Labyrinths of Reason (42 page)

“Random” phenomena are governed by the same physical laws as everything else. What makes them unpredictable is this fact: In chaotic phenomena, the error in our measurement of their initial state grows exponentially with time. Jules Henri Poincaré anticipated chaos when he wrote in 1903:

Any measurement is a little bit off. If your driver’s license says you are six feet one inch tall, it doesn’t mean you are precisely that tall. The measurement was rounded off to the nearest inch; the measuring stick warped slightly since it was calibrated; you slouched a little during the measurement; your height has changed minutely since the measurement. You accept the fact that a measurement of human height is liable to be as much as 1 percent off, and leave it at that. We can live with such errors of measurement because they don’t increase. In other contexts, a small error compounds until it is so huge we no longer have any knowledge of the measured quantity.

Chaos is the unstated rationale behind shuffling cards. After a hand of poker, the dealer collects the cards to shuffle. Inevitably, some people see which cards go where in the reassembled deck. One person notes the two of spades on the bottom; another sees that his hand, a straight, went on top. Each person has some knowledge, and some uncertainty, about the composition of the deck. Shuffling multiplies this uncertainty.

Suppose you had a straight flush, the 6-7-8-9-10 of hearts, which you arranged in that order, and saw that the dealer picked up the hand intact while reassembling the deck. If the dealer
didn’t
shuffle before dealing, this would give you information about the other players’ new hands. If in the next hand you were dealt the 8 of hearts, you could conclude that the player before you in the deal got the 7 of hearts, the player after you got the 9 of hearts, and so on.

On the average, a riffle shuffle inserts one card between cards that were originally adjacent. The sequence 6H-7H-8H-9H-10H becomes 6H-?-7H-?-8H-?-9H-?-10H and then 6H-?-?-?-7H-?-?-?-8H-?-?-?-9H-?-?-?-10H.
The space between originally adjacent cards doubles with each shuffle. By the second shuffle the first and last cards of the original straight flush are sixteen cards apart; chances are good that the cut for the third shuffle will split them into different packets. Then the cards will be scattered throughout the deck.

This understates the confusion, for of course no one shuffles with perfect interleaf. Sometimes two cards fall instead of one; sometimes several cards fall. The uncertainty about the shuffling process hikes the total uncertainty with each shuffle. Try this experiment: Put the ace of spades on top of a deck and riffle shuffle a few times. The ace of spades quickly migrates down through the deck. (It may remain on top a few shuffles depending on how the cards fall.) Were the deck infinitely large, the position of the ace of spades from the top would approximately double with each shuffle. So too would any small uncertainty about the card’s position double with each shuffle. In a finite deck, once the card is shuffled below the midpoint of the deck, it goes in the bottom packet on the next shuffle and thereafter can be anywhere in the deck. To completely lose the card in a standard deck requires about six or seven shuffles.

Chaotic phenomena are said to be
irreducible
. They cannot be reduced to a model that is any simpler than themselves. A “model” can be many things: an equation, a working scale model, the set of neural circuits in your brain corresponding to your thoughts about the phenomenon. A stable orbit can be represented by a few equations or a planetarium. It is impossible, however, to build a shoe box-size scale model of a deflating balloon in a room so precise it allows prediction of where the full-sized balloon in the full-sized room will go. It is all the more impossible to model fully a river, a tornado, or a brain. The simplest representation of a chaotic phenomenon is the phenomenon itself. There is more to a cuckoo than a cuckoo clock can contain.

The irreducibility of the brain is suggested by this experiment: Think of an obscure past experience; think of a person you were with at the time, someone you have not thought of in a long time; count the number of letters in that person’s first name; then,
if and only if
that number is odd, dog-ear the corner of this page. Could even your closest friend predict whether you will dog-ear the page? In a situation like this (and in many others), one minute part of your memory, a few neurons perhaps, can zoom to the forefront and determine your train of thought. No one could hope to predict what you would do in that situation unless they shared all your
memories in cellular and even molecular detail. Nothing simpler than yourself can be expected to behave exactly as you would.

Chaos is distinct from quantum uncertainty. A world made of fully deterministic atoms would still have chaos. Together, chaos and quantum uncertainty make prediction all the more difficult. Even in ideal situations where there are no other sources of error, there is always quantum uncertainty. Chaotic phenomena magnify it over and over. Quantum uncertainty bubbles up to the everyday world and renders it unpredictable.

Philosophers make much of the conflict between free will and determinism. How can there be free will in a deterministic world? This question has occupied philosophers ever since the mechanistic philosophy gained sway. It is a big part of the puzzle of Newcomb’s paradox.

There are at least three ways of dealing with the question. You may decide that there is no such thing as free will, and that’s that. Free will is an illusion.

The trouble with that is, everyone feels like he has free will in most things. In plain everyday life, not having free will means you want to do something and some outside agency prevents it. You want to speak your mind about the Premier, but here in Transylvania they send you to the salt mines if you do that. You probably
wouldn’t
feel your free will compromised if you learned that the states of quarks and gluons in your brain are strictly determined by physical law.

Alternatively, you can say that determinism is the illusion. The world, or at least the human mind, is not completely determined by the past. This option is unattractive to most contemporary thinkers. You have to turn your back on the science of the past five centuries to deny that events are constrained by natural law (quantum theory notwithstanding) and don’t just happen any which way.

The “compatibilist” position says that there is no essential contradiction between free will and determinism. Determinism does not necessarily imply predictability (much less absence of free will). Our growing appreciation of the role of chaos in the universe lends a plausibility to this position.

Free will means doing as you please, even if what you please is predetermined by the states of neurons in your head. If your actions are predetermined but neither you nor anyone else can ever learn
what is going to happen before it happens, the seeming conflict is avoided. You might well ask what difference that kind of determinism makes. The future is still unknowable. Do what you will, no one is ever looking over your shoulder and muttering with certainty, “Yep, he’s going to take both boxes.”

The only way that determinism can impinge on our sense of free will is when we learn of our predestination. Presumably God knows whether or not you will squeeze the toothpaste tube from the middle tomorrow morning. No problem—as long as God doesn’t tell you. The unacceptable case is
knowing
that you are destined to make such and such a choice, and being “forced” by all those unfeeling atoms to do it. Only then is deterministic physical law the sort of coercive agency that prevents us from having free will.

The problems of predicting irreducible phenomena are many. One thought experiment sometimes advanced in connection with Newcomb’s paradox goes like this: A hermetically sealed chamber contains a super-computer with an exact knowledge of all the atoms in the room. All the laws of physics, chemistry, and biology are programmed into the computer, so it can predict anything that is going to happen in the room. (The room must remain sealed so that outside agencies do not affect any prediction.) A terrarium in the room contains a few frogs and plants. The computer predicts the births, deaths, matings, territorial struggles, and mental states of the frogs: All these predictions amount to charting the courses of a large but finite number of atoms in the terrarium. No light bulb can burn out, no coat of paint can blister without the computer predicting it.

In the room are also several people. Again, the computer knows every atom of their makeup. One person gets annoyed at the seeming violation of free will and asks the computer this question: “Will I be standing on my head at midnight tonight?” She then announces: “Whatever the computer predicts, I’m going to do the opposite. If it says I
am
going to be standing on my head at midnight, then I will do everything within my power to make sure that I don’t stand on my head. If it says I won’t stand on my head, I will.” What would happen in this situation?

There are ways the computer could avoid being wrong. It could decline to answer, not answer until one minute after midnight, or answer in a language unknown to the room’s inhabitants. It could
predict the subject would not be standing on her head, and she could fall asleep early that night, forgetting the whole thing. But just because such scenarios avoid paradox does not mean they must occur.

Assuming a timely prediction is forthcoming, there is no reason why the person
couldn’t
make good on her promise. Call free will an illusion if you like; any of us can resolve to do a headstand (or not). The computer’s prediction is not going to make anyone any less free to do so.

In fact, the computer cannot make a valid prediction. To see why, ask how the computer makes its prediction. Does it use a shortcut: a rule, a gimmick, a mathematical formula? It is beyond belief that any
simple
rule tells whether a specific person will be standing on her head at a given moment! It is one thing to predict days of the week, seasons, or comet returns. There is regularity in these phenomena. There is no regularity to standing on one’s head. Even if there was (if the subject was in the habit of standing on her head every second Tuesday at midnight), her promise to be contrary invalidates it.

Evidently, the computer predicts by modeling the situation in the room. It was stated that the computer predicted the very courses of the atoms to foresee the actions of the frogs. Here we come to the crux of the paradox. Since the subject certainly will be influenced by the computer’s prediction, the computer must predict its own prediction as well as the subject’s reaction to it. The computer’s model must represent the computer itself
in full detail
.

This paradoxical requirement recalls the map Borges and Adolfo Bioy Casares describe in
Extraordinary Tales:

The computer wants to set aside a certain portion of its available memory to simulate its own actions. Unfortunately, no part of the
computer’s workings any smaller than the whole computer can do this. The most efficient way the computer can model itself is to
be
itself. That, like Borges and Casares’s map, leaves no room for anything else.

Even if the computer is highly redundant, you run into trouble. Some computers, such as those used for spacecraft navigation and life support, have two or more separate subsystems doing the same thing. This greatly reduces the chance of an error. Potentially, it also allows each of the redundant subsystems to “predict” what the computer as a whole will do.

Compare this to a Borges-Casares map at a scale of 1:2. The map is half as wide as the country it represents. A 1:2 map of the United States stretches from San Francisco to Kansas City, blanketing the mountain states. A map that big is itself a significant man-made feature worthy of inclusion on all maps of the country. That means the 1:2 map must show itself. And the map on the map must contain a map of itself, and so on, ad infinitum.

For the same reason, a redundant computer modeling itself would contain a model of the computer, a model of the model, a model of the model of the model … Nothing prevents you from imagining this. But no real computer made of atoms can attain an infinite regress. The models and models in models must have some physical reality as states of memory chips, and memory chips cannot be infinitely small. The prediction is therefore impossible.