Tuesday, January 26, 2016

OK, so you do/not have free will. So what?

We previously proved that you (do or do not) have free will. What are the consequences of that fact?

Before we answer, let's review the conclusion. There are many definitions of free will. It is worth asking (and I have been asked) whether we would reach the same conclusion based on other definitions.

One definition asks the question: if there were an infinitely powerful being who knew all facts about the present, would it be capable of predicting the outcome of your future decision? I cannot tell if this question has a counterfactual premise, like the question we previously answered, or if it is a theological question. If it is the former, the same proof form would apply to demonstrate the same result. If it is a theological question the answer probably depends on which holy book you consult. A related question asks: if we could build a sufficiently powerful computer and feed it the complete state of the universe, would it be capable of computing our future choices? Again, the proof form we used before can be applied. A team of rogue scientists (rogue because they used "borrowed" equipment) claim to have constructed such a computer, and they claim that its prediction rate is 100% so far. I am suspicious of this claim, as their computer appears not to be capable of predicting a decision until after it has been made. The computer is called World Of Real Life Determinator, with the nice-sounding acronym WORLD.

A different definition asks the perhaps more sensible question regarding your future decisions, rather than your past decisions: for your future decision, is more than one option a possible future? In other words, can you do "otherwise" for a future decision? This question can be answered scientifically! There are three scientific approaches to the question, but unfortunately the result is somewhat ambiguous. The first approach is simple: from experience it is obvious that every decision we made previously was not made "otherwise", but was instead made in precisely the way we made it. If we assume the future to be similar to the past - that is a basic assumption of science, after all - then we should expect that future decisions to similarly not be "otherwise". We can test this theory too and we observe, as we expect, that any further decision we make is not made "otherwise". This approach to the scientific question clearly points out that we do not have free will. This is a well-respected proof form called retrospective determinism.

The second approach is to consult the physicists. The most widely accepted interpretations of quantum physics teach us that the results of quantum interactions are not determined by the previous state of the universe. That leaves room for small random fluctuations at the quantum and microscopic levels to affect our behavior over time as differences are amplified by chaotic processes in nature. In other words, the future is not determined. The incompatibilists have shown that this does not allow for free will, as we are not fundamentally in control of the random processes that affect our decisions; this is the well-respected proof form moving the goalposts. On the other hand, there are interpretations of quantum mechanics that, though not widely accepted, say the opposite.

The final scientific approach to this question gets to the root of the issue. The point of the question regarding free will is really about moral judgments of others. If nobody has free will, the argument goes, it would be absurd to blame or praise other people, for they were unable to freely choose how to behave. This is a well-respected argument form called argumentum ad lapidem. Can we scientifically test whether blame and praise are absurd or useful?

It turns out such an experiment had already been performed. In 1954 a team of scientists put together one of the largest controlled scientific experiments involving people that had ever been conducted. Eight hundred thousand people were selected for participation in the experiment, and randomly assigned to one of two groups. One group was assigned to live in the newly constructed city Dexter, which was operated under the assumption that blame and praise are proper to the conduct of a society. The other group was assigned to the new city Sinister, which was operated under the assumption that blame and praise are useless. Most people were not told which group they were assigned to; though they knew the name of their city, they did not know under which set of assumptions it was operated. Of course it became apparent after a time. The experiment ran for a full year. It led to a large number of PhD theses, research results, conferences, and scholoarly scientific debates. As we'll see, the results of the experiment were somewhat ambiguous.

The differences between Dexter and Sinister were as follows. In Dexter, police detectives were given the duty of assigning blame for crimes to individuals living in the city based on the individual's presence late in the causal chain leading to the criminal act. In other words, in the usual way. These people, called "criminals", were then subjected to what we would consider a typical criminal justice system. Rather than eliminating the criminal justice system in Sinister entirely, scientists eliminated only the assignment of blame based on the person associated with the criminal act. Instead, the police detectives of Sinister were responsible for conducting a kind of lottery for each crime committed. In this way blame would be assigned to a random citizen, or nobody at all (at a rate comparable to the conviction rate in Dexter). The conduct of the criminal justice systems in Dexter and Sinister mirrored each other, with the exception that blame in Sinister was assigned randomly.

