther age, did espouse the imaginaries, and was crucified head-downwards.

            It turns out, though, that the problem of indefinitely iterated addition  can be tamed, at least partially.  Doing so involves a conceptual detour, to the notion of “absolute convergence”.  Once that is understood, infinite sums separate essentially into sheep and goats:  the sheep-series converge as nicely as you like, with shepherding (re-arrangements) allowed; the goat series stink, and are to be shunned (save as further techniques may allow us to herd a few of these).

            This fable is reasonably reflective of the actual developments, though it has been compressed, and truncated before subsequent ingeniosities like Cesaro-summation.  An even clearer instance of a case, in which we thought we understood a concept, and had even become rather adept at slinging it around, only to discover that we didn’t know how to proceed when we came to the edge of a certain cliff, is provided by integration.  Here the infinite process was present from the start: the ever-shrinking rectangles of the Riemann integral.  Thus, the case is not like that of mere addition, where indeed we might have planted our flag without ever crossing over the line (and how soon it came! right in the grocery store!) into infinite collections of rule-schemata or infinite processes.  (There is, so to speak, no analogous Downs Syndrome Theory of Integration, and those suffering from that affliction  are advised to steer clear of integrals.) Yet some otherwise-lovable functions  are not Riemann-integrable.   The perplexities led to a new kind of integral, called the Lebesgue integral: and thus to a realization that what we had thought was integration tout court, was really only one species of what turns out to be a more general idea.
            The latter saga has an after-fable: for it was this general perplexity that impelled Cantor down a path that led eventually to something quite unlooked-for, and which still lifts the bristles at the back of the neck:  the topless tower of uncountable infinities.  These were thus sired   not of a fever, nor an opium-dream,  but from (at origin) the quite practical matter of adding stuff up.  If you open Cantor’s closet, that’s what falls out.

            And so, to reply to the imaginary speaker of the title:  You might like to add just one thing, but, unless you shut your eyes to the bloom and truth of the Creation, you’ll wind up adding much more.  In for a penny, in for a pound; in for an integer, in for the infinite.

Tales from the Woods


 [In the never-ending struggle to keep our “humble woodchuck” search stats up, in the face of repeated attacks by the Illuminati, we offer this. ]

            A humble woodchuck, an even humbler woodchuck, and the humblest woodchuck of all, walked into a bar.
            “What’ll it be?” boomed the bartender.
            First woodchuck:  “Um… I’m good.”
            Second woodchuck:  “Yes, nothing for me.”
            Third woodchuck:  “Pass.”

Now -- that’s humble!

Monday, January 24, 2011

VISIBILIA VEL SENSIBILIA



[This is a continuation of a thread begun here.]

Credo in unum Deum, Patrem omnipotentem,
factorem cœli et terrae, visibilium omnium et invisibilium.
-- The Creed

The theory of knowing  is modeled after what was supposed to take place in the act of vision.  …  A spectator theory of knowledge is the inevitable outcome.
-- John Dewey, The Quest for Certainty (1960)


