Adventures in Lineland

It is to be hoped that you have all had occasion, at some point in your childhood or thereafter, to meet the marvelous mathematical allegory Flatland, by Edwin Abbott, followed (in adolescence or later) by the masterfully pedagogical The Shape of Space, by Jeffrey Weeks, which takes you further into higher dimensions.

Here we do the opposite:  We take as our starting-point, a starting-line, either the continuum or some more manageable countable dense subset -- seemingly too exiguous to be interesting.  But remember:  Mind pervadeth all spaces of every sort;  and so we examine what it might be like, to live in this one.   A Leibnizian-Wolframian fantasy, fleshed-out.

[Note:  A prior exercise in Lineland physics was that of Ernst Ising in the 1920’s.   The one-dimensional Ising model attempted to model ferromagnetism, using the simplest possible assumptions.  It proved disappointing -- no phase transitions.   A two-dimensional model produced much more interesting behavior; a three-dimensional model  has resisted exact solution.]

ADVENTURES IN LINELAND

            At first blush, Lineland would seem a constrained sort of place.  As, if its population includes Tom, Dick, Harry, and Mary, arrayed along the line in that order, then Tom can never enjoy the immediate propinquity of Mary; and Harry is stuck forever next door to Dick, even if they don’t get along.

Yet Lineland  can actually be experientially rich, though the creatures cannot move or switch places.  And this, despite a certain apparent simplicity of its citizens. Its residents are monads (the term is due to Leibniz): physically, cells.  Their instantaneous expressivity is maximally limited: the only signal they can emit is “On” or “Off”; but internally, they have enormous storage.  The (public) state of all Lineland is identical with the pattern of On & Off along the line; what are the thoughts of the individual monads, we cannot know (they are 'noumena', 'Dinge an sich' -- Kant this time).  These publically inspectable states  evolve stepwise through quantized time, as in cellular automata. The physics is such that an “On” state of my neighbor  n cells down  emits a signal of strength ½^n (that is, falling off another half-strength with each cell) Hence the perception of a given monad  at any instant  consists of two real numbers (indeed, rational numbers, since n is always finite), one for the world to his right  and one for the left. The numbers may be represented as binary expansions, exactly corresponding to the pattern of On’s & Off’s. Thus, if my right-hand neighbor is Off, and the next beyond him On, the next two Off, and the rest On, the signal-strength is .0100111111….  The sensitivity and storage-capacity of the perceiver determines how many of these digits will actually be perceived by a given monad at each reception.  

            As time advances, the pattern unfolds: we, from above, out of time, see the whole thing spread out like a carpet. 

            The states of Lineland evolve according to some single given rule, of the sort familiar from cellular automata: thus predictably, so long as the world is merely “kicking over”.  However – and this is key --  at least some of the monads have (a very elementary sort of) free will. 

            Now there is, of course (lest you think this concession too grand), only one thing they can do with it: namely, refrain from turning “On” when the ground rule says they should (or vice versa).  These rare but bold interventions are then perceptible ‘from above’  as a switch in the pattern – which then propagates automatically for all time (in all the surprising ways that cellular automata can toss up, albeit deterministically), long after the rogue monad has fallen back into step.

This free will, though it is defined as the ability to resist the dictation of the master pattern, and though unambiguously displayed  only when that ability is exercised to produce a contradiction to that dictation, need not be exercised always as contrarian (like the youth who, to demonstrate their freedom from some convention, unanimously and invariably adhere to some other convention).  Any given monad might indeed have a rule (the rule being fixed, but its adoption free), that (to take a random example), whenever you have laid down five Zeros in succession, you output a One – whether or not this would be in accordance with the master program.

 

            Further, the sentient monads can ascertain  whích of their fellows have free will and whén they exercise it.  This, even though the capacity of any monad be finite.  For, suppose the reception capability sufficiently capacious to perceive the signal-strength  N-cells-out  on either side, and to retain a record of these states for T instants.  And suppose that the ground rule allows for determination  upon next output of a given cell  from no farther than K cells away.  Then the ambient band N on either side  produces an output in the next band N minus K on either side  that is completely determinate, providing it follows the ground rule.  (In the side-band, influences from outside the perceivable 2N bleed in, so the evolution there is anybody’s guess.)  If, within this narrower band, any cell does not manifest the predicted next state, it has exercized its free will on that step. 

