SimCity: the foundations

Background:   Topology is familiarly, informally characterized as “rubber-sheet geometry”.  That is, unlike the Euclidean geometry that we learned in school, which applies to flat rigid surfaces,  you’re allowed to stretch and bend the space, so long as you don’t tear it or let it intersect itself.

But at some point, we might like -- without returning quite to the simplicities and rigidities of the Euclidean picture -- to make our space… a bit less rubbery.   As the godfather of Calabi-Yau manifolds puts it:

We start with some raw topological space, which is like a bare patch of land that’s been razed for construction.  On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 77

[The above is an update to this post:

https://worldofdrjustice.blogspot.com/2011/08/urysohn-metrization-theorem-for-real.html  ]

Pucker Up

The world took notice when some old mathematical conjectures  were finally solved in our own lifetimes:  Fermat’s Last “Theorem”, and the Poincaré Conjecture.   To somewhat less fanfare, a conjecture concerning sphere packing, dating back to Johannes Kepler in the seventeenth century (yes, that Kepler -- the planet guy) was finally solved in the twenty-first.   A breakthrough came in 1998, but a formal proof was not recognized until this very year.

George Szpiro wrote an unusually readable account in his book Kepler’s Conjecture (2003).   It is an example of a sphere-packing problem,  which is a global problem, with applications stretching from the fruiterer’s tray to error-correcting codes. Among the related matters is the kissing number problem, which is a local problem:  how many spheres (or, in two dimensions, discs) can you pack around a given central sphere.  The answer is the Newton number or kissing number for that dimension.   

In two dimensions, the answer is six, as you can verify by ringing a penny with six of its fellows;  that no further coin could butt in, can be seen “by inspection”.  (That, incidentally, is the only known use for pennies, in our own day.)   For three dimensions (the everday space we live in), the answer turns out to be 12.  That is more difficult to see, though you can accomplish it using a baker’s dozen of beachballs and a dozen undergraduates.

Somewhat surprisingly, for dimensions four, five, six and seven, the answer is not known exactly, only upper limits.  Very surprisingly, we then suddenly get a break :

In eight dimensions, all of a sudden the Newton number is known exactly:  240 white balls can kiss the black ball in the center.  Why is that number known precisely?  In this case the upper bound came out to be 240;  but a certain well-known lattice arrangement, called E₈, also allows 240 balls to touch the central sphere.   Since the actual kissing number of E₈ coincides with the theoretical upper bound, 240 must be the highest kissing number. -- From dimensions nine to twenty-three, again, only bounds are known.   (p. 96)

What is so gratifying about this is that what is in essence a simply, concretely conceptualizable problem  should be solved by the testimony of so fiercely abstract an algebraic object as E_8.  (Szpiro calls it “well-known”;  depends what circles you move in.  I just asked the fellow next to me in the pub, and the poor fool confused it with the Leech lattice.)  I first learned about this enormous structure from a 2007 article in the New York Times.   Or rather, learned virtually nothing about it at all, beyond its mere name, since the journalist, and all hands asked, concurred that the thing is indescribable.  A sketch of that encounter, back from when we were shilling for Platonism, can be read here.

Language note:  This metaphor of ‘kissing’ (for tangency) is used elsewhere in mathematics: osculating plane, osculating circle.

The Ladder of Abstraction (with added rungs)

The following  logically belongs in the “Abstraction” section of our essay Consilience in Mathematics.  But as that effort is growing overlong,  we begin to cultivate here a particular idea  building upon that of abstraction simpliciter :  namely, the tendency, in modern mathematics -- and indeed this may serve virtually as the defining characteristic of modern (even: modernist) mathematics -- to abstract from any given abstraction, layer upon layer, rise upon rise, to a virtual (topless/cloud-topped) Babel, reaching to the Beyond.

(Oh, and here again we have a term from the arts, Modernism, which, as it includes “abstract art”, metaphorically applies to mathematics.  Compare our earlier essay on Minimalism in Mathematics.)

In normal practice, mathematicians mostly talk to one another -- and indeed, mostly just to those within their own hyperspecialized neck of the woods.  But occasionally, one writes an undergraduate textbook, and thus must descend to earth, if only for the nonce, and address the laity.  Thus:

This “intrinsic” formulation of Calculus, due to its greater “abstraction”, and in particular  to the fact that, again and again, one has to leave the initial spaces, and to climb  higher and higher  to new “function spaces” (especially when dealing with the theory of higher derivatives), certainly requires some mental effort, contrasting with the comfortable routine of the classical formulas.  But we believe that the result is well worth the labor, as it will prepare the student to the still more general idea of Calculus on a differentiable manifold.