Similarly, praise and reward in Dexter would be assigned to individuals on the basis of their presence late in the causal chain of events leading to outcomes considered useful or desirable. Productive employees would receive a raise and perhaps a promotion. In Sinister, however, praise and reward would be assigned randomly, in a way unrelated to the behavior of the individual. Parents in Sinister were taught to love and praise their children unconditionally, no matter the child's behavior.

The experiment was originally intended to run longer than a year, but had to be cut short due to funding issues. By the end of the year there were severe problems in Sinister that led many of its citizens to want to quit the project. There was, however, sufficient funding to analyze the results. It was clear to all that there were deeply disturbing differences between the two cities. Dexter, on the one hand, evolved in the way one would expect of a civilized society. Sinister, on the other hand, experienced rampant crime, surprisingly low worker productivity, and many competing gangs and militia. A significantly larger number of people survived the experiment in Dexter rather than in Sinister. But what to make of these results?

There emerged two camps of scientists who differed in their interpretation of the experiment's results. They called themselves the conflationists and the inconflationists.

The inconflationists believed that the experiment had improperly conflated "fundamental, moral" blame with blame in the usual sense, and therefore was not useful for answering any question about whether or not blame and praise are appropriate. While the experiment had demonstrated differences between the two cities, the inconflationists argued that all such differences were easily explained by virtue of the usual mechanisms of the laws of physics, psychology, economics, and other sciences. Many of the inconflationist researchers went on to take prestigious and influential positions as Philosophy professors.

The conflationists believed that the experiment had solidly demonstrated the role of blame and praise in the conduct of a civilized society, and that the results on their face were a clear demonstration of their utility. Many of the conflationist researchers went on to take prestigious and influential positions as researchers in the Social Sciences.

Despite their differences, the conflationists and the inconflationists agreed, for the most part, on appropriate conduct for individuals.

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Friday, January 22, 2016

Proof That We Have Free Will. And Don't Have Free Will!

A key question philosophers ask is whether the decisions and actions that a person takes could have been otherwise. If so, we say they have free will. If not, we say they do not have free will. The battle between these two camps of philosophers is fierce and bloody, and responsible for the severe shortage of philosophers around the world. I will attempt to finally settle the issue logically, in a way that each side will be able to claim victory. Finally, peace for philosophers!

If you're a compatibilist, and believe that free will is compatible with the laws of physics, you're in luck! Think of any decision you made in the past that turned out to be a good decision. Did you accept just the right marriage proposal? Pick the right stock at the right time? Maybe your dessert choice at the restaurant turned out to be even more delicious than you imagined. I will prove, logically, that you could have done otherwise, and therefore that the choice was up to you. You are therefore entitled to all the praise and enjoyment that may result from your decision. Hurray for you!

If you're an incompatibilist, and believe that free will is incompatible with the laws of physics, you're in luck! Think of any decision you made in the past that turned out to be a bad decision. Did you pick the wrong job? Cheat on your significant other, causing a breakup? Maybe you just picked something from the menu at the restaurant that turned out to be tainted and made you sick? I will prove, logically, that you could not have done otherwise, and therefore that the choice was not up to you. You are therefore excused from any moral responsibility for the choice. It wasn't your fault!

If you're a logician, and fear that these two positions are fundamentally incompatible, you're in luck too! If we parse the meaning of the two positions carefully, we will see that they are not logically opposite to each other. Perhaps you selected the most delicious dessert, but it also made you sick. Should you take credit for the decision or not? As we'll see, the answer is yes, your should take credit for the decision... or not. You will not need to believe impossible things, like the Queen in Alice Through the Looking-Glass, to believe both at the same time. I will prove that these two positions are not merely logically compatible, but they are both mutually necessary consequences of our understanding of the world. Yay logic!

Background, Assumptions, and Definitions

The thing to be proven is

(if you're an incompatibilist):
If we could rewind the world to precisely the state before any particular decision you made that you regret, then you would make exactly the same decision.

(if you're a compatibilist)
If we could rewind the world to precisely the state before any particular decision you made that you were happy with, then you would make a different decision.
If you are a logician we will expect a little more from you. You will have to follow two proofs at once. You're up to it.