Why anyway this fetish with *seeing* things?  There are some things which you can see, but which nobody else can; and you can’t touch them, and you can put your hand through them; and when you blink, they’re gone, or have shifted position. In such a case, we say: “You’re seeing things.”  These things are called “hallucinations”, or “muscae volitantes”, and most of them probably are.  If being “visible” is all you’ve got going for you, you’ll need something further on your resumé.
            Now try this thought-experiment.  In a universe (which might even be our own), there is a race of rational beings, known as the Blorks:  blind, deaf, immobile (and quite spherical), who nonetheless can sense objects around them with incredible precision, using a kind of echo-location, but only when those objects move.  (Vision in some animals is rather like this.)  Now, things don’t move much in their friction-ridden landscape, unless you nudge them.  And the only way the memberless Blorks have to nudge them, is via a biological analog to the technology currently being developed for quadriplegics:  you think an appropriate thought, and it has an effect on your computer, and thence, via robotic coupling, on the physical world.  Chez the Blorks, the action is direct, cutting out the middle-machine.  Now, in that world, the way to induce a certain movement of a body, is to discover (whether by calculation or direct insight) a solution to the equations of motion, and then to “think it aloud”.  Fortunately, hardwired into this species are the laws of dynamics, in their elegant Lagrangian form.  (We too have a few of these hardwired, though they don’t go very far, being practically exhausted in the infant’s experiment of repeatedly dropping her spoon.)  So, to calculate these trajectories  is virtually second-nature, rather like the abilities of an experienced driver, to calculate a merge.  (There are, to be sure, some drivers among us, who never manage this and similar skills, attempting to pass you on the right and then zip in and cut you off, in a way wholly inconsistent with present projected trajectories;  such I desire might become perfectly spherical, and then roll down some infinite inclined plane.)  If your solution is correct, the results are very real: the object moves as directed, and every detail of its motion registers on your echolocator, your nerves thrill at the inrush of sense-data, with a precision which beggars our own retinal amblyopia.  (If your solution is incorrect, God administers a mild – a very mild, not at all dangerous (for, Boshaft ist er nicht) – though admittedly rather sharply painful, electric shock.)  The boulder so moved may go crashing into other boulders, and if you miscalculated, it may roll right over you, thus reducing you from a ball to a disk. 
            It seems clear that, to these creatures, the equations of motion, and the mathematics of their solution, will have a lived reality, a concrete sensuousness, similar to  or exceeding  that of our own, dim, barely binocular judgment of distances, which we cannily calculate as best we can, before reaching for the coffee-cup.  (Dang. Spilled another one.)
            Upshot:  Discount sight.

            Problem to tackle next:  Not visibilia (trivial), but… observables.  Much more problematic.

Sunday, January 23, 2011

The Urysohn Metrization Theorem: an alternate account



[from: The Journal of Proctoscopic Philosophy, (to appear)]

            We have seen that belief in the intellectual curiosity known as the “Urysohn Metrization Theorem”, oddly widespread, can be adequately accounted for in adaptationist terms.  However, an even simpler explanation lies ready to hand.

            The underlying puzzle is known in cosmology as the Horizon Problem.  To wit:  How can far-flung regions of the universe, well outside one another’s event horizons, share certain arbitrary and contingent values of various parameters -- for instance, the uniform temperature of the Cosmic Microwave Background Radiation?    In mathematics, the problem is the same:   How can some random fragment of patriarchal European ideology -- in this case, the U.M.T. -- be held to so tenaciously by scholars who have never met one another?
            The solution to both problems is the same:  Cosmic Inflation.  You see, many billions of years ago, all the world’s mathematicians were packed together tightly into a tee-ny, ti-ny region of space, no bigger than a thimble.  One of them (possibly Urysohn) somehow took it into his head that, if a space is regular and second-countable, it must be metrizable (whatever that is supposed to mean).  All the other compactified mathematicians nodded their little heads vigorously and followed like sheep.   Then the cosmos went -- VROOM ! -- and that’s where we are today.
           Simple, really.

M(or)e(is)men(it)to


[a follow-up to this and this ]

            Come spring, you carefully fold your winter sweater, and set it on its proper shelf in the linen closet.
            The wheel of seasons turns, the cold winds blow; you unlatch the cupboard door … and find the garment in its place.  All is well.
            Suppose now rather, that at the first blast of winter, the sweater were to appear before you in midair.  Convenient to be sure; but disconcerting.
            Yet such is memory.  We do not choose where to store what we remember, nor can we fold these items to size.  Our experience vanishes backwards down a deep well, much of it never to reappear, some of it to pop back, bidden or unbidden, partially preserved or distorted or we know not what.  Our own experiences have become estranged from us, fallen into a vat of alchemy, from which they emerge (it may be) changed, we know not how or whether, for we have nothing against which to compare them – these insolent ‘memories’, these soi-disant relations to our own vital past, spewn forth from out the cauldron of the unconscious, like some unknown Australian self-proclaimed cousin showing up on our doorstep, bearing a letter of reference and a leer…
            How much   how far much   better  that, which our Memento hero hews to: It Is Written, on Mine Own Flesh.  He chooses which limb to scribe it on, what the wording, which the facts.   – Too, our standard memory is a cloaca, stuffed with every stray sensation, useless wadding stifling the essential.  Yet Leonard sets down only what will fit on his skin: Yea yea, nay nay; the rest -- away with it.