            Different monads have different personalities.  Some exercize their free will sparingly, some often; and a few contrarians (who might as well be dead) invariably do exactly the opposite of what the ground rule tells them.  Monads exercise their will in a variety of entirely different styles.  Some are absolutely random; others, absolutely determinate; others exercise it when they darn well feel like it (whether this last option is anything more than some blend of stretches of pattern and stretches of randomness, is not immediately clear).  Those that are determinate (forever, or for a stretch) may be so in an infinite variety of ways.  One may defect from the ground rule  only at even instants; another, only at multiples of three; another, at prime numbers.  Others are determinate but not predeterminate:  Lineland’s loveslaves, these defect at t = T+1 if and only if their beloved neighbor  M cells to the right  defected at t = T.   Of course, the monads may fall in and out of love, mimicking another’s pattern  only for a time. Other monads blend all these strategies (in a bewildering variety of proportions).  As: Defect when and only when t is a Fermat prime, unless some specified defection pattern of one’s neighbors occurs  (as, iff an even number of neighbors at positions -3, -7, -22 to the left  and 5, 9, 220, 5555 to the right  have defected at t; this pattern itself may be fixed, or may evolve – deterministically or otherwise), unless one happens to feel contrary that day and does otherwise.

Monads have a psychic lifespan.  Before their soul is instilled, and after they die, they never exercise their will, but turn on and off as predicted by the ground rule.  Lifespans are a connected subset of the timeline, and may (exceptionally) be infinite, either into the future or into the past, or both.

            Internally to the monad, this lifespan is (we conjecture) entirely determinate: the soul is either present, or it is not.  But from outside, it is in principle difficult to tell  when life ends or begins (cf. our own inglorious extremal stretches, of blastula and senility).  For, suppose that a given monad has been inanimate for all time (that is, always following the ground rule), then suddenly at a given instant (call it t = 1) defects for the first time, thereafter defecting precisely at Fibonnaci numbers.  It hews to this pattern for a quintillioan iterations, then falls forever silent.  We may say that it has died – by definition, never defecting is “as good as dead” – but we cannot say when, just as we cannot say actually when it was born, even approximately. For its free-will pattern may have been: Follow the ground-rule for a quintillion iterations; then Fibonnaci for a quintillion; then the ground-rule for three quintillion; then blink on and off alternately for as long as you remain alive.  It’s actual lifespan may be anything from one quintillion to five quintillion iterations (it definitely died before it could implement its plan to blink alternately during its golden years of retirement).

            Reincarnation seems to be possible.  This consists in a personality (i.e. a temporal pattern of defection) reappearing after it has been extinguished for a time; it may reappear at the same cell, or in a different one.  If (as is usually the case) the personality in question lived only a finite time, then its pattern is largely indeterminate, and the reincarnation therefore only approximate or probable.   Still, if a monad, during its recorded life, emits a quintillion-long Fibonacci pattern followed by a sextillion-long pattern of Fermat primes followed by a digital representation of “Yankee Doodle” before falling forever silent; and then another monad (having been forever silent itself), exhibits exactly the same pattern before it expires, then we may certainly say that, while they lived, they exemplified the same spirit.  (Of course, had they both lived longer, they might have in time diverged – we’ll never know.)

            Such reincarnation can, of course, be multiple, and (if in different monads) temporally overlapping.

            Monads can also get married and have babies.  As, a monad defined by the keystream “Emit Fibonacci”, and one defined by the keystream “Emit Fermat Primes” ,may – at any offset (say, staring at the 17^th Fibonacci number for Mom, and the 19^th Fermat prime for Pop) – blend their instructions:  Junior (who may be born at any specific delay later) emits the mod-two sum of his parents’ keystreams.  Sometimes the results are rather beautiful, as when a digital recording of the violin part of a sonata (Mom) weds and procreates with the piano part (Dad).

            Polygamy, we regret to report, is permitted: a child may have any finite number of parents (its keystream being the mod 2 sum of all its progenitors).  There is even incest: a child may be the mod 2 sum  of two identical keystreams at some nonzero offset.  Par-polyploidal self-cloning is, however, fortunately self-stultifying: should some self-important monad attempt to bud off an offspring parthenogenetically with the mod-2 sum of 2k copies of its own genome (with no offset), the result is identically zero: the child is stillborn.

