Labyrinths of Reason (26 page)

Charlier showed that, under an endless chain of hierarchies, paradox can be avoided even if the number of stars is infinite. You might have, for instance, a super-super-supercluster of galaxies so remote that its image in our sky could fit behind the tiny orb of Arcturus or Betelgeuse. There would be super-super-super-superclusters and super-super-super-super-superclusters so much farther away that they appear smaller yet. Under Charlier’s scheme, you could travel endlessly in most directions and never come to a star. Thus the night sky is dark.

Charlier’s explanation is geometrically possible. It fails only in that it does not seem to describe the relative distances and dimensions of the observed cosmological hierarchies. Nearby galaxies loom much larger than nearby stars. Though extremely faint, the Andromeda galaxy is several times the apparent diameter of the sun or the full moon. The Magellanic Clouds of the southern sky (the two galaxies nearest our own) are about the size of lemons held at arm’s length. And nearby clusters of galaxies are bigger yet. The Virgo Cluster, invisible to the eye, sprawls over an entire constellation.

Current thought on Olbers’s paradox invokes a fact not suspected until this century: The universe is expanding. All the distant galaxies we can see are receding from our own galaxy at great speed. We cannot measure this motion directly, of course, but it produces a telltale shift in the light we receive from the galaxies, and attempts to explain this shift in any other way have failed. The galaxies in any given part of the sky are moving away from us; and the galaxies in the opposite part of the sky are also moving away from us
in the opposite direction
.

One interpretation of this is that our galaxy is “special,” the center of the universe. The facts are equally well explained by postulating that the whole universe is expanding. That way of putting it is convenient but a bit misleading. It is not a uniform expansion like the one Poincaré talked about, but an expansion at the very top of the scale of distances. Neither the earth nor the Milky Way is getting bigger—and perhaps not even the Local Group. But the distances between galactic clusters
are
getting bigger. In principle, we can measure the ever-widening intergalactic gulfs with our yardsticks, which have not expanded.

Under the hypothesis of universal expansion, there is nothing unique about our galaxy or its place in the universe. The inhabitants of those distant galaxies would find themselves the “center” of expansion too. Since this hypothesis does not demand the extraneous assumption that our galaxy is special, it is favored.

The most distant galaxies known are hurtling away from the earth at near the speed of light. The light given off from a rapidly receding object undergoes a “red shift.” This increases the light’s wavelength and reduces its energy. Energetic visible light is red-shifted to low-energy microwaves. When a luminous object is moving away at nearly the speed of light, the energy is attenuated almost to the vanishing point. Thus the light we receive from very distant galaxies is of such low energy as to be invisible.

Let’s see how this affects Olbers’s reasoning. Think of the universe as being divided into a series of concentric “shells” of space centered on the earth. Because light attenuates with the square of the distance, the amount of light we receive from each shell should (on the average) be equal. All the stars within 10 light-years of the solar system should generate about as much light as the stars between 10 and 20 light-years, or between 30 and 40 light-years, or for that matter, between 1,000,000 and 1,000,010 light-years.

If the universe is infinite, the total amount of light we receive is the sum of an infinite series: something like
x + x + x + x + x +
…, where
x
is the light from each shell. This type of infinite series does not converge but rather adds up to infinity.

When the light from the more distant shells is weakened by the red shift, it changes everything. The more distant the galaxy, the faster it recedes, and the less energetic its light. The infinite series might look more like this: x + 0.9x + 0.81x + 0.729x + 0.6561x + … An infinite series like this, where each term is smaller by a fixed factor, does converge. An infinity of stars could shine in the earth’s sky and still produce only a finite amount of light.

Few cosmologists doubt that the expansion of the universe is an acceptable explanation of the paradox, but there is a simpler explanation. In 1720 Edmund Halley wrote that the darkness of the sky argued against an infinity of stars. Today, many cosmologists believe the universe
is
finite (though for reasons other than Olbers’s paradox). General relativity provides a way for the universe to be finite without ever coming to a worrisome “end.” Space could curve back on itself, a three-dimensional analogue of the surface of a sphere. If you could walk far enough in any direction on the earth, you would come back to where you started. Space itself might be like that: A rocket traveling far enough in a straight line would return to its point of launch.

Most current cosmological models predict just such a finite universe, provided the density of matter in the universe equals or exceeds a certain limit. The observed density of visible matter (stars) is below the limit, but it is conjectured that there is enough invisible matter (intergalactic hydrogen, black holes, neutrinos?) to create a finite universe. Recent studies of “gravitational lens” effects of distant galaxies and quasars support the belief that there is much invisible matter.

There is a subconscious double standard: Infinities of time seem a little different from infinities of space. It is natural to think that space extends out in all directions forever (or is this a culturally instilled belief?). Time is supposed to be infinite
only
in the future direction. We ask when time began but rarely where space began.