מנא ,מנא, תקל, ופרסין


            Such is the sanguine and level-headed view.  But now consider.  He awakens each day  to a world new-made; he spots the writing  with the same surprise  that Crusoe spotted footprints in the sand.  He cannot really recognize it as his own: even those he wrote himself, he stippled into the skin – it won’t resemble his normal cursive – and others he left to the tatoo-artist.   The writing must therefore confront him like that at Belshazzar’s feast.  It is otherwordly.  He is wreathed in cryptic admonitions, some penned in a Gothic script like that of Scripture.  He might almost be forgiven for fancying himself a prophet.  And yet – for here the story is bleakly modern.  He pays no mind to the source of these writings, just takes them for granted.  He simply takes the next step forward, in his appointed task.  That Mene, Mene, Tekel Upharsin  has no resonance, divine or diabolocal.



            The image of a man whose living skin is parchment, is striking in itself, apart from whatever story might be attached.  It is not a widespread motif.  One finds it in  Bradbury’s story “The Illustrated Man”, whose title character sports moving tatoos, each with a tale to tell.  These depictions are multifarious, and look outward, concerning the onlookers more than the bearer.  The situation in “Memento” is the opposite, the exact inversion: all the inscriptions prowl around the same central obsession; they are addressed to the bearer in the imperative; they hold nothing for anyone else.  Even for the bearer they are enigmatic, as he strains to find the goal at which they point.  In this respect the situation recalls that of Kafka’s “In the Penal Colony”, where the skin of the tormented man is slowly inscribed with the secret of his own guilt.  The parallel is exact if, in fact, Leonard is his wife’s murderer.

Saturday, January 22, 2011

End of World Postponed for Now


Here is a much nicer California traffic-story;  humanity is redeemed.

Shortly after the proof of the Heawood conjecture had been announced, Ringel was driving along the California expressway and was stopped by a traffic cop for a minor traffic violation.  As soon as he found that the culprit’s name was ‘Ringel’, the cop asked, ‘Are you the one that solved the Heawood conjecture?’  Ringel, surprised, admitted that he was, and was duly let off with only a warning.

[Recounted in Robin Wilson, Four Colors Suffice (2002), p. 172]

REALIA & IRREALIA


[This is a continuation of a thread begun here.]

            How, then, are the prime numbers – or Hilbert space – different from a unicorn?
            (That reads like a joke, to which the reply would be:  “They lack a horn.”
Or:  Hilbert and this unicorn  walk into a bar… Yet the question is meaningful enough.)
            Unicorns do not (alas) exist, or so I’m told; but people do talk about them, and some (such as myself) may even believe in them.  Descriptions and depictions do exist, which purport (wrongly; or tongue-in-cheek; or in some extended sense, accessible only to those who believe in snow-bunnies) to depict them – in all their monocerous, silver-shanked magnificence.  (There are worse purposes, to which silk or ink might be put.)
A picture-of-a-unicorn (so goes the rote) is as much an unfissionable entity as a hotdog, which does not split neatly into a hot and a dog:  what at first glance seem  components (<picture> + <unicorn>)  are in reality synsemantic (“incomplete symbols”).  A picture-of-a-unicorn differs from a picture-of-a-phoenix, but you cannot extract the quantifier and give it wide scope:  there exists no thing which eíther is a picture of.  When we say (straining the usage of “exists” just a bit) that “The unicorn exists in heraldry”, we mean, nothing ontological, but merely: If you wish to find a picture-of-a-unicorn, go to the heraldry section of your local library,  and not to the section on pets.
            Okay, so no unicorn; darn.  What about its Meinongian doppelgänger, the “Idea of a Unicorn”?  The modern consensus, to which I overtly subscribe (with certain arrière-pensées concerning the unicorns themselves, whose hooves, like flint, strike fire from the stones  whereon they gallop, never heeding the wind) declines to admit that either unicorns or the “idea” of a unicorn does exist, in any concrete or meaningful sense – that is, as a thing with some independence, an idea that is there, somewhere, even if no-one is presently thinking it; a thing, so to speak, with a front and a back.  (And here I do truly, and passionately, agree:  if the forests lie sad and silent, unroamed by unicorns, then away with such rubbish as the “unicorn-idea”!)   I may be thinking of unicorns (as indeed I usually am), or entertaining (second-order) beliefs about them (their mane is like silk, like sand), but this is more like a verb than a thing.  (Again: There exists no unicorn, whereof I am thinking; I am merely daydreamicorning, unicontemplating…)  People may well continue to entertain ideas about unicorns (and by now you can supply the punctilious punctuation: entertertain-[ideasaboutunicorns]) so long as the (foolish, folk) tradition is passed down:  but once let the Earth disappear in a puff of smoke (which, bear in mind, it may do at any instant), and all ideas of unicorns die with it. (And here, though at my most sentimental, I make no objection:  They lie, with Thor, in a common grave.) Travelers from another dimension, roaming the ruins of our world, will never encounter a trace.