We will prove these propositions using only first-order predicate logic and a few "facts" about the world that you are asked to agree to. Don't worry, we are not asking for much. These will be treated as axioms in our proofs. We'll label them so that we can refer to them later.
(A1) Axiom 1: Increasing Entropy. Future states of the universe have a higher total entropy than past states.
The time scales of interest here are those meaningful to a person: seconds, minutes, months, years. The intent of this axiom is that nothing may occur that would cause total entropy of the universe to decrease over any meaningful time scale. This axiom is consistent with every widely-held scientific view of the physical world. It is one form of the second law of thermodynamics.
(A2) Axiom 2: (In)Determinism. The world is deterministic, in the sense meant by physicists. (Unless you're a compatibilist.)
The future state of the world is fully determined by the present state, without the possibility of any variation. For those of you who believe in any of that hooey "quantum physics science" nondeterminism nonsense, or the soul, or god's influence over our actions, or karma, we won't allow any of that in our model of the world. Actually, no, wait, scratch that. If you're an incompatibilist, this axiom is that the world is deterministic. If you're a compatibilist, this axiom is that the world is nondeterministic, and you're allowed whichever of these odd beliefs make you happy. (If you're a logician, take your pick. We don't actually use this axiom.)
(A3) Axiom 3: No Time Travel. The future state of the world is only affected by the past state of the world.
The future can be affected by the past, and whatever other things we allowed you to believe through Axiom 2. But none of those things are allowed to carry information, or matter, of any kind from the future into the past. You're not allowed to whisper stock picks into the ear of your past self, or send an iPad to 1960, or kill your grandfather. These are things that would modify the state of the world after rewinding it to a previous state, so it isn't allowed. This axiom is consistent with the currently known laws of physics. (If you're a logician and aren't sure what Axiom 2 allowed you to believe, don't worry. We don't actually use this axiom either.)

The Experimental Method

One approach we could take to addressing this question is to just try it. Surely, as scientists, that would be the most rational thing to do. Unfortunately, reality rears its ugly head:
  • We do not currently have the technology to "reset" the complete state of the world to some previous state.
  • If you were to actually reset the world to a previous state, you would probably end up going through the day and then getting to a time that you do the same experiment again. You would, essentially, find yourself in some strange version of Bill Murray's Groundhog Day, living the same day of your life over and over again forever.
  • Since we are resetting the entire state of the world, as experimenters we would not know what you did the first time because our memory would have been erased. So we would have no way to judge whether or not you made the same decision the second time.
Unfortunately, this will remain a thought experiment for now.

Proof Technique

We use the ordinary proof techniques of first-order predicate logic. Specifically, if we want to prove a statement of the form "If P then Q" also sometimes written "P implies Q" for some proposed statements P and Q, then we can proceed using the technique of proof by contradiction. To do that, we assume all of our axioms, and also assume P, and also assume NOT Q, and then attempt to derive a contradiction from this set of assumptions. If we can derive a contradiction, then we consider the proposition "If P then Q" to be proven. Our original thing to be proven is precisely in this form, so this proof technique can be applied directly.

The Proof

Along with the axioms, we proceed to use proof by contradiction by assuming P (for "premise") and NOT Q (we will call this S) from the thing to be proven, which we recall is
(if you're an incompatibilist):
If we could rewind the world to precisely the state before any particular decision you made that you regret, then you would make exactly the same decision.

(if you're a compatibilist)
If we could rewind the world to precisely the state before any particular decision you made that you were happy with, then you would make a different decision.
(P) Premise: we could rewind the world to precisely the state before a particular decision that you made.
(S) You would not make
  • (for incompatibilists) exactly the same decision;
  • (for compatibilists) a different decision.
Now, we need a lemma, derived from Axiom A1. Note that the entropy of the universe is, by A1, a property of the total state of the universe that is monotonically increasing when viewed on human time scales. Rewinding the state of the universe from some time after a decision to some time before that decision would be changing the state of the universe from a higher entropy state to a lower entropy state, which would violate A1. Therefore, as a corollary to A1 we have the lemma L1:
(L1) Lemma: we could NOT rewind the world to precisely the state before a particular decision you made.
We now have two facts, (P) and (L1) that are directly contradictory to each other. As we have derived a contradiction, we have completed a proof by contradiction of "if P then Q", quod erat demonstrandum.

