Archive for December, 2005

Mathematics and Reality

In the first part of Roger Penrose’s book The Road to Reality (which I have not yet finished) Penrose argues that mathematics has an objective reality. It’s not entirely clear to me what he means by this, but I suspect that he is wrong. He points out that everybody who studies math comes to the same conclusions, and that seriously studying math has the feel of uncovering deeper and deeper truths. He also brings in Mandelbrot’s set, though I’m not sure why since it doesn’t seem to support his thesis.

It is undeniable that math is a deep subject, and that mathematics is an investigation shared by the human race across time. However, this does not mean that math has an objective reality beyond its existence in humanity’s shared knowledge. I think the easiest way to see this is to compare it to newer branch of knowledge, such as cellular automata.

The study of cellular automata is not mathematics. Cellular automata are abstract, but they are not based on numbers or geometry. Cellular automata are nevertheless a complex field of study, as can be seen by the extensive studies of just one instance, Conway’s Game of Life. There are undoubtedly general laws and insights waiting to be found in the study of cellular automatons.

However, I think it is silly to argue that cellular automatons have any sort of objective existence in a Platonic ideal space. The only way to support that is to become a thoroughgoing Platonist and believe that there exists an ideal object for all abstract concepts. So since I don’t see any essential difference between the study of mathematics and the study of cellular automata, I don’t see any reason to believe that mathematics has any objective reality.

There is a related issue, which is why reality is apparently describable in mathematical terms. Of course this is in part probably just an effect of how we observe things: we describe reality in terms of mathematics because those parts of reality which can not be described that way tend not to be described at all.

For example, consider the physics of gases. There are simple equations describing the relationship between the volume, temperature and pressure of an ideal gas. These equations do a good job of describing reality. However, we know that these equations are really just a way of describing the interactions of many many individual molecules. In this case, we are able to reduce the complexity of tracking individual molecules with simple mathematical formulas. Note that the gas laws are perfectly real: they correctly predict and describe the behaviour of gases. However, there is an underlying, more complex, system which produces the results the gas laws describe.

The weather on the Earth is also a complex set of interactions of many many individual molecules. However, in this case we have so far completely failed to describe the weather using mathematical formulas. Our best attempts to describe the weather involve complex models–in fact, a form of cellular automata. So when we say that reality can be described by mathematics, we evidently don’t mean the weather. Are we simply picking and choosing the things which we can describe? Also, when we find a simple mathematical formula in physics, I think we always have to ask: is this reality, or is it a formula which manages to capture underlying complex behaviour?

There is another way to consider the relationship between mathematics and reality, based on the weak anthropic principle. We are complex creatures, and our complexity was formed by evolutionary processes over a long period of time. Evolution is a very flexible technique, but it can only create complex objects when it operates in a relatively static reality. If the laws of physics changed unpredictably, it’s difficult to see how evolution would ever be able to build up complexity. Therefore, since we exist, we can conclude that the reality we inhabit must be relatively static. And mathematics is well suited to a description of many static systems. This does not prove that reality must be describable mathematically, but it suggests that we need not be surprised that it can.

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Meta jokes

First, a brief digression to a recent family experience. My wife and 4-year old daughter were looking through a children’s magazine. They came to a page of jokes. My wife read a few of them. Sample joke: “Q: What did the lamb do for her birthday party? A: She had a sheep-over.” My wife and I chuckled a little bit, in part to to encourage my daughter. However, she simply didn’t understand the joke. So we tried to explain it, saying that “sheep” and “sleep” sound similar, and so forth; of course the joke isn’t very funny to start with, and naturally any explanation made it completely unfunny. However, my daughter seized on what appeared to be the salient point, which was that a joke is asking a question and responding with a meaningless answer. She then told several jokes of her own, for example: “Q: What can you cut with a knife? A: A table!” This was followed by much laughter. We laughed along. At the same time I was picturing the reaction of one of her pre-schoolmates to this riddle: a start of incomprehension, followed by a change of subject.

OK, digression over. What makes this sort of joke funny (if it is funny) is the confusion between two concepts which are not normally related but which become related in the telling of the joke. I think this confusion underlies most jokes that ordinary people tell (comedians tell this sort of joke too, but they also tell more personal stories which are more interesting and usually funnier.) My daughter’s joke wasn’t funny at all because there was no conceptual confusion. The sheep joke was a little bit funny, but it wasn’t very funny because it was simply a bad pun. Good puns expose unexpected congruence; bad puns just rely on words that happen to sound alike.

