Labyrinths of Reason (22 page)

But what if a committee of Nobel Prize-winning scientists supervises the study, taking the greatest possible pains to ensure its validity? A system of scrupulous controls, statistical checks, and double checks such as never before seen in an experiment are instituted. Then this undeniably valid study truthfully asserts that all psychological experiments on humans (of which the study is one) are invalid because of subconscious experimenter bias.

This, the heart of the paradox, is the liar paradox with validity substituting for truth. A valid study that asserts its own invalidity just can’t be; we have entered the realm of the impossible.

One well-known idiom of philosophy is “possible worlds.” It is natural to wonder why the world is as it is. Why is there evil? That we even ask that question demonstrates that we can imagine a world without evil, a world much different from the one that exists.
There is reason to believe that the ability to conceive of possible worlds is a fundamental part of human intelligence. All the thousands of choices we make in our lives, momentous and trivial, are acts of imagination. You imagine the world in which you get your car washed this afternoon and the world in which you don’t, and decide which you would rather live in.

The first Western writer to use the idea of possible worlds extensively was the German mathematician and philosopher Gottfried Leibniz (1646–1716). Leibniz wondered why, out of all the possible worlds, God chose to create this one. His peculiar answer was that this is in fact the best of all possible worlds. Leibniz imagined that the pain and suffering of the world was at an absolute minimum; any tweaking on the part of the Creator, any attempt to right a wrong here or there, would make things worse globally. This incredible point of view is remembered for inspiring the character of Dr. Pangloss in Voltaire’s satire
Candide
. Candide could not see how a world in which the Lisbon earthquake (which killed about 40,000 in 1755) had not occurred would fail to be better than ours.

Possible-worlds philosophy was revived in the 1960s by such philosophers as Saul Kripke, David Lewis, and Jaakko Hintikka. Lest there be any confusion, let’s clarify what a “possible world” is. It is
not
another planet out there in space. A possible world is a complete universe unto itself with a past, present, and future. You can talk about the possible world in which Germany won World War II, and even about the year 10,000 A.D. in that possible world. People often use the singular to denote what is actually a class of possible worlds. There must be trillions upon trillions of possible worlds in which Germany won World War II, each differing from one another in some detail. There are, or seem to be, an infinite number. The one possible world that we live in is called the “actual” world.

Even so metaphysical an idea as this has its limits. The concept would not be very useful if any and every effort of imagination constituted a possible world. Most philosophers allow that it is possible to talk about worlds that are not possible worlds.

Though we string these words together, “a world in which 1 plus 1 does not equal 2” does not describe a possible world. Nor could there be a world in which 6 is a prime number; a world in which pentagons have four sides; a world in which the Lisbon earthquake occurs
and
the Lisbon earthquake doesn’t occur; or a world in which Abraham Lincoln is taller than Joseph Stalin, Joseph Stalin is taller than Napoleon, and Napoleon is taller than Abraham Lincoln.

(Some dispute this. Though no one has any idea how there could be a world in which 1 and 1 don’t equal 2, a dyed-in-the-wool skeptic can doubt our certainty that no such world is possible. Most philosophical discussion of possible worlds takes as a ground rule that our logic at least applies to other possible worlds. If
not
, we are in no position to reason about them.)

To say that something is impossible—as opposed to merely false—is to say that there is no possible world in which it could be true. One of the deepest questions in philosophy is how diverse the possible worlds are.

Saul Kripke argued that such facts as “the atomic number of gold is 79” are true in
any
possible world. Most find this hard to accept. It seems easy to imagine a world in which gold’s atomic number is 78 or 80 or 17. Quite possibly, you’ve lived your whole life not knowing or much caring what gold’s atomic number is. Envisioning a different atomic number for gold appears little different from imagining a world in which your phone number or license plate is different. But is it?

The properties of elements can be predicted from their position in the periodic table. Gold falls beneath silver and copper in the table and resembles them in many ways. It is a dense, soft, unreactive metal that conducts electricity very well. Were gold’s atomic number even one more or less, it would occupy a different position and would be expected to possess different properties.

Suppose gold’s atomic number was 78. It would fall beneath nickel and palladium in the table and resemble them. It would still be a dense metal, but its properties ought to be more like platinum (which in fact has atomic number 78). Would “gold” that resembles platinum in all respects be gold at all?

You could contend that the other elements would be shifted one atomic number down in the periodic table so that gold could still occupy the same relative position. Gold would be element 78, platinum would be element 77, and so on. But then you’d drop an element at the beginning of the periodic table. The dropped element would be hydrogen, which makes up stars and is by far the most common element in our universe. A universe without hydrogen would be so different that we are unable even to guess how different it would be.

To a chemist, Kripke maintained, the elements have properties
that follow more or less inexorably from their atomic numbers. The idea of a world in which helium is not an inert gas is not so much different from the idea of a world in which 2 is not 1 + 1. Deciding if a world is possible is trickier than it looks!

There may come a day when our knowledge of physics is as complete as the current state of chemistry. It is conceivable that the properties of electrons, quarks, and photons have the same underlying justification as the properties of chemical elements. “Super-string” theories try to provide just that. If they are right, many exotic worlds that seem to be possible (a world in which protons are more massive than neutrons; a world in which electrons are the size of golf balls) may actually be ruled out. Physicists have even speculated that the actual world is the only one possible. The laws of physics and even the initial state of the world may be preordained with a logical rigor we can scarcely imagine.