Counterfactual What?

The disagreement between the compatibilists and the incompatibilists is based entirely on a fundamentally false, or counterfactual, premise. It is not necessary to interpret such a position as literally requiring that the counterfactual be true when it isn't. Another way of interpreting this argument is to consider a universe as similar as possible to ours, but with the minimum possible changes such that the counterfactual statement is true. The question then becomes whether such a universe is more similar to the one understood by the compatibilist, or the one understood by the incompatibilist. Unfortunately, in this case the second law of thermodynamics is so deeply rooted in our understanding of the way the world works that it does not make sense to imagine what such a world would look like at all. A world like that would just not make sense to us, as the usual rules of cause and effect would not apply. So whichever camp you're in: congratulations, you're right!

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Thursday, January 21, 2016

Free Will is a False Dilemma

There is an age-old debate among philosophers: do people have free will? The question is considered important for moral judgment. If people don't have free will, it seems they shouldn't be held responsible for their actions. Why should we praise or scold a person for actions they were unable to control?

I submit that the question isn't as complex as it seems, and neither is the answer important to help us decide how to behave. I believe the sides aren't as diametrically opposed as it would seem.

The key question these philosophers are asking is whether the decisions and actions that a person takes could have been otherwise. If a person's decisions could have been otherwise, then clearly they had free will to choose one or another option, and it makes sense to hold them responsible for the choice they actually made. But if, on the other hand, they could NOT have done otherwise, what sense would it make to give them credit, or blame, for their actions?

The question of free will is usually posed with the assumption that the world is deterministic, despite what we know about the inherent unpredictability of quantum physics. My intent isn't to question that assumption, at least not today. Let's take it as given that the world unfolds as a sequence of events, one after another, each the cause of the next in a well-defined way, even though mere mortals aren't able to predict the future in practice. The two sides in this debate are the compatibilists and the incompatibilists, reminiscent of the big-endians and little-endians of Gulliver's Travels. Recall that the big-endians cracked their eggs on the big end, and the little-endians cracked them on the little end. Because they could not agree on such an important issue, they were forever at war.

The incompatibilists believe free will is impossible ("incompatible" with determinism), as people are not in control of the ultimate causes of their actions. By "ultimate cause" they mean the cause of the cause of the cause... ad infinitum. Or as far as time can be said to have existed. Because a person cannot influence any link in the causal chain, they "could not have done otherwise." The incompatibilists deem something like the big bang as the ultimate cause of everthing that has happened since. Because we did not control the big bang, we can't reasonably be held responsible for anything since then, such as our actions. To an incompatibilist, the concepts of moral blame and worth are nonsensical. A person can't reasonably be said to be ultimately responsible for their actions, and therefore they should not be held moraly responsible, blamed, or praised. Many incompatibilists believe that an understanding of this truth will lead to a more humane treatment of our fellow man.

The compatibilists, on the other hand, believe that we have free will (it is "compatible" with determinism) because people think, feel, make moral judgments, and decide how to act based on their thoughts and feelings. It is obvious that we have free will. The compatibilists don't care whether we have control over the chemical reactions in our bodies, or our genes, or the limited set of choices we have to select from. To a compatibilist, it isn't relevant whether or not we are the ultimate cause; we are the proximate cause of our actions, and they arise from us. That is all that's necessary to say that free will exists. Compatibilists believe that the kind of free will that the incompatibilists say we don't have wouldn't be useful even if we had it. When a compatibilist wonders if we "could have done otherwise", the question is really whether one is likely to repeat the same decision in situations that are the same only in morally relevant ways. A murderer has demonstrated that he is likely to murder: clearly, under the specific circumstances that actually occurred, he did murder. But the compatibilist looks at all of the relevant circumstancess to see if at the time and place of the act there were circumstances - such as being threatened by the victim - that would have forced the hand of any reasonable person. Because people respond to them, we can use praise and blame to influence others to act for the benefit of society. Parents use blame and praise to teach children how to behave: for things such as intentionally poking someone in the eye, or taking the time to do a good job on one's homework. Laws and punishment for violating laws are necessary for a functioning society.

