Labyrinths of Reason (3 page)

Descartes would probably assert that the beings running the brains-in-vats laboratory couldn’t have dreamed up
everything
from scratch. There would be, let us say, eyes and fur out there in the “real” external world outside the laboratory, even if they were not arranged like a dog. Descartes also remarked that the colors of the most fanciful paintings are
real
colors no less. For this reason, he felt justified in believing that the color red exists, even if he was deceived by an evil genius. (Do you agree? Or is it conceivable that the “real” world is black and white, and that color is a neurological illusion created by a remarkably inventive brains-in-vats research department?)

In saying that colors are real, Descartes meant the subjective sensations of color, rather than pigments, wavelengths of light, or anything else that we have come to associate with those primal sensations. In fact, Descartes concluded that the one thing one can be certain of is subjective feelings—specifically, one’s own subjective
feelings (for who can be sure that others think and feel as one’s self?).

Suppose you doubt that your own mind exists. Then you doubt that you are doubting—which is to say, you
are
doubting after all. Something must be doing the doubting. You may be deluded in many ways, but there must at least be a mind that is being deluded. Hence Descartes’s famous conclusion: “I think, therefore I am.”

Idealism
is the belief that only mind is real or knowable. Though not properly an idealist, Descartes inspired the movement. An idealist says that when you eat a chili pepper and burn your mouth, the
sensations
of pain or heat are indisputably real. The chili pepper itself may be an illusion: a marzipan fake doctored with Tabasco sauce, or part of a bad dream brought on by indigestion. Because pain and flavor are purely subjective, the fact of the pain or the flavor is beyond dispute. Subjective feelings transcend the physical reality of their cause.

Another example: Almost everyone has been frightened by horror movies, horror novels, and nightmares. Although it’s only a movie/story/dream, the momentary fear is
real
fear. Penfield’s patient J.V. was genuinely frightened by the man with the bag of snakes, even if he was (in neurological replay on the operating table) an illusion. Likewise, one cannot doubt that one is happy, sad, in love, in grief, amused, or jealous, if such mental states apply.

Subjective feelings are a very limited basis for reasoning about the external world. Descartes nevertheless believed he could deduce many significant conclusions from the fact of his own mind. From “I am,” he concluded that “God exists.” Every effect must have a cause, Descartes reasoned, and thus he must have a creator. From “God exists” Descartes jumped to “The external world exists” because, as a perfect being, God could not deceive us into believing in an illusory external world: He would not permit an evil genius.

Few modern philosophers accept this chain of reasoning. All things may seem to have a cause, but do we know this with total certainty? Again, cause and effect could be a fiction put in our minds by an evil genius.

Even allowing that there is a cause for one’s existence, it is misleading to call that cause “God.” “God” means a lot more than a cause for one’s existence. Perhaps Darwinian evolution is a cause for our existence, but that is not what most people mean by “God.” And even allowing that God exists, how do we know that He wouldn’t countenance an evil genius?

None of this means that Descartes was wrong, but only that he
was not true to the spirit of his original skepticism. One of Descartes’s severest critics was Scottish philosopher and historian David Hume (1711–76). At the height of his renown, Hume was a celebrity in London and Paris but could not teach at any university because of his outspoken atheism. For a time he made a difficult living as private tutor to the Third Marquess of Annandale, who was insane. Hume doubted every step of Descartes’s argument, even the existence of one’s own mind. Hume said that when he introspected, he always “stumbled on” ideas and sensations. Never did he find a self distinct from those thoughts.

Hume argued that there are only two types of revealed truth. There are “truths of reason,” such as 2 + 2 = 4. Then there are “matters of fact,” such as “The raven in the aviary of the Copenhagen zoo is black.” This double-pronged conception of truth is called “Hume’s fork.” A question not of either type (such as “Does the external world really exist?”) is unanswerable and meaningless, maintained Hume.

To derive useful conclusions about the real world, we must work from premises that are (in the exacting sense of the philosophical skeptic) uncertain. Science and common sense are forever building structures of belief on uncertain foundations. No scientific conclusion is utterly certain.

There are two ways by which we know (or think we know) things, and they are closely related to Hume’s distinction. One way of knowing is deduction, the “logical” way of drawing conclusions from given facts. An example of deductive reasoning is:

The first two lines are premises, facts assumed to be given. Deduction is the act of deriving the third line from the two lines above it. Valid deductions are truths of reason in Hume’s terminology.

Descartes wanted to use deduction to derive new facts from certain premises. The new facts would then be equally certain. Fortunately, deduction may also be applied to less than certain premises. A hard-core skeptic can insist that neither of the two premises above is certain. There may be immortal men somewhere, and Socrates could have been a being from another planet. These uncertainties
are transmitted to the conclusion. The deduction itself, however, is as certain as any statement of logic can be.
Whenever
“all A are B” and “C is A,” it follows that “C is B.” You can just as well conclude:

This type of reasoning is called a
syllogism
. An irony of deduction is that the “subject matter” of the premises makes no difference to the deductive process. Socrates, Rockefeller, or Poe’s raven, it’s all the same.

The other basic way of knowing is induction. Induction is the familiar process by which we form generalizations. You see a raven. It’s black. You see other ravens, and they’re black too. Never do you see a raven that isn’t black. It is inductive reasoning to conclude that “all ravens are black.”

