Jach's personal blog

(Largely containing a mind-dump to myselves: past, present, and future)
Current favorite quote: "Supposedly smart people are weirdly ignorant of Bayes' Rule." William B Vogt, 2010

Possibility and Free Will

What is possibility? What do humans mean when they say "This is possible" or "That's impossible!"?

There are many definitions of "possibility", so in polite conversation it's useful to use them precisely. Let's start with "physical possibility" as one form. This is basically saying "The known Laws of Physics either permit or do not forbid this outcome from occurring." As an example, suppose you're pondering the possibility of picking up a rock and throwing it such that it lands on the moon. This is physically impossible because your body and your arm do not contain the required energy to get the rock there. As another example, suppose you are considering the possibility of sustained nuclear fusion in the lab providing a huge supply of cheap energy. This is physically possible because we have physical proof of this process already happening--that is, the sun does it already.

Another form of possibility is "Logical possibility". This is basically saying "Using two-valued True/False logical propositions and their rules, we can logically deduce whether some proposition questioning the logical possibility of something is true (that is, it's possible) or false (that is, impossible)." As an example, suppose you assume for the sake of argument that a human arm can generate a million petajoules of energy on a whim, then you assume the fact that it takes some amount of joules to throw a rock to the moon. Then you can logically deduce that because you have a sufficient amount of energy to throw a rock to the moon, you can do so. It is logically possible under this set of assumptions. As another example, suppose for the sake of argument that penguins can fly. Then it's logically possible for a penguin in a zoo to take off and fly back to its home in wherever it came from. As another, assume for the sake of argument that crows can't fly, and so it's logically impossible for it to escape a glass container you put it in.

My three examples are all on their face absurd. This is intentional. Logical possibility and logical impossibility are only as valid as the assumptions used to arrive at those conclusions. If you assume nothing supernatural can exist, then it's logically impossible for most definitions of God to exist. But this has no bearing on the empirical question of whether God exists, supernatural things exist, the empirical question of whether penguins or crows can fly, the empirical question of whether a human arm actually can generate petajoules of energy within seconds. Therefore physical possibility is stronger than logical possibility. This doesn't mean logical possibility is useless: you can pick less controversial assumptions that most people agree are obvious and use the many tools of logic at your disposal from there to find out interesting facts. You can even use the very useful probability theory tools. If you as a cop know and assume that people who break store windows late at night are likely robbers, you can conclude that you should arrest first and ask questions later if you see two thugs break a store window.

You can even intermingle logical and physical possibility by assuming physics as we know it is completely correct and working out logical implications that you can then experiment. If your logical implication says something is impossible, that's great news! Now you can go test it and see if your assumption that physics is correct is empirically sound.

There's another form of possibility I want to talk about, which is the most general kind. Sometimes people qualify it with "That's theoretically possible or theoretically impossible", but that assumes a theory and often logic so it's not the precise form of possibility I'm after. I don't really have a handy qualifier for it apart from "general", so I'll have to explain it. This most general form of possibility is closer to "I can imagine it to be the case."

"General possibility" is basically saying "this abstract world-configuration is a way out of all the available ways some state-space can be configured." More concretely, it's kind of saying "This universe is made of atoms. This particular arrangement of atoms, with one here, and one there, is a way to arrange the atoms. It is therefore possible." Thus it is generally possible that the Eiffel Tower could hover over the Golden Gate Bridge tonight at 6pm and stay there indefinitely, because I can imagine the universe having the same arrangement of atoms that is the Eiffel Tower but in a different place--namely above the bridge.

Thus it is generally possible for me to throw a rock to the moon in this sense: at one particular slice of time, the arrangement of the atoms in the universe has me holding the rock. In another slice of time (not necessarily in the past or future, just an arbitrary readout of the state of the universe's atoms), the rock could be 100m in the air and my arm in the follow-through motion of a throw. At another slice, the rock could be on the moon and I, still on earth, would have just completed a throw. If you took these three pictures together you could interpret them as "I threw a rock to the moon".

So this general notion of possibility isn't that useful, either from a logical or empirical standpoint. Logical possibility at least has useful tools, and physical possibility actually tells you about the world we live in. At best this general possibility is like the existential quantifier in math: general possibility says that something exists in the set of ways the universe could have its atoms arranged, and general impossibility says that something cannot exist in the set of ways the universe could have its atoms arranged.

Unfortunately this general sense of "possibility" seems to be the most commonly used in everyday discourse when people say "That's possible!" or "That's impossible!"

Okay, maybe not quite as general, but close. And so people are easily deceived (lotteries), scared (password cracking), or confused (quantum physics), or argumentative (free will). Let's look at each of those in turn.

Here's a lottery bet for you. I have a fair coin, I will flip it twice. If heads comes up twice (25% of the time), I pay you $1000. Otherwise, you pay me $500. (75% of the time.) Do you want to take this bet? You are free to take it as many times as you like.

Las Vegas' popularity indicates that there are indeed people who will take this bet. They are fools and Sure Losers, by both empirical observation and logical deduction.

But they can grandstand and say "Ahh, but it's possible for heads to come up twice 200 times in a row and thus it's possible for me to win $200,000!"

Logic says this: your expected earnings is mathematically calculated by multiplying the probability of your wins times the value of your winnings, then subtracting the probability of your losses times the value of your losings. In this case, $1000*0.25 - $500*0.75 = -$125. This is a losing strategy, you are more likely to have losses than gains, so you shouldn't take this bet even once let alone multiple times.

A combination of logic and physics says this: if we assume the math is correct, and we actually found a sucker to accept this bet 10 times, then on average, over those 10 rounds, the sucker will have lost around $1250 dollars at the end of the 10th round.

For all intents and purposes, you winning after multiple rounds is impossible by the two important versions of possibility we discussed. Don't take the bet! Don't get deceived!

Let's talk about password breaking now briefly. Most people don't get scared about this particular topic, but some nerds fall for it. There's a program called "bcrypt" that securely stores your password in a way such that if a cracker got access to that stored piece of information, it would take them some configurable number of seconds to see if any particular password corresponds with the information they hacked. An overkill configuration might be "it takes 10 minutes to take the string of text 'foo' and convert it into the string of text 'FJIi3020jfFS2030j2f'." So suppose the attacker got hands on the random-looking string, they want to brute force what the password is since they don't know if it's "foo" or "apple" or "cat" or something more complicated. A common brute force method is to try every word in a dictionary sequentially. In my digital dictionary, "foo" is the 71,635 entry. It would take about 1.4 years of computer time to get to it. If you require more complicated passwords (such as needing to be over 8 characters, needing to have lowercase and uppercase letters and numbers), the time required quickly exceeds the age of the universe to compute. This effectively makes it impossible to crack such a password by brute-force, even though a scared person might grandstand and say "But you could just choose randomly and you might get it on the first try!" Yes, that's generally possible, it's even logically possible because you have an imperceptibly greater-than-zero probability of that happening, but it's not physically possible. If you perform the experiment you will not crack it in the age of the universe, let alone on your first try. Relax, trust in bcrypt.

Without going too much into quantum physics, I think it's easy to agree that it has confused a lot of people. I'm not even going to argue for one of the "interpretations", I'll stick with what's known and not controversial, though it is heavily simplified. Quantum physics says that atoms aren't fundamental objects in the universe, rather it's information about energy. So you might have a light particle (photon) not as a particle but as a "packet" of information. You might also have a "packet" of information describing an electron, and the physical laws describe how these two "packets" interact. Quantum physics says that the photon can be absorbed by the electron then emitted, or the electron can emit another photon then absorb the other one. These are the two "possibilities" physics says are available. They are physical possibilities.

For some reason people seem to interpret this as "Quantum physics says all things are possible! It's possible for me to pass through this wall!" But the thing is, it's not physically possible in the sense that you can try passing through the wall for ages of the universe and never do it. There are so many electrons and photons involved that make up you, the air, the wall... that for all the right electrons and all the right photons to go just the right way is absurd in a way far more absurd than the lottery or the password breaking examples. It won't happen to you. Just because it's a valid state of the universe to be in where you are on one side of the wall, and a valid state of the universe to be in where you are on the other side of the wall, doesn't make it physically possible, it only makes it "generally possible". It's physically impossible, don't get confused and think quantum physics means anything can happen.

At last I think I can now explain the free will problem to you in a way that's intuitive, provided you understand most of the above. Once explained, you should no longer be troubled by questions like "Do humans have free will?" or "What is free will?" or "Is free will compatible with determinism?" Free will is a solved issue. If you do not understand after reading, it is a failure on my part as an explainer to convey my intuition properly.

Let's get this notion of determinism out of the way quickly. Physics is deterministic. Quantum physics is deterministic. Some people like to say quantum physics is inherently random, but it's not. We are just uncertain about the starting conditions--the ignorance is only in our minds, not in reality itself. In the example above where I gave two interactions for the photon and electron, there were only two because physics says there are only two. It's not a random number. It's not going to be 1 today and 2 tomorrow and 10 the next day then back at 1. It's 2 always and forever. Physics says so. (Okay, again, my simplified representation of physics says so.)

You and your brain are made of electrons and photons (and quarks). This means that you are a part of the physical system and the physical system is a part of you. When you consciously command your body to move, physical processes are happening in your brain and later on physical processes result in your body moving. All deterministically--if I tell my arm to move up, it moves up. It doesn't vanish to the moon.

Free will says: I could go to the store if I wanted to. Some people interpret this to mean "Until I decide you can't predict whether I will with any decent accuracy because I can always change my mind, especially if you tell me what you predicted!"

Okay, so instead of predicting your precise actions, we make predictions on your decision making process. If I discover data that shows you have no food at home and you're hungry, I can predict with very high accuracy that you will go to the store soon. I can predict with such high accuracy that you can make a bet out of it and I would always be a winner.

You are predictable. You are deterministic.

"But it's still generally possible that you could be wrong once even if you're most likely to never be wrong in ten trillion separate trials."

That's what free will feels like. It feels like we could always reach a state where you don't actually go to the store despite deciding to earlier, or despite being out of food and hungry. Except, if you're a normal human and avoiding other confounding factors like money (so I'm assuming you have money for food and time to shop and a host of other reasonable assumptions about people), you're going to go to the store, period. I predict so and I wouldn't ever be wrong if we did the experiment over and over 10 trillion times. That's our setup. But it still feels like I could be wrong every time, and again, that's what free will feels like.

It's because humans do not apply physical possibility or physical impossibility to their selves very well. We have this notion that we're apart from physics, and that we have this "free will" component in our brains that determines our future outside of physics. But this "free will" component is itself made of physics, and so really it's the same quantum physics doing the work as always, it just feels different to you.

I want to take a somewhat abstract but different angle on identifying the feeling but really nailing down the impossibility of certain actions.

Suppose you lived in a world full of lots of robots, and you yourself were one of these robots. There are two types of robots in this world, ones that can only yell even numbers, and ones that only can yell odd numbers. None of the robots have a notion of what an even number is or what an odd number is, they just know that they and all the robots around them shout various numbers a lot and sometimes when they try to imitate certain other robots by saying what they said they can't--nothing comes out of their voicebox.

You are an odd robot, but don't realize it. You stat your day saying "1, 3, 889, 2923, 13, 17." Your neighbor, who is an even robot, greets you by saying "4, 8, 98892, 9331930, 893289328."

Free will is like you the odd robot thinking "My neighbor said 8. I feel like I could say 8, if I wanted to."

But you can't. You're an odd robot and odd robots cannot physically say 8. Furthermore any odd robots who have ever tried to say 8 ended up not saying anything, or worse saying an odd number completely different like 7. It is physically impossible for you to say 8, even though you can hear it, see it, think it, and even believe it should be generally possible for you to say 8 because you see other robots just like you (apart from the fact that they're even robots) saying it with some frequency.

I'll leave you with that. Just because you can imagine yourself doing something, just because something may indeed be "generally possible", does not have any say on whether that is physically possible which is the main form of possibility you should care about. If you stop thinking of general possibilities as anything more than interesting philosophical considerations, and start putting a lot of emphasis on physical possibility and what can actually happen as far as we know physics, you'll find that your free will notions will melt away over time. You don't lose any sense of control in your life, for the physics are deterministic, but you are part of the physics, and thus you in part determine your life to the physically possible extents allowed by reality. That's all there is.

Posted on 2012-02-04 by Jach

Tags: free will, philosophy, probability, rationality


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