These two camps don't really disagree with each other about the facts. They just disagreee about what words we should use to describe them. The compatibilists agree that we aren't the ultimate cause of our actions, they just don't happen to care about that. The incompatibilists agree that it's useful to act as if we have free will, as long as we recognize that it's an illusion. What they disagree about is which we should call "free will". The incompatibilists agree that we should influence others to behave in ways beneficial to society. The compatibilists aren't too concerened about making absolute "moral" judgments, and agree that we should treat people humanely. What they disagree about is which we should call "praise" and "blame".

Of course I am oversimplifying both positions. Many people believe, for example, we can make absolute moral judgments, sometimes by reference to a deity (or deities). When people ask me what I believe, I ususally say that I am agnostic on free will, or that it depends on which definition you want to use. I agree with both camps, and I don't really care which words we elect to attach to which definitions, as long as we're clear about what we're trying to say. The two camps mostly agree with each other about the facts, though not always about what conclusions we should draw from them. The fight is more about the right to choose the "correct" meaning for words such as "free will," "cause", and "responsibility", because those words have emotional baggage attached to them that influence our thinking.

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Tuesday, January 19, 2016

You should be random, so carry dice!

Why should you act in a random, unpredictable, or arbitrary way? It seems no rational adult would want to do that. But as we'll see, there are situations that arise in our lives — surprisingly often — where acting randomly is exactly the right thing to do.

What is random?

What does random mean? By this we mean unpredictable. A random variable is some quantity that cannot be predicted in advance. A good example is the result of the roll of a six-sided die.

If randomness is useful at all (which is yet to be seen) then dice may be good enough for some purposes. But for others you might want something that is truly random: something that is impossible to predict in principle. Many modern computers can produce random numbers. For the purposes of this essay we will pretend that dice are truly random.

Acting randomly

Would it make sense to act in a random manner? You'd be howling at the moon, bouncing off the walls, jumping through windows, and generally making a nuisance of yourself. We'd probably have to keep knives far away from you. We'd probably have to keep people far away from you! No, I think we agree that this is not the kind of random behavior any of us would want.

Perhaps it would make sense to act randomly sometimes. Perhaps there are specific, well-defined situations in your daily life where you would benefit from making some decision that cannot be predicted. Perhaps it makes sense to keep dice in your pocket.

Philosophers have proven that this is impossible. Suppose you have a decision to make, and you want to make it rationally. If there is a rational reason to choose one option over the other, then that is, of course, what you should do. There would be no need to act randomly in that case. On the other hand if there is no rational reason to choose one option over the other, throwing dice would just be a waste of your time. You might as well just pick the first option, whichever that is. In either case, the dice would remain in your pocket. And since you will never use them, you might as well travel light and just leave them at home. Or at the game store. So philosophers have "proven" that we should never want to act randomly.

But philosophers are wrong.

Board Games

The most obvious time that you would want to act randomly is when you are playing a board game like Monopoly or Backgammon. The rules of the game require it. You simply cannot play these games without rolling dice and basing your actions on the outcome. Admittedly, this is a trivial example. Are there serious situations where rolling dice would help us in our daily lives? Yes, there are.

Shopping for toothpaste

Imagine that you are shopping for toothpaste. You go to the corner store and head down the toothpaste aisle. There are hundreds of brands to choose from. Every one of them is different from every other in some way. Every advertised feature seems relevant. Extra flouride? Whitening? Tartar control? Baking soda? Do you care what sweetener is used, or even if a sweetener is used at all? There are dozens of flavors. How will you make a rational selection? Surely, unless you inspect each and every choice, it seems impossible that you will select the optimal choice.

This is a difficult problem. If you only have fifteen minutes to make your selection, you might not be able to make the optimial choice. You could spend your fifteen minutes gathering what information you can, and return at a later time (or several times) to complete the job. In the meantime you'll just have to settle for having bad breath and not caring for your teeth. No, that doesn't sound like a very attractive option. Perhaps you should just get the same brand that you got last time - even though you aren't really very happy with it. Difficult.

