sex with ducks

Thanks to a Nadder, I just discovered the music of Garfunkel and Oates. You really need to listen to the (need I say NSFW) "sex with ducks" track.

How Many Worlds?

The Many-Worlds Interpretation of quantum mechanics (MWI) has become popular in recent years. Among non-physicists, this popularity may be due to the woo factor - imagining a parallel universe in which I won the Nobel Prize, or I didn't eat the last donut, can be lots of fun. However, it has also achieved a certain popularity among physicists - this I find less easy to understand. Let me explain.

In quantum mechanics, we often have to deal with superposition states. These are states in which more than one outcome for some measurement is possible, and we only know the probabilities of each outcome. For example, the state
(|+> + |->)/sqrt(2)
is a superposition state of a spin 1/2 particle in which the "spin-up" state (represented by |+>) and the "spin-down" state (represented by |->) are equally probable. (Here "up" and "down" are defined with respect to some arbitrarily chosen reference direction - not with respect to gravity.)

In the MWI, a superposition state is taken to mean that the world has split into two branches. In one of these branches, the particle really is in the spin-up state, and in the other branch, the particle really is in the spin-down state.

To discuss the MWI it will be helpful to have an alternative interpretation to refer to. My preferred interpretation (at present) is the statistical interpretation (SI). In the SI, any state refers to a collection of identically prepared systems. A superposition state like the one above represents the case where half of the collection is in the up state and the other half is in the down state.

Now, the weird thing about quantum mechanics is that a superposition state is fundamentally different from merely saying there is a 50% probability of spin up and a 50% probability of spin down. That's because a superposition state can exhibit interference, as in the two-slit experiment.

OK, now that we've got some of the basic ideas down I can start to explain what's wrong with the MWI.

Metaphysical excess: In quantum mechanics, superposition states are formed all the time. According to the MWI, that means that the universe is constantly splitting into new branches. So the MWI posits a near-infinity of additional universes, none of which can be detected from whatever branch we happen to land in. This seems to run afoul of Occam's razor.

Compare this to the SI, in which we simply talk about many runs of the experiment. (We need to do many runs in any case, in order to test the probabilities that result from quantum mechanical computations.)

Worlds don't really split in the MWI: We might expect that, once the universe has split into two branches, those branches no longer have anything to do with one another. However, if that were the case, we would never observe quantum interference! In order to retain interference, the MWI must suppose that the different branches of the universe share a ghostly connection that allows interference to occur.

Compare this to the standard interpretation, in which we say that particles sometimes "behave like a wave." For me, it's much easier to imagine a wave moving through real space than a mysterious connection between universes that are separate (but not really).

The basis problem: When do worlds split? If the answer is "whenever there is a superposition", then we have a problem. In quantum mechanics, we can write a state in any basis we like. So in one basis, a given state will have only one term, while in another basis the same state might have two, five, or an infinite number of terms! So how does the universe know how many branches to split into?

The MWI is not so clear on this. The answer is sometimes given as "when a measurement occurs." Then the appropriate basis for branching is taken to be the measurement basis. But how do we decide what sorts of quantum interactions constitute a measurement? Deciding when a measurement has occured is one of the things the MWI was supposed to save us from. But it hasn't.

Even worse is the answer "when decoherence has occurred." Because, if the evolution of the universe is completely quantum mechanical, as the MWI claims, then decoherence is never complete.

Probability problems: If we have a superposition state that has a 50-50 split, then saying the universe branches in two seems to make sense: we have equal probability of ending up in each branch. But what if the probabilities are 2/3 and 1/3? Or 1/sqrt(2) and (1-1/sqrt(2))? The way MWI deals with this is to assume we start with a large number of identically prepared systems, so that the branching can occur according to the usual quantum mechanical probabilities. But this is just the statistical interpretation, sneaking in through the back door!

The bottom line: The MWI doesn't actually solve the problems it is supposed to solve. It doesn't explain interference, it doesn't solve the measurement problem, and it needs to be supplemented with a statistical interpretation in order to reproduce quantum mechanics. So we have paid a huge price in metaphysical multiplication, and have nothing to show for it!

In my opinion, the MWI is an unnecessary and unhelpful load of metaphysical baggage.

Free Will Index

An index to my series on free will:

1. Free will and quantum mechanics
2. The Consequence Argument
3. Some Uninteresting Questions
4. Sphexishness and Other Concerns About Predictability
5. Electrons R Us!
6. Quantum Mechanics Revisited
7. The Problem of Free Will - Solved!
8. A Libertarian's View  
9. Varieties of Libertarianism
10. Caused But Not Determined
11.  Ekstrom's Solution
12. My Gut Says, "Meh..." 
13. Indeterminism Where?
14. Not All Things Considered 
15. Command or Description?  
16. Level With Me
17. Level With Me Again

I make no claim to originality in any of this. My views have been largely shaped by Daniel Dennett's books, but my take on the importance of quantum mechanics for free will is somewhat different from his. A nice collection of articles about free will is linked at this FRDB thread. See especially the articles there from the Stanford Encyclopedia of Philosophy, and this set of articles from the Routledge Encyclopedia of Philosophy.