            The case with integers is quite otherwise.

            Let all sentient beings perish in the Big Crunch (this is meant, not as an optative, but as a [contrafactual] hypothesis); then let the universe evolve anew.  There may or may not, in that cycle, evolve any rational beings at all; or they may evolve, but be such sobersides as to have (like your boss) no interest in fables and fairy tales; or they may develop some folklore of their own – in all likelihood, nothing resembling a unicorn (these new rational beings  being themselves, for one thing, perfect spheres;  mothers frighten their children with ellipsoids):  yet let any one of them turn his hand (or flipper) to enumerating the stars of the firmament, or the ways in which he loves his sweetheart:  and the precise same integers as before  will stream to his aid.

(Yet let us pause for a bit, for I am seized with sudden sadness.  In principle very glad, that our sturdy friends the integers  survived that  cosmo-catastrophic transition;
and yet I do lament,
and sorely miss,
the brightness and that brashness,
of those   proud
                      white
                                 steeds ….)

*

Let us look  at the same thing  from another angle.  (A nice diagnostic, for things that are really realin the round.)
People have been tempted, and were scolded by Russell for so doing,  to say that Hamlet is real  in Shakespeare’s imagination, the way (or at least rather like) Napoleon in ours.  Russell’s retort is classic:

       When you have taken account of all the feelings roused by Napoleon in writers and readers of history, you have not touched the actual man; but in the case of Hamlet  you have come to the end of him. If no-one thought about Hamlet, there would be nothing left of him; if no one had thought about Napoleon, he would have soon seen to it that some one did.

Spoken like an Englishman!

Stop ...  being ... silly ..... ..... ...

I still have a sneaking sympathy with the reality of the gloomy Dane, in the sense that there are propositions about him (/it/whatever) that are true or false in our world, never mind in Platonic paradise, or in that of the late Shakespeare.  If, on a test, you identify Hamlet as the prince of Macedonia, you will be marked down; nor is his ladylove named “Buffy”.  (In southern California, that answer might get you half-credit.)  So, Hamlet is not real (unlike “Hamlet”, a play by the bard of Avon), but…real-ish.  (Compare the concept, discussed elsewhere, of truthitude.)  And contrariwise, the constructs of our world are – mostly  exactly that:  constructs, and thus ideal or even -- fake-ish.  -- I won’t emphasize this latter point, as it has been  if anything  overemphasized of late  by post-modernists and deconstructionsists, not to mention earlier and more honorable  skeptical attacks.  The upshot is simply that the two levels of Things  tend to edge towards  meeting in the middle.  We can still, faced with the following examination-question,

            Choose the odd man out:   
            (A) unicorn   (B) dog    (C) robin

correctly pick (A).  But we must concede that “dog” is a concept held together with duct tape, consisting as it does of an ever-evolving medley of breeds, beginning with one insensibly different from the ancestral wolf, and fanning out into some that may be no longer interfertile, and differing the one from another, in both appearance and behavior, more than do some allied species among themselves.  And as for “robin”, the term in lay use designates polysemically (though not really homophonically, the way “pen” designates both a female goose and a writing implement) a variety of superficially similar species, depending on where you hail from.  In terms of being a “natural kind”, the unicorn may have more uniformity than the canine.

