Scenes from Office Life

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

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

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

For more mathy funnies,  here.]

Achilles and the Tortoise: the inside dope

Famed indeed is the fable of Achilles and the Tortoise, known since Antiquity to every schoolchild.  And yet the telling of it has steadily decayed, until contemporary versions have become equivalent to the old Irish Bull about how you could never walk from your table to the bar  for another pint, since in doing so, you would first have to traverse half the distance, and then half the remaining half, and then half of that remaining segment, and so on “to infinity”.    Anyone who has every enjoyed a cold, foaming pint of Guiness ™  knows how absurd that is.  -- Bartender!  Same all round!

Anyhow, the tale of Achilles racing the tortoise, as told by the mathematician Zeno in ancient times, recalls an actual contest, held on the plains of Troy ca. 1278 B.C. (Historians differ as to the precise date.)  For a bar bet, the wily Thersites bet Achilles than he couldn’t outrun a tortoise, if he gave the tortoise a head start.  The Achaean hero snorted and said, “Ha!  I’ll grant the creature a thousand cubits.  Gentlemen, place your bets!”

Achaeans and Trojans alike foregathered at the appointed time, Achilles arrogantly lounging on his shield, the tortoise waiting humbly, a thousand cubits in front.   Unfortunately, Achilles was better famed for his fleet foot and his ferocity, than for brains (you had to go to his countryman Odysseus  for that), and had neglected to inquire the length of the race.  In fact, it had been set at precisely … one thousand and one cubits.

Achilles, realizing belatedly that he’d been had, set off like lightning at the sound of the bell:  but when he finally reached the finish-line, there stood the tortoise, contentedly munching on grass.

Thahhhhh ...  Winnnnahhhhhhh !!!!

Zeno’s arithmetical point was that, although Achilles was much faster than the tortoise, he was not a thousand times as fast, and thus lost.  Indeed, even had he been a thousand times as fast, he’d have been vanquished: since in the time he covered the first thousand cubits, the tortoise would have traveled one cubit, to the finish-line.  (Left as an exercise for the reader:  How fast would Achilles have to be, to beat the tortoise under these conditions?)

In the event, Achilles had the last laugh, since he supped on the tortoise, in the form of soup.

The tale then takes a darker turn, as arcane mathematico-philosophical disputes led many who had lost the wager  to refuse to pay up;   a free-for-all ensued, which led directly to the Trojan War.

[Note:  You were probably told some foolish tale in school, about how the cause of it all was a woman.   As if.  We shall not deign to refute  that account.]

~

As the centuries went by, the lessons of that fateful day  were lost, finally issuing in a folkloristic version for children, “The Tortoise and the Hare”.  Once again, the tortoise wins;  but the mathematical substructure has been completely discarded.  Proving once again that the level of geometrical sophistication has sadly declined since the Age of Troy.

Constructivist Angelology

But yet when considered, may help us to enlarge our thoughts  towards greater perfections of it  in superior ranks of spirits. … The several degrees of angels  may probably have larger views.

-- John Locke, An Essay Concerning Human Understanding (1690)

Man’s understanding, though allied to the angelical, operates differently.  The angels understand intuitively, man by the painful use of the discursive reason.

-- E. Tillyard, The Elizabethan World Picture (1942)

It is presumably not obvious to the chimpanzee (or, if this be setting his smarts too low, to the humble woodchuck) that for all m, n in Z, m + n = n + m.  Nevertheless, in his daily scurryings and burrowings, he will repeatedly meet up with particular instantiations of this modest truth.

            For the woodchuck (at any event the southern northeastern lesser striped variety) builds a number of nests and other temporary dwellings, each of which has the framework of a variously triangulated  polyhedron, built tinkertoy-fashion from a fixed number of sticks.  Now, gathering them one by one would take too long, nor can the tidy woodchuck stand to have any sticks left over.  So when constructing his summer dwelling -- an icosahedron, which needs thirty sticks (did I get that right? My calculating powers are not much beyond those of a woodchuck) -- he normally harvests a jubjub bush, which has twenty-two sticks of exactly the right specs and which blooms in the spring, then rounds it out with the eight-sticked glubglub bush, which sprouts slightly later. 

But then one year, the blooming of the jubjub was delayed, and the woodchucks despaired.  All but one, the enterprising Willie, who went doggedly (or groundhoggishly) ahead  and harvested the available glubglub, supplementing this  when the jubjub arrived slightly later.  This remarkable exploit was recorded in the annals: for 22 then 8, one may substitute 8 then 22.