This is so serious and common a problem that it has been extensively studied. Overchoice, also referred to as "choice overload", describes a cognitive process in which philosophers have a difficult time making a decision when faced with many options. From Wikipedia:

The phenomenon of overchoice occurs when many equivalent choices are available. Making a decision becomes overwhelming due to the many potential outcomes and risks that may result from making the wrong choice. Having too many [approximately] equally good options is mentally draining because each option must be weighed against alternatives to select the best one. As the number of choices increases, people tend to feel more pressure, confusion, and potentially dissatisfaction with their choice. Although larger choice sets can be initially appealing, smaller choice sets lead to increased satisfaction and reduced regret. Another component of overchoice is the perception of time. Extensive choice sets can seem even more difficult with a limited time constraint.
Fortunately, those of us who carry dice have an easy solution to this problem. Here is what you do. First, roll a die. If the result is any number one through five, just buy your favorite brand from among all those that you have previously tried. If the die shows a six, use your dice again to select a small random sampling from among all of the available choices. Two or three choices will do. Then look at those and if any of them seem plausible alternatives — if one might be your new favorite if only you had the chance to try it — then buy the best from among that small selection. If none of them seem likely, just buy your old favorite.

This strategy doesn't guarantee that you will get the optimal choice. But it does give you a very good chance of selecting a nearly optimal choice. Over time you will be more and more satisfied with the toothpaste that you end up using because you will be trying new ones from time to time. As the toothpaste industry produces new innovations, you will find yourself enjoying them. Most importantly, it reduces the amount of time you spend selecting toothpaste, leaving you more time for important choices, such as Ice Cream.

Now you know why philosophers have bad breath. It is because they do not carry dice.

The narrow bridge problem

Consider a two lane road that has one lane running in each direction. There is a place where the road narrows to a single lane to go over a bridge. Traffic on this road is not very heavy, and visibility is good, so there is no traffic signal. The obvious thing is to enter the bridge only when there is no traffic heading in the opposite direction. But what if two cars arrive at opposite ends of the bridge at about the same time? Each can wait for the other, but then they will both wait forever. Each can try going, and then back off if the other does the same. But then they will be going and backing off again and again, forever.

Philosphers have specified that the drivers are to exit their vehicles, meet in the middle of the bridge, and play chess until one of them has won a game. The winning driver is entitled to cross the bridge first. Philosophers find this a just solution, as it gives an advantage based only on the driver's ability to apply logic. Unfortunately, philosophers are perfect chess players. When they play each other, they always reach a stalemate.

The local engineers have devised a different solution. When you reach the bridge, you stop and roll a die. You then wait a number of minutes indicated by the number on the die. Then you enter the bridge. If you encounter another car on the bridge, you back out to the beginning and try again. This is the purpose for the fuzzy dice hanging from the rear-view mirror of many cars.

A more familiar example occurs when cars reach an intersection in which there is a stop sign for each of the four roads entering the intersection. It is customary — in fact a requirement under the law — that if you arrive at the intersection around the same time as another car on the cross street, the car to the right goes first. This works fine except when cars arrive on all four roads. In that case obeying the rule, even though mandated by law, would lead to deadlock. That is when the procedure used on the bridge works very well.

Now you know why philosophers do not drive. It is because they do not carry dice.

The Scientific Method

There are applications in science where the intentional use of randomness is necessary to the proper conduct of an experiment. Chief among them is the double-blind trial. This is an especially stringent way of conducting an experiment that eliminates subjective bias both on the part of the scientist and on the part of the subject of the experiment. Double-blind studies are frequently used to test drugs for their effectiveness. It is the gold standard for scientific rigor.

In order to conduct a double-blind study, a population of test subjects is divided into two groups: those who will receive the drug to be tested, and those who will receive a placebo in its place. The subjects are assigned to one or the other group randomly. Neither the scientific investigator nor the test subjects know who is in which group. It is only after the outcome has been evaluated for all test subjects does it become known to the investigator which was which. Consequently, there is no opportunity for any bias to arise due to such knowledge.
Now you know why philosophers do not conduct scientific experiments. It is because they do not carry dice.