-- Jean Dieudonné, Foundations of Modern Analysis (1960), p. 141

We dub this the “ladder of abstraction”, taking the phrase from our teacher of yore. Referring likewise to ascent into functions-of-functions, and function spaces, and functions from one function space to another, and to the duals of all that:

Detached from any context, this construction is a pointless formality.  But as we move up the ladder of abstraction, we find that constructions such as this  become commonplace …

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 43

The metaphor of ascent is well attested.  Gödel speaks of

the infinite series of ever stronger axioms of infinity, each of which expresses a new idea or insight.

-- quoted in Hao Wang, From Mathematics to Philosophy  (1974), p. 325

A mathematician writes of

the inferential staircase  leading from the laws of physics  to the world that lies about us …

-- David Berlinski, “The End of Materialist Science”, collected in:  The Deniable Darwin (2009), p. 160

~

Saunders MacLane,  in his book Mathematics:  Form and Function (1986), p. 36ff, has a section called “Mathematical Activities”, structured somewhat like our own in the Consilience essay.  Some of the topics are the same (analogy, abstraction, generalization), while others, not relating to consilience especially, differ (conundrums, axiomatization, proof).  One, intrinsic structure, seems to relate to consilience, but is only briefly developed; and the last, completion, we have treated under the more Quinean label of rounding out.

Now, abstraction and generalization are related notions, but neither entails the other.  MacLane acutely adduces the example of group theory.  Originally, this grew out of the concrete examples known as groups of transformations.  Later, algebraists abstracted into abstract groups.  Whether a generalization has thereby been achieved, is (as Chomsky likes to put it) “an empirical question”;  and in this case, it turns out, it has not. “No new groups turn up in this process, in view of the famous theorem of Cayley, which asserts that every (abstract) group is isomorphic to a group of transformations.”  Thus, in this case, the ladder of abstraction has only one rung.  (By contrast, abstract rings do turn out to generalize upon their original model, rings of integers.)

~

Seeking analogues of the Ladder of Abstraction  outside of mathematics proper, I happened upon this:

Quine suggests that levels of abstractness, modeled on Russell’s Theory of Types, might be established.  “In the beginning  there are only concrete objects.”  These constitute type zero and are the values of bound individual variables.  “To be is to be a value of a variable.”  Next comes first-order classes and relations:  they constitute entities of type 1 and are the values of bound predicate variables.  Classes of classes, and relations, constitute entities of type 2;  and so on.

-- Harold Lee, “Discourse and Event”, in: Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 297

The resemblance to Russell’s Theory of Types had not escaped me, but I rejected mention of it, since, rather than leading -- as the Ladder does -- to ever greater depth (the metaphor is here in distress -- maybe think of it as a ladder down a mineshaft), it seems to lead mostly to More of the Same.  In other words, forming those strata, as described above, is less like the dizzying and ethereal Abstract Ascent of mathematics, than simply forming new sets via the Power Set operation (a new and larger set consisting of all the subsets of the original set).  Now this, if we start with a finite set, leads absolutely nowhere.  It’s just like counting.  If you start with the whole of the Natural Numbers, now the Power Set operation does become more powerful, leading to new and incomparable levels of infinity.    Whether this leads to true new depth, or is rather a mere formal exercise, I do not know, since I lack all intuition of any infinities beyond the countable, let alone the Power of the Continuum or Measurable Cardinals.  Perhaps it does;  espresso-sodden Berkeley conversations about Quality emerging out of Quantity, return to mind.

Still, I am inclined to doubt it.  The very fact that the fellow can say “and so on”  virtually proves as much.  For there is no “and so on” to true mathematical abstraction.   There is nothing mechanical about such ascent -- it is more like a miracle.  You can proceed only one step -- nay rather, one leap at a time;  and the interval between leaps may take decades or even centuries.   Above the calculus lies Function Theory;  above that, Topology.  Above that, Algebraic Geometry. Far, far above us, hovers Category Theory, unreachably aloft.  And far, far above and beyond that, soars Topos Theory.  What comes next  is known only to angels.

To vary Nestroy’s celebrated epigram -- “Bis die Topologie gehts noch, aber von da bis sheaf theory  zieht sich der Weg.”

Additionally, Quine introduced the term semantic ascent.  There is some similarity to Gleason’s ladder of abstraction, but the ascent doesn’t go very high, and Quine himself -- perhaps surprisingly for a logician -- is wary of the upper reaches, preferring basic-level entities  behaviourally grounded.

Here the Russian author A. D. Aleksandrov, instead of envisaging a ladder,  uses the metaphor of layers  or (appropriately enough) of nesting, like Russian dolls, in the procession to affine or projective geometry and on to topology:

The properties of space are stratified … with respect to their depth and stability.  The ordinary Euclidean geometry was created by disregarding all properties of real bodies other than the geometrical;  here we perform yet another abstraction within geometry.

-- Aleksandrov et al, eds, Mathematics: Its Content, Methods, and Meaning (publication in the original Russian: 1956;  Eng. tr. publ. 1963), vol. III, p. 133

~

The more I think about it, the more this Ladder of Abstraction idea seems possibly fruitful.  Not so much as in the Theory of Types, but as in the scala naturae, which encompasses angelology.  (Compare also graded algebras.)