There is another trait, though, which distinguishes more sharply between unicorns and zoologically more respectable species, than the existence of detailed descriptions (some fictional, some factual) of each.  And that is, the ability to reason with them, inductively.  If, on your trip to Shropshire, you meet a particular bird, let us call him Hoppy, which on visual inspection prompts you to call it a “robin”, an ejaculation which in turn prompts vigorous assent from the circumambient peasants, then, even though Hoppy is not actually quite the same sort of bird as what you used to call a “robin” back in Illinois, nonetheless, you’ll be correct in predicting that Hoppy can fly, and probably that he has a palatal penchant for worms.   Whereas, should you chance upon a monocerous equine in your ramble through Sherwood forest, you can conclude almost nothing.  Whether it would, like the unicorns in books, meekly lie down at the sight or scent of a virgin, is anybody’s guess.  (I’m betting:  Yes.   I mean -- I would….)

So now, let’s try our hand at this one:

            Choose the odd man out:
            (A) unicorn  (B) dog   (C) compact Hausdorf space

The answer is still (A); you can reason with the latter two.  And indeed, (C) is in a sense more tangible, and more reliable, than is (B).  Should you ever wake up one morning in a compact Hausdorf space, you can be quite sure that, should you chance upon an infinite sequence of points running along the path of your morning stroll, then that sequence will infallibly converge to a point in the space. And this would remain true, were the space somehow embedded (this time preferably without your presence) as a subspace of Hell.  For though the Dark Prince (whom God defeat) may have power over reprobates, and even the power  from time to time  to tempt the righteous, yet he has no power over topology.

            This reminds us of the old conundrum:  How can there be an omnipotent Supreme Being, who yet has no power to refute the truths of arithmetic, nor indeed any necessary proposition, such as those of topology?  The simplest answer may be, that topology is part of God.  You don’t refute your own arms and legs.

Straightaway (for conscience pinches) let me hasten to explain what I mean and do not mean by “Topology is part of God”.  It’s a nifty epigram (assonance and all), but it can mislead.  What I mean is – well, all that was said above, which is hard to summarize, but in a nutshell: If you consider the Creation to be part of what characterizes the Creator, then (actually, a fortiori) you should consider the principles by which that creating is regulated (topology among them) to characterize Him.  A simple point.  What I do not mean is – anything a breathless journalist might make of all this.  In particular, it does not directly say anything about what matters most to most of us, day to day, when it comes to God:  Does He bid us do this, or that? Might I be damned? Can I be saved? What must I believe?  Need I believe anything? --  If anything, contemplation of the multitude of invisibilium distracts our attention from such questions:  The more our mind must focus on God as Creator (since, as we come to learn, there is so much more to the Creation than Levittown, so much more still than we could ever behold with our eyes), the fewer neurons are left over for contemplating God as Judge, God as Comforter, God as Redeemer.  Fact is, He is infinite, we’re finite. Only so much bandwidth down here.
            Nonetheless, I must insist on this point -- trivial though, in a moral or eschatological perspective, it may be; if only to help correct the imbalance that has gone before. The Bible (meaning: the sum of the Old and New Testaments) dispatches the Creating in a couple of paragraphs (curiously prescient paragraphs though they be).  All the rest of the text just takes the visible world as given, asking after nothing more; and busies itself with guides to conduct, awful warnings, things admirable but not to be tried at home, before finally culminating in Christ, who if anything (despite a philosophical-sounding, if vague, “In the beginning was the Word” in the odd-man-out of the Gospels, John; an apophthegm  in any case  never really developed) is more centered than ever on the relationship of God to Man, not God and the Plan.