When we say that “This sentence is false” is paradoxical, we mean that there is no possible world in which that sentence accurately describes itself. The situation can be broken into two parts: (1) if the sentence is true, then it is false; and (2) if the sentence is false, then it is true. We are free to imagine worlds in which the sentence is true or false, but both alternatives lead to contradiction.

Jaakko Hintikka defined knowledge via possible worlds. To increase one’s knowledge is to decrease the number of possible worlds compatible with what one knows. For instance, everything we know is compatible with there being life in the Alpha Centauri star system, and everything we know is compatible with there
not
being life in Alpha Centauri. Such is our ignorance that we cannot distinguish the real world from a merely possible world identical to ours in every way except in whether there is life in Alpha Centauri. If and when we find out whether there is life in Alpha Centauri, one set of possible worlds will be ruled out.

Scientific discovery decreases the number of compatible possible worlds. It is natural to ask how far this process may be carried. In Hintikka’s view, total knowledge would mean paring away all the possible worlds until just
one
remains—the actual world.

Notice the slender distinction between omniscience and paradox. To someone utterly ignorant, the number of possible worlds compatible with his knowledge is infinite. To someone gifted with total knowledge, the number of possible worlds is narrowed to one. What
if the field was narrowed to
zero?
That would be the predicament of someone who has discovered that no possible world is compatible with what he knows. His set of known facts includes a contradiction. The best paradoxes seem to prove that
this
is not a possible world.

In the essay “Avatars of the Tortoise,” Borges speculated that paradoxes were clues to the unreality of the world:

We have all seen those overly self-effacing prefaces in which the author (after thanking spouse and typist) takes responsibility for the “inevitable” errors. You’ve probably wondered why, if the author is so sure there are errors, he doesn’t go back and correct rather than acknowledge them. Inspired by these disclaimers, D. C. Makinson developed the “paradox of the preface” (1965). Related to both the expectancy paradox and the unexpected hanging, the paradox of the preface “proves” that there is no such thing as nonfiction.

An author writes a long book he believes to be nonfiction. It makes many statements that he has carefully checked. A friend reads the book, shrugs, and says, “Any book
that
long contains at least one error.” “Where?” the author demands. The friend avers that he didn’t catch any error, but still, virtually all long nonfiction books do have an error or two. Reluctantly, the author agrees. “Then,” says the friend, “your readers are not justified in believing
any
statement in your book.”

“Look,” the friend says. “Pick a statement.” He opens the book at random and points to a declarative sentence. “Ignore this statement for the moment. I’ll put my finger over it so you can’t see it. Do you believe that every
other
statement in the book is true?”

“Of course. I wouldn’t have made the statements unless I believed them, and had good justification for believing them.”

“Right. And you agree that the book must contain at least one error, even though neither you nor I have spotted it. If you believe the book contains at least one error, and further believe that every statement other than this one is true, then you
must
believe that this statement I am covering with my finger is false. Otherwise your beliefs are self-contradictory. And I just chose this statement as an example. I could have chosen any statement and said the same thing of it. You can’t legitimately believe that any of the statements in your book are true,” the friend concluded.

Not wanting to mislead his readers, the author wrote a preface to the book warning: “At least one of the statements in this book is false.”

If the book contains one or more errors, then the prefatory statement is accurate. If the book exclusive of the preface contains no errors, then the prefatory statement is in error. Then there
is
an error in the book after all, and the prefatory statement is correct. But if the prefatory statement is correct, then there is no error and it is wrong … A series of errata sheets inserted in later editions of the book did little to resolve the matter!

Many real prefaces do admit errors. Kurt Vonnegut, Jr.’s novel
Cat’s Cradle
is prefaced with the statement: “Nothing in this book is true.” This is not Makinson’s paradox of the preface but a more directly contradictory relation. Insofar as Vonnegut’s book is a work of fiction, the prefatory statement is accurate
except
as concerns the prefatory remark itself. The preface, presumably coming from the real Kurt Vonnegut and not a fictional character, is nonfiction. Speaking of itself, it creates a liar’s paradox.

The paradox of the preface also brings to mind mathematician William Shanks’s tragic lifework of computing pi and making an error in the 528th decimal place that invalidated all subsequent work. Imagine you are writing a book called
The Digits of Pi
. On page one you write: “The first significant digit of pi is 3.” Each succeeding page asserts the next digit in pi’s decimal expansion. You derive the digits by hand calculation. You are a competent mathematician using an accepted algorithm. Therefore you have justification for believing each and every digit you derive.

By the time you get to the 1000th digit, you realize that, very
likely, you have made at least one error in your math. Oops! Now things are much worse than in Makinson’s paradox. Calculating each new digit depends on the values of the previous digits (as in long division). You do not determine the 1000th digit of pi directly; you must first determine the 999th digit, and before that the 998th digit, and so forth. Any error in determining a given digit will render all succeeding digits invalid. It is like setting up a line of 1000 dominoes: When the 307th domino falls to the right, all the dominoes after it follow. If you’ve made at least one error in the first 1000 digits, the 1000th digit must be wrong.
¹

So, very likely, is the 999th digit, the 998th digit, and a long string of the digits before them.