By contrast, mere ungraded “abstractness” in itself is of little interest. Thus, to take MacLane’s Group Theory example:  the so-called “abstract” groups (MacLane himself uses the sneer-quotes here) mean to lift aloft from Groups of Transformations, in that they retain the laws (associativity, inverses, and all that) while becoming agnostic as to the nature of the elements of the group.  But, first of all, groups of transformations are, compared with, say, pickles, already quite Abstract;  so the word adds, really, nothing.  Indeed, as soon as you say that two apples plus two apples are four apples, and that in the same sense  two penguins plus two penguins make four penguins (well, and a few more, after a while, if the sex mix is right), you are already indulging in such abstraction.

~

I tried looking up “abstraction” in the index of the various math textbooks and philosophy treatises on my shelves, and basically came up with  bupkes.  Thus, in Dummett’s omnibus volume, Truth and Other Enigmas (1978), we find no reference to abstraction per se, let alone to the Ladder of Abstraction, but only to “abstract objects” -- i.e., pickles versus the Meaning of ‘Pickle”,  the Idea of a Pickle, the set-containing-a-pickle, the… sandwich containing a pickle, the -- but enough.  Mathematics is so far beyond this, no comment is required.

~

The ethic -- even, the aesthetic -- of abstraction for its own sake, sociologically chronicaled here (“On Vulgar Numbers”), eventually evoked a backlash.

The Bourbaki group sought to present the entire abstract structure of all mathematical concepts in one set of volumes, the Eléments de Mathématique. In that treatise, the real numbers, which most of us regard as a starting point, only appeared midway into the series, as a special “locally compact topological group”.

An opposing idea, promoted especially in the Russian school, is that a few well-chosen examples can illuminate an entire field.

-- David Mumford, Forward to Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. xv

~

For the latest in fine reading, check this out:

http://www.linguasacrapublishing.com/justice.html

For more about abstraction, here:
         http://worldofdrjustice.blogspot.com/search/label/abstraction

Differential-geometric monostich

∫ ω        =    ∫   dω

  ∂D               D

[Such is the generalized Stokes Theorem, crown and culmination of our apprenticeship in Math 55.  Pronounce:  “The integral of omega over dee-dee  equals the integral of dee-omega  over dee.”   Chant to the calypso rhythm of  “First there is a mountain, then there is no mountain, then there is.”]

Semantics, local and global

[The distinction between the local and the global view, with increasing enrichment of the latter, has brilliantly characterized geometry (and cosmology) for the past half-century or so.  Addressing a different problematics, we proceed in that spirit  here.]

In math, the familiar notion of a function  normally (and only-ever in one’s earlier education) assigns some value to each of a set of points.  Later, one studies function-like entities, or functions sensu lato, whose mandibular gape is more capacious, taking in as argument, not merely a point, but a variable region, or even another function:  thus we come to study things called functionals, densities, and distributions.

In the simplest popular literature, such as the Hardy Boys on which my generation of masculine rascals was raised, semantic interpretation is local, even punctual.   The interpretation may well be more than literal, and require certain background conventions for proper appreciation, but the meaning may nevertheless be derived directly from the sentence in question -- it does not require any sort of “contour integral” through surrounding text, mapping its textual neighborhood into a richly layered semantic space.

Thus, consider:

            Joe’s jaw dropped.

This, our legent lad is to understand, denotes, primo, that Joe’s mouth widened somewhat, involuntarily; and that, segundo, this reflex was caused by emotionally tinged surprise. 

Such associations are to some extent conventional, and thus vary across cultures.  Part of the value of such digestible and repetitious reading as the Hardy series, is to inculcate such simple mappings.  To accomplish this does initially require the learner to pay attention to immediate context:  as, a preceding

There beneath the Christmas tree stood a shiny red roadster.

followed by

“Jumpin’ Jehoshaphat!   This is the best present ever!”

in which case the jaw-drop manifests surprise plus delight;  or else

There on the kitchen floor lay Aunt Gertrude, bound and gagged.

followed by

“WTF ?!??!!!”

(well, not that exactly), in which case the concommittant to surprise is rather dismay.

Surprise is the common factor, and persists for all later instances of falling jaws, even in the absence of such circumambient clues.   That there exists some attendant emotion  is likewise a given, with some (surprise, dismay) being much the most common, and others being almost ruled out (perplexity, embarrassment, boredom).  What objectively occasions the surprise in each new instance can be described as a matter of simple statistics, derived from such considerations as the relative likelihood of encountering a beribboned roadster on Christmas morn, versus a fruitcake, say, or a set of Lincoln logs (or, for adults of a Glengary Glen age, a set of steak-knives);  and whether Aunt Gertrude is a conventional reserved sort of elderly lady, or whether, rather, she is given to bouts of self-bondage, like that MI6 agent whose corpse was found bound in his bathtub.  Such frequency-profiles are of course relative to culture, and their calculation and application go beyond the content of the immediate sentence under interpretation, but they are soon solidified into background assumptions tacitly available to anyone competent in the culture, and need not be specifically keyed by the ambient text.