Game theory

Even if you playing a "deterministic" game like Chess, there may be value in making some choices randomly. As when selecting toothpaste, you usually cannot select the optimal move because you are operating under a time constraint. A technique called Monto Carlo Game Tree Search, which uses randomness in selecting moves to consider, has resulted in a revolution in the quality of computer play for Go, a board game that is widely played in Korea, Japan, and China.

Consider the game rock-paper-scissors. What if you want an optimal strategy, or procedure, for playing the game? What we mean by "optimal" is that there is no other strategy that has a winning edge over it, in the long run. There is a field of mathematics called game theory that studies games, and the game rock-paper-scissors is well understood. One strategy is to choose "rock", "paper", or "scissors" randomly, and it has been proven, formally, that this is an optimal strategy. It has also been proven that any optimal strategy for rock-paper-scissors must, necessarily, use randomness. In fact many games require randomness to be played optimally.

Game theory has wide applications in the real world, including war strategy, political science, economics, and negotiation. Now you know why philosophers are not rich. It is because they do not carry dice.


How can the philosophers have been so wrong? After all, didn't they prove that basing a decision on the roll of the dice is irrational? Yes, but they made a hidden assumption: that the decision is a one-time decision made in isolation. If you only have one decision to make in your whole life, the philosophers may have been right. But most decisions we make are part of a series of decisions, and the logic that the philosophers used does not work in that case. When we have a series of decisions to make, we need a strategy for making them, and it has been shown that randomness is a necessary part of the strategy for many problems.

While it seems that basing your behavior on the roll of the dice might be irrational, there are situations that arise every day where it is the most rational thing to do. Those people who refuse to do so are systematically weeding themselves out of the human gene pool. Don't be one of them. Carry dice.

"View source" and search for "Dining Philosophers" to find out why philosophers are hungry.
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Tuesday, January 12, 2016

Feynman on Philosophy of Science

This is one of my favorite Feynman quotes, from The Feynman Lectures on Physics
(Volume 1 page 2-6)

Another most interesting change in the ideas and philosophy of science brought about by quantum mechanics is this: it is not possible to predict exactly what will happen in any circumstance. For example, it is possible to arrange an atom which is ready to emit light, and we can measure when it has emitted light by picking up a photon particle[...]. We cannot, however, predict when it is going to emit the light or, with several atoms, which one is going to. You may say that this is because there are some internal "wheels" [variables] which we have not looked at closely enough. No, there are no internal wheels; nature, as we understand it today, behaves in such a way that it is fundamentally impossible to make a precise prediction of exactly what will happen in a given experiment. This is a horrible thing; in fact, philosophers have said before that one of the fundamental requisites of science is that whenever you set up the same conditions, the same thing must happen. This is simply not true, it is not a fundamental condition of science. The fact is that the same thing does not happen, that we can find only an average, statistically, as to what happens. Nevertheless, science has not completely collapsed. Philosophers, incidentally, say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong. For example, some philosopher or other said it is fundamental to the scientific effort that if an experiment is performed in, say Stockholm, and then the same experiment is done in, say, Quito, the same results must occur. That is quite false. It is not necessary that science do that; it may be a fact of experience, but it is not necessary. For example, if one of the experiments is to look out at the sky and see the aurora borealis in Stockholm, you do not see it in Quito[...]. "But," you say, "that is something that has to do with the outside; can you close yourself up in a box in Stockholm and pull down the shade and get any difference?" Surely. If we take a pendulum on a universal joint, and pull it out and let go, then the pendulum will swing almost in a plane, but not quite. Slowly the plane keeps changing in Stockholm, but not in Quito. The blinds are down, too. The fact that this happened does not bring on the destruction of science. What is the fundamental hypothesis of science, the fundamental philosophy? [...] the sole test of the validity of any idea is experiment. If it turns out that most experiments work out the same in Quito as they do in Stockholm, then those "most experiments" will be used to formulate some general law, and those experiments which do not come out the same we will say were the result of the environment near Stockholm. We will invent some way to summarize the results of the experiment, and we do not have to be told ahead of time what this way will look like. If we are told that the same experiment will always produce the same result, that is all very well, but if when we try it, it does not, then it does not. We just have to take what we see, and then formulate all the rest of our ideas in terms of our actual experience.

Saturday, January 09, 2016

The Many-Worlds Interpretation is a realist interpretation of the universe, but not a realist interpretation of the world.

The Many Worlds Interpretation (MWI) is a deterministic, realist interpretation of quantum mechanics (QM). MWI starts with two postulates

  1. The universe is described by a quantum state, which is an element of a kind of vector space known as Hilbert space.
  2. The quantum state evolves through time in accordance with the Schrödinger equation, with some particular Hamiltonian.
From the point of view of MWI, the quantum state of the universe (also known as the Universal Wave Function) is the thing that is "real". It evolves in a locally deterministic way.

What happened to the "other worlds"? Why is it even called the "many worlds" interpretation? Other worlds are not postulated by MWI, rather they arise naturally from an understanding of the behavior of the system based on just the two things we do postulate. And what about "decoherence"?

In MWI, decoherence is said to occur when the phase angle between components of the quantum state are sufficiently orthogonal that, for practical purposes, they do not exhibit interference. The fact that this occurs is a consequence of the underlying math. This happens naturally when information about quantum interactions (e.g. the result of a quantum experiment) spreads into the environment through further interactions (e.g. because the result is displayed on the measurement instrument, and photons from the instrument's display reach the experimenter's eyes, the walls, etc). Once that occurs, we can analyze the orthogonal components of the quantum state in isolation. These orthogonal components can be interpreted as independent worlds, or alternative futures of the world, each representing the future following one possible outcome of the interaction (e.g. measured result).

In practice the phase angles are never completely orthogonal, because the spread of information into the environment is limited by the speed of light; there are sufficiently distant regions of the universe where the components may interact. So the meaning of decoherence is interpretational: it depends on what we mean by "sufficiently orthogonal" and what the "practical purposes" are. If we are only interested in what happens in our experimental laboratory, the behavior of distant reaches of the universe in the distant future can be treated as irrelevant.

This is no different from saying that, for sufficiently small velocities, mechanical systems obey classical rather than relativistic behavior. What is "sufficiently small"? It depends on the context. Nature does not care what we mean by sufficiently small, it always obeys the relativistic rules. But the concept of classical behavior allows us to simplify our calculations (at the expense of introducing a small inaccuracy) to improve our understanding of the system.

So it is for decoherence. It is not a term that is rigidly defined in the theory, but (like "non-relativistic velocity") is rather a concept for a simplifying assumption that we use to understand the behavior of the quantum state. Decoherence cannot properly be said to occur at some particular time, like quantum collapse in the Copenhagen interpretation. It is not an "event that happens", but rather a change in the way we interpret the meaning of the quantum state from one time to another.

One accepts MWI at the expense of rejecting objective reality as we know it. When we open the box to see whether Schrödinger's cat is alive or dead, we become entangled with the cat's quantum state. If we see that the cat is alive (as we hope), we cannot say that the cat's status of being alive is a fundamentally true fact about the universe. Rather in the quantum state of the universe, there are nearly orthogonal components that can be interpreted as two versions of our world, one in which we observe the cat being dead, and one in which we observe a living cat. One of them feels more real, somehow, but each component describes a version of us who thinks it is he who is observing the true world. MWI doesn't designate one of these components as somehow more real than the other, and thus we can think of them as separate worlds, or futures.
[MWI] predicts that we will think and claim, that we do not observe superpositions at all, even when our own states are highly indefinite, and that we are simply mistaken in the belief that we see a particular outcome or other. That is, it preserves unitary [deterministic] QM – at the expense of a skepticism that "makes Descartes’s demon and other brain-in-the-vat stories look like wildly optimistic appraisals of our epistemic situation" [The Ashgate Companion to Contemporary Philosophy of Physics page 43]
This is like Einstein's principle of relativity in another way, too. In MWI, the meaning of the world is relative to the observer. If you ask whether the cat is alive or dead as a property of the universe, the simple answer is that the cat is in an indefinite state. To give a more definite answer we would need to know which (mostly) orthogonal component of the quantum state you're asking about. Which world did you mean?