Both science and common sense are founded on induction. Despite his renown for “deduction,” most of Sherlock Holmes’s reasoning is more induction than deduction. Induction is reasoning from “circumstantial evidence” or Hume’s “matters of fact.” It extrapolates from observations that are not understood on a deeper level. You don’t know
why
all the ravens seen have been black. Even after seeing 100,000 ravens, all black, the 100,001st raven just might be white. A white raven isn’t inherently absurd, like a triangle with four sides. There is no logical necessity to an inductive conclusion.

For this reason, induction has always seemed less legitimate than deduction. Hume for one was skeptical of it. As he complained, we use inductive reasoning to justify inductive reasoning. (“Induction has stood the test of time. Therefore it should be reliable in the future.”) Philosopher Morris Cohen quipped that books on logic are divided into two parts: a part one, on deduction, in which fallacies are explained, and a part two, on induction, in which fallacies are committed. (Note the modified plan of this book!)

Induction is working backward, like solving a maze by backtracking from the goal. Instead of taking a general law (“All ravens are black”) and applying it to specific cases (“This bird is a raven;
therefore this bird is black”), induction goes from specific cases to a general law. Induction is founded on the belief—the hope—that the world is not essentially deceptive. From the fact that every raven ever examined was black, we conclude that
all
ravens are black, even the ones no one has ever seen. We assume that the unobserved ravens are similar to the observed ravens, that the seeming regularities of the world are genuine.

It could be that the world teems with unseen white ravens, forever behind your head, never straying into sight. Lingering uncertainty plagues every inductive conclusion. Why do we bother with inductive reasoning, then? We use it because it is the only way of getting broadly applicable facts about the real world. Without it, we would have only our trillions of experiences, each as separate and meaningless as a bit of confetti.

Induction provides the fundamental facts from which we reason about the world. Empirically tested generalizations take the place in science that Descartes hoped certain axioms would in his philosophy. The teaming of induction and deduction is the basis of the scientific method.

The problem of knowledge has intrigued many of the keenest minds of philosophy, science, and even literature for as far back as we have records. Philosophers call this study
epistemology
. A newer term, applied in more strictly scientific contexts, is
confirmation theory
. Each is a study of how we know what we know; an investigation into the business of drawing valid conclusions from evidence.

Investigating the very process of knowing is different from investigating butterflies, nebulae, or anything else. Confirmation theory is largely a study of logic puzzles and paradoxes. To the uninitiated, this probably sounds as peculiar as a study based on mirages. By their nature, paradoxes expose the cracks in our structures of belief. Bertrand Russell said, “A logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science.”

The past few decades have been a very fruitful time for paradoxes of knowledge. This book discusses a selection of recent paradoxes that are so significant and mind-bending that they deserve a place in the mental bestiary of any broadly educated person.

It is best to start by explaining what is meant by a paradox. The word is used in different ways, of course, but at the heart of all the uses is contradiction. A paradox starts with a set of reasonable premises. From these premises, it deduces a conclusion that undermines the premises. It is a travesty of the notion of proof.

One thing not immediately apparent (if the paradox is clever enough) is the reason for the contradiction. Is it possible for a perfectly valid argument to lead to contradiction, or is that “guaranteed” not to happen?

Paradoxes can be loosely classified according to how and where (if anywhere) the contradiction arises. The weakest type of paradox is the fallacy. This is a contradiction that arises through a trivial but well-camouflaged mistake in reasoning. We’ve all seen those algebraic “proofs” that 2 equals 1, or some other absurdity. Most are based on tricking you into dividing by o. One example:

The fatal step is dividing by
(x – x)
, which is o. Line 5,
x(x – x) = (x + x) (x – x)
, correctly asserts that 1 times 0 equals 2 times o. It does not then follow that 1 equals 2; any number at all times 0 equals any other number times o.

In a fallacy, the paradox is an illusion. Once you spot the error, all is right with the world again. It might seem that all paradoxes are like this deep down. The error may not be as obvious as in the example above, but it is there. Rout it, and the paradox vanishes.

Were that all there is to paradox, confirmation theory and epistemology would be simpler and less interesting fields. We will not be concerned with simple fallacies. Many paradoxes are valid and disturbing.

More powerful paradoxes often take the form of a
thought experiment
(sometimes known by the German name
Gedankenexperiment)
. This is a situation that may be imagined but which (usually)
would be difficult to carry out in practice. Thought experiments typically show that certain conventional assumptions can lead to an absurdity.

One of the simplest and most successful of thought experiments was that devised by Galileo to demonstrate that heavy objects do not fall faster than light ones. Suppose (as was the belief in Galileo’s day) that a 10-pound lead ball falls faster than a 1-pound wooden ball. Imagine that you connect the balls with a string and drop them from a great height. The wooden ball, being lighter, will lag behind the lead ball, pulling the string taut. Once that happens, you have a wooden ball weighted down by a lead ball: an 11-pound system that, being heavier yet, ought to fall faster than either ball alone. Does the system speed up once the string is taut? Although not strictly impossible, that conclusion is suspect enough to cast doubt on the original assumptions. Unlike most thought experiments, Galileo’s was easily carried out. Galileo dropped objects of different weight (but not, as story has it, from the leaning Tower of Pisa) and found that they fell at the same rate. The uniformity of gravitational acceleration has been so well accepted, in fact, that we see nothing paradoxical about Galileo’s thought experiment today.