In time, one accumulates great hordes of such things, no longer needing any contextual guidance to interpret such conventional gestures as

Frank slapped his forehead.

Iola blanched.

Clint’s eyes narrowed.

Wolfe frowned and polished his glasses.

and so on and so forth.   As one’s literary experience proceeds, other, no longer culture-wide but character-specific physical indicia may be acquired:

Merlin’s thumb tingled.

(Cf. “By the pricking of my thumbs, something wicked this way comes.”)  But once acquired, their interpretation is straightforward, and purely local.

~

Consider, now, a character  not from any series, and whose acquaintance we have only just begun to make, in a character-crowded tapestry, the novel Martin Chuzzlewit:  to wit, Tom Pinch.  And consider this line, appearing isolated in a paragraph all by itself:

Mr Pinch opened his eyes wider, and looked at the fire harder than he had done yet.

Whatever can this mean?

[Note:  I shall pause here, pending future leisure to expand, and meantime let you ponder this, and to seek its context if you will, in Chapter 6.]

Tom has recently been sent to fetch Martin Chuzzlewit fils at the inn, to be Mr Pecksniff’s new architectural apprentice.  When we first meet him, coming in from the cold on a frosty night, he too engages in a bit of hearth-staring:

The stranger became thoughtful, and sat for five or ten minutes  looking at the fire in silence.

Here the significance of the action is quite different from that which applies in Pinch’s case.  Yet, different in a deep way, which requires psychological excavation to uncover:  not different in the merely algebraic way whereby x may denote, now 2, now ½, or his may be co-indexed, now by Frank (“…stroked his chin”), now by Rico (“… hefted the Lugar in his hand”).

Though born to wealth and privilege, Martin has recently been evicted from the good graces of his elderly uncle and theretofore-presumed bequeather-to-be.  And since (like half the characters in Dickens, it sometimes seems) he is otherwise an orphan, he has seen his “Great Expectations” (Martin’s phrase; where have we heard that before?)  evaporate before his eyes (which, had he been a Hardy Boy, would have “popped” at the news, along with his plummeting jaw).  He is accordingly a great fire-starer, by way of absorbing the warmth he feels to be his due, and of brooding upon his wrongs.  Here we see him later, at it again, with poor Pinch huddling on a neighboring footstool:

Beyond that, Martin is not (yet) a deep character, further than what we have seen, and indeed is typical of Dickens as being sharply (though, to a beginning reader, somewhat subtly) delineated, repeatedly bodying-forth a certain trait of character.  This is the sort of thing that has led some critics (unjustly I believe) to dismiss the Dickensian menagerie as caricatures.   It is likewise part of Dickens’ craft, to throw this character’s essence into greater relief, by means of characteristic physical gestures;  further, in the case of Martin, by the juxtaposed contrasting figure of Tom Pinch, who is something like Martin’s dual or inverse.  Where Martin was born to privilege, Tom was born to none.  Whereas Martin continually frets at any crumb that might be missing from his own bounty, Tom is grateful -- truly, deeply grateful -- for any scrap that might fall from the table of his betters.  (Dickens offers a literal picture of this, in Tom’s delighted feasting upon the wilted leftovers of the departed Misses Pecksniff.)

Tom Pinch is more of a puzzle -- more of a mystery.   On the face of it, he is a simple fellow, almost a simpleton;  certainly that is the opinion of the various Pecksniffs.  But there are a deep roots to Tom’s humility, to which the instincts of a Christian will quicken.  He is less a village idiot, than a Holy Fool.

And now these polar opposites are confronted.  Martin speaks casually, heedless of his snubs towards Pinch.  Pinch, from his good heart, never quite perceives these, just as he does not from the Misses Pecksniff.   Martin’s pretentions are likewise dutifully seconded, just like those of Pecksniff.  (Martin on his family’s failings, which fortunately “haven’t descended to me”; he must “be very careful that I don’t contract ‘em.”  “’To be sure,’ said Mr Pinch. ‘Very proper.’”)   But at length, Martin’s account of his haughty rebuke to his elder relative, is too much for poor Tom to swallow whole.  It is his glimpsing that flash of Satanic pride -- that non serviam -- which sets Pinch to staring, wide-eyed, sightlessly, into the fire:  into the depths of the fiery pit.

To resume the mathematical metaphor:   This ignispective image is visually equivalent for Martin as for Tom, but means something a bit different for each.  It represents, if you will, the punctual intersection of two life-curves;  and its meaning in momentum-space is given  not by the locus alone, but by the tangent.