The infinity of past time is an unpopular belief. Yet it would “answer” questions of when or how the world was created by throwing them out as meaningless. In contrast, the infinity of future time appears to be universally accepted, even by those religions that postulate an apocalypse. After the millennium, the good live on
forever, or the cycle starts over with a new creation. Few if any doctrines are nihilistic enough to believe in a
real
end to time, where things revert to exactly the same nonexistence as before the beginning of time, only this time for good.

Bertrand Russell’s “paradox of Tristram Shandy” plays with the idea of an infinite future. Tristram Shandy is the raconteur narrator of Laurence Sterne’s rambling novel of the 1760s,
The Life and Opinions of Tristram Shandy, Gentleman
. Russell wrote: “Tristram Shandy, as we know, took two years writing the history of the first two days of his life, and lamented that, at this rate, material would accumulate faster than he could deal with it, so that he could never come to an end. Now I maintain that, if he had lived for ever, and not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten.”

Russell’s reasoning goes like this: Say that Shandy was born on January 1, 1700, and began writing on January 1, 1720. The first year of writing, 1720, covers that first day, January 1, 1700. The progress would go like this:

There is a year for every day, and a day for every year. Were Shandy still writing today, in 1988, he would be up to the events of September 1700. In turn, this immortal Shandy would get to setting down
today’s
events circa the year 106,840. You cannot single out a day for which it
isn’t
possible to schedule a future year for recording its events. Therefore, said Russell, “no part of his biography would have remained unwritten.” Even so, Shandy gets further and further behind in his writing. With every year of writing he falls 364 years further from completion!

Russell’s reasoning was based on Georg Cantor’s theory of infinite numbers. If two infinite quantities can be placed in one-to-one correspondence to each other, they are equal. For instance, mathematicians hold that the number of whole numbers (0, 1, 2, 3, 4, 5
…) equals the number of even numbers (0, 2, 4, 6, 8, 10 …)—rather than being twice as much, as you might think. The two are equal because every whole number
n
can be paired with one even number 2
n
, and the pairing will leave no even numbers left over.

More mind-bending is a reversal of the paradox discussed by W. L. Craig. Suppose that there is an infinity of
past
time, and that Shandy has already been writing for an eternity. Then, suggested Craig, there is the same Cantorian correspondence between years and days. Shandy would have just finished the last page of his autobiography. But that’s ridiculous. How could Shandy have chronicled yesterday’s events already, when it should have taken him a whole year?

Craig and others have used the reverse paradox to demonstrate, not very convincingly, the impossibility of a past eternity. A reasonable resolution of the reverse Tristram Shandy paradox was supplied by Robin Small. It is, in fact, impossible to establish a correspondence between
specific
days and
specific
years.

Pretend it is midnight, December 31, 1988, and Shandy has just finished the last page of his manuscript. What day has Shandy been writing about this past year? It can’t have been any day of this year. (Otherwise he would have spent the early part of the year writing about a day that hadn’t happened yet.) The most recent day he could have been writing about in 1988 is December 31, 1987.

If indeed Shandy spent 1988 recounting December 31, 1987, then he must have spent 1987 writing about December 30, 1987. That again is impossible. Actually, Shandy couldn’t have written about any day later than December 31, 1986, in 1987.

But if he wrote about December 31, 1986, in 1987, he would have had to write about December 30, 1986, in 1986 … Any proposed correspondence crumbles beneath our feet. The alleged day Shandy has been writing about recedes into the infinite past. There is no way of singling out any day.

Conclusion: If there is an eternity of past time, and Shandy has been writing since the beginning, he will have an infinitely long
unfinished
manuscript. The most recently completed page will describe events of the infinitely remote past.

Russell’s and Craig’s versions of the paradox are not so different after all. Russell does not claim that Shandy will “ever” finish the manuscript. Rather, no specific day you can mention goes unrecorded. Tristram Shandy’s “last” page is forever a mirage.

J
ORGE LUIS BORGES’S STORY “The Garden of Forking Paths” describes a labyrinth so intricate that none escape from it. Upon receiving road directions, the narrator digresses:

The term “labyrinth” is of uncertain and very ancient origin. In classical times, a labyrinth was a building, at least partly underground, of intentionally confusing design. Herodotus rated the Egyptian labyrinth near Crocodilopolis (completed in 1795
B
.c.) a greater wonder than the pyramids. It contained 3000 chambers, half above and half below ground level. A forest of pillars stretched as far as the eye could see. Herodotus toured the upper half but was not permitted to descend below; there, he was told, sacred crocodiles guarded the tombs of kings. The progressive decay of this labyrinth is chronicled by a number of ancient writers, and its site was never lost. The foundation, unearthed in 1888, measures 800 by 1000 feet.