Labyrinths of Reason (25 page)

Thomas Malthus’s famous tract was motivated by the realization that the world’s population was increasing geometrically but that food production was increasing at only an arithmetic rate. Malthus had reason to believe that the number of new acres open to agriculture each year was approximately fixed. Thus the food supply could grow something like this: 100, 102, 104, 106 … On the other hand, the rate of population growth (depending mainly on the number of babies born each year) grows with the size of the population itself. The more people of child-bearing age, the more babies. The world’s population tends to double every so many years, growing like this: 1, 2, 4, 8, 16, 32 … This, like Sissa’s reward, is a geometric progression. It is destined to outstrip the available food supplies, resulting in global famine, Malthus warned.

“Geometric” is a poor term for these series, the analogy to geometry being weak and confusing. A better term is “exponential,” from the word “exponent.” Exponential growth is characteristic of living organisms. Whether it is a bacteria culture or the population of the human race, the number of new individuals is proportional to the total number. Savings accounts with compound interest grow exponentially—a circumstance that evidently has something to do with the fact that it is living organisms that lend and borrow, create an economy that grows exponentially, and trade in currency that inflates exponentially.

Exponential growth may be described by a simple mathematical
function. A function is a procedure that transforms one number into another. Think of it as a special key on a pocket calculator. You punch in a number, press the key, and get a new number. The square root function (which is a key on many calculators) gives a number that when multiplied by itself, yields the number punched in. If you enter 36 and press the square root key, you get 6.

A function does not have to be one of the functions you find on calculators. Any clear and exact procedure for constructing new numbers from old will do. You can define a function as 67 times
n
plus 381 (for any number
n)
, and it will be a valid function. A function is often described in an equation like this:

“f(
n
)” is read “the function of
n.”

Just as it is natural to wonder which animal is the largest or the fastest, mathematicians have wondered which functions are the largest or fastest-growing. Some functions overtake other functions. A function is said to be “bigger” or “faster-growing” than another if its values are always greater
for big enough values
of
n
. Notice that there is no limit on “big enough.” If function A is
A(n) =
1,000,000,000,000,000 and B is
B(n) =n
, B will take a long time to catch up to A. For any
n
greater than 1,000,000,000,000,000, though,
B(n)
will be greater than
A(n)
. B is therefore faster-growing than A.

Neither of these functions is remarkable. Any constant function—where
f(n)
equals a fixed value—will eventually be surpassed by any function that is proportional to
n
. It is also evident that any function proportional to
n
²
will outgrow either type of function. A function proportional to
n
³
will ultimately grow bigger yet, and likewise for functions in
n
⁴
,
n
⁵
,
n
⁶
, and so on.

A
polynomial
is an expression, such as
n
³
+ 8
n
²
– 17
n +
3, combining powers of a variable. A polynomial describes a function, and the relative rate of growth of a polynomial function is, loosely speaking, a matter of the highest power. The function
n
³
+ 8
n
²
+ 17
n
+ 3 grows much larger than any function whose highest power is
n
²
;
in turn it is surpassed by a function containing
n
⁴
or a higher power.

Many functions grow yet faster. Malthus’s pessimism was founded on the fact that exponential functions grow faster than any polynomial functions. In an exponential function, you set a certain constant number to the power
n
(rather than
n
to a constant power).
f(n)
= 3
ⁿ
is an exponential function. This means 3 multiplied
by itself
n
times. If
n
is 2, 3
ⁿ
is 3
²
or 9. When
n
is 1, the result is just the base (3 in this case), and when
n
is 0, it is defined to be 1 no matter what the base. So the values of 3
ⁿ
for 0, 1, 2, 3, 4 … are 1, 3, 9, 27, 81 … Each successive value is 3 times bigger than its predecessor. The higher the base, the faster the function grows. The successive values of 10
ⁿ
are 10 times bigger, and the values of 1000” are a thousand times bigger.

In complexity theory, the difficulty of problems is most commonly measured by the time required to solve them. It goes without saying that not all persons work at the same rate. Neither do computers. Just as important, there can be more than one algorithm that solves a problem, and some algorithms are faster than others. The differences in time requirement for various classes of problems are so vast, however, that they dwarf the differences in calculating speed between computers (or people).

In particular, some problems can be solved in “polynomial time” while others require “exponential time.” This means that the time required to solve a problem can be expressed as a polynomial (or exponential) function of the size or complexity of the problem. Usually, a problem requiring polynomial time is practical to solve. A problem requiring exponential time is often hopeless. Infinity machines may be chimeras, but exponential-time problems are real and ubiquitous. Solving them can require a “practically” infinite number of steps, even in a finite universe.

The distinction between polynomial-time and exponential-time problems, and the way it relates to paradox, will be explored in the next chapter. For now let’s look at two paradoxes that question the infinity of space and time.

In 1826, German astronomer Heinrich Wilhelm Olbers realized that something is wrong with the universe. Among the sciences, astronomy cannot ignore infinity. Either the physical universe is endless or it is finite. Neither possibility is easy for most people to accept.

“When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there,” Blaise Pascal wrote. A finite universe may be
even harder to believe in. The mind rebels at imagining how space can end.