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

The Theometry of Paul Erdős

Paul Erdős was no atheist; rather, he had an adversarial relationship to God.  He pictured the Creator as hoarding all the best math proofs in The Book, not wanting to share.   Life is basically a board game played against this miser;  and though we cannot win, we can strive to keep his winnings to a minimum.

The adversarial relationship recalls the tragic case of Lucifer; yet nothing else about Erdos seems actually diabolical.   For Erdős does not oppose the Lord head-on, but at a bias.  Simply, Erdős sees the Deity, not exactly through a glass darkly, but under some distortion -- perhaps a projective transformation.

Now, the same transformation must apply to the rest of the celestial denizens.  We may surmise the results.  The angels get recast as mathematicians (not much of a stretch).  And the Holy Spirit turns into the Friendly Ghost, the S.F.’s mischievous sidekick.   So soon as the Big Guy isn’t looking, Casper scurries off, say to rural India, where he finds the lad Ramanujan puzzled beside a hayrick.

“Psst! You know how, you take the reciprocals of the squares of all the integers, and sum them all up?” (Whisper whisper.)
Ramanujan, suddenly:  “I see it!  Pi-squared over six!”

[Update]  It turns out Heine already wrote a sort of sotie on much this theme:
Die Götter im Exil

[Morphological appendix]

He had studied theology.  But if theology and theosophy, then why not theography and theometry;  why not theognomy, theotrophy, theotomy, theogamy?  Why not theophysics and theo-chemistry?  Why not that ingenious toy, the theotrope?

-- Aldous Huxley, Antic Hay (1923), first page

 

The Urysohn Metrization Theorem: for real this time

Background:   Topology is familiarly, informally characterized as “rubber-sheet geometry”.  That is, unlike the Euclidean geometry that we learned in school, which applies to flat rigid surfaces,  you’re allowed to stretch and bend the space, so long as you don’t tear it or let it intersect itself.

But at some point, we might like -- without returning quite to the simplicities and rigidities of the Euclidean picture -- to make our space… a bit less rubbery.   As the godfather of Calabi-Yau manifolds puts it:

We start with some raw topological space, which is like a bare patch of land that’s been razed for construction.  On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 77

[Note from Jan 2011]  We were nonplussed to learn that this site comes up on the first page of Google search on “Urysohn Metrization Theorem”. 
[Note:  This has since changed, owing to a hack attack by the Nominalist Internationale.]
[Metanote:  It's back again, thanks to a counteroffensive by the Realist Underground.]
And ill at ease, since our post of that title is a satire on sociobiological/ultraDarwinistic  overreach, a satire of a sort practiced almost a century ago by G.K. Chesterton in his book The Everlasting Man.  Pity the unsuspecting physicist or math major who winds up there in hopes of learning the first thing about metrization, Urysohn or otherwise.   So we feel we owe it to these blameless Internauts to offer them at least a little something for their trouble.   Here, then, for the non-mathematician, or (God willing) the mathematician-to-be, is a thumbnail sketch of what led to this theorem in the first place. 

The intuitive content of the theorem is as follows.   If you have a space with enough structure to keep things apart which ought to be, and if the space itself is not too huge, then you can define a distance between any pair of elements.  The function that specifies this distance is called the “metric”, from the Greek word for 'measure'.

Thus, you mightn’t be able to do this if you lived in an oozy sort of world, where the minimal entities were like blobs with sometimes inextricably intertwined tentacles; nor if your world were scattered among separate universes.

(Footnote:  the idea is that you can come up with a nontrivial metric.  After all, any set whatever can be regarded as a (trivial) metric space, given the discrete topology.)

As for formal statements, these vary somewhat.  Here is a sampling.  (The following assemblage is an atavism from my days as a lexicographer at Merriam-Webster;  we worked from piles of attestation-slips, called "cites".)

John Kelley, General Topology (1955), p. 125:

Metrization Theorem (Urysohn)

A regular T1-space whose topology has a countable basis  is homeomorphic to a subspace of the [Hilbert] cube and hence metrizable.

James Dugundji, Topology (1965), p. 195, formulates it as

In 2-countable spaces, regularity is equivalent to metrizability.

and he labels this merely a “corollary” of

Theorem (Nagata and Smirnov) A topological space is metrizable if and only if it is regular and has a basis that can be decomposed into an at most countable collection of neighborhood-finite families.

The same can be said for the Bing metrization theorem, which likewise sharpens the sufficient condition into one both sufficient and necessary:  “a topological space X is metrizable if and only if it is regular and T0 and has a σ-discrete basis.” (Wiki)

Other formulations:

Stephen Willard, General Topology (1970), p. 166:

Urysohn’s metrization theorem.  The following are equivalent for a T1-space X:

(a)  X is regular and second countable

(b) X is separable and metrizable

(c )  X can be embedded as a subspace of the Hilbert cube.

