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 whรญch of their fellows have free will and whรฉn 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 17th Fibonacci number for Mom, and the 19th 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:
