Labyrinths of Reason (24 page)

If the first digit requires 30 seconds of computation, the machine will calculate all the digits of pi in a minute.
¹

What’s more, at the end of the minute, the machine will be displaying the actual, bona fide “last” digit of pi! Of course, the latter is pure moonshine, for pi has no last digit.

Rounding out this trio of impossible machines is the “Peano machine.” This is something like an automated slide whistle. The whistle is calibrated like a ruler. One end is labeled “0,” the other end “1.” A plunger travels from the “1” end to the “0” end at constant velocity in one minute’s time. As the plunger passes a point whose inverse is a whole number, a pair of mechanical lips announces the number. The machine’s voice gets higher as the plunger moves down, allowing it to recite ever faster and faster.

For instance, at the very start of the minute, the plunger is at 1, and 1/1 is 1. The machine recites “One” in a rich baritone. Thirty seconds later, the plunger is at 0.5. The inverse of that is 2, and the machine says “Two” (now in tenor). Ten seconds later comes a contralto “Three.” Five seconds after that, a soprano “Four.”

Toward the end of the minute, the recitations come fast and furious. The crescendo becomes too high-pitched for anyone to hear.
Dogs whine and paw the ground frantically a few moments … then even they can’t hear the machine. By the end of the minute, the name of every natural number will have been spoken.

The infinite, a symbol of the vast world which cannot be fully grasped, is a common motif of paradoxes. Frequently, the paradox has the infinite impinging upon and threatening the complacent everyday world.

Among the oldest paradoxes of the infinite are those attributed to Zeno of Elea (lived fifth century B.C.). Zeno described his paradoxes in a book (written about 460 B.C.?) that has been lost. The paradoxes are known to us only via the often abbreviated accounts of them by other ancient writers. Zeno was a curmudgeon who delighted in demonstrating that time, motion, and other commonplaces cannot exist. His best-known paradox goes like this: Swift Achilles races a tortoise; the tortoise has a head start of, say, a meter. In order to overtake the tortoise, Achilles must run the meter to the tortoise’s starting point. In the time it takes him to do that, the tortoise will advance a shorter distance, 10 centimeters. Now Achilles must run 10 centimeters to gain the lead. Meanwhile the tortoise pulls out 1 centimeter ahead. This analysis can go on forever; the tortoise’s lead dwindles but never does Achilles overtake it.

Zeno denied the reality of infinite series or quantities. He felt that if you could show that something involved an infinity, you proved it couldn’t be. To modern minds, some of Zeno’s arguments are less compelling. Zeno is apt to come off as a mathematical crank who never got the hang of infinite series. The series of intervals Achilles must run adds up to a finite total: 111.1111 … (or 111 and 1/9) centimeters. The “infinity” is more in Zeno’s analysis than in the physical situation.

A more puzzling invention of Zeno’s is the arrow paradox. An arrow flies through the air. At any point in time the arrow is motionless. The instantaneous arrow is like a still photo of the arrow or one frame of a movie of it. Time is made up of an infinity of these instants, and in each instant the arrow is stock-still. Where is the arrow’s motion?

The arrow paradox repays further thought. Put it in a more modern context. We have an arrow of atoms; it moves in the space-time of relativity and is measured in an inertial frame of reference. Even
in that context, “an instant of time” has something of the informal meaning it did to Zeno, We still believe in cause and effect (except at the quantum level: negligible here?), that the future is determined by the present, and the present by the past. Now, in that frozen instant of time, what distinguishes a moving arrow from a stationary one? It would seem there must be some information attached to a moving arrow that identifies it. Otherwise, how does it “know” to jerk forward in the next instant?

More within the scope of this book are the contemporary “infinity machine” paradoxes above. Inspired by Zeno, they question knowledge rather than kinematics. Nor does the modern concept of an infinite series do anything to dispel them. The operation of each machine is a supertask, an infinity of action that, while perhaps impossible, can be described unambiguously. In each case, the supertask promises us a glimpse of the Medusa—something that seems unknowable.

The practical-minded may question the point of infinity machines. Philosophic discussion of supertasks is like a doctor looking for a cure for a nonexistent disease. However, there is an analogy between supertasks and certain real-world processes. The peculiar status of questions that can be answered only through an infinite (or “practically infinite”) series of discrete actions is worth exploring.

Some discussion of infinity machines has focused on their nuts-and-bolts operation. Although their practicality seems irrelevant, a slightly more detailed analysis may point up logical difficulties. Adolf Grünbaum analyzed all three machines.

One objection to the Thomson lamp might be that a light bulb can’t be switched on and off infinitely quickly. Past a certain point in the process, the filament won’t have time to heat up fully when the current is on, or cool off when the current is off. Possibly the filament will remain half incandescent throughout the last moments.

Besides, everyone knows that switching lights on and off is a good way to burn out bulbs. The Thomson lamp’s bulb would burn out for sure.

Grünbaum argued that these issues are not crucial. The riddle is, will the light be on or off at the end of the minute? After the minute is up, you can always unscrew the burnt-out bulb and pop in a fresh one. Will it light up?

The real problem is with the switch. The on-off button in a Thomson lamp evidently travels a distance
x
each time it is turned on or off. Therefore the button must travel an infinite distance in a finite time. To mention just one physical objection: Near the end of the minute, the button must be traveling faster than the speed of light, which is impossible.

It isn’t essential that the button traverse an infinite in-and-out route—after all, it’s not going anywhere. Grünbaum and Allen Janis tinkered around a bit and came up with a modified Thomson lamp that would be more plausible.

Picture the button as a vertical cylinder with an electrically conducting base. When the button is fully depressed, its base makes contact with two exposed ends of the circuit. The current flows through the base and lights the bulb.