James Munkres, Topology: a First Course (1975), p. 217:

Urysohn’s metrization theorem.  Every regular space with a countable basis is metrizable.

Michael Henle, A Combinatorial Introduction to Topology (1979), p.  283:

Metrization Theorem (Urysohn)

A compact Hausdorff space that is second countable is a metric space.

That one uses a stronger condition to reach the same conclusion, and is thus a weaker theorem; the same version appears here:

Boto von Querenburg, Mengentheoretische Topologie (3^rd edn.  2001):

Ein kompakter Hausdorff-Raum is genau dann metrisierbar, wenn er eine abzählbare Basis besitzt.

Something of an odd-man-out, possibly importing the stronger “normality” condition from the Urysohn Lemma, is this:

Seymour Lipschutz, General Topology (1965), p. 142:

Urysohn’s metrization theorem. Every second-countable normal T1-space is metrizable.

But cf. this:

George Simmons, Introduction to Topology and Modern Analysis (1963), p. 138, which offers a slightly stronger version, and names it differently:

Urysohn Imbedding Theorem.  If X is a second-countable normal space, then there exists a homeomorphism of X  onto a subspace of R-to-the-infinity, and X is therefore metrizable.

And indeed, the Lipschutz formulation is echoed much more recently in the October 2010 American Mathematical Monthly (“A Tale of Topology”, by Gerald Folland):

    Every second-countable normal space is metrizable.

If all that  already makes sense to you and seems obvious, you’re done.  If not, read on.

*

            The first order of business is to motivate the theorem.   What does it mean for a space to be metrizable, and why should we care?

            The space we’re best familiar with is the one we live in;  but the one we have studied most analytically, traditionally in high school geometry class, is the nice flat one, called the Euclidean plane.  This we studied  first by Euclid’s own methods, which date back over two thousand years, with axioms and proofs that justify each step -- the best possible mental exercise -- and lots of diagrams.  Later (if we stay the course) we take up a new approach, using analytic methods, which largely began with Descartes, in the seventeenth century.   Here we add a grid of axes, which measures exactly where each point is and how far apart they are, and prove things about figures: now not just triangles and circles and rectangles, but hyperbolas and cycloids and any shape we want, by means of equations.  You don't have much in the way of equations with Euclid;  for that, you need numbers -- given by the metric.

            And lo -- already, in these simple memories of high school, we have, in miniature, a picture of what has happened at the forefront of mathematical research over the past century or so.    For geometry,  in the sense with which you are all familiar, came to be generalized to a new subject, topology (originally called analysis situs -- both mean ‘the study of place’, as geometry means ‘the measuring of the earth’).   Whereas the Euclidean plane is rigid, we let these spaces get all stretchy and bendy.  In that case  we can no longer say what the circumference of a circle is, because by the time we wake up in the morning it may have stretched and drooped like one of Salvador Dali’s watches (in his painting, “The Persistence of Memory”):  but some things do remain true, such as the fact that that curve has an inside and an outside, meaning you can’t get there from here without crossing that curve.  (Note:  Such entirely general, almost naively simple-sounding statements are typical of topology.  The content of that one is called the Jordan Curve Theorem, and it's a real bear to prove.)

            Now topology was originally point-set topology, which mentally is rather like Euclid’s geometry:  you set up the ground rules for a space, then you ponder and visualize and reason things through, using pictures if you possibly can, and your own intuition.    Meanwhile, behind the scenes, a new view of topology was taking shape, somewhat analogous to what Descartes did for (or to) geometry:  instead of reasoning, half-intuitively, with spaces and shapes, you come up with an algrebra whose structures manage to reflect what is going on in those spaces in more detail, yielding numbers and equations and things you can calculate with.   It could have been called “analytic topology” by analogy with “analytic geometry”, but instead it is called algebraic topology.    Though very powerful, it is somewhat bloodless (at least for the beginner), and requires different habits of mind.   (Habits I alas lack.  Readers of my tales of woe will recall my bruising encounter with that subject;  the spot still smarts  in frosty weather yet.)

            So:  Cartesian geometry takes us, from shapes,  to the antecedently familiar realm of equations involving numbers.   Homology (a part of algebraic topology) takes us from more general shapes to the relatively tractable algebraic structures called Abelian groups.  

            The distance function in Cartesian geometry is what you get from the Pythagorean theorem.  The criteria for the general topological notion of a metric are a straightforward abstraction from this:  mainly, if you make a beeline from here to there, and another beeline from there to yonder, the distance traveled must be at least as much as had you simply gone straight to yonder from here.  As to what-all can meet the criteria -- ah, there lie surprises.