Friday, January 21, 2011

"Memento": a meditation


            Acutely aware of my own limitations of understanding and even simple memory (I take *lots* of notes), I identified keenly with the plight of the protagonist.  I’ve had somewhat similar experiences, dusting off a math text from freshman year, remembering nothing of it -- but see, there in the margin, copious notes in my own hand…
            (I cannot count how many times I have gone back to basics, painfully relearning the calculus.  Just as, each spring, Suzanne takes me on a tour of the garden, pointing out the various flowers and naming them; I enjoy them with the freshness of a child, of a new-bloomed bud, but never remember their names.  My mind is so constituted, that only language sticks – many languages, reams and reams of that! – but without their referents.)
            But I digress.  The --  Where was I?

[Update July 2014] Acutely aware of ....    Where was I ??

Blogblogblogblogblogblogblog


 While a search on “Urysohn Metrization Theorem” or “humble woodchuck”  or even "Sit fides penes authorem"  takes you right here  [well, did, until the Illuminazis hacked our site] -- despite the fact that, so far, we have had nothing of interest to say upon either of those first two worthwhile topics, and the third is a misspelling --  a search on  “david justice blog” takes you, well, everywhere else. 
Distressing.
Probably the problem is that we’ve never actually used the word “blog”.  Accordingly, we do so now.
I blog; thou bloggest; he blogs (or: bloggeth), y’all blog.
Blog; blogger; bloggest.
I have blogged (he hath bloggen), thou art blogging, she shall blogoscopy
A priest and a rabbi walk into a blog.   A blog on ice.  The blog of war. 
“Drop that blog!” he blogged, bloggily.
(and as a strong verb):  blig; blag; blog;bluggen.
Blogissimo.  Blogosity.  Blogonics.

That should do it.

[Update 4 May 2011:   It did not.
Not merely do -- understandably -- sites devoted to the baseball slugger predominate,  but even some astrological dreck shows up before this site (if it even ever does; I lost patience paging).
For Google he is a Jealous God;  great is his wrath; unpitying.]

World Ends; Pork-Belly Futures Hold Steady

Facilis descensus Averno.
Herewith some recent waymarks on the declivity.


http://latimesblogs.latimes.com/lanow/2011/01/hit-and-run-accident-devolves-into-vigilante-beating.html

[Update 1 Feb 2011]
Oof, and now this:

http://www.philly.com/philly/news/114994669.html?cmpid=15585797


[Update March 2011]
...and this:

http://www.nydailynews.com/news/national/2011/03/08/2011-03-08_police_arrest_16_in_gang_rape_of_11yearold_girl_in_cleveland_texas.html

http://nymag.com/daily/intel/2011/03/a_lot_of_people_in_cleveland_t.html?imw=Y&f=most-viewed-24h5

http://www.whataboutourdaughters.com/waod/2011/3/11/three-ring-circus-arrives-in-cleveland-texas-angry-local-wom.html

[Update 14 July 2011]
Remarkable even for Baltimore:
http://www.washingtonpost.com/blogs/crime-scene/post/mother-holding-baby-dives-from-carjacked-car/2011/07/14/gIQAJEXrDI_blog.html

Remarkable even for D.C.:
http://www.washingtonpost.com/local/trial-for-brother-sister-linked-to-green-card-slaying-begins/2011/07/13/gIQAaF4DDI_story.html

[Update 27 July 2011]
Beneath the barrel-bottom lies...
http://latimesblogs.latimes.com/lanow/2011/07/woman-allegedly-snatches-baby-from-stroller-slams-him-against-rail.html
("The woman later told police she wanted to eat the baby's arm...")

[Update 27 October 2011]
Noteworthy even for West Philadelphia:

http://www.msnbc.msn.com/id/45063859/ns/us_news/#.TqoCKjuo__M

[Update 19 Dec 2011]  Oh well oh-kay then...
http://latimesblogs.latimes.com/nationnow/2011/12/man-arrested-in-elevator-torching-says-victim-owed-him-money.html

[Update 29 Mar 2012] Mirror image of a currently celebrated Florida case:
http://news.sky.com/home/world-news/article/16198359

[Update -- nolens volens -- 30 May 2012]

The porn star and the body parts:

Naked man chows down on hobo’s face:

[Update 12 Oct 2012]
http://www.latimes.com/news/nation/nationnow/la-na-nn-mother-sentencing-hands-glued-20121012,0,6904031.story
Texas mom gets 99 years for beating, super-gluing girl's hands
Escalona, a mother of five, said she was molested, abused by boyfriends, and was a recovering marijuana and cocaine addict. She was raising her children on child support payments.
She glued the toddler to the wall of their apartment and beat her.
It’s too early to tell if she has lasting brain damage.

[Update 17 Oct 2012]


Parents ran exotic strip club in home as children watched
Investigators found seven adopted children, all under 11 years old, living there and at least five ecstasy pills sitting on the kitchen counter  
The children told officers that LaQuron Lacy, 43, would hit them with “fists, belts, hangers and metal objects, which caused them traumatic injuries and scarring,”
A 7-year-old girl also told officers Gregory Lacy had recently sexually assaulted her on a bathroom floor.

[Update 25 Nov 2012]

An alleged drunk driver plowed into a pedestrian and then careened along with the fatally struck man remaining on her windshield for more than two miles before stopping. Wilkins continued driving for about 2.3 miles with the man "embedded in her windshield.”
“O tempora,” you cluck, disgusted.
Ahh… but here’s the beauty part:
Wilkins, whose blood-alcohol level was more than twice the legal limit, told officers she was coming home from work.
From … work !?   What is she like when she’s leaving a party?


*     *    

Tired of marinating in a rehash of such squalid shenanigans?
Then bail out and read something timeless instead:
(For those not opting to do so --
We now return you to the folly of life here below.)

*     *     *

[Update, 26 November 2012]  Uh-oh, it gets better, i.e. worse.
First, she liked about the “work” part:
Wilkins told officers she was on her way home from work at the time of the crash, but Lisonbee said the facility only had daytime meetings on Saturdays and was closed.

But further, what was her “work” exactly?  Bartender?  Hostess in a strip joint? No-o-o….
The woman accused of fatally hitting a pedestrian and carrying him on her windshield more than two miles worked at a Torrance recovery center as a substance-abuse counselor.

[Update 27 July 2013]
http://www.latimes.com/local/lanow/la-me-ln-girl-held-prisoner-in-metal-box-was-runaway-from-la-20130727,0,3548542.story
 

This Will Count for your Final Grade


Essay question:

            God is Necessary.                         We are Contingent.
            Math is Necessary.                        Pickles are Contingent.

Discuss.

Thursday, January 20, 2011

ONTOLOGIA NUMERORUM


[This is a continuation of a thread begun here.]

“Die ganzen Zahlen  hat der liebe Gott gemacht…”
    -- L. Kronecker, in a rare moment of candor



So, God made the integers, good job; but what *are* they, exactly?  -- The question may be ill-posed, as unanswerable as what Matter really 'is', or Time.  A mathematician will be less troubled by this, than a physicist, since mathematicians are accustomed to things being characterized only up to isomorphism.  Even so, a thoughtful mathematician is aware that there is an ontological problem:

“The” natural numbers  have a conceptual existence which is quite independent of set theory [dbj:  Certainly it is cognitively prior], and it is an act of faith on our part  to agree that the set of “the” natural numbers  endowed with the “count-one-more” function  is a configuration satisfying the postulates for a simple chain.
-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p.  153


The Intuitionist philosopher Michael Dummett points out that, even in this most basic and most intuitive object, unexpected depths and terrors lurk:

The notion of ‘natural number’, even as characgterised by the formal system, is impredicative.
-- Michael Dummett, “The Philosophical Significance of Gödel’s Theorem” (1963), repr. in  Truth and other enigmas (1978), p. 199.

(Impredicativity is a petri-dish for paradox.)