Whenever the lamp is supposed to be on, the button is resting on the open circuit. Whenever the lamp is supposed to be off, the button makes an up-and-down excursion at unvarying speed. Each time, the button ascends only as far as is permitted by the time available and the fixed speed.

During the first 30 seconds the button is flush against the circuit ends and the bulb is lighted. During the next 15 seconds, the bulb is off. The button travels up for 7.5 seconds, then down for 7.5 seconds. Then it stays put for another 7.5 seconds, closing the circuit and lighting the bulb again. Next, the button ascends for 1.875 seconds and descends for 1.875 seconds, keeping the bulb dark for 3.75 seconds.

The button ascends and descends an infinite number of times, but
each time it ascends only a quarter of the previous distance. It is like a not very bouncy ball. The total distance traversed is as tidily finite as the total time for the operation. The speed is constant and much less than the speed of light.

Unfortunately, Grünbaum and Janis’s modified Thomson lamp isn’t completely housebroken. Its back-and-forth motion must also entail arbitrarily great acceleration and deceleration. Presumably, infinite acceleration is easier to swallow than infinite velocity. Still … any physical object can stand only so much acceleration. At some point the acceleration would destroy the button as surely as if you smashed it with a hammer.

A worse problem with the modified lamp is that there is no question whether it would be lighted at the end of the minute. The base of the button remains ever closer to the open circuit and finally ends up right on top of it (just as a bouncing ball ends up flush against the floor). The modified lamp would definitely be on at the end of the process. Unsatisfyingly, this is due to the modified switching arrangement. What if any relevance this has to the original Thomson lamp is problematic.

Other problems, some similar and some not, face the builder of a pi machine or a Peano machine. (The latter, by the way, was named by Grünbaum in honor of Italian number theorist Giuseppe Peano.) The pi machine raises the question of how it could calculate the digits of pi so quickly. As we will see, there are limits on computation as binding as those on motion. To avoid infinite velocities, the numerals would have to pop up and down ever smaller distances. Eventually you wouldn’t be able to tell which digit was being “displayed.” An alternative model of the pi machine prints the digits of pi in a surreal typeface where every digit is half as wide as its predecessor. The complete printout fits on an index card, but not even the most powerful electron microscope will reveal the last digit.

A problem unique to the Peano machine is the ever-lengthening names of numbers. It takes a long time to rattle off the name of a hundred-digit number. Janis suggested that the machine dispense with English names and “whistle” the numbers in a code that pairs every number with a tone of a certain frequency.

The energy required to produce a sound depends on the frequency (pitch) and amplitude (loudness). The amplitude of the tones must decrease in step with the increase of the frequencies to avoid an infinite requirement of energy. By the end of the minute, the volume control on the mechanical lips will be down to zero. You
couldn’t hear the last whistle, even if you were capable of hearing tones of infinitely high pitch.

Notice this: The attempt to render any of the three infinity machines more physically realizable leads to the conclusion that the result would be invisible (or inaudible). Many philosophers think there is something fishy about infinity machines, supertasks, and “facts” knowable only through supertasks.

The literal infinite is inconceivable, but that which verges on infinity is everywhere. An Indian legend tells how King Shirim was bested by his grand vizier, Sissa Ben Dahir, the inventor of chess. The king was so grateful for the new game that he offered to reward Sissa with a gold piece for each of the 64 squares of a chessboard. The vizier politely declined and asked for an alternative reward. He asked the king to place a single grain of wheat on the first square of the chessboard, then place two grains on the second square, four grains on the next square, and so on, doubling the number of grains on each succeeding square until each square of the chessboard was covered.

The king was taken aback by the modesty of Sissa’s demand, and called for a bag of wheat. The grains were counted out carefully as Sissa asked. When the king’s servants came to the 12th square, they had trouble fitting all the grains on the square, so they continued by placing the vizier’s allotted grains in a heap to the side of the chessboard. The king saw to his amazement that the bag of wheat ran out before the 20th square could be accounted for. He sent for more bags of wheat … and finally gave up. Not all the wheat in his kingdom, or in India, or the
world
would fulfill Sissa’s request.

The moral, oddly mathematical for a folktale, is never to underestimate a geometric progression. The king’s original offer of gold pieces was directly proportional to the number of squares on the board. Had Sissa designed the chessboard to have 81 squares or 49 or some other number, it wouldn’t have made all that much difference to the king’s grand gesture. What are a few gold pieces more or less to the wealth of a king?

Geometric progressions, however, grow beyond all worldly limits of wealth or anything else. The fact that the unit of the vizier’s request, a mere grain of wheat, was so trifling compared to a gold piece scarcely changed matters.

Let’s see how many grains would be required to satisfy Sissa’s
request. It is 1 + 2 + 4 + 8 + … Another way of writing this is 2
⁰
+ 2
^l
+ 2
²
+ 2
³
+ … 2
⁶²
+ 2
⁶³
. (The series ends with 2
⁶³
, not 2
⁶⁴
, because the first square has 2
⁰
or 1 grain.)

The sum of a series of consecutive powers of 2 is always 1 less than the next greater power of 2. That is, 2
⁰
+ 2
¹
+ 2
²
(= 1 + 2 + 4) is one less than 2
³
(= 8). The total number of grains of wheat needed works out to 2
⁶⁴
– 1. That equals 18,446,744,073,709,551,615.

There are something like 100 million grains in a ton of wheat, so this amounts to about 200 billion tons. The current annual production of wheat is only about 460 million tons. The king owed Sissa about four centuries’ worth of present world wheat production. Obviously wheat production was a lot less back then. (Just how far back is uncertain, for the date of the invention of chess is unknown. Like baseball, it went through several incarnations, and whether a historic Sissa Ben Dahir existed is also unknown.)