            Euclidean geometry is described as what you can do with a straight-edge and compass.   Sometimes people say “ruler” and compass, but that is a mistake:  we have no measurement-markings on our straight-edge; there are no pre-established units of measurement.   We can still determine that two different line-segments are the same length -- just take our trusty compass, measure the first segment with it, and now see if that compass-setting matches the endpoints of the second segment.  You might say that, in this world, length itself is not absolutely defined, whereas being-as-long-as is.   (This observation could be pursued in a syntactic direction -- that of incomplete symbols -- with interesting results.)

            Now, in the Cartesian approach -- analytic geometry -- we want to work with actual numbers, because that speeds things up.   So we turn the plane into a metric space -- “metric” just means ‘measurement’.   And the reason it is possible to do so is that the (pre-Cartesian) Euclidean plane was already rigid:  you do not change the length of something simply by moving it about; you can slide one triangle over to another one and see if they’re congruent.   

            Furthermore, the way we shall conveniently measure things  was already suggested to us by the celebrated truth of Euclidean geometry, expressed in the Pythagorean Theorem.   The earlier formulation of this was:  “the square on the hypotenuse is equal to the sum of the squares on the other two sides”:   meaning, the area of a square figure,  one of whose sides is the long side (the ‘diagonal’) of a right-angled triangle,  is equal to the sum of the areas of two other triangles likewise sticking off the shorter sides that are perpendicular to each other.    This was still a ‘point-set’ geometric view.   But now we start writing it in symbols, saying that, if x is the length of the one leg, and y is the length of the other, then the length of the diagonal is the square root of the sum of x-squared plus y-squared.  (This is known as the “Euclidean metric”.)  That’s algebra.  And the new viewpoint is reflected in the way you’ll here the theorem quoted nowadays: “the square of the hypotenuse is equal to the sum of the squares of the other two sides”:

            The rest  you are familiar with.  We briskly mark off a bunch of equal lengths along one direction, which we call the x-axis, and likewise along the y-axis that is perpendicular to it, and from this we get graph-paper, with its familiar grid.   And now our old friend the plane, which we first came to know as a tabula rasa -- the plane itself, and our own childish minds -- wears its Metric Space status on its sleeve, so to speak.

            This is all so familiar, that we are in danger of letting our memories do the thinking for us.   For in fact there are many different ways of deciding to set of a scheme of measurement on a flat surface.  We might stipulate that the ‘distance’ between two points shall be simply whichever is larger, the difference in the x-value or the difference in the y-value.  Or we might say instead that the distance shall be the square root of the difference of x-squared and y-squared, rather than their sum.   This is what Minkowski did, and it turns out to be the key to uniting space and time into a single Metric Space -- spacetime.   These and others are alternative possible metrizations of a plane.  (In one of them, it is possible to draw a round square -- the paradigm example of what philosophers tell us is impossible.   You can read about one here.)

            Of course, we don’t ourselves live inside of piece of paper (as Flatlanders do), we live in nice big rooms -- three dimensions rather than two.   We set up a third axis, the z-axis, like a tent-pole, to give us some breathing-space:  and now the grid shapes are little cubes instead of little squares.  Using this Euclidean metric, it turns out that an analog of the Pythagorean Theorem holds here as well:  we can consistently define the distance as the square root of the sum of x-squared plus y-squared plus z-squared.   Analytic geometry proceeds as before, with barely any change in methods (I originally wrote, "since the ones we used in the two-dimensional case were already so powerful"; but actually that puts the cart before the horse:  it is precisely such (unexpected) generalizability of a method that leads us to call that method 'powerful'.).    So now, instead of just circles and parabolas and so forth, we have a richer world of shapes, like cones and spheres and ellipsoids and helices, and on and on.  (Actually these were already known to the Greeks, though how they managed it with the pre-Cartesian methods they had, is something of a miracle.)  And it continues to be easy to prove things about these, since we still have basically the same metric, which is well adapted to equations and their numerical solutions.    Likewise in four dimensions, and on up as high as you like.
(Note:  People get all spooky when they hear things like 'fourth dimension', but these metric methods absolutely tame them. -- Children:  Study math.)

            Bottom line:  A metric space is a very convenient thing to work with.  You can pretty much know where you are and do what you want, even when the space gets hairy in other ways, like being infinite-dimensional, or very curvy.

            But.

            In the inexhaustible splendor of the Creation, there are many many different spaces, more numerous than the stars.    They sprang full-blown from the Creator’s brow, and it is up to us to discover their structure.  Unfortunately, when we first meet up with one of these, it may not be wearing a nice convenient metric on its sleeve.  It may be a very confusing, huge, menacing, squishy blob.  -- Recall that when we first met the plane, it too came without a pre-drawn grid.  But a particular grid was already implicit, because Euclid’s axioms imply the Pythagorean theorem.   (There are other metrics you could adopt where that theorem wouldn’t be true, but these would not conform to our everyday local experience, which is why Euclid chose the one he did, and why it took two millennia to generalize the naive notion of "distance" to the mathematical notion of "metric".)

            So:  Faced with such a blob, can we come up with a consistent measuring-scheme that will tame it, by turning it into a metric space?  That is, is it metrizable?  -- And the answer is:  Sometimes you can, and sometimes you can’t.   So, we want to come up with ways we can tell, whether the project is doable or hopeless.  It’s a bit like figuring out whether you can tame a given kind of animal.   Long ago, people figured out that you could do that with dogs, and later with horses, to a huge extent, so that instead of being wild beasts they are actually useful.  Cats, it turns out can be tamed to a lesser extent -- tamed to tolerate us, so long as we are not late with the cheeseburgers. They’re not useful, but they’re decorative.  (Meanwhile in Catland, the lecture reads:  “Peeps are redonkully e-z 2 tame;  goggies, not so much.”) So, you see a puppy and, no matter what breed it is, the mere fact that it is Canis familiaris tells you that you have an excellent chance of taming it:  though just how to do so may vary with the breed.  In a similar fashion, when we first meet an untamed space at the Space Store, we can know, by certain signs (which Urysohn specifies in his theorem) that the thing is in principle metrizable -- though it doesn’t give us a useful metric just for free;  for that we still have to do some work.

*

            If you and I are points in a metric space, the metric tells us how far apart we are:  that’s like analytic geometry.   But topology lets out the sails a bit.  In the most general topological space, you can’t say how far apart two points are -- but you can always say what bunks with what.  The bunks are known as “neighborhoods” or (roughly synonymous) “open sets”, and are given as part of the very definition of the space.   Thus, the most featureless space of all, justly (in this metaphor) called the “indiscrete” space, everybody bunks with everyone else, no privacy at all.  In the opposite, the “discrete” space (discrete, not discreet), each man is an island.  But most spaces, and every space of interest, is in between.

            To describe just where they fall on this in-between spectrum, we name various “separation” properties.   The simplest common one is that any two points can be separated (any two people can sleep in separate bunks).  In a metric space, this is easy.  If you and I are a certain distance apart, then if I draw (if we’re in a plane) a circle or (if we’re in a fatter space) a sphere  around me, with a radius less than that distance, then I’m in my own special bubble and you’re outside it.   Such elementary capacity of separation is also available in most non-metric spaces. This basic degree of separation is called T1.  If you and I can each draw such a neighborhood simultaneously, even better -- the space is “Hausdorf”.

            A more demanding requirement is that I can fix around myself a bubble (a neighborhood) which keeps me clear, not of just a single point, but of any collection of points called a “closed” set.  A set is closed if, for any point you can creep up on, in an infinite sequence, that point is in the set.  So for instance, on the number-line, take all the reciprocals of the natural numbers, ½, 1/3, ¼, etc:  these creep up on zero -- they get as close as ever you please, so the set of these reciprocals is not closed:  to close this set, you have to add zero.  Or, take everything inside a circle.  You can creep up to any point on the boundary, from within the circle, so the interior is not closed.  Add the boundary, now it’s closed.   -- So:  given a point, can that point stay clear (hide inside a bubble), not just from any other point (that’s easy), but from any other closed set that doesn’t contain that point -- no matter how pushy and encroaching?   Well, if that set is closed, it can’t keep creeping up on me indefinitely:  at some stage, it can’t come any closer, otherwise I’d be a limit point of that set, and since I’m not in that set, it wouldn’t be closed.  So at some point it keeps its distance -- I’m safe in my neighborhood (say, Beacon Hill), where the menacing set cannot encroach.  (In a metric space, this is easier to visualize:  it keeps its distance -- say, d.  So I draw a little bubble round me, of radius smaller than d, and I’m safe.(  -- Spaces that are like this are called regular.  Obviously, every regular space is Hausdorf, but not vice versa.

            (The next step up is:  Can any two closed sets -- not just one closed set and a point -- be kept apart by disjoint neighborhoods (neighborhoods that don’t intersect)?  If so, that space is called normal.  Normality is used in the Urysohn lemma, basically unrelated to the UMT.)

  

            So, being regular suffices to keep things separate enough to define distances between things.  But regularity by itself is not sufficient -- the space might be too ‘big’ to fit in a metric.  How big is too big?  Bigger than second-countable, the other premise of the U.M.T.  In that case, points can be just too far apart to have a finite distance.

*

None of these definitional and formal considerations  gets across the real power of the metric-space idea.   This arises when we begin to consider a more abstract sort of space, in which the “points” are not characterless, dimensionless ideal dots, but … functions.   (This is an example of the “Ladder ofAbstraction”.)   Andrew Gleason puts the matter well:

The assignment of a metric to a set of functions  gives this set an intuitively geometric character.  The success of the theory of metric spaces in analysis  can be attributed to the remarkable insight into the nature of functions  which has come from exploiting the geometric point of view.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p.  226

[ continued here …]