            Life in Lineland, I hope I have shown you, is a perpetual festival, a riot of laughs. Perhaps that world strikes you nevertheless as more monochrome than our own.  But then consider.  How many different personality types does our own world show – how many (however intricately varied) personal individual quiddities?  So far, only finitely many, at most as many as the number of people who have lived.  -- But I mean, in principle.   Fifty billion? umpety jillion? infinitely many?  Well, in Lineland, there is a one-to-one correspondence between personalities and real numbers: thus, there are uncountably many.  Nor are these, though admittedly numerous, as blandly indistinguishable as the real line seen “from afar”, all unnumbered.  There are, as we have seen, a variety of styles – actually, of classes of styles – or should we say, collections of classes of styles… -- in personality, each one of which has an unbounded number of variations.  And any one variation – any one little perfect little round little self-sufficient monad – contains any amount of evolving variety – sometimes an unbounded amount; sometimes infinite in both directions.  And despite the fact that all one monad can learn about any other monad at any given instant is whether that monad is Off or On, over time it can know an unbounded amount, up to the capacity of its own memory, about an unbounded number of fellow monads. 

            Furthermore, despite the solipsistic flavor of the basic metaphor, life in Lineland is boundlessly social.  Your own actions are affected (though not determined, owing to your ready reserve of free will) by the states of your neighbors.  At any given instant, K of them on either flank  are inputs to your next step, and mK on either side  to your action m steps later.  This can lead to cooperation, but also, alas, to conflict.  As, Mike the Monad likes to be flanked by neighbors (up to some depth; or, as many as possible) that are On as often as possible.  So he cleverly varies his output in ways that will tend to effect this (of course, the effect is not always immediate, but may take time to make itself felt).  If his immediate neighbors are inanimate, he can eventually build up a pretty cosy little neighborhood for himself, all On and atwinkle like Christmas lights, most of the time. And if his next animate neighbor, Marvin, twenty doors down, is like-minded, they can together do even better at keeping alive a nice On column between them (Marvin fending off contrary propagations from the right, Mike from the left). If, however, they have opposite desires for the real estate between them, then life is an endless battle of once-twice-three-shoot.

Such, then, is life  among the merry little monads.  And, as it may be, among ourselves.

~

For a comparable Minimalist  mathematical fable, check this out:

http://worldofdrjustice.blogspot.com/2010/12/constructivist-angelology.html

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.

The Urysohn Metrization Theorem: an Apology

The other day, we learned to our distress, that, owing to an earlier post, a Google search on the phrase 

 => “ Urysohn Metrization Theorem “ <=

brings up our own Cantor-cum-woodchucks site  on the very first page.   This is embarrassing, since we have never said anything substantive  specifically about this well-known theorem.   Innocent youngsters searching on this phrase may be led to a page that only purports to deal with that celebrated result of point-set topology, but which actually is simply a satire on sociobiological overreaching, a genre practiced by G.K. Chesterton a good hundred years ago (in The Everlasting Man and elsewhere).
[We interrupt this post to bring you a sobering update, 11 I 11:  that first-page hit has been axed, probably by the Nominalist International.  For an an earlier such episode, click here.]

By way of atonement, we present a picture of the saintly man himself: 

"Golly, I sure do hope y'all enjoy my swell theorem!"

            To be sure -- the Urysohn Metrization Theorm was a spot-on choice, among many possible such choices, to represent something for which an adaptationist account must collapse of its own absurdity.    Nor can the U.M.T.  readily be smuggled into a “spandrel”, as an exaptation of some more basic skill.  It would be a stretch even to demonstrate the differential, for successful procreation, of the… conscious… ability to count up to twenty.   All sorts of complex non-conscious, instinctual behaviors can be of such value, as witness those mighty engineers, the spiders.   (Note:  I became fonder than ever of spider-kind, upon reading of the sheer variety of their engineering styles, in the pages of Richard Dawkins.   Let that be noted here, in case I am ever driven, nolens volens, to say something not-nice about the man.)  But conscious mastery is requisite, though not sufficient, for extension of our curious mathematical faculty  to such attainments as proving -- nay, even conceiving -- the Poincaré Conjecture. 