The unease was not new. Greek philosopher Lucretius felt he could prove the infinity of space with this argument: If space is finite, it has a boundary. Let someone go to that ultima Thule and throw a dart past it. Either the dart will fly past the boundary or something will stop it—something that must itself be just beyond the boundary. Either way, there is something beyond the boundary. This demonstration can be repeated any number of times to push back the putative boundary ad infinitum.

Most astronomers of Olbers’s time took the infinity of space for granted. Olbers’s reaction was a haunting fantasy that bears his name. Assume that the universe is infinite and that the stars (and galaxies, which Olbers and his contemporaries didn’t know about) extend out in all directions forever. In that case, a straight line extended in any direction from the earth must hit a star.

The line may, of course, have to be extended billions of light-years. The point is that in an infinite universe of scattered stars, the line must hit a star
eventually
. This is no more exceptionable than the observation that if you spin a roulette wheel long enough, any given number must come up.

The sun is a star, the only one in our sky of perceptible breadth. Were the sun ten times farther away, it would cover a mere one-hundredth of the apparent area it does and be a hundredth as bright (this by the long-established formula for the attenuation of light). Were the sun a million times farther away, it would be a trillion times dimmer and its disk in the sky would be a trillion times smaller. Notice that the brightness
per area of sky
would be the same. It would be the same no matter what distance the sun was from the earth. On this simple fact, Olbers realized, rests a paradox.

The other stars are pinpricks in the bowl of night, but those pinpricks are (on the average) as dazzling as the sun’s surface. Light travels in a straight line. If a straight line extending from the earth hits a star, we see that star’s light. And if
every
straight line extended from the earth hits a star, the entire sky should consist of the overlapping disks of stars—each as blindingly bright as the solar disk—fusing into an all-enclosing celestial sphere. It should be as if the sun were a hollow sphere, with us in the middle. There should be no such thing as a shadow, including the shadow we call night.

From this panoramic sun there would be no respite. You might think that dark objects would block some of the starlight from our gaze. But nothing could be dark under these circumstances. All
objects must absorb, transmit, or reflect light (usually a combination of all three). Anything that absorbs light (the moon, cosmic dust, this book, your eyelids) should heat up until it is the same average temperature as the stars themselves. Then it should shine with the same intensity. Anything that transmits light perfectly (an ideal pane of glass) is by definition transparent and provides no shade. Objects that reflect light (a mirror) should cast back a glare identical to the background and be invisible.

This reasoning, which is clearly wrong, is Olbers’s paradox. Olbers got the credit, but he was not the first to think along these lines. The idea had been kicked around for centuries, attracting the attention of Thomas Digges, Edmund Halley, and Edgar Allan Poe, among others. Like the infinity machines, the paradox apparently supplies a tantalizing cosmic truth (whether the universe is infinite) in short order.

Looking through the other end of the telescope yields a companion paradox, an updated version of Zeno’s “argument against plurality.” We are told that the shortest line segment contains an infinity of points. Then even the shell of a walnut can embrace a spatial infinity as imponderable as intergalactic space.

“Solid” matter is made of atoms—which are mostly empty space. The part that
isn’t
empty space is protons, neutrons, and electrons. But these particles are mostly empty space too. If space is infinitely divisible, there may be an endless hierarchy of particles, subparticles, and sub-subparticles—all of which are
mostly empty space
. Then everything would be 99.999999 + percent nothing. It should be impossible to see anything—like Gertrude Stein’s Oakland, there’s no there there.

Physics provides a quick resolution for this paradox. Visible light scatters off electrons in atoms, which can act like waves extended in space. The electrons “smear out” and cover atoms, in effect. The fact that electrons can also act like infinitely small particles never enters into it. Nor does the nucleus of protons and neutrons play any role in scattering ordinary light.

For the paradox to work, you would have to have some kind of magic X-ray vision that allows you to see something if and only if a geometrically perfect line connects a point occupied by matter to your eye. Then you would not see a walnut, but the myriad pointlike electrons and quarks composing it (or the ultimate subparticles
of which electrons and quarks are made). Everything would be a fractal dust. Since you can’t see a single infinitely small point, everything should be invisible.

That leaves Olbers’s macroscopic paradox. Any resolution must lie in the premises: that the universe is infinite, that stars are scattered randomly, that nothing prevents the light of distant stars from reaching us. All three explanations have been advanced.

One approach is to suppose that the distribution of stars is something like the distribution of subatomic matter in the discussion above. The two complementary paradoxes annihilate each other. C. V. L. Charlier, a Swedish mathematician, resolved Olbers’s paradox by proposing that stars are not scattered haphazardly but cluster in ever-greater hierarchies. We now know that all the nearby stars are part of a galaxy, the Milky Way, and the Milky Way is itself part of a cluster of galaxies called the Local Group. The Local Group is part of a yet greater hierarchy called the Local Supercluster. The Local Supercluster is part of the Pisces-Cetus Supercluster Complex … If and when someone announces that the Pisces-Cetus Supercluster Complex is part of something even bigger, no one will be much surprised.