            It was subsequently found that a mubmub bush (18 sticks) followed by a nubnub bush (12) would do just as well – und zwar, in either order!  This fact too was recorded.

            The years went by, then the centuries, and the millennia, and the annals grew to seven times seventy stout volumes, densely filled with such arcana as: a cube-for-cubs may be constructed of a lublub (7) plus a rubrub (5), and this in either order; and so on for billions of examples.  All this was considered a branch of botany, a purely empirical science.

            By this means, the woodchucks arrived at an analogue of Babylonian mathematics.

Interlude:   A physicist depicts the arithmetical state-of-play in a papyrus from Egyptian/Babylonian times:

It records the resolution of a great number of fractions  into a sum of aliquot parts,  the original numerator always being 2:  as, for instance,

2/97 = 1/56 + 1/679 + 1/776

But no rules are given for effecting such resolutions, and the whole treatise seems to be a mere compendium of results obtained by repeated trials.

-- James Jeans, The Growth of Physical Science (1947 [posthum.]; 2^nd edn. 1951), p. 11

            Until one day one Wisedome Woodchuck, a distant descendant of Willie, figured the whole thing out, and in a remarkable demonstration of only eighty pages (rather hard to follow, but sound), showed that m + n = n + m  was a perfectly general fact, replacing the seven-times-seventy volumes at a stroke, and freeing up his brethren for yet further architectural innovations, which previously had been shunned, as their particulars were not yet in the book.  The annals were placed in a museum, which the elder woodchucks might still visit, marveling at favorite exhibits (as who could forget that remarkable winter, when 5,878 + 519 turned out to be equal to 519 + 5,878?  A tour de force!). Meanwhile generations of young woodchucks (the pride and despair of their parents, who could not follow them into Canaan, with their aging brains) studied Wisedome’s proof, breaking their little heads against it.

           

Meanwhile in Metropolis… The humans, learning of this, politely saluted Wisedome’s modest accomplishment, and experienced a pang of sympathy for woodchuck-kind; yet felt no inclination to visit their Museum of Particular Results: for which they felt, indeed, a kind of horror.  And even the general result, while true, is somehow to us not truly interesting. In any case we are all too busy wrestling with the Riemann Hypothesis, to have time to look back.

Meanwhile in Elysium, where throne the angels sensu strictior, the lowest order of angelic beings sensu lato, a mock compliment is paid to Andrew Wiles, who finally figured out that little Fermat puzzle, with which the angel-kind  are wont to amuse the nursery.  Not that the angels arrived earlier at his proof, nor any refinement thereof.  They simply scoop up a few infinities of integers with their fractal fingers, twist them this way and that—and see, it doesn’t fit!  Simple.

            Moreover, all facts about all structures of ordinal type omega, whether or not deducible by any finite axiomatization, are equally transparent to the angels. They just look.

            So, is Elysium the mathematical Paradise?  Not quite…

            In a remarkably lucid and accessible article*, which should be packed into every pupil’s lunchbox by a considerate mom, Gödel observes that our continuing failure to resolve Cantor’s continuum problem, left over from the previous century, is quite an embarrassment.  It means that we are unable to wrap our minds around the very simplest multiplication problem possible, beyond the finite ones that these days can scarcely stump a woodchuck. Namely, two times two (times two, times two – keep going).  He writes:

            “It is easily proved that the power of the continuum is equal to 2^(aleph-nought). So the continuum problem turns out to be a question from the ‘multiplication table’ of cardinal numbers: namely, the problem of evaluating a certain infinite product (in fact the simplest non-trivial one that can be formed).  There is, however, not one infinite product (of factors > 1) for which so much as an upper bound for its value can be assigned. […] It is not even known whether or not m < n implies 2^m < 2^n.” 

            We are  so to speak  staring helplessly  at a pile of sticks.

            Nor does the subsequent Cantor+Cohen demonstration of the independence of the continuum hypothesis  from a particular system of axioms for set theory   set the matter aside. Gödel had already anticipated Cohen’s result, and wrote:

A proof of the undecidability of Cantor’s conjecture from the accepted axioms of set theory (in contradistinction, e.g., to the proof of the transcendency of pi) would by no means solve the problem.  For if the meanings of the primitive terms of set theory … are accepted as sound, it follows that the set-theoretical concepts and theorems describe some well-determined reality, in which Cantor’s conjecture must either be true or false.