            So:  Instinctual unconscious knowledge  by no means suffices  for further conscious exfoliation of its possibilities.   Neither birds nor bats will ever invent the moon rocket.   But note further -- at this point  simply as a curiosity -- that our instinctual arithmetic armamentarium  is pretty paltry.   Some languages don’t even have words for integers above three or so.   (I am leaving a lot out at this point, in the logical sequence;  but were it filled in, it would only strengthen the case.)   I came across a passage recently in which one scholar wished to project all of mathematics from our instinctual ability to recognize groups of two objects, as such, and of three objects, as such.   But he misses a subtle logico-linguistic point.  That ability, whatever it may be, is not in itself evidence of the ability to count -- that is to say, to have mastered the successor-function, and have grasped the resulting structure.   The instinctual world of such peoples is populated by pairs, and by triads, and perhaps quartets and, for some, maybe even a quincunx:  but these no more form an intrinsically and extendably ordered sequence than do squares and triangles.   Pairs and triads populate the tribe’s ontology -- but qualitatively, not quantitatively:  much in the manner of tigers and lions.  Such ability (which scarcely extends beyond half a dozen or so, apart from supposed cases of idiot-savants) is not an instance of actual counting, but a qualitative (non-quantitative) substitute for it -- and which will help you get by, so long as your horizon does not extend much beyond the six-pack.

            Other adaptationist accounts of advanced (and by no means universal) human capabilities -- say, Mozart is to birdsong as wings-for-flying are to winglets-for-heat-exchange -- though really just just-so-stories, are still ingenious enough, and have a certain plausibility:  birdsong is, after all, demonstrably utilized in courtship, and likewise in our own species:  the swain beneath the balcony, with his lute.    You’ll have a harder time with Schoenburg -- dodecaphony ne’er won fair maid.   But mathematics is far more recalcitrant, because vastly more developed.  To say it’s all just a spandrel -- well, here the spandrel would be bigger than the entire cathedral.

*

            Actually I am quite sympathetic to the program of trying to discover as much as we can about the causal-temporal structure of the biosphere along Darwinian lines, as I am sympathetic to every serious attempt at explanation which employs -- always bounded by common sense -- a due reductionism.  Thus, parts of chemistry have indeed been illuminated by (if not quite “reduced to”) parts of physics:  quantum mechanics and the structure of the simpler atoms; statistical mechanics and aspects of thermodynamics.   The complexity of the biological Creation becomes only more beautiful when lit up by the insights of evolutionary science -- what had seemed random and unrelated, even absurd (what’s the whale doing with that tiny useless floating bonelet where a hipbone ought to be?), stands revealed as a splendidly articulated and unfolding (“e-volving”) panorama:  no longer mere complexity or complication, but intricacy, worthy of a watchmaker.

*

            To return to the Urysohn Metrization Theorem and to our heartfelt apology.   We feel a responsibility to those unsuspecting souls, astray in the Googlewoods, who in quest of enlightenment concerning the Urysohn Metrization Theorem, should wind up at our own humble cottage door.   Accordingly we have formed the intention to make good, and to come up with a narrative that may illuminate this result (the Urysohn Metrization Theorem, I mean), beginning with no more than the reader may recall from high school geometry, yet winding up with some intuitive feel for that theorem (namely, the Urysohn Metrization Theorem).   This represents a serious undertaking, since first we’ll have to understand the darn thing  (the Urysohn Metrization Theorem, that is).  But we are inspired by the splendid example of Sir Jeffrey Weeks (we have just now decided to knight him) in The Shape of Space.   If we can do, for the <<Urysohn Metrization Theorem>>, even a part of what he did in bringing the geometry and topology of three-manifolds within the grasp of the suckling undergraduate, we shall not have lived in vain.

Note:  Anyone who fancies that this post about the Urysohn Metrization Theorem, which bears the phrase “Urysohn Metrization Theorem” right in its title, is simply a ploy to keep our stats up (namely as regards a phrasal search on “Urysohn Metrization Theorem”, or perhaps “Urysohn’s Metrization Theorem”, or “The Metrization Theorem of Urysohn”, or even “Urysohn:  Whither the Metrization Theorem?”, “The Urysohn Metrization Theorem for Dummies”, “The Urysohn Theorem:  Metric or Menace?” and “The Urysohn Metrization Theorem explained for the Tired Business-Man”, usw.), is just plain cynical.