Wishing to dish up something quotable  and a little less evasive than what St. Augustine said when asked the nature of Time, we coin this epigram:

Z  is the substrate  on which we grow the truths of number-theory. (*)

~

The best-known approach to 'reducing' numbers to something supposedly more basic, is that of set theory, in any of several varieties.
The world waited for Russell, for them to be declared “classes of classes”.   Thus ‘four’ is -- not merely may be analogized to, but is -- a great bulging bag of examples, one of these being the number of the Evangelists (or, in more modern terms, the Beatles).  -- So:  are we commited to an ontology of classes of classes?
            Maybe not.  Even Russell says that “classes…” (let alone classes of classes) “cannot be regarded as part of the ultimate furniture of the world.”  We can in fact, for the present, afford to be completely agnostic as to the best mathematical or philosophical characterization of integers, just as we can afford to be agnostic as to the best characterization of rocks.  For, nothing so far hangs on this characterization.  You can build castles out of rocks, and that didn’t change when rocks were discovered to be atoms surrounded by mostly empty space; they didn’t suddenly become porous, or bounce.  And seven will always be prime, whether it turns out to be best understood as a class of classes, a mess of masses, or the representative on earth of His Holiness Septimus the Seventh. In a sense, I’m not saying we need believe anything abstract or ethereal or (to begin with) even ‘mathematical’ about numbers:  I’m saying they’re already real as rocks.  But if you believe in rocks – if you don’t think you’re just a brain in a vat and are hallucinating them – then you must believe what you discover about the nature of rocks: some are hard, some are friable, etc.  And so it is with numbers: some are composite, some are prime, some are zeros of the Riemann zeta function.  Mathematically, these things just irresistably follow – follow without our pushing them along, like those now-unemployed angels behind the planets-- follow irrespective of philosophy.  Philosophically, we are at liberty to contemplate the integers with the breezy, boozy indolence of the chimp: one banana good, two banana more good, three banana more gooder still.  What we are not at liberty to do, is to exclude them from our ontology.
            Nothing really hangs on what numbers themselves “are”, or whether the question even has an answer, or has meaning.  Everything hangs on their objective transcendent fixity as positions in a pre-existent pattern.

            Another parable, on the inflexible reality of the integers.  Even if no-one in your entire universe ever ever counts anything, the numbers are present, like the stone guest at the feast.  (Rather as electromagnetic radiation is present, even if everyone is blind.)
            Take for instance planet Xymol.  There, all reference to anything quantitative, or logical, has always been taboo.  The activity of ‘counting’, which (by report) exists in other galaxies, they hold to be a merely contingent and local (and rather disgusting) custom, much like cannibalism.  The inhabitants of Xymol are purely “artistic”, purely emotional and qualitative; numbers, they maintain, are non-existent, and they have no need of them.  And one of the emotional and qualitative things that little Timmi von Xymol loves to do with his cubical blocks, is to arrange them into perfect rectangles (non-trivial ones: each side greater than a unit length).  As Timmi’s guardian, you want him to be happy.  But despite your best intentions, you’d better be careful what you give him.  If you give him 24, or 93784668225, or RSA-576 blocks, he’ll be fine, since these are composite.  But if you give him 17, or Mersenne-41, then after a great deal of fretting and fussing, the poor lad will break down in tears. For these will ever be prime, on the peaks as in the valleys,  though you deny the existence of numbers.

            The question, “What are numbers?”, is  strictly in itself  not very interesting: neither mathematically, theologically, nor philosophically.  Though Carnap’s outlook is in general foreign to the views developed here, his deflationary remarks anent that question, in “Empiricism, semantics, and ontology” [1950; repr. in Benacerraf & Putnam 1983] may stand.
            It’s like – What are penguins, really?  Godlike birds?  – to be sure.  Fusiform instantiations of ornithological perfection? – freilich, freilich.  But what are they quantum-mechanically, cosmologically, mathematically?  -- Oh, away with you, let us simply admire them, as they slide on their tummies along the ice!

 
-->
(*) Footnote, lest that splendid epigram be taken in a Nominalist sense.
The metaphor is actually fairly exact.  We don’t invent tomatoes:  we discovered them in nature, and now help them propagate their pre-existent kind.  The ‘truth’ of the tomato -- its genetic type -- existed prior to our discovery.   When we grow any individual tomato, it may be stunted, or waterlogged -- we simply do our best to approach the magnificent pre-existent Platonic Tomato.   And likewise with math:  our (imperfect) constructions and theorems are peasant-style approximations to the Type.

God grant that, in Paradise, my resurrected body may feast on a Platonic BLT.

[Further reflections here.]