            Indeed Gödel suspects that the Cantor conjecture is actually, factually false: which means that somewhere, among the actual literal real numbers, there is hiding a set of cardinality intermediate between aleph-nought and its power set, with definite members which the angels could name.  Not, however, the lowest order thereof; this lies beyond them.  But at the next step up, the archangels hang these sets from mobiles over their infants’ cribs.  In fact a woodchuck may somewhere inadvertantly have used one of these sets for nesting materials, and even now lies sleeping on it – a night of troubled dreams.

            So much for a simple pancake-stack of omega-many deuces – the limit of the lower-angels’ ken.  What about the square root of omega-to-the-omega; or cross sections of fibre bundles on toroidal cap-omega-cross-theta space? For each level of angels, there will be something beyond them that they just don’t get.

*

There are two poles of the range of approaches to the problem of infinities.  One is that of the badger-like Brouwer, who simply sweeps the chessmen to the floor, folds up the board and goes home.  (An only somewhat more amenable figure, says Gödel, is Weyl, who allows as how there might be something to board games, but suggests we play checkers – or Chutes ‘n Ladders – rather than chess.)  The other pole says:  Infinities are tricky, but they all exist, and are present to the Infinite Mind. Gödel himself uses that term, e.g. noting that Ramsey’s admission of formulae of (countably) infinite length  might be constructivistic for an infinite mind  but not for our own.  Gödel does not, however, seem to feel much need for any desperate appeal to such a mind, in the course of an ordinary day, since he -- like Badger’s amiable friend the Water-Rat-- is a thoroughgoing Realist, and comfortable as such in his own skin.  For him the assumption of infinite classes “is quite as legitimate as the assumption of physical bodies, and there is quite as much reason to believe in their existence.”  The outwardly gloomy Austrian  is really the jolly Dr. Johnson of set theory.

            Only now there’s a problem, of a sort which did not confront the schoolmen, who never counted on the uncountable:  the Infinite Mind is all very well, but -- Which infinity did you have in mind?

            Who comprehends *everything*? God does, by definition. Yet He cannot be simply the crown on a tower of constructively ascending intelligences.  He is like an “inaccessible cardinal” – and not the first.  Nor perhaps ‘the last’, if there is no last.  Whatever He might be, there is Cantor in the wings, grinning, waiting to perform a Power Set on God, yielding – what?  -- Nothing one can begin to commence to pretend that we can approach with our sadly finite understanding.

            All of which suggests, if nothing else does,  that God is something more and other than an alternately wrathful and affectionate granddad  with a perfectly enormous white beard – however much longer that beard might be, than the stubble which disfigures your chin or mine.  Who one day, apparently from sheer idleness, as one might choose chocolate, chose the Jews.  Who later, some say, cast a Jove-like eye  on a certain Palestinian virgin.  And who at present is very angry indeed with the Democrats (or the Ravens, or whomever).  Yet what He in fact might be, we cannot even begin to imagine anyone’s beginning to conceive.  (Cf. the suggestion of 1 Kings 8:27  that the heavens themselves have heavens (and so on up); and that the whole omega-tower of them  cannot encompass God.)

*     *     *

~ Commercial break ~

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

We now return you to your regularly scheduled essay.

*     *     *

            We actually wind up with a sort of hamstringing of the Ontological Argument. Notoriously its conclusion does not really follow from its premise;  but now even its premise limps: “Since we can imagine a Perfect Being…”  But that’s just it, we can’t!  Not even little infinite bits of one! Yet paradoxically (and God reportedly loves paradox – at least Chesterton does, His publicity agent on Earth), this seeming stomping on the prostrate corpse of the offspring of Anselm, this despairing cry that somehow even Infinity does not suffice, so far from opening the agora  to legions of snickering atheists chanting “Toleja so!”, points somehow upward, -- outward,   -- onward ….  Praise Him!

Postscript:

John Locke himself, normally regarded as the Poster Boy for Empiricism, of I'm-from-Missouri common-sensicality, yet delivers himself of this (Essay, III.vi.12):

That there should be more species of intelligent creatures above us, than there are of sensible and material below us, is probable to me from hence:  that in all the visible corporeal world, we see no chasms, or gaps.

That is to say:  The gap between ourselves, and God, must somehow be filled, according to the Principle of Plenitude.

And again (IV.iii.23):

He that will consider the infinite power … of the Creator of all things, will find reason to think, it was not all laid out upon so inconsiderable, mean, and impotent a creature, as he will find man to be;  who  in all probability, is one of the lowest of all intellectual beings …

Angels of all sorts are naturally beyond our discovery, and all those intelligences, whereof ‘tis likely there are more orders than of corporeal substances, are things, whereof our natural faculties give us no certain account at all.