The Problem of Free Will - Solved!

I decide to pick up my pen - I reach for my pen - and my hand moves in accordance with my will. The connection between my conscious thought and my actions is an empirical fact. If this is free will, there are few who would deny we have free will. And this, I think, is the only sort of free will that really matters.

(Daniel M. Wegner, in The Illusion of Conscious Will, argues that the connection is not nearly so close as our intuition suggests. He points to phenomena such as automatic writing, hypnosis, and multiple personalities that indicate it's possible for there to be action without the feeling of will, and will without a corresponding action. This may well be true, but the "normal" connection of will and action persists nonetheless.)

To summarize what I've been writing about:

• - We need not fear determinism, for the world is not deterministic.
• - We need not fear the fixity of the future; that is the fatalist fallacy.
• - We need not worry that strict microscopic laws of physics allow our every action to be predicted, because chaos and quantum mechanics will not allow such predictability.
• - We need not worry that the electrons are in control, because the electrons are us.

So it seems there's nothing much left to worry about. As Daniel Dennett puts it in Elbow Room (p.169):

When we look closely at the sources of our suspicion and dread, we find again and again that they are not indisputable axioms or overwhelmingly well-supported, empirical discoveries, but unfocused images, hastily glanced at - like the shadows on the bedroom wall that take on an apparent robustness and menace precisely because we do not look at them closely.

There are other varieties of free will discussed by philosophers. Some would prefer an immaterial spirit that somehow affects our material bodies. Others attempt to create free will out of the very quantum jumps that (perhaps) leave our actions unpredictable. 

But the plain ability to decide to reach for my pencil, and then to do so, seems to me a variety of free will "worth wanting," in Dennett's phrase. If I can make my body do what my thoughts choose to do, what more could I want? To defy the laws of physics? Wishing for a supernatural miracle, or a natural, quantum-mechanical miracle, is as pointless as wishing I could leap tall buildings in a single bound.

I do not need such miracles to be content. I can glory in the notion that all of the human race's complex thought is a result of purely physical processes, and that those processes are the result of millions of years of evolution. From simple one-celled organisms whose only need was to find the next organic molecule to devour, through uncountable small changes in the ability to respond to the environment, and hence the ability to survive, came the amazing information-processing creatures we call humans. That's a heritage we can be proud of.

Quantum Mechanics Revisited

Looking back over my free will posts, I see that sometimes I seem to be saying that quantum mechanics is important for free will, and sometimes I seem to be saying the opposite. Let me see if I can clear things up.

I have addressed several different issues regarding the free will debate, and quantum mechanics relates to each issue differently.

Determinism and the Consequence Argument: To the extent to which the Consequence Argument depends on the premise of deterministic laws of nature, quantum mechanics is certainly relevant. Since quantum mechanics is an indeterministic theory, if quantum mechanics is a true description of the world, then the Consequence Argument fails. However, we have seen that the fear behind the Consequence Argument is not really the fear of determinism, but of lack of control: If the electrons are in control, then I am not in control. The problem then is not deterministic laws of nature, but any sort of microscopic laws at all.

Predictability: Suppose someone were to write a computer program that accurately predicts everything I will do and say tomorrow. That would be a huge blow to free will (though perhaps not a fatal one). How can I be anything more than a machine if a mere machine can duplicate my actions? Here, I claim quantum mechanics is relevant. Accurate prediction of a classically chaotic system will run up against the quantum limit in a very short time, forcing us to adopt a quantum description. But a quantum description will never produce perfectly accurate predictions: first, because the computer cannot completely replicate my quantum state (thanks to the no-cloning theorem), and second, because quantum predictions are only probabilistic (even if the exact quantum state were known).

Control: Here it gets a bit tricky. We have seen that for me to be in control of my actions, we need a fair amount of determinism in the world. I have been arguing that quantum mechanics has enough indeterminism to make effective prediction impossible. Does quantum mechanics leave us enough determinism for moral responsibility? I suspect it does, but this point clearly needs more investigation.

For now, let me just say that I think the way to proceed is by distinguishing the lower-level and higher-level laws of nature, as I began to do in the previous post.

Electrons R Us!

An imaginary scenario of NASA headquarters as a new robotic Mars rover begins its mission on the surface of that planet: Evelyn, the software engineer who wrote the code that allows the rover to pick its way across the Martian surface without running into rocks and cliffs, has invited her philosopher friend, Pete, to watch the rover's progress. Evelyn exults as the rover successfully negotiates one obstacle after another, and brags to Pete about how well her control program is working. Pete, however, complains, "It's not your program that controlling the rover; it's just electrons in those electronic circuits following the laws of physics!"

At this point, I can only imagine Evelyn standing there agape, thinking of the years she spent developing that software, unable to come up with any kind of non-insulting reply.

Pete is guilty of level confusion: his objection amounts to the claim that if a physical description of the rover at the level of electrons is correct, then any other description of the rover's behavior must be incorrect. I think the free will deniers are guilty of the same thing.

(Note: I am not making the claim that human brains are like computers in any particular way. I am only proposing the rover story as an analogy to help us understand different levels of description.)

Let's recall Argument #2 from this post:

Argument #2 - Causal Determinism is a Necessary Condition for Moral Responsibility Premise 1: Unless there are extenuating circumstances, persons are (to be) held morally responsible for their actions. Premise 2: Being unable reasonably to have foreseen the consequences of their actions is one such extenuating circumstance. (Recall that young children who cannot reasonably foresee the consequences of their actions are not to be held morally responsible for the consequences.) Premise 3: In order to be able to anticipate or foresee the likely (or even the remotely likely) consequences of one's actions, the world must not be random, i.e. the world must be fairly regular (or causally determined). Thus: Moral responsibility requires that there be causal determinism.

This argument, I think, is sound. In order for there to be moral responsibility, I want my actions to be causally determined. Determined by what? By me, of course! But what am I? I am my thoughts, feelings, desires, beliefs, and so forth. But if we are right in assuming that the mind is a purely physical entity (no magic allowed!), then thoughts, desires, etc., are all the results of some sort of electro-chemical activity in my brain.. ("We are electrons, and electrons are us.") And if my actions are to be determined by my thoughts and desires, then they ought to be determined by those patterns of electrical activity that are the physical counterpart of those thoughts and desires.

So if free-will deniers claim that we can't do other than what we did because "it's all just electrons" following the mindless laws of physics, they are committing the same sort of level confusion as Pete.

There can be a low-level description of an action that consists of electrons following physical laws. And there can be a high-level description of the same action that consists of mental states like thoughts and desires. And there need not be any contradiction between the two descriptions, any more than there is a contradiction between the electron-level description and the program-level description of the rover. At the physical level, all we see is electrons affecting the motion of other electrons. But at the mental level, we have desires affecting thoughts, thoughts influencing desires, and, crucially, a mind capable of reflecting on its own mental state.

Free will is something that exists at the mental level; it is a characteristic of the relationship between thoughts and actions. To try to look for it at the level of electrons is to commit a category error - just as it would be to look for a FOR loop inside the rover. ("All I see is electrons!")

Sphexishness and Other Concerns About Predictability

Philosophers of free will give the issue of predictability short shrift, in my opinion. Typically they are worried about whether, given the past state of the universe and the laws of physics, the future of the universe is completely determined in principle. They aren't particularly worried about whether it is predictable in practice. I think they should be.

Consider the curious habits of the Sphex wasp. She lays her eggs underground, then flies off to get a cricket, which she paralyzes and places in the burrow for the hatchlings to eat when they emerge. In doing so, she follows a strict routine: after dragging her prey to the burrow, she goes inside to check on the eggs, then comes back out and finishes the task of dragging the cricket inside. Now, if some devious human moves the cricket away from the entrance to the burrow while she is inside checking on the eggs, she comes out, drags the cricket to the entrance again, and once again goes inside the burrow to check on the eggs.

If again the cricket is removed a few inches while the wasp is inside, once again she will move the cricket up to the threshold and re-enter the burrow for a final check. The wasp never thinks of pulling the cricket straight in. On one occasion this procedure was repeated forty times, always with the same result.

(Wooldridge, quoted in Dennett,  Elbow Room, p. 11.)

The first thing to note about the strange behavior of Sphex is that this is not a case of an agent doing the exact same thing under the exact same microscopic conditions. It doesn't matter if the experimenter moves the cricket three inches or four, the "same behavior" is repeated. And by "same behavior", we don't mean the microscopically exact same behavior, rather we mean the same general class of behavior. This is, I think, a much more interesting question than the question of whether we would do the exact same thing under the exact same conditions. Is there some general class of conditions under which I, as a human, would necessarily produce the same general response? If so, this is a more serious threat to free will than the possibility that I would do the exact same thing under the exact same conditions.

What is so striking about Sphex is the mindless predictability of her behavior. She apparently never stops to think, "Hey, didn't I just do this? Do I really need to do it again?" She is apparently pre-programmed to execute a series of steps. If those steps are interrupted, the series must be re-started at the point of interruption.

Are humans at all sphexish? Are there circumstances under which we will mindlessly repeat a certain behavior? We seem to be exempt from such behavior because of our ability to reflect on our own actions. Yet, if we behave according to (reasonably) deterministic laws of nature, there ought to be some (perhaps very small, yet not microscopically exact) set of conditions under which we would produce nearly the same behavior.

We can imagine someone, a burglar, say, who somehow gets hold of a complete physical description of  someone - down the quantum state of her very molecules. The burglar uses this information to predict when the victim will be out of her house, and then goes in and robs her.

This seems a reasonable concern, if the universe runs according to strict physical laws. But the laws of physics have been invoked to conjure up the spectre of sphexishness, and now the laws of physics rush in to the rescue. In a sufficiently complex, deterministic system, something called "chaos" comes into play. In a chaotic system, the set of conditions needed to produce with certainty a particular outcome is exponentially small. If we stipulate that humans are sufficiently complex, then the set of conditions needed to guarantee a particular human behavior will be smaller than the quantum limit allows, even for very short times. In other words, it will be impossible, without violating the laws of quantum mechanics, to predict the future behavior of a human from physical principles for more than some very short time.

If we move to a quantum mechanical description of our human (supposing such a thing to be possible, which is doubtful), one thing improves: quantum systems are never chaotic, as long as the Hamiltonian (the function that describes how the system changes with time) is completely known. So at first glance we seem to be better off with a quantum description, in terms of being able to predict outcomes. However, there are several problems with the quantum approach.

First problem: the Hamiltonian is never completely known as long as the system interacts with its environment. So, we could apply quantum mechanics to a person who was completely isolated from everything (including air, water, food, etc.)  - an unattractive prospect - or we could try to include the entire environment in our quantum description, greatly magnifying the difficulty of the task.

Second problem: Even with an exact quantum description (of, say, the person plus her environment) we will only get probabilistic results. That is, we can say she will have eggs for breakfast with 38% probability, jam and toast with 12% probability, etc. This is no more useful - and no more a violation of free will - than taking a survey of her eating habits.

Finally, I should point out that the problem I glossed over, of obtaining an exact quantum description of someone (to say nothing of her environment) is not just difficult, it is physically impossible. This is because of the no-cloning theorem I mentioned last time: it's impossible to take an unknown quantum system and make an exact copy of it.

So it seems that, from the point of view of predictability, free will has nothing to fear from the fundamental laws of physics. That is not to say that there couldn't be some higher-level laws that provide more predictability. After all, the same quantum considerations apply to Sphex, who remains quite predictable. (Not to mention a rock - also, presumably, a quantum system - whose behavior is extremely predictable.) So there is no way at this point to rule out the possibility of a nefarious burglar who exploits some as-yet-undiscovered psychological principles to generate useful predictions. But from what we have seen, there is no reason to think that the existence of such laws follows necessarily from the physical description of humans.

Next time: Electrons R Us.

Some Uninteresting Questions

As promised last time, I'm going to tackle some uninteresting questions about free will. What's interesting about these uninteresting questions is the amount of time and effort philosophers have spent on them. Well, I suppose they have to earn their salaries somehow.

Are Zombies Possible?

I'm not talking about the arm-ripping, brain-eating type of zombie, I'm talking about the philosophical zombie. In philosophy, a zombie is something that looks, acts, and talks in every way like a human, but there's no one home. These creepy entities have no thoughts, feelings, desires, or sense of self. If you think zombies could exist, then it's possible that everyone you know (including your spouse, parents, and children) is a zombie. As a physicist, I feel justified in ignoring these fantastical creatures: by assumption, there is no detectable difference between a zombie and a human. So what's the use in talking about them?


Would You Do It All Again?

One way of phrasing the free will issue is to ask, "If you were in the exact same situation, would you do the exact same thing, or could you have made a different choice?" This is often illustrated with Austin's Putt. Philosopher J. L. Austin wrote

Consider the case where I miss a very short putt and kick myself because I could have holed it. It is not that I should have holed it if I had tried: I did try, and missed. It is not that I should have holed it if conditions had been different: that might be so, but I am talking about conditions as they precisely were, and asserting that I could have holed it. 

The problem with this sort of discussion is in the phrases "exactly the same" or "conditions as they precisely were." Presumably we are talking about conditions being microscopically the same: same position of the ball, of every blade of grass and every air molecule. Further, I must be in the exact same state myself: every atom and electron in my brain and my body must be in the exact same state.

If that is what is meant, then I think this is a profoundly uninteresting question.First of all, there is clearly no way to test this question on the macroscopic level. Even if we could get the ball in the right place and the external conditions precisely the same (which we can't, of course) my own internal state will be different on a second go-round, for I will have a memory of the first try. Secondly, we already know the answer: from a physical point of view, having the exact same (quantum) state of the system is no guarantee of the same outcome. Finally, it is not at all clear what this has to do with free will. Let's humor the questioner and suppose we were able to do the experiment: make ten absolutely identical copies of me and my environment, and observe what the ten copies do. Suppose we found that I did the exact same thing each time. What have we learned? Only that identical copies do identical things - not particularly surprising or interesting. Suppose, on the other hand, that the copies do different things. On the physical level, this is possible if some quantum randomness is getting amplified into behavioral differences, as already noted. This would indeed be an interesting result - but it's not clear whether we've learned anything about free will. We have merely slipped back to the question of whether the underlying physics is deterministic or indeterministic. And, as I argued in the previous post, (in)determinism isn't really a crucial issue for free will.

(As an aside, I note that our inability to perform the experiment is not just a matter of technological sophistication: it is actually impossible in principle to put someone or some thing into the exact same state if the world runs according to quantum mechanical principles. This is the famous no-cloning theorem, to which I will return in the next post.)

What About The Nefarious Neurosurgeon?

Dennett, paraphrasing Frankfurt, presents this scenario (Elbow Room, p. 132):

Jones hates Smith and decides, in full possession of his faculties, to murder him. Meanwhile Black, the nefarious neurosurgeion, who also wants Smith dead, has implanted something in Jones's brain so that just in case Jones changes his mind (and chickens out), Black, by pushing his special button, can put Jones back on his murderous track. In the event, Black doesn't have to intervene; Jones does the deed all on his own.

Is Jones responsible for the murder, Frankfurter asks, even though he could not have done otherwise?

My question: why would anyone waste time worrying about such ridiculous stuff?

Next time: All sphexish things considered.

Thou Shalt Kill Everybody And The Horse They Rode In On

Michael Fridman is blogging the Bible. Today he presents 1 Samuel 15, in which God demands that Saul kill all the Amalekites, down to the last woman, child, and farm animal. Saul fails in this: he manages to massacre all the babies, but can't bring himself to kill the livestock. So God gets mad at him.

You can't make this stuff up.

William Lane Craig’s Argument Against Actual Infinities

William Lane Craig is a prominent defender of the Kalam Cosmological Argument (KCM) for the existence of God. (Nowadays philosophers tend to discuss “arguments for” – rather than “proofs of” – the existence of God.) The basic outline of the KCM is as follows:

1. Whatever begins to exist has a cause of

   its existence.

2. The universe began to exist.

3. Therefore, the universe has a cause of its

   existence.

Now, there are a lot of points about this argument that could be questioned, and many philosophers have objected to various points of Craig’s argument. But one part that stands out for me, and that seems to have been missed by those attempting to answer Craig, is his discussion of the nature of infinity.

As part of his version of the KCM, he asserts that

2.11. “An actual infinity cannot exist.”

To support this assertion, he offers the example of Hilbert’s Hotel.

Perhaps the best way to bring home the truth of (2.11) is by means of an illustration. Let me use one of my favorites, Hilbert's Hotel, a product of the mind of the great German mathematician, David Hilbert. Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? The proprietor just added the new guest's name to the register and gave him his keys-how can there not be one more person in the hotel than before? But the situation becomes even stranger. For suppose an infinity of new guests show up the desk, asking for a room. "Of course, of course!" says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room number twice his own. As a result, all the odd numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were full! And again, strangely enough, the number of guests in the hotel is the same after the infinity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infinitely many times and yet there would never be one single person more in the hotel than before.


But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds! Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel. But suppose instead the persons in room number 4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infinite converted to finitude. And yet it would remain true that the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out. Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things.

He ends with a rhetorical question. As Daniel Dennett reminds us, (Freedom Evolves (2003) p. 8) the rhetorical question usually marks the weakest point in an argument. It certainly does so here, for Craig has offered no proof that the features of Hilbert’s Hotel cause any real problems. In fact, in other writings, Craig has admitted that these counterintuitive properties of infinity do not lead to any logical contradiction. However, he insists that they nevertheless constitute a proof of metaphysical impossibility. But all he offers in support of this claim is the counterintuitive (but not self-contradictory) features of Hilbert’s Hotel.

In fact, Craig has done nothing more than appeal to his own personal incredulity: “I can’t imagine how such a thing could be true, therefore it’s impossible.” This type of appeal is, in fact, a well-known logical fallacy, and as such this part of Craig’s argument falls apart. That is really all that needs to be said to demolish Craig’s argument for the impossibility of an actual infinity; nonetheless, let’s proceed and show how a little imagination can help sort out Craig’s difficulties. Here, then, is a different version of Hilbert’s Hotel, one which will show that, while these unusual features of infinity may seem awkward at first, there is nothing metaphysically impossible about them.

Rather than an infinite hotel, suppose we have an infinite universe containing an infinite number of galaxies. From our location here on Earth, we choose a particular direction in space and imagine extending a ray infinitely outward in that direction. (Note I am talking about a conceptual ray, not anything physical, like a light beam or a particle beam.) Now, for a reasonably dense and random arrangement of galaxies in the universe, this ray will intersect an infinite number of galaxies. Let us label the galaxies by calling the one closest to us along the ray #1, the next galaxy #2, and so forth.

As an aside, note that I am assuming that the property of “intersecting the ray” is a binary one: each galaxy either intersects or doesn’t intersect, and there is a clear dividing line when a galaxy goes from non-intersecting to intersecting.  

Now let us assume that the galaxies of the universe are in motion, in such a manner that on a particular day (say Monday), all the galaxies that were intersecting on Sunday are still intersecting, but one new galaxy has moved into intersection with the ray, in a position closer to us than galaxy #1. When this happens, the new galaxy immediately becomes galaxy #1, the old galaxy #1 becomes galaxy #2, and so forth. We have added one to infinity, and still remain with an infinite number of galaxies.

Now, I don’t think any of what I suggested above is in any way metaphysically impossible. (Apart, possibly, from the assumption of an infinite number of galaxies – but Craig can’t object to that because the impossibility of an actual infinite is what he’s trying to prove.) The process can be repeated infinitely many times: just let all the galaxies drift away from us, so that there is room for another galaxy to slip into the closest position. But this situation is exactly equivalent to Hilbert’s Hotel: the numbers are the “rooms” and the galaxies are the “guests.” So Craig’s first objection to Hilbert’s Hotel is swept away.

Craig also seems disturbed by the idea that, even though we have added one galaxy/guest, we still have “the same number” of guests as before, namely, an infinite quantity. Here Craig seems to have confused the ideas of number and cardinality. He expects that infinite cardinalities can be dealt with in the same way as natural numbers, so that (infinity + 1) is somehow a bigger “number” than infinity. But in this Craig is simply demonstrating his mathematical ignorance: no mathematician thinks that infinite cardinalities obey the rules of addition for finite numbers. (It is well known that there are, in fact, some infinite cardinalities that are bigger than others. You can’t obtain a bigger cardinality simply by adding a few more elements to your set, however. The correct way to get a bigger infinity is to take the “power set” of an infinite set: the set of all subsets of the original set.)

Now imagine that every other galaxy along the ray is moving in such a manner that, on Tuesday, all the even numbered galaxies become non-intersecting.  This is exceedingly unlikely in any actual physical arrangement of galaxies, but not, surely, metaphysically impossible. This is equivalent to having an infinite number of guests check out of the hotel. Clearly, there will still be an infinite number of galaxies that intersect the ray. So we have subtracted infinity from infinity and are left with infinity! So we have disposed of Craig’s second “absurdity.”

Finally, we imagine all the galaxies numbered 4 and higher become non-intersecting (on Wednesday, say) and we are left with just three galaxies that intersect the ray. Here Craig’s objection is about the “number” of guests: he complains that “the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out.” His objection, it seems, is that in one instance (infinity – infinity = infinity) and in another instance (infinity – infinity = 3). Once again, it seems Craig has confused regular addition of integer numbers with operations on infinite cardinalities. It is simply not legitimate, mathematically speaking, to assume that the arithmetic rules for ordinary integers can be extended to rules for infinite cardinalities. As we all learned in math class (or should have learned, anyway), the expression (infinity – infinity) is “undefined.” That’s for a very good reason: the same reason Craig pointed out, that (infinity – infinity) can have different values depending on how the operation is conceived.

In fact, there is one definition of infinite cardinality that runs like this: “A set of elements S is said to be infinite if the elements of a proper subset S' can be put into one-to-one correspondence with the elements of S.” So Craig’s complaint – that the infinite subset of even-numbered galaxies is of the same size as the infinite set of the galaxies themselves – is actually the very definition of infinity.

To Craig’s rhetorical question, “Can anyone sincerely believe that such a hotel could exist in reality?” there is a clear answer: “Yes!” I suspect that Hilbert originally chose the hotel example specifically to emphasize the counterintuitive nature of infinity. But when we change the context, what once seemed absurd now seems perfectly natural. The claimed absurdities of Hilbert’s Hotel are not, in fact, absurd after all. 

(A more recent, and more technical, paper by Craig and Sinclair actually takes note of the definition of an infinite set given above, so it seems he has learned a little more math. But not enough: Craig continues to use the subtraction-of-infinities argument and the Hilbert Hotel "absurdities" argument.)

Sam Harris Again

Sam Harris has posted a FAQ that summarizes his claim to derive "ought" from "is." I further summarize his proposal as follows:

1. There is a state that we can call "the worst possible misery for everyone."
2. We ought to avoid the worst possible misery for everyone.
3. "There are behaviors, intentions, cultural practices, etc. which potentially lead to the worst possible misery for everyone."
4. We ought to avoid those behaviors, intentions and cultural practices.

In my version (and you can check for yourself if you think I've represented Harris's argument correctly) it is clear that Harris does not, in fact, derive "ought" from "is." Rather, he presents a fundamental "ought" which he thinks no one can reasonably deny. In this he is following a tack taken many years ago by Mortimer Adler, who proposed in his essay "Moral Values" (in Ten Philosophical Mistakes) that we can form a basis for morality by taking as an axiom some "ought" statement that is self-evidently true. (I don't know if Adler was the first to make this proposal.)

Adler's suggestion was

We ought to desire whatever is really good for us and nothing else.

Harris, in essence, turns this around and says we ought not to desire that which is really bad for us.

The problem that Adler faces, which is really the great problem of ethics going all the way back to Aristotle, is to answer the question "What is really good?" Harris thinks he can avoid this by asking instead, "What is really bad?" I don't think he succeeds.

It seems obvious (to Harris, anyway) that we should avoid the worst possible misery for everyone. But we first have to ask, what is the worst possible misery for everyone? Is there indeed such a state? We can imagine conditions that are really horrible for nearly everyone, but terrific for a few (a small group enslaving the majority to live in luxury, perhaps). Maybe even worse is a state where each group thinks it is enslaved by the other group, leading to continual conflict, violence, and revolt - not unlike the U.S. Congress. But is there truly a state that is the the worst possible misery for everyone?

Now, you might be thinking, "We don't really need to define the worst possible misery for everyone, Harris is really talking about avoiding some set of states that we can all agree are bad." I think this is basically right. In fact, Harris needs not only a set of states to be avoided, but some sort of gradation of those states, for he writes:

FACT #5: It is possible to be confused or mistaken about how the universe works. It is, therefore, possible to have the wrong values (i.e. values which lead toward, rather than away from, the worst possible misery for everyone).

and

FACT #7: In so far as our subsidiary values can be in conflict—e.g. individual rights vs. collective security; the right to privacy vs. freedom of expression—it may be possible to decide which priorities will most fully avoid the worst possible misery for many, most, or even all sentient beings.

So we can't just avoid  the worst possible misery for everyone, we must be able to do it more fully or less fully, to move toward or away from the worst possible misery for everyone. Harris is sneaking in a utilitarian concept of "greatest good for the greatest number" (or something like that) by way of "the least bad for the greatest number."

On the face of it, then, Harris's argument makes no sense. First of all, he doesn't derive "ought" from "is" - he introduces a fundamental (and, he thinks, undeniable) "ought." Secondly, if our only goal is to avoid the worst possible misery for everyone, and if we are not in the state of the worst possible misery for everyone, then there is nothing more to do. There is no imperative, for instance, to alleviate the misery of a small group, as long as that misery will not bring about the worst possible misery for everyone. His whole program only makes sense if we reinterpret it in some quasi-utilitarian framework. But he hasn't told us what that framework is.

There Is No Life After Death

For a long time I was more or less agnostic about life after death. "No data," I thought, "so no conclusions can be drawn." (Actually, there's a fair amount of data, if you include ghost sightings, visions of spirits and gods, mediums, and such like. But I think all these are more easily explained psychologically than supernaturally. When photography was young there were often claims of ghosts captured on film. Now that cameras are nearly ubiquitous, one doesn't hear of such claims very often. Strange that the ghosts have gotten so shy....)

After an (embarrassingly long) time, it occurred to me that there may not be any reliable reports from those who have died, but there is actually quite a bit of positive data about how the mind works. And what all this data points to is that the mind is entirely dependent on the brain, which is a machine that functions according to electro-chemical principles. As strongly as we may feel that there is an "I" that exists separate from the body, that feeling is an illusion.

If you have ever had a glass of wine you have done the experiment (on) yourself. Introduce the chemical alcohol into the bloodstream, and your mental function is affected. Of course there are many more such mind-altering chemicals, with greater or lesser effects. Psychological experiments have revealed much more dramatic changes occur when the brain is physically altered. In split brain patients (whose corpus callosum - the bundle of connections between right and left brain hemispheres - has been surgically cut to prevent seizures), the left hand literally does not seem to know what the right hand is doing. Or consider the case of Phineas Gage, a railroad worker who had a three-foot-long iron rod blasted through his skull. Afterward, he was able to function more or less normally, but his doctor noted a distinct personality change. Even our most cherished aspects of ourselves - our personalities that, we feel, make us who we are - are tied to the material structure of the brain.

EEGs record the electrical activity of the brain, and are so well correlated with brain functionality that brain activity is now what determines who is legally alive and who is legally dead. A newer technique, known as functional MRI, is helping scientists map out what brain functions are carried out in what regions of the brain. At a much cruder level, electroconvulsive therapy demonstrates the influence of electricity on brain function, as, of course, does electrocution.

In short, everything we know about mental processes and the brain indicates that physical processes are responsible for everything that makes us us. It follows that when those physical processes stop, we stop existing.

This may seem terribly obvious and trivial to you, as it does to me now. But, as I said, it took me a long time to make this connection, so I'm laying it out for anyone else who is trying to think these things through.

Free will and quantum mechanics

I never used to spend much time thinking about the problem of free will. If I can think, "I'm going to move my hand now," and then move my hand - well, that seems like enough free will for me.

Recently, though, I started reading up on the free will debate. Here are two of the arguments philosophers use. (The first argument is courtesy of the Stanford Encyclopedia of Philosophy, the second is cribbed from Norman Swartz.)

Argument #1 – There is No Moral Responsibility if Determinism is True
Premise 1.	No one has power over the facts of the past and the laws of nature.
Premise 2.	No one has power over the fact that the facts of the past and the laws of nature entail every fact of the future (i.e., determinism is true).
Premise 3.	Therefore, no one has power over the facts of the future.
Thus:	If determinism is true, it appears that no person has any power to alter how her own future will unfold, and therefore no moral responsibility for it.

Argument #2 - Causal Determinism is a Necessary Condition for Moral Responsibility Premise 1: Unless there are extenuating circumstances, persons are (to be) held morally responsible for their actions. Premise 2: Being unable reasonably to have foreseen the consequences of their actions is one such extenuating circumstance. (Recall that young children who cannot reasonably foresee the consequences of their actions are not to be held morally responsible for the consequences.) Premise 3: In order to be able to anticipate or foresee the likely (or even the remotely likely) consequences of one's actions, the world must not be random, i.e. the world must be fairly regular (or causally determined). Thus: Moral responsibility requires that there be causal determinism.

So the first argument (known as the consequence argument) seems to show that if the universe is deterministic, then there is no free will (because everything in the future is determined by events that happened long ago), and hence no moral responsibility. But the second argument (I don't know if it has a name) says that if there is no determinism, then there can't be any moral responsibility, either. Taken together, the two seem to imply that there can be no moral responsibility. I don't think this conclusion is correct, and neither does Swartz. You can read his notes to see how he resolves the problem; here, I want to consider a different aspect.

The current best models of how the universe works are based on quantum mechanics, and quantum mechanics is not deterministic. Or perhaps I should say quantum mechanics is partially indeterministic. That is, the quantum state of a closed system determines the quantum state of the system at a later time,  but the quantum state of the system doesn't determine all of the interesting aspects of the system. In fact, for any quantum system, there will always be questions I can ask about the system that don't have a definite answer - even if the quantum state of the system is exactly known. However, quantum mechanics can give us the probabilities of the various possible answers to any question, and so the future is partially determined: determined up to the range of possible values allowed by quantum mechanics.

How does quantum mechanics relate to the two arguments outlined above? The consequence argument is right out: Premise 2 is simply false if quantum mechanics gives a true picture of the world. What about Argument #2? According to Premise 3 of this argument, the world must be "fairly regular," which, according to quantum mechanics, it is. But the indeterminacy of quantum mechanics provides some elbow room (in Daniel Dennett's phrase), so that the future is not completely determined by the past. Quantum mechanics thus seems to walk a fine line between a clockwork universe in which every action of every person is, in principle, completely predictable, and a chaotically random universe in which nothing is predictable and so there can be no moral responsibility.

Quantum mechanics may not be the explanation of free will, but it does seem to allow for free will.

You Can't Be Good WITH God

The flip side of the previous post: John D at Philosophical Disquisitions summarizes an article by Stephen Maitzen that purports to prove that morality is impossible if you believe in God:

• (1) God exists (or "I believe that God exists").

• (2) Necessarily, God only allows undeserved, involuntary suffering if it produces a net benefit for the sufferer (principle of Theodical Individualism).

• (3) Therefore, we never have an obligation to intervene to prevent suffering.

Is Morality Objective?

Sam Harris, author of The End of Faith and Letter to a Christian Nation, gave a recent TED talk in which he argues, against what seems to be a common consensus, that morality and moral values can be put on a firm "scientific" - meaning fact-based - ground.

Many bloggers have responded, including one of my favorite physics bloggers, Sean Carroll. Sean points out the saw, going all the way back to Hume, that one can't derive moral truths from factual statements about the world. This is often shortened to, "You can't get 'ought' from 'is'."

Harris responded to Carroll and other critics in a post at Project Reason. He quite reasonably points out that he only had 18 minutes to present his case (per TED requirements), and lays out his view in more detail. (He also points to a forthcoming book in which he will make the case in greater length.) Harris says Hume is by no means the last word on "is" and "ought", and not all philosophers accept Hume's conclusion. Luke Muelhauser makes a similar point on Common Sense Atheism.

Carroll answered Harris again here. Russell Blackford came down on Carroll's side, too, citing Peter Singer.

I have no idea who, if anyone, is right in all this. I know I would like to have an objective definition of morality,  both because I would like a rational way of ordering - and defending - my own actions and because I would like a rationally defensible way of requiring others to do the same. If it's all just a matter of personal preference, then no progress can ever be made in the moral arena. But I'm skeptical of Harris's approach precisely because of that desire: my desire for an answer might mislead me into thinking that there is an answer.

At any rate it is a fascinating, and, I think, valuable debate.