The jokes which I tend to think are funniest, and the ones which I actually remember, are actually meta-jokes: they are jokes in which the humor results from presenting something in the form of a joke which is not actually funny. The humor lies in the confusion between the concepts of humor and non-humor.

The classic meta-joke is the well known “Q: Why did the chicken cross the road? A: To get to the other side.” This joke has lost its humor through repetition, but it expresses the idea of the meta-joke perfectly: it is funny because it is not funny.

Other meta-jokes that I like:

There was a young man from St. Bees,
Who was stung on the arm by a wasp.
When asked, “Does it hurt?”
He replied, “No, it doesn’t.
I’m so glad it wasn’t a hornet.”
(I first saw this in a Metamagical Themas by Doug Hofstadter.)

Q: Why is a raven like a writing desk?
A: I don’t know.
(From Alice in Wonderland.)

Let me tell you a knock-knock joke; you start:
“Knock knock”
“Who’s there?”
“…”
(I’m not sure where I saw this first, although it was recently in the film Mirrormask. I’ve observed that this joke generally doesn’t work with children. They just quickly make something up, or recycle another knock knock joke they’ve heard before.)

As you can probably tell, I’m not a very funny person. When I told jokes in the schoolyard, and they fell flat, I was usually baffled, and tried to explain them. This was naturally (in retrospect) ineffective, and generally lead to eye-rolling and subject-changing. But, again in retrospect, it’s easy to see where good comedians come from: they are the people who, when a joke falls flat, follow up quickly with another one. They learn from their mistakes and steadily improve. Eventually, if they are clever enough, they become genuinely original and funny.

I think the funniest joke I ever told was itself a meta-joke. In high school a group of us, some twenty people or so, were at a restaurant. Somebody decided to try the old game of Telephone, in which the first person whispers something to the second person, he or she whispers it to the third, and so on, and you compare the final result to the original saying. When it got to me, I listened to the person on my right, and then said something completely different to the person on my left. The final result was thus bizarrely different from the original saying. Of course they figured out that it was me pretty quickly, but it was still pretty funny. At least to me.

(I don’t remember what I heard, but I think that what I said was an old limerick I saw in a Martin Gardner column:

There was a young lady named Bright,
who travelled much faster than light.
She left home one day,
in her relative way,
and returned home the previous night.)

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The Ontological Proof

I’ve always felt that the ontological proof of the existence of God was one of the more compelling and interesting arguments about God. The proof, which was originally composed by Saint Anselm, amounts to a simple syllogism:

1) God is, by definition, the most perfect entity.
2) Something which exists is more perfect than something which does not exist.
3) Therefore, God exists.

It’s a natural first response to dismiss this argument as mere logical legerdemain, confusing words with their meanings. But I think that sort of thinking misses the real power of an omnipotent entity. For God, being omnipotent and encompassing the whole universe, thoughts are indistinguishable from actions. The universe as a whole is simply than the thoughts of God. If God thinks “what if Ian went to bed instead of writing in his blog” there would in effect be two different universes, albeit perhaps only temporarily. I don’t think we can casually dismiss the ontological argument without understanding that for God, words and meanings are the same thing. Or, to put it another way, the syntax is itself semantics.

I think the weak point in the argument is the notion that existence is more perfect than non-existence. I think existence implies change. If something truly never changes, then in what sense does it exist? The only way to not change is to not be subject to time. But how can we speak of something existing without speaking of some time in which it exists? (The observant reader will note that I am resorting to rhetorical questions, a sure sign of a shaky argument. These are definitional issues, but the definitions are the key to the ontological proof.)

The next step is to observe that if God changes, then it follows that God is not always the single perfect entity. If there are multiple possible perfect entities, then it follows that there are multiple Gods, which is not satisfactory. So there must be only one perfect entity. And it must never change.

So, since perfection implies not changing, and existence implies changing, it follows that non-existence is more perfect than existence. And thus we see that the ontological proof actually proves that God does not exist. At least not if we define God as the most perfect entity.