Since theism is far from central to Locke’s Essay, it is curious to see the emphasis on this scala naturae idea.

--------------

*”What is Cantor’s Continuum Problem?”, repr. Benacerraf & Putnam, eds., Philosophy of Mathematics.

~

Postscript:  For the possibility that the structure of certain mathematical truths relating to an infinite domain  might resist any but a case-by-case “Babylonian” approach, cf. the quotation from Michael Dummett towards the end of this post:

http://worldofdrjustice.blogspot.com/2012/01/what-is-truth.html

Compare further (re ascending ranks of abstraction and generality):

"The Ladder of Abstraction"

.

Mathematical hamsters

The following hamsters are mathematical:

Fluffy

Fuzzy

The proof is left as an exercise.

For extra credit:  Show that Fluffy and Fuzzy constitute an orthonormal vector basis for the entire mathematical hamsterspace.

[Hint:  Sad;  Happy;  get it?]

A menace to penguins

We interrupt your weekend merry-making  with a red alert:

Let us imagine a planet covered with calm water. If you drop a large rock into the water at the North Pole, a wave will propagate out  in a circle of ever-increasing radius.  In due course, however, this circle will reach the equator, after which it will start [inexorably] to shrink,  until eventually the whole wave raches the South Pole at once, in a sudden burst of energy.

-- “Manifolds and Differential Geometry”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 44

Moral of the story:  DO NOT DROP A LARGE ROCK INTO THE WATER AT THE NORTH POLE !  It would swamp the penguins!

[Note for connoisseurs:  This effect is an analog of what S|G|NTers call "antip*dal recepti*n", one variety of the 'whispering-gallery' phenomenon.]

Incidentally -- For an example of a manifold in the form of a differentiable penguin, click here.

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

Infallible: A Parable

[a continuation of this]

<Bi-smi llâh :>

This mini-essay concerns, not the unfailing truths of mathematics, which predate us, but the human practice of mathematizing, which (like everything else situated within History) is in principle contingent, and thus eminently fallible. (For more on the distinction, click here.)

As a matter of curious historical fact, our mathematical progress has been much less contingent than you might expect.  There have been no real false-starts, and many simultaneous independent discoveries.  Why that should be, would be a subject for a dissertation in itself, except that no human could write it, as it would require a God’s-eye view of the mathscape, across which we mortal/mental Flatlanders  grope our way …

And while we are on -- or rather off the subject, not having yet commenced the actual mini-essay itself, nor so much as hinted at its topic (today is Sunday, you see, and we are at leisure): 

A God’s-eye view -- truly?  Possibly not.  Perhaps an angel could write it:  much as a three-dimensional Fatlander can tell the Flatlanders what’s what, without requiring yet higher dimensions.  But if an angel were to write such -- could we understand what he wrote?  (“Wenn der Löwe sprechen könnte, wir könnten ihn nicht verstehen.”  Angeli, a fortiori.)   More generally:  Could we ever understand something that  in principle  we ourselves could never have discovered unaided? 

One is inclined to think, not.  -- Which in turn suggests a very depressing prospect of the Afterlife, to anyone who envisions the latter  more in terms of Topological Pataphysics than in terms of harps, and gowns, and tennis played on clouds (the New Yorker-cartoon school of eschatology.)   The Resurrected Body -- a shimmering mystery, but --  soit.  Yet what of the Resurrected Brain?

Yet it may be, that we have already had a taste -- here below -- of how we might handle this.  For we have had Revelation, both Scriptural and mathematical (though the mathematicians modestly call it Intuition).   The result?  Some of us understand some of it, somewhat, some of the time.

[End excursus.  Or precursus.  Whatever.]

<Ammâ ba`d:>

And so to the Parable.

(1) The closest we come to innovative external guidance towards what we think or hope might be the Truth, in the moral sphere, is from a pronunciamento from our spiritual leader, be he Pope, or imam, or (for Babarites) Babar.  Otherwise we rely on traditional community consensus -- ijmâ`, to use the useful Muslim term for this.

(2)  In mathematics, there is no Pope or mufti, nor could there be.  There is usually a “dean” of mathematics at any one time, and occasionally a “Prince” (Gauss).   The latter we admire;  but we wouldn’t take their word for anything -- anything at all.
(Andrew Wiles proved Fermat's Last 'Theorem';  I grovel in the gravel should his shadow pass.
Yet let Wiles opine, that Diet Pepsi is the beverage of choice;  and I reply,  "Oeww.... reahhhhlehhh....")

Thus, the proposition erroneously known as “Fermat’s last theorem” would doubtless never have received all the effort and attention it did, had it been known merely as “Schlumpfnerd’s Conjecture”.   But for all that, the prestige of Fermat did not in itself incline mathematicians to reckon that the proposition was probably true, and they soon came to the conclusion that Fermat had been deluding himself as to possessing a(n unwritten) proof.   The real reason the proposition is so celebrated is because of that endlessly retold anecdote about his having a proof but the margin was too small to contain it.

Outsiders might imagine that Proof is Pope, but that’s not quite right either.  Though we  from time to time  do use explicit Proof,  here too  consensus is the ultimate arbiter  -- e.g., re whether a purported proof is actually valid.  This is not to say that mathematical truth is socially constructed, merely that our growing/groping understanding of it  is partial, and historically conditioned.

Occasionally -- although, in general,  mathematics has been blessedly bereft of fitna -- occasionally  consensus is lacking,  not merely on the technical status of a purported proof (think:  de Branges), but about the philosophical status of a means of proof:  thus, Intuitionists rejecting the non-constructive.

(3) Both theological and mathematical understanding may be revised and enlarged  in light of subsequent discovery or revelation:  and indeed, in such a way that the previous understanding is not so much refuted, as sublated -- that is, transcended, as a saint transcends the ovum.

Such a state of affairs does not mean that the concept of Truth is no more than: “true for our time”, on the slippery slope down to:  True for the Trobrianders;  True for Solipsistic Intersexuals;  True for J.Q. Ortcutt, sole member and founder of Ortcuttism, a new religion.  Truth, more than beauty, is aere perennis.

Now:  I have mentioned that the historical course of mathematical invention (presumably because it is largely a matter of mathematical discovery) has run more smoothly than one might have anticipated, given the repeated futilities of all human activities else.   Yet let us, in this Parable -- it is almost a sotie -- imagine that the muse of mathematical history, Clio -- or as it might be, Clotho:  for we are in that dim time, before the gods -- had willed things differently (wise, yet wilful, is She):  and that initially, there had been no Parallel Postulate.   That is to say, mathematicians began with a slightly different Euclidean Creed.   The postulate was by no means denied;  it simply was not generally noticed;  and by those (and they were most of them) who tacitly took it for granted, they took it for granted as well, that others too took it for granted.

And indeed, one can prove enormously much without ever invoking the thing -- all the truths that go over to the general Gaussian case.   Yet for certain particular propositions -- as gradually became clear -- the thing is needed.

At which point (so our parallel parahistorical fantasy runs), mathematicians fell to quarreling, dividing into warring camps;  and discord reigned in the land.   Practitioners fell into sects:  the Parallelists;  the Intersectionists; and the Diversionists (the latter prefiguring Riemann resp. Lobachevsky).  Yet none could convince the other, and the acrimony waxed hot.   Many the bloody broil  that ensued:  and many the skald, lyre-strumming atop the hill, who sang brave deeds of battle.

Now:  among the sectarians, none could  by any means  point to any concrete grounds for his belief -- or call it, an intuition -- yet only the more stubbornly, and with an inmost light, held fast to what he took as Truth.

At length -- the wide world  weary of the fray, the fields in ruins, sheep scattered amid the tares -- a great Conclave was called.  And after hearing-out each side, Pope Euclid the Great stood up at last, and, spreading his robes before the multitudes, proclaimed, “infallibly”, that the Parallel Postulate should hold sway.  It was duly added to the Credo.

Plain men rejoiced, since this axiom, after all, matches our everyday intuitions.  And so the matter remained for over 2,000 years.

Then lo, there arose in the West, not one prophet but two:  Saint Riemann, aye, and e’en Saint Lobachevsky (known to the faithful of another pastorate as Saint Bolyai), each proclaiming a new truth, axiomatized and elaborated with startling clarity;  and the scales fell from our eyes.  Then the Church of Mathematics welcomed this advance; and the Riemannian Postulate, and the Lobachevskian Postulate, took their honored places in the geometric tryptich, alongside that of Saint Euclid:  which remains, in our own day, in its own sphere and as far as it goes, absolutely True.

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, beyond 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 (even for a bounded function on a compact domain): 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; others still are, yet their collection can converge to a function that is not.   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.