On a related topic, here is a brief consideration of the old schoolyard riddle: “Could God create a rock so heavy that He (or She) could not lift it?” As mentioned above, since God is omnipotent, any thought is reality. As Dante put it, “this has been willed where that which is willed must be.” So another way to put the question is “could God choose to never think a particular thought (namely, to lift that rock)?” Or, to put it yet another way, “could God lose the quality of being omnipotent?” I think the answer to that question has to be no: I don’t believe that an omnipotent being can cease being omnipotent. God might never, as a matter of fact, lift the rock. But it is not possible for God to be unable to lift the rock.

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The Suburbs

I grew up in a small city next to a large city, and I have always lived in cities. My grandparents lived in small town (3000 people), and we often spent summers with them when I was a child. My parents now live in a rural area, in a town that has no traffic lights and perhaps three stop signs.

From this experience, when I am in a city, I may not know where I am, but I feel that I always know more or less what kind of place I am in. This is true not only in the U.S., but also in my travels in countries like Japan and India. Similarly, when I am in the country, while I am not a part of the local society, I feel that I more or less understand how that society works.

On the other hand, I do not have these feelings in the suburbs, or the recently christened exurbs. I recently happened to drive through Fremont, California, a classic suburb on the outskirts of Silicon Valley. I once again felt the feeling I always feel in the suburbs, which is that I don’t know where I am. I don’t understand how the society works. Some Europeans in the African bush have describe the situation of being lost a hundred steps from the caravan, simply because they were unable to recognize the features in the terrain which distinguished one path from another; that is how I feel in the suburbs.

The suburbs are essentially the city writ large: neighborhoods become subdivisions; shopping districts (squares, as we say where I grew up in Massachusetts) become shopping malls and strip malls; the downtown area where most people work becomes the city proper. But recreating the city in the large changes its essential character. In particular, people are forced to drive everywhere. It is hardly an original observation that the suburbs were created by the car and depend upon the car. And it is likewise unoriginal to observe that driving everywhere puts you in complete control of your environment. You go only the places you want to go, you see only the places you want to see.

In the city, the other inhabitants of the city are part of your environment, for better or for worse. You can not avoid seeing people who are very different from yourself, although you certainly can and do avoid talking to them. You can not avoid passing through places that are not your destination. Chance meetings of acquaintances are routine. Serendipitous discoveries, of a new restaurant, an interesting house, a small shop, are commonplace.

In the countryside, you deal with the same people day in and day out, and you come to know them, at least superficially. There are only two people who run the cash register at the local store. There is only one auto mechanic. There are no plumbers. There are several town eccentrics of one stripe or another. While a newcomer is rarely truly accepted, it does not take long to make a nodding acquaintance with many of the locals. Gossip covers everybody who lives in the town.

The suburbs are in between. You control who you see, but the people at the supermarket and the mall are by and large anonymous. The most uncontrolled situation, the place where you might meet anybody, is at church. This ability to control who you see and to preserve your privacy and anonymity is attractive, but I think the price is a tendency to feel disconnected from modern society in all its complexity, to feel that all ordinary people are much like yourself, to start to think that people who seriously disagree with you must be not merely mistaken or misguided but actually ignorant and/or evil.

I don’t want to overstate the case. These tendencies exist in all places for all people, and my feeling that the suburbs encourage them, and my general discomfort in the suburbs, may be due mostly to lack of familiarity. I continue to struggle to understand why people choose to live in the suburbs. I can see the advantages in space, in having places for your children to roam safely, but the price seems so high. And raising children in a safe but constricted environment seems like a risky choice in general, not conducive to their future happiness, though certainly one that is appropriate for some children.

I want to close with an observation. I don’t like to drive, and I prefer to use public transportation. Most suburbs are in fact accessible by buses, at least if you are willing to walk up to a mile. I’ve used buses in suburbs in Massachusetts, California and Washington (state). I am normally the only middle aged male on these buses, at least the only who isn’t hunched in a corner muttering to himself. The other people on the buses are usually teenagers (the buses normally travel to and from malls) and women who I suspect work as house cleaners or other jobs along those lines. For most people in the suburbs–the people I am visiting–the ability to travel without a car is akin to stage magic: wholly baffling until explained, at which point it seems more like a waste of time for a grown man. For me, though, riding the bus in the suburbs is not merely an eccentricity; it is the only time in the suburbs that I feel that I actually know where I am.

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Quantum Mechanics

The theory of quantum mechanics poses a well-known problem: at the atomic level, particles must be described as waveforms, but at the macroscopic level, they must be described as particles. The well-known double-slit experiment shows that if we, at the macroscopic level, remain ignorant of the location of the particle, it acts as a wave. However, if we measure the location of the particle, it acts as a particle. This is often described as “collapse of the wave function” or the “measurement problem.” As far as we can tell, this is reality, despite the complete defiance of common sense. The question that needs to be solved is why it looks different at the microscopic and macroscopic levels.

The traditional explanation of the measurement problem is known as the “Copenhagen interpretation,” so named because it was developed by Neils Bohr and Werner Heisenberg in Copenhagen. In this explanation, the wave function collapses when it is “measured.” Unfortunately, the nature of “measurement” is left undefined. It is often assumed that “measurement” occurs when a conscious observer becomes aware of the occurrence. However, if we take a materialistic world view, as I do, this does not make sense. Consciousness is created by atoms like other atoms, and they obey the laws of quantum mechanics just like everything else. It does not make sense to assign consciousness any special powers.

As far as I can see, there are two supportable theories to describe waveform collapse (ignoring various detailed differences between theories, and just focusing on the highest level approach). The first is one described in Roger Penrose’s book Shadows of the Mind (which is, by the way, in general a terrible book, continuing and expanding the deep confusion about the nature of consciousness which he first demonstrated in The Emperor’s New Mind). In this theory the waveform collapses when there is too much energy involved in it. That is, as particles interact with each other, the waveform grows to encompass more particles and hence more energy, until it eventually collapses. I have no idea if this is completely coherent, but let’s assume that it is.

The second theory is the Many Worlds Interpretation, originally proposed by Hugh Everett, which simply says that the wave function never collapses. In this theory, our consciousness gets bound up in one version of the wave function, and so it appear to us that the waveform has collapsed. This process of getting bound up is now known as “quantum decoherence.”

These theories can be experimentally distinguished in principle, so we could in principle determine that one of them was wrong. We can determine an upper limit on the amount of energy which would cause a waveform collapse in Penrose’s theory–evidently once a human brain is entangled the waveform collapses. We can then in principle construct an isolated quantum entanglement, make it contain more energy than a human brain, and then try to replicate waveform effects. In practice the isolation requirement would be fiendishly difficult.

The Many Worlds Interpretation is often assumed to suggest that all possible worlds exist. For example, the suggestion is that in some other path of the universal wave function I started this essay with the word “the” instead of the word “a.” While this argument may make for some good science fiction stories, I believe it is an error. The supposition that in some sense every choice is made at the subatomic level does not imply that every choice is made at the macroscopic level. There are many choices that I will never make, even though they are technically possible. It is not the case that my consciousness depends critically on quantum waveforms in my brain, or that my decisions are in way controlled by wavefunction collapse (Penrose might disagree, but that is because he is seeking a last redoubt for his confused view of consciousness). It is not the case that everything that we can imagine is actually possible.

On the other hand, one could argue that perhaps all the atoms in my brain will simultaneously change position, thus giving me a different personality, while still retaining my memory, causing me to make different choices. Of course, this would be a vanishingly improbable event, much less likely than my sudden spontaneous explosion. The most disturbing aspect of the Many Worlds Interpretation is that it suggests not only that there is a vanishingly small chance that I will suddenly explode–all interpretations of quantum mechanics suggest that–it suggests that in some sense I am in fact exploding all the time. I don’t know of any way to deal with this infinitesimally small possibility other than the time honored technique of not worrying about it.

Sticking to higher probability branches, the main problem with the Many Worlds Interpretation is trying to explain why we humans don’t see an entangled wave function, but only see a collapsed one. There is no immediately obvious reason why we could not operate on a quantum view of the universe, and even simultaneously choose different actions. That might sound odd, but it is more or less what quantum computers do, and they seem to work under laboratory conditions. Why can’t we do it?

If the Many Worlds Interpretation is true, I think the answer has to be that for some reason it was evolutionary advantageous to our remote ancestors to only perceive a single version of the wave function. Perhaps it is as simple as that when there are two slightly different high probability locations of food, choosing to move toward one causes the entire organism to act as though a single instance of the wave function were reality, effectively causing quantum decoherence.

I believe that life in some form is bound to be common in the universe, though I see no reason to believe that intelligence is common. It is interesting to speculate about the possibility of discovering a life form which was able to simultaneously choose different actions. Would such a being appear to us to behave almost randomly? Or would it seem to be able to